diff --git "a/data/validation.jsonl" "b/data/validation.jsonl"
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+++ "b/data/validation.jsonl"
@@ -0,0 +1,500 @@
+{"id":"openstem_003994","subject":"physics","topic":"mechanics","subtopic":"work_energy_theorem","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A 5 kg block initially moves at 11 m s⁻¹. A constant net force does 17 N of work over 19 m in the direction of motion. Find its final kinetic energy.","options":{"A":"-625.5 J","B":"625.5 J","C":"938.25 J","D":"623.5 J"},"correct_answer":"B","correct_answer_value":"625.5 J","solution":"By the work–energy theorem, K_f=K_i+W=½mu²+Fs=½(5)(11)²+(17)(19)=625.5 J.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 625.5 J, not -625.5 J.","B":"Correct. The worked calculation gives 625.5 J.","C":"Incorrect. Substituting the stated values into the governing relation gives 625.5 J, not 938.25 J.","D":"Incorrect. Substituting the stated values into the governing relation gives 625.5 J, not 623.5 J."},"concepts_required":["work–energy theorem","kinetic energy","work by constant force"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_work_energy","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_003625","subject":"physics","topic":"mechanics","subtopic":"projectile_range","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A projectile is launched from level ground at 79 m s⁻¹ at 40° to the horizontal. Take g=10 m s⁻². Find its horizontal range.","options":{"A":"307.30926 m","B":"614.61852 m","C":"-614.61852 m","D":"921.92778 m"},"correct_answer":"B","correct_answer_value":"614.61852 m","solution":"Range R=u²sin(2θ)/g. Substituting u=79, θ=40° and g=10 gives R=(79)²sin(80°)/10=614.61852 m.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 614.61852 m, not 307.30926 m.","B":"Correct. The worked calculation gives 614.61852 m.","C":"Incorrect. Substituting the stated values into the governing relation gives 614.61852 m, not -614.61852 m.","D":"Incorrect. Substituting the stated values into the governing relation gives 614.61852 m, not 921.92778 m."},"concepts_required":["projectile motion","range equation","trigonometry"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_projectile_range","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_000633","subject":"mathematics","topic":"sequences_and_series","subtopic":"arithmetic_progression","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"An arithmetic progression has first term 9 and common difference 9. Find the sum of its first 26 terms.","options":{"A":"6318","B":"3159","C":"2369.25","D":"-3159"},"correct_answer":"B","correct_answer_value":"3159","solution":"Use Sₙ = n/2[2a+(n−1)d]. Substituting a=9, d=9, n=26: Sₙ=26/2[2(9)+(26−1)(9)]=3159.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 3159, not 6318.","B":"Correct. The worked calculation gives 3159.","C":"Incorrect. Substituting the stated values into the governing relation gives 3159, not 2369.25.","D":"Incorrect. Substituting the stated values into the governing relation gives 3159, not -3159."},"concepts_required":["arithmetic progression","finite series"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_ap_sum","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_008618","subject":"chemistry","topic":"physical_chemistry","subtopic":"boiling_point_elevation","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A non-volatile solute is dissolved in water to produce a solution of molality 0.39 mol kg⁻¹. If its van't Hoff factor is 1 and K_b for water is 1.0 K kg mol⁻¹, find the boiling-point elevation.","options":{"A":"-1.61 K","B":"-0.39 K","C":"0.2925 K","D":"0.39 K"},"correct_answer":"D","correct_answer_value":"0.39 K","solution":"ΔT_b=iK_bm. Substitution gives ΔT_b=1×1.0×0.39=0.39 K.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.39 K, not -1.61 K.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.39 K, not -0.39 K.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.39 K, not 0.2925 K.","D":"Correct. The worked calculation gives 0.39 K."},"concepts_required":["colligative properties","van't Hoff factor","molality"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_boiling_elevation","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_002895","subject":"mathematics","topic":"vectors","subtopic":"dot_product","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Given vectors u=(7, -3, 5) and v=(8, 9, -6), calculate u·v.","options":{"A":"0","B":"-4","C":"-1","D":"1"},"correct_answer":"C","correct_answer_value":"-1","solution":"The dot product is the sum of corresponding component products: (7)(8)+(-3)(9)+(5)(-6)=-1.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives -1, not 0.","B":"Incorrect. Substituting the stated values into the governing relation gives -1, not -4.","C":"Correct. The worked calculation gives -1.","D":"Incorrect. Substituting the stated values into the governing relation gives -1, not 1."},"concepts_required":["three-dimensional vectors","scalar product"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_vector_dot","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_003711","subject":"physics","topic":"mechanics","subtopic":"work_energy_theorem","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A 6 kg block initially moves at 6 m s⁻¹. A constant net force does 14 N of work over 16 m in the direction of motion. Find its final kinetic energy.","options":{"A":"330 J","B":"498 J","C":"332 J","D":"334 J"},"correct_answer":"C","correct_answer_value":"332 J","solution":"By the work–energy theorem, K_f=K_i+W=½mu²+Fs=½(6)(6)²+(14)(16)=332 J.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 332 J, not 330 J.","B":"Incorrect. Substituting the stated values into the governing relation gives 332 J, not 498 J.","C":"Correct. The worked calculation gives 332 J.","D":"Incorrect. Substituting the stated values into the governing relation gives 332 J, not 334 J."},"concepts_required":["work–energy theorem","kinetic energy","work by constant force"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_work_energy","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_009073","subject":"chemistry","topic":"physical_chemistry","subtopic":"nernst_equation_298K","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"At 298 K, a galvanic cell has E°cell=1.07 V, transfers 2 electrons, and has reaction quotient Q=10^3. Using E=E°−(0.0591/n)log₁₀Q, calculate Ecell.","options":{"A":"-0.01865 V","B":"-2.01865 V","C":"1.09814 V","D":"0.98135 V"},"correct_answer":"D","correct_answer_value":"0.98135 V","solution":"Apply the Nernst equation: E=1.07−(0.0591/2)(3)=0.98135 V.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.98135 V, not -0.01865 V.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.98135 V, not -2.01865 V.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.98135 V, not 1.09814 V.","D":"Correct. The worked calculation gives 0.98135 V."},"concepts_required":["electrochemical cells","Nernst equation","logarithms"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_nernst","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_000644","subject":"mathematics","topic":"sequences_and_series","subtopic":"arithmetic_progression","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"An arithmetic progression has first term -9 and common difference 7. Find the sum of its first 21 terms.","options":{"A":"2562","B":"1921.5","C":"1281","D":"1537.2"},"correct_answer":"C","correct_answer_value":"1281","solution":"Use Sₙ = n/2[2a+(n−1)d]. Substituting a=-9, d=7, n=21: Sₙ=21/2[2(-9)+(21−1)(7)]=1281.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 1281, not 2562.","B":"Incorrect. Substituting the stated values into the governing relation gives 1281, not 1921.5.","C":"Correct. The worked calculation gives 1281.","D":"Incorrect. Substituting the stated values into the governing relation gives 1281, not 1537.2."},"concepts_required":["arithmetic progression","finite series"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_ap_sum","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_007715","subject":"chemistry","topic":"physical_chemistry","subtopic":"first_order_half_life","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A first-order reaction has a half-life of 20 minutes. If the initial reactant concentration is 10 mmol L⁻¹, what concentration remains after 140 minutes?","options":{"A":"0.07812 mmol L⁻¹","B":"0.03906 mmol L⁻¹","C":"0.15625 mmol L⁻¹","D":"1.07812 mmol L⁻¹"},"correct_answer":"A","correct_answer_value":"0.07812 mmol L⁻¹","solution":"140 minutes corresponds to 7 half-lives. The remaining concentration is [A]₀(1/2)ⁿ=10/2^7=0.07812 mmol L⁻¹.","option_explanations":{"A":"Correct. The worked calculation gives 0.07812 mmol L⁻¹.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.07812 mmol L⁻¹, not 0.03906 mmol L⁻¹.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.07812 mmol L⁻¹, not 0.15625 mmol L⁻¹.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.07812 mmol L⁻¹, not 1.07812 mmol L⁻¹."},"concepts_required":["first-order kinetics","half-life","exponential decay"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_first_order_halflife","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_001772","subject":"mathematics","topic":"combinatorics","subtopic":"binomial_coefficient","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"How many distinct subsets containing exactly 5 elements can be selected from a set of 53 distinct elements?","options":{"A":"-2869685","B":"2869684","C":"2295748","D":"2869685"},"correct_answer":"D","correct_answer_value":"2869685","solution":"Order does not matter, so the count is C(53,5)=53!/[5!(48)!]=2869685.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 2869685, not -2869685.","B":"Incorrect. Substituting the stated values into the governing relation gives 2869685, not 2869684.","C":"Incorrect. Substituting the stated values into the governing relation gives 2869685, not 2295748.","D":"Correct. The worked calculation gives 2869685."},"concepts_required":["combinations","factorials"],"reasoning_steps":2,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_combinations","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_003843","subject":"physics","topic":"mechanics","subtopic":"work_energy_theorem","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A 13 kg block initially moves at 8 m s⁻¹. A constant net force does 13 N of work over 4 m in the direction of motion. Find its final kinetic energy.","options":{"A":"936 J","B":"468 J","C":"469 J","D":"47.8 J"},"correct_answer":"B","correct_answer_value":"468 J","solution":"By the work–energy theorem, K_f=K_i+W=½mu²+Fs=½(13)(8)²+(13)(4)=468 J.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 468 J, not 936 J.","B":"Correct. The worked calculation gives 468 J.","C":"Incorrect. Substituting the stated values into the governing relation gives 468 J, not 469 J.","D":"Incorrect. Substituting the stated values into the governing relation gives 468 J, not 47.8 J."},"concepts_required":["work–energy theorem","kinetic energy","work by constant force"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_work_energy","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_008069","subject":"chemistry","topic":"physical_chemistry","subtopic":"faradays_law_copper","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"An aqueous Cu²⁺ solution is electrolysed using a current of 13 A for 18001 s. Assuming 100% current efficiency, calculate the mass of copper deposited. Use M(Cu)=63.5 g mol⁻¹ and F=96500 C mol⁻¹.","options":{"A":"92.3927 g","B":"38.49696 g","C":"76.99391 g","D":"78.99391 g"},"correct_answer":"C","correct_answer_value":"76.99391 g","solution":"Cu²⁺+2e⁻→Cu, so z=2. By Faraday's law m=MIt/(zF)=63.5×13×18001/(2×96500)=76.99391 g.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 76.99391 g, not 92.3927 g.","B":"Incorrect. Substituting the stated values into the governing relation gives 76.99391 g, not 38.49696 g.","C":"Correct. The worked calculation gives 76.99391 g.","D":"Incorrect. Substituting the stated values into the governing relation gives 76.99391 g, not 78.99391 g."},"concepts_required":["electrolysis","Faraday's laws","redox stoichiometry"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_faraday_copper","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_004327","subject":"physics","topic":"mechanics","subtopic":"inclined_plane_with_friction","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A 37 kg block slides down a 37° rough incline. The coefficient of kinetic friction is 0.19. Taking g=10 m s⁻², find the net force down the plane. Use sin37°=0.6 and cos37°=0.8.","options":{"A":"198.912 N","B":"165.76 N","C":"166.76 N","D":"124.32 N"},"correct_answer":"B","correct_answer_value":"165.76 N","solution":"Net force=mg(sinθ−μcosθ)=37×10×(0.6−0.19×0.8)=165.76 N.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 165.76 N, not 198.912 N.","B":"Correct. The worked calculation gives 165.76 N.","C":"Incorrect. Substituting the stated values into the governing relation gives 165.76 N, not 166.76 N.","D":"Incorrect. Substituting the stated values into the governing relation gives 165.76 N, not 124.32 N."},"concepts_required":["Newton's laws","inclined plane","kinetic friction"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_incline_friction","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_003307","subject":"mathematics","topic":"complex_numbers","subtopic":"real_part_of_complex_quotient","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Let z=(7+8i)/(1+3i). What is Re(z), the real part of z?","options":{"A":"4.1","B":"-3.1","C":"3.1","D":"2.325"},"correct_answer":"C","correct_answer_value":"3.1","solution":"Multiply numerator and denominator by the conjugate (1-3i). Then Re(z)=(ac+bd)/(c²+d²)=((7)(1)+(8)(3))/(1²+(3)²)=3.1.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 3.1, not 4.1.","B":"Incorrect. Substituting the stated values into the governing relation gives 3.1, not -3.1.","C":"Correct. The worked calculation gives 3.1.","D":"Incorrect. Substituting the stated values into the governing relation gives 3.1, not 2.325."},"concepts_required":["complex conjugate","complex division","real and imaginary parts"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_complex_quotient","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_006760","subject":"chemistry","topic":"stoichiometry","subtopic":"limiting_reagent_water","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Hydrogen reacts with oxygen according to 2H₂ + O₂ → 2H₂O. If 41 mol H₂ and 8 mol O₂ are mixed and the reaction goes to completion, how many moles of H₂O form?","options":{"A":"14 mol","B":"15 mol","C":"18 mol","D":"16 mol"},"correct_answer":"D","correct_answer_value":"16 mol","solution":"From 2 mol H₂ and 1 mol O₂, 2 mol H₂O form. H₂ can produce 41 mol H₂O; O₂ can produce 16 mol H₂O. The limiting condition is O₂; product=16 mol.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 16 mol, not 14 mol.","B":"Incorrect. Substituting the stated values into the governing relation gives 16 mol, not 15 mol.","C":"Incorrect. Substituting the stated values into the governing relation gives 16 mol, not 18 mol.","D":"Correct. The worked calculation gives 16 mol."},"concepts_required":["balanced chemical equations","limiting reagent","mole ratios"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_limiting_reagent","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_000434","subject":"mathematics","topic":"sequences_and_series","subtopic":"arithmetic_progression","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"An arithmetic progression has first term -4 and common difference 12. Find the sum of its first 31 terms.","options":{"A":"-5456","B":"5458","C":"6547.2","D":"5456"},"correct_answer":"D","correct_answer_value":"5456","solution":"Use Sₙ = n/2[2a+(n−1)d]. Substituting a=-4, d=12, n=31: Sₙ=31/2[2(-4)+(31−1)(12)]=5456.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 5456, not -5456.","B":"Incorrect. Substituting the stated values into the governing relation gives 5456, not 5458.","C":"Incorrect. Substituting the stated values into the governing relation gives 5456, not 6547.2.","D":"Correct. The worked calculation gives 5456."},"concepts_required":["arithmetic progression","finite series"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_ap_sum","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_004306","subject":"physics","topic":"mechanics","subtopic":"inclined_plane_with_friction","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A 15 kg block slides down a 37° rough incline. The coefficient of kinetic friction is 0.18. Taking g=10 m s⁻², find the net force down the plane. Use sin37°=0.6 and cos37°=0.8.","options":{"A":"-68.4 N","B":"69.4 N","C":"82.08 N","D":"68.4 N"},"correct_answer":"D","correct_answer_value":"68.4 N","solution":"Net force=mg(sinθ−μcosθ)=15×10×(0.6−0.18×0.8)=68.4 N.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 68.4 N, not -68.4 N.","B":"Incorrect. Substituting the stated values into the governing relation gives 68.4 N, not 69.4 N.","C":"Incorrect. Substituting the stated values into the governing relation gives 68.4 N, not 82.08 N.","D":"Correct. The worked calculation gives 68.4 N."},"concepts_required":["Newton's laws","inclined plane","kinetic friction"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_incline_friction","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_004990","subject":"physics","topic":"electricity","subtopic":"series_parallel_circuit","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Resistors 11 Ω and 11 Ω are connected in series; this combination is connected in parallel with a 15 Ω resistor across a 30 V ideal battery. Find the total current supplied by the battery.","options":{"A":"0.36364 A","B":"6.36364 A","C":"3.36364 A","D":"5.04545 A"},"correct_answer":"C","correct_answer_value":"3.36364 A","solution":"Series branch resistance=11+11=22 Ω. Parallel equivalent R_eq=(22×15)/(22+15)=8.91892 Ω. Total current I=V/R_eq=30/8.91892=3.36364 A.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 3.36364 A, not 0.36364 A.","B":"Incorrect. Substituting the stated values into the governing relation gives 3.36364 A, not 6.36364 A.","C":"Correct. The worked calculation gives 3.36364 A.","D":"Incorrect. Substituting the stated values into the governing relation gives 3.36364 A, not 5.04545 A."},"concepts_required":["series resistors","parallel resistors","Ohm's law"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_series_parallel","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_005068","subject":"physics","topic":"electrostatics","subtopic":"coulombs_law","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Two point charges of +8 μC and +6 μC are separated by 5 cm in vacuum. Using k=9×10⁹ N m² C⁻², find the magnitude of the electrostatic force.","options":{"A":"345.6 N","B":"172.8 N","C":"259.2 N","D":"173.8 N"},"correct_answer":"B","correct_answer_value":"172.8 N","solution":"Convert charges to coulombs and distance to metres. F=k|q₁q₂|/r²=9×10⁹(8×10⁻⁶)(6×10⁻⁶)/(5×10⁻²)²=172.8 N.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 172.8 N, not 345.6 N.","B":"Correct. The worked calculation gives 172.8 N.","C":"Incorrect. Substituting the stated values into the governing relation gives 172.8 N, not 259.2 N.","D":"Incorrect. Substituting the stated values into the governing relation gives 172.8 N, not 173.8 N."},"concepts_required":["Coulomb's law","SI unit conversion","inverse-square law"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_coulomb","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_005050","subject":"physics","topic":"electrostatics","subtopic":"coulombs_law","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Two point charges of +2 μC and +8 μC are separated by 2 cm in vacuum. Using k=9×10⁹ N m² C⁻², find the magnitude of the electrostatic force.","options":{"A":"432 N","B":"360 N","C":"361 N","D":"359 N"},"correct_answer":"B","correct_answer_value":"360 N","solution":"Convert charges to coulombs and distance to metres. F=k|q₁q₂|/r²=9×10⁹(2×10⁻⁶)(8×10⁻⁶)/(2×10⁻²)²=360 N.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 360 N, not 432 N.","B":"Correct. The worked calculation gives 360 N.","C":"Incorrect. Substituting the stated values into the governing relation gives 360 N, not 361 N.","D":"Incorrect. Substituting the stated values into the governing relation gives 360 N, not 359 N."},"concepts_required":["Coulomb's law","SI unit conversion","inverse-square law"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_coulomb","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_005602","subject":"physics","topic":"electrostatics","subtopic":"capacitor_energy","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A 58 μF capacitor is charged to a potential difference of 6 V. Calculate the energy stored, expressed in μJ.","options":{"A":"-1044 μJ","B":"1566 μJ","C":"1252.8 μJ","D":"1044 μJ"},"correct_answer":"D","correct_answer_value":"1044 μJ","solution":"U=½CV². Since C=58 μF, U in μJ is ½×58×(6)²=1044 μJ.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 1044 μJ, not -1044 μJ.","B":"Incorrect. Substituting the stated values into the governing relation gives 1044 μJ, not 1566 μJ.","C":"Incorrect. Substituting the stated values into the governing relation gives 1044 μJ, not 1252.8 μJ.","D":"Correct. The worked calculation gives 1044 μJ."},"concepts_required":["capacitance","energy stored in a capacitor"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_capacitor_energy","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_008181","subject":"chemistry","topic":"physical_chemistry","subtopic":"faradays_law_copper","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"An aqueous Cu²⁺ solution is electrolysed using a current of 8 A for 20115 s. Assuming 100% current efficiency, calculate the mass of copper deposited. Use M(Cu)=63.5 g mol⁻¹ and F=96500 C mol⁻¹.","options":{"A":"53.94518 g","B":"42.35615 g","C":"52.94518 g","D":"63.53422 g"},"correct_answer":"C","correct_answer_value":"52.94518 g","solution":"Cu²⁺+2e⁻→Cu, so z=2. By Faraday's law m=MIt/(zF)=63.5×8×20115/(2×96500)=52.94518 g.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 52.94518 g, not 53.94518 g.","B":"Incorrect. Substituting the stated values into the governing relation gives 52.94518 g, not 42.35615 g.","C":"Correct. The worked calculation gives 52.94518 g.","D":"Incorrect. Substituting the stated values into the governing relation gives 52.94518 g, not 63.53422 g."},"concepts_required":["electrolysis","Faraday's laws","redox stoichiometry"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_faraday_copper","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_001668","subject":"mathematics","topic":"calculus","subtopic":"definite_integral_polynomial","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Evaluate the definite integral ∫₀^2 (0x² +4x +4) dx.","options":{"A":"16","B":"15","C":"32","D":"19.2"},"correct_answer":"A","correct_answer_value":"16","solution":"An antiderivative is (0/3)x³+(4/2)x²+(4)x. Evaluating between 0 and 2 gives (0/3)(2)³+(4/2)(2)²+(4)(2)=16.","option_explanations":{"A":"Correct. The worked calculation gives 16.","B":"Incorrect. Substituting the stated values into the governing relation gives 16, not 15.","C":"Incorrect. Substituting the stated values into the governing relation gives 16, not 32.","D":"Incorrect. Substituting the stated values into the governing relation gives 16, not 19.2."},"concepts_required":["definite integration","power rule","limits of integration"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_integral_poly","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_002183","subject":"mathematics","topic":"probability","subtopic":"same_colour_without_replacement","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A bag contains 6 red, 12 blue and 11 green balls. Two balls are drawn uniformly at random without replacement. What is the probability that both have the same colour?","options":{"A":"0.33498","B":"0.16749","C":"1.33498","D":"-1.66502"},"correct_answer":"A","correct_answer_value":"0.33498","solution":"Favourable pairs = C(6,2)+C(12,2)+C(11,2)=136. Total pairs=C(29,2)=406. Therefore P=136/406=0.33498.","option_explanations":{"A":"Correct. The worked calculation gives 0.33498.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.33498, not 0.16749.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.33498, not 1.33498.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.33498, not -1.66502."},"concepts_required":["classical probability","combinations","without replacement"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_probability_colours","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_006558","subject":"physics","topic":"oscillations","subtopic":"mass_spring_period_over_pi","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A mass of 8 kg is attached to an ideal spring of force constant 195 N m⁻¹ and oscillates horizontally without damping. What is the time period divided by π?","options":{"A":"0.4051 s","B":"2.4051 s","C":"-1.5949 s","D":"1.04051 s"},"correct_answer":"A","correct_answer_value":"0.4051 s","solution":"For a mass–spring oscillator, T=2π√(m/k). Therefore T/π=2√(8/195)=0.4051 s.","option_explanations":{"A":"Correct. The worked calculation gives 0.4051 s.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.4051 s, not 2.4051 s.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.4051 s, not -1.5949 s.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.4051 s, not 1.04051 s."},"concepts_required":["simple harmonic motion","mass–spring oscillator","time period"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_shm_period","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_006209","subject":"physics","topic":"ray_optics","subtopic":"convex_lens_magnification","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A real object is placed 39 cm in front of a thin convex lens of focal length 11 cm. Using the Cartesian sign convention, what is the magnitude of the linear magnification?","options":{"A":"-2.60714","B":"0.19643","C":"0.39286","D":"-0.60714"},"correct_answer":"C","correct_answer_value":"0.39286","solution":"Using distances as positive magnitudes, 1/f=1/v+1/u gives v=uf/(u−f)=(39)(11)/(39−11)=15.32143 cm. Thus |m|=v/u=15.32143/39=0.39286.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.39286, not -2.60714.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.39286, not 0.19643.","C":"Correct. The worked calculation gives 0.39286.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.39286, not -0.60714."},"concepts_required":["thin lens equation","linear magnification","real image"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_lens_magnification","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_002156","subject":"mathematics","topic":"probability","subtopic":"same_colour_without_replacement","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A bag contains 5 red, 5 blue and 3 green balls. Two balls are drawn uniformly at random without replacement. What is the probability that both have the same colour?","options":{"A":"-1.70513","B":"0.44231","C":"-0.29487","D":"0.29487"},"correct_answer":"D","correct_answer_value":"0.29487","solution":"Favourable pairs = C(5,2)+C(5,2)+C(3,2)=23. Total pairs=C(13,2)=78. Therefore P=23/78=0.29487.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.29487, not -1.70513.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.29487, not 0.44231.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.29487, not -0.29487.","D":"Correct. The worked calculation gives 0.29487."},"concepts_required":["classical probability","combinations","without replacement"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_probability_colours","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_008361","subject":"chemistry","topic":"physical_chemistry","subtopic":"boiling_point_elevation","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A non-volatile solute is dissolved in water to produce a solution of molality 1.04 mol kg⁻¹. If its van't Hoff factor is 4 and K_b for water is 0.52 K kg mol⁻¹, find the boiling-point elevation.","options":{"A":"0.1632 K","B":"1.0816 K","C":"2.1632 K","D":"-2.1632 K"},"correct_answer":"C","correct_answer_value":"2.1632 K","solution":"ΔT_b=iK_bm. Substitution gives ΔT_b=4×0.52×1.04=2.1632 K.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 2.1632 K, not 0.1632 K.","B":"Incorrect. Substituting the stated values into the governing relation gives 2.1632 K, not 1.0816 K.","C":"Correct. The worked calculation gives 2.1632 K.","D":"Incorrect. Substituting the stated values into the governing relation gives 2.1632 K, not -2.1632 K."},"concepts_required":["colligative properties","van't Hoff factor","molality"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_boiling_elevation","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_008541","subject":"chemistry","topic":"physical_chemistry","subtopic":"boiling_point_elevation","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A non-volatile solute is dissolved in water to produce a solution of molality 1.42 mol kg⁻¹. If its van't Hoff factor is 3 and K_b for water is 0.512 K kg mol⁻¹, find the boiling-point elevation.","options":{"A":"0.18112 K","B":"4.36224 K","C":"2.18112 K","D":"4.18112 K"},"correct_answer":"C","correct_answer_value":"2.18112 K","solution":"ΔT_b=iK_bm. Substitution gives ΔT_b=3×0.512×1.42=2.18112 K.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 2.18112 K, not 0.18112 K.","B":"Incorrect. Substituting the stated values into the governing relation gives 2.18112 K, not 4.36224 K.","C":"Correct. The worked calculation gives 2.18112 K.","D":"Incorrect. Substituting the stated values into the governing relation gives 2.18112 K, not 4.18112 K."},"concepts_required":["colligative properties","van't Hoff factor","molality"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_boiling_elevation","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_000481","subject":"mathematics","topic":"sequences_and_series","subtopic":"arithmetic_progression","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"An arithmetic progression has first term -13 and common difference 4. Find the sum of its first 16 terms.","options":{"A":"272","B":"326.4","C":"217.6","D":"270"},"correct_answer":"A","correct_answer_value":"272","solution":"Use Sₙ = n/2[2a+(n−1)d]. Substituting a=-13, d=4, n=16: Sₙ=16/2[2(-13)+(16−1)(4)]=272.","option_explanations":{"A":"Correct. The worked calculation gives 272.","B":"Incorrect. Substituting the stated values into the governing relation gives 272, not 326.4.","C":"Incorrect. Substituting the stated values into the governing relation gives 272, not 217.6.","D":"Incorrect. Substituting the stated values into the governing relation gives 272, not 270."},"concepts_required":["arithmetic progression","finite series"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_ap_sum","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_007271","subject":"chemistry","topic":"physical_chemistry","subtopic":"ideal_gas_volume","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Assuming ideal behaviour, what volume is occupied by 11 mol of gas at 209 K and 0.57 atm? Use R=0.082057 L atm mol⁻¹ K⁻¹.","options":{"A":"661.92647 L","B":"330.96323 L","C":"329.96323 L","D":"397.15588 L"},"correct_answer":"B","correct_answer_value":"330.96323 L","solution":"PV=nRT, so V=nRT/P=(11)(0.082057)(209)/(0.57)=330.96323 L.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 330.96323 L, not 661.92647 L.","B":"Correct. The worked calculation gives 330.96323 L.","C":"Incorrect. Substituting the stated values into the governing relation gives 330.96323 L, not 329.96323 L.","D":"Incorrect. Substituting the stated values into the governing relation gives 330.96323 L, not 397.15588 L."},"concepts_required":["ideal gas equation","unit consistency"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_ideal_gas","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_004107","subject":"physics","topic":"mechanics","subtopic":"inclined_plane_with_friction","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A 41 kg block slides down a 37° rough incline. The coefficient of kinetic friction is 0.12. Taking g=10 m s⁻², find the net force down the plane. Use sin37°=0.6 and cos37°=0.8.","options":{"A":"103.32 N","B":"309.96 N","C":"206.64 N","D":"154.98 N"},"correct_answer":"C","correct_answer_value":"206.64 N","solution":"Net force=mg(sinθ−μcosθ)=41×10×(0.6−0.12×0.8)=206.64 N.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 206.64 N, not 103.32 N.","B":"Incorrect. Substituting the stated values into the governing relation gives 206.64 N, not 309.96 N.","C":"Correct. The worked calculation gives 206.64 N.","D":"Incorrect. Substituting the stated values into the governing relation gives 206.64 N, not 154.98 N."},"concepts_required":["Newton's laws","inclined plane","kinetic friction"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_incline_friction","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_006886","subject":"chemistry","topic":"stoichiometry","subtopic":"limiting_reagent_water","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Hydrogen reacts with oxygen according to 2H₂ + O₂ → 2H₂O. If 6 mol H₂ and 5 mol O₂ are mixed and the reaction goes to completion, how many moles of H₂O form?","options":{"A":"5 mol","B":"6 mol","C":"1.6 mol","D":"8 mol"},"correct_answer":"B","correct_answer_value":"6 mol","solution":"From 2 mol H₂ and 1 mol O₂, 2 mol H₂O form. H₂ can produce 6 mol H₂O; O₂ can produce 10 mol H₂O. The limiting condition is H₂; product=6 mol.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 6 mol, not 5 mol.","B":"Correct. The worked calculation gives 6 mol.","C":"Incorrect. Substituting the stated values into the governing relation gives 6 mol, not 1.6 mol.","D":"Incorrect. Substituting the stated values into the governing relation gives 6 mol, not 8 mol."},"concepts_required":["balanced chemical equations","limiting reagent","mole ratios"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_limiting_reagent","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_009093","subject":"chemistry","topic":"physical_chemistry","subtopic":"nernst_equation_298K","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"At 298 K, a galvanic cell has E°cell=1.07 V, transfers 3 electrons, and has reaction quotient Q=10^6. Using E=E°−(0.0591/n)log₁₀Q, calculate Ecell.","options":{"A":"-0.0482 V","B":"0.9518 V","C":"3.9518 V","D":"2.9518 V"},"correct_answer":"B","correct_answer_value":"0.9518 V","solution":"Apply the Nernst equation: E=1.07−(0.0591/3)(6)=0.9518 V.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.9518 V, not -0.0482 V.","B":"Correct. The worked calculation gives 0.9518 V.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.9518 V, not 3.9518 V.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.9518 V, not 2.9518 V."},"concepts_required":["electrochemical cells","Nernst equation","logarithms"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_nernst","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_008803","subject":"chemistry","topic":"physical_chemistry","subtopic":"equilibrium_degree_dissociation","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"For A(g) ⇌ 2B(g), the initial concentration of A is 1.29 mol L⁻¹ and initially B is absent. At equilibrium, 40% of A has dissociated at constant volume. Find K_c.","options":{"A":"3.376","B":"1.376","C":"0.688","D":"2.752"},"correct_answer":"B","correct_answer_value":"1.376","solution":"Let α=0.4. At equilibrium [A]=c₀(1−α) and [B]=2c₀α. Thus K_c=[B]²/[A]=4c₀α²/(1−α)=4(1.29)(0.4)²/(1−0.4)=1.376.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 1.376, not 3.376.","B":"Correct. The worked calculation gives 1.376.","C":"Incorrect. Substituting the stated values into the governing relation gives 1.376, not 0.688.","D":"Incorrect. Substituting the stated values into the governing relation gives 1.376, not 2.752."},"concepts_required":["chemical equilibrium","equilibrium constant","degree of dissociation"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_equilibrium_alpha","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_008723","subject":"chemistry","topic":"physical_chemistry","subtopic":"equilibrium_degree_dissociation","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"For A(g) ⇌ 2B(g), the initial concentration of A is 0.25 mol L⁻¹ and initially B is absent. At equilibrium, 47% of A has dissociated at constant volume. Find K_c.","options":{"A":"0.83358","B":"0.41679","C":"0.31259","D":"0.62519"},"correct_answer":"B","correct_answer_value":"0.41679","solution":"Let α=0.47. At equilibrium [A]=c₀(1−α) and [B]=2c₀α. Thus K_c=[B]²/[A]=4c₀α²/(1−α)=4(0.25)(0.47)²/(1−0.47)=0.41679.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.41679, not 0.83358.","B":"Correct. The worked calculation gives 0.41679.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.41679, not 0.31259.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.41679, not 0.62519."},"concepts_required":["chemical equilibrium","equilibrium constant","degree of dissociation"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_equilibrium_alpha","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_006893","subject":"chemistry","topic":"stoichiometry","subtopic":"limiting_reagent_water","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Hydrogen reacts with oxygen according to 2H₂ + O₂ → 2H₂O. If 18 mol H₂ and 22 mol O₂ are mixed and the reaction goes to completion, how many moles of H₂O form?","options":{"A":"18 mol","B":"36 mol","C":"-18 mol","D":"20 mol"},"correct_answer":"A","correct_answer_value":"18 mol","solution":"From 2 mol H₂ and 1 mol O₂, 2 mol H₂O form. H₂ can produce 18 mol H₂O; O₂ can produce 44 mol H₂O. The limiting condition is H₂; product=18 mol.","option_explanations":{"A":"Correct. The worked calculation gives 18 mol.","B":"Incorrect. Substituting the stated values into the governing relation gives 18 mol, not 36 mol.","C":"Incorrect. Substituting the stated values into the governing relation gives 18 mol, not -18 mol.","D":"Incorrect. Substituting the stated values into the governing relation gives 18 mol, not 20 mol."},"concepts_required":["balanced chemical equations","limiting reagent","mole ratios"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_limiting_reagent","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_007658","subject":"chemistry","topic":"physical_chemistry","subtopic":"strong_acid_base_mixture","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"83 mL of 0.10 M HCl is mixed with 82 mL of 0.10 M NaOH at 25°C. Assuming complete dissociation and additive volumes, find the final pH.","options":{"A":"2.21748","B":"3.21748","C":"5.21748","D":"6.43497"},"correct_answer":"B","correct_answer_value":"3.21748","solution":"Excess H⁺ moles=0.10×(83−82)/1000=0.000100 mol. Total volume=0.165 L, so [H⁺]=0.000606 M and pH=−log₁₀[H⁺]=3.21748.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 3.21748, not 2.21748.","B":"Correct. The worked calculation gives 3.21748.","C":"Incorrect. Substituting the stated values into the governing relation gives 3.21748, not 5.21748.","D":"Incorrect. Substituting the stated values into the governing relation gives 3.21748, not 6.43497."},"concepts_required":["strong acid–strong base neutralization","molarity","pH and pOH"],"reasoning_steps":5,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_acid_base_pH","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_006385","subject":"physics","topic":"oscillations","subtopic":"mass_spring_period_over_pi","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A mass of 18 kg is attached to an ideal spring of force constant 451 N m⁻¹ and oscillates horizontally without damping. What is the time period divided by π?","options":{"A":"3.39956 s","B":"0.39956 s","C":"-0.39956 s","D":"0.59933 s"},"correct_answer":"B","correct_answer_value":"0.39956 s","solution":"For a mass–spring oscillator, T=2π√(m/k). Therefore T/π=2√(18/451)=0.39956 s.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.39956 s, not 3.39956 s.","B":"Correct. The worked calculation gives 0.39956 s.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.39956 s, not -0.39956 s.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.39956 s, not 0.59933 s."},"concepts_required":["simple harmonic motion","mass–spring oscillator","time period"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_shm_period","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_007887","subject":"chemistry","topic":"physical_chemistry","subtopic":"first_order_half_life","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A first-order reaction has a half-life of 20 minutes. If the initial reactant concentration is 72 mmol L⁻¹, what concentration remains after 20 minutes?","options":{"A":"36 mmol L⁻¹","B":"54 mmol L⁻¹","C":"43.2 mmol L⁻¹","D":"37 mmol L⁻¹"},"correct_answer":"A","correct_answer_value":"36 mmol L⁻¹","solution":"20 minutes corresponds to 1 half-lives. The remaining concentration is [A]₀(1/2)ⁿ=72/2^1=36 mmol L⁻¹.","option_explanations":{"A":"Correct. The worked calculation gives 36 mmol L⁻¹.","B":"Incorrect. Substituting the stated values into the governing relation gives 36 mmol L⁻¹, not 54 mmol L⁻¹.","C":"Incorrect. Substituting the stated values into the governing relation gives 36 mmol L⁻¹, not 43.2 mmol L⁻¹.","D":"Incorrect. Substituting the stated values into the governing relation gives 36 mmol L⁻¹, not 37 mmol L⁻¹."},"concepts_required":["first-order kinetics","half-life","exponential decay"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_first_order_halflife","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_004627","subject":"physics","topic":"rotational_mechanics","subtopic":"disc_angular_acceleration","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A uniform solid disc of mass 4 kg and radius 2 m is acted on by a constant torque of 7 N m about its central axis. Find its angular acceleration.","options":{"A":"-1.125 rad s⁻²","B":"0.875 rad s⁻²","C":"1.3125 rad s⁻²","D":"3.875 rad s⁻²"},"correct_answer":"B","correct_answer_value":"0.875 rad s⁻²","solution":"For a solid disc, I=½MR²=½(4)(2)²=8 kg m². Using τ=Iα, α=7/8=0.875 rad s⁻².","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.875 rad s⁻², not -1.125 rad s⁻².","B":"Correct. The worked calculation gives 0.875 rad s⁻².","C":"Incorrect. Substituting the stated values into the governing relation gives 0.875 rad s⁻², not 1.3125 rad s⁻².","D":"Incorrect. Substituting the stated values into the governing relation gives 0.875 rad s⁻², not 3.875 rad s⁻²."},"concepts_required":["moment of inertia","rotational dynamics","torque"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_disc_torque","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_001226","subject":"mathematics","topic":"calculus","subtopic":"polynomial_differentiation","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Let f(x)=3x³ -6x² -5x+7. Evaluate f′(-4).","options":{"A":"93.5","B":"187","C":"186","D":"188"},"correct_answer":"B","correct_answer_value":"187","solution":"Differentiate: f′(x)=3(3)x²+2(-6)x+(-5). At x=-4, f′(-4)=3(3)(-4)²+2(-6)(-4)+(-5)=187.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 187, not 93.5.","B":"Correct. The worked calculation gives 187.","C":"Incorrect. Substituting the stated values into the governing relation gives 187, not 186.","D":"Incorrect. Substituting the stated values into the governing relation gives 187, not 188."},"concepts_required":["power rule","polynomial derivative"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_derivative_cubic","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_003416","subject":"physics","topic":"mechanics","subtopic":"projectile_range","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A projectile is launched from level ground at 30 m s⁻¹ at 70° to the horizontal. Take g=10 m s⁻². Find its horizontal range.","options":{"A":"57.85088 m","B":"86.77633 m","C":"55.85088 m","D":"6.78509 m"},"correct_answer":"A","correct_answer_value":"57.85088 m","solution":"Range R=u²sin(2θ)/g. Substituting u=30, θ=70° and g=10 gives R=(30)²sin(140°)/10=57.85088 m.","option_explanations":{"A":"Correct. The worked calculation gives 57.85088 m.","B":"Incorrect. Substituting the stated values into the governing relation gives 57.85088 m, not 86.77633 m.","C":"Incorrect. Substituting the stated values into the governing relation gives 57.85088 m, not 55.85088 m.","D":"Incorrect. Substituting the stated values into the governing relation gives 57.85088 m, not 6.78509 m."},"concepts_required":["projectile motion","range equation","trigonometry"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_projectile_range","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_008599","subject":"chemistry","topic":"physical_chemistry","subtopic":"boiling_point_elevation","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A non-volatile solute is dissolved in water to produce a solution of molality 0.11 mol kg⁻¹. If its van't Hoff factor is 3 and K_b for water is 0.52 K kg mol⁻¹, find the boiling-point elevation.","options":{"A":"0.2574 K","B":"0.1716 K","C":"1.1716 K","D":"0.1287 K"},"correct_answer":"B","correct_answer_value":"0.1716 K","solution":"ΔT_b=iK_bm. Substitution gives ΔT_b=3×0.52×0.11=0.1716 K.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.1716 K, not 0.2574 K.","B":"Correct. The worked calculation gives 0.1716 K.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.1716 K, not 1.1716 K.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.1716 K, not 0.1287 K."},"concepts_required":["colligative properties","van't Hoff factor","molality"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_boiling_elevation","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_003883","subject":"physics","topic":"mechanics","subtopic":"work_energy_theorem","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A 2 kg block initially moves at 6 m s⁻¹. A constant net force does 11 N of work over 19 m in the direction of motion. Find its final kinetic energy.","options":{"A":"294 J","B":"-245 J","C":"247 J","D":"245 J"},"correct_answer":"D","correct_answer_value":"245 J","solution":"By the work–energy theorem, K_f=K_i+W=½mu²+Fs=½(2)(6)²+(11)(19)=245 J.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 245 J, not 294 J.","B":"Incorrect. Substituting the stated values into the governing relation gives 245 J, not -245 J.","C":"Incorrect. Substituting the stated values into the governing relation gives 245 J, not 247 J.","D":"Correct. The worked calculation gives 245 J."},"concepts_required":["work–energy theorem","kinetic energy","work by constant force"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_work_energy","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_004439","subject":"physics","topic":"rotational_mechanics","subtopic":"disc_angular_acceleration","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A uniform solid disc of mass 9 kg and radius 5 m is acted on by a constant torque of 8 N m about its central axis. Find its angular acceleration.","options":{"A":"-1.92889 rad s⁻²","B":"3.07111 rad s⁻²","C":"0.07111 rad s⁻²","D":"2.07111 rad s⁻²"},"correct_answer":"C","correct_answer_value":"0.07111 rad s⁻²","solution":"For a solid disc, I=½MR²=½(9)(5)²=112.5 kg m². Using τ=Iα, α=8/112.5=0.07111 rad s⁻².","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.07111 rad s⁻², not -1.92889 rad s⁻².","B":"Incorrect. Substituting the stated values into the governing relation gives 0.07111 rad s⁻², not 3.07111 rad s⁻².","C":"Correct. The worked calculation gives 0.07111 rad s⁻².","D":"Incorrect. Substituting the stated values into the governing relation gives 0.07111 rad s⁻², not 2.07111 rad s⁻²."},"concepts_required":["moment of inertia","rotational dynamics","torque"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_disc_torque","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_006506","subject":"physics","topic":"oscillations","subtopic":"mass_spring_period_over_pi","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A mass of 5 kg is attached to an ideal spring of force constant 1053 N m⁻¹ and oscillates horizontally without damping. What is the time period divided by π?","options":{"A":"-0.13782 s","B":"0.13782 s","C":"1.13782 s","D":"0.27563 s"},"correct_answer":"B","correct_answer_value":"0.13782 s","solution":"For a mass–spring oscillator, T=2π√(m/k). Therefore T/π=2√(5/1053)=0.13782 s.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.13782 s, not -0.13782 s.","B":"Correct. The worked calculation gives 0.13782 s.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.13782 s, not 1.13782 s.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.13782 s, not 0.27563 s."},"concepts_required":["simple harmonic motion","mass–spring oscillator","time period"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_shm_period","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_001962","subject":"mathematics","topic":"combinatorics","subtopic":"binomial_coefficient","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"How many distinct subsets containing exactly 2 elements can be selected from a set of 69 distinct elements?","options":{"A":"2346","B":"2345","C":"1876.8","D":"-2346"},"correct_answer":"A","correct_answer_value":"2346","solution":"Order does not matter, so the count is C(69,2)=69!/[2!(67)!]=2346.","option_explanations":{"A":"Correct. The worked calculation gives 2346.","B":"Incorrect. Substituting the stated values into the governing relation gives 2346, not 2345.","C":"Incorrect. Substituting the stated values into the governing relation gives 2346, not 1876.8.","D":"Incorrect. Substituting the stated values into the governing relation gives 2346, not -2346."},"concepts_required":["combinations","factorials"],"reasoning_steps":2,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_combinations","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_001221","subject":"mathematics","topic":"calculus","subtopic":"polynomial_differentiation","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Let f(x)=-2x³ -9x² -8x+7. Evaluate f′(-2).","options":{"A":"2","B":"7","C":"3","D":"4"},"correct_answer":"D","correct_answer_value":"4","solution":"Differentiate: f′(x)=3(-2)x²+2(-9)x+(-8). At x=-2, f′(-2)=3(-2)(-2)²+2(-9)(-2)+(-8)=4.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 4, not 2.","B":"Incorrect. Substituting the stated values into the governing relation gives 4, not 7.","C":"Incorrect. Substituting the stated values into the governing relation gives 4, not 3.","D":"Correct. The worked calculation gives 4."},"concepts_required":["power rule","polynomial derivative"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_derivative_cubic","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_000449","subject":"mathematics","topic":"sequences_and_series","subtopic":"arithmetic_progression","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"An arithmetic progression has first term 2 and common difference 14. Find the sum of its first 13 terms.","options":{"A":"2236","B":"559","C":"1119","D":"1118"},"correct_answer":"D","correct_answer_value":"1118","solution":"Use Sₙ = n/2[2a+(n−1)d]. Substituting a=2, d=14, n=13: Sₙ=13/2[2(2)+(13−1)(14)]=1118.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 1118, not 2236.","B":"Incorrect. Substituting the stated values into the governing relation gives 1118, not 559.","C":"Incorrect. Substituting the stated values into the governing relation gives 1118, not 1119.","D":"Correct. The worked calculation gives 1118."},"concepts_required":["arithmetic progression","finite series"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_ap_sum","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_000744","subject":"mathematics","topic":"sequences_and_series","subtopic":"geometric_progression","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A geometric progression has first term 8 and common ratio 5. What is the sum of its first 12 terms?","options":{"A":"488281247","B":"-488281248","C":"390624998.4","D":"488281248"},"correct_answer":"D","correct_answer_value":"488281248","solution":"For r≠1, Sₙ=a(rⁿ−1)/(r−1). Thus Sₙ=8(5^12−1)/(5−1)=488281248.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 488281248, not 488281247.","B":"Incorrect. Substituting the stated values into the governing relation gives 488281248, not -488281248.","C":"Incorrect. Substituting the stated values into the governing relation gives 488281248, not 390624998.4.","D":"Correct. The worked calculation gives 488281248."},"concepts_required":["geometric progression","finite geometric sum"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_gp_sum","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_009701","subject":"chemistry","topic":"inorganic_chemistry","subtopic":"complex_ion_oxidation_state","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"For a hypothetical coordination ion [Cu(OH)_4]^-4, each OH ligand has charge −1. Determine the formal oxidation state of Cu.","options":{"A":"3","B":"0","C":"1","D":"-1"},"correct_answer":"B","correct_answer_value":"0","solution":"Let the formal oxidation state of Cu be x. Charge balance gives x+(4)(−1)=-4; hence x=-4+4=+0.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0, not 3.","B":"Correct. The worked calculation gives 0.","C":"Incorrect. Substituting the stated values into the governing relation gives 0, not 1.","D":"Incorrect. Substituting the stated values into the governing relation gives 0, not -1."},"concepts_required":["coordination compounds","oxidation state","charge balance"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_complex_oxidation","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_008595","subject":"chemistry","topic":"physical_chemistry","subtopic":"boiling_point_elevation","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A non-volatile solute is dissolved in water to produce a solution of molality 0.5 mol kg⁻¹. If its van't Hoff factor is 4 and K_b for water is 1.0 K kg mol⁻¹, find the boiling-point elevation.","options":{"A":"1 K","B":"-1 K","C":"2 K","D":"3 K"},"correct_answer":"C","correct_answer_value":"2 K","solution":"ΔT_b=iK_bm. Substitution gives ΔT_b=4×1.0×0.5=2 K.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 2 K, not 1 K.","B":"Incorrect. Substituting the stated values into the governing relation gives 2 K, not -1 K.","C":"Correct. The worked calculation gives 2 K.","D":"Incorrect. Substituting the stated values into the governing relation gives 2 K, not 3 K."},"concepts_required":["colligative properties","van't Hoff factor","molality"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_boiling_elevation","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_002124","subject":"mathematics","topic":"probability","subtopic":"same_colour_without_replacement","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A bag contains 5 red, 11 blue and 8 green balls. Two balls are drawn uniformly at random without replacement. What is the probability that both have the same colour?","options":{"A":"0.16848","B":"3.33696","C":"1.33696","D":"0.33696"},"correct_answer":"D","correct_answer_value":"0.33696","solution":"Favourable pairs = C(5,2)+C(11,2)+C(8,2)=93. Total pairs=C(24,2)=276. Therefore P=93/276=0.33696.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.33696, not 0.16848.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.33696, not 3.33696.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.33696, not 1.33696.","D":"Correct. The worked calculation gives 0.33696."},"concepts_required":["classical probability","combinations","without replacement"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_probability_colours","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_005725","subject":"physics","topic":"thermodynamics","subtopic":"ideal_gas_isochoric_pressure","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A fixed amount of ideal gas is kept in a rigid container. Its pressure is 4.3 atm at 423 K. What is its pressure at 761 K?","options":{"A":"15.47187 atm","B":"8.73593 atm","C":"7.73593 atm","D":"5.80195 atm"},"correct_answer":"C","correct_answer_value":"7.73593 atm","solution":"At constant volume, P/T is constant. P₂=P₁T₂/T₁=(4.3)(761)/(423)=7.73593 atm.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 7.73593 atm, not 15.47187 atm.","B":"Incorrect. Substituting the stated values into the governing relation gives 7.73593 atm, not 8.73593 atm.","C":"Correct. The worked calculation gives 7.73593 atm.","D":"Incorrect. Substituting the stated values into the governing relation gives 7.73593 atm, not 5.80195 atm."},"concepts_required":["ideal gas law","isochoric process","absolute temperature"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_isochoric_gas","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_001503","subject":"mathematics","topic":"calculus","subtopic":"definite_integral_polynomial","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Evaluate the definite integral ∫₀^7 (-1x² +5x +5) dx.","options":{"A":"51.8","B":"-43.16667","C":"64.75","D":"43.16667"},"correct_answer":"D","correct_answer_value":"43.16667","solution":"An antiderivative is (-1/3)x³+(5/2)x²+(5)x. Evaluating between 0 and 7 gives (-1/3)(7)³+(5/2)(7)²+(5)(7)=43.16667.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 43.16667, not 51.8.","B":"Incorrect. Substituting the stated values into the governing relation gives 43.16667, not -43.16667.","C":"Incorrect. Substituting the stated values into the governing relation gives 43.16667, not 64.75.","D":"Correct. The worked calculation gives 43.16667."},"concepts_required":["definite integration","power rule","limits of integration"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_integral_poly","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_008001","subject":"chemistry","topic":"physical_chemistry","subtopic":"first_order_half_life","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A first-order reaction has a half-life of 20 minutes. If the initial reactant concentration is 80 mmol L⁻¹, what concentration remains after 140 minutes?","options":{"A":"-2.375 mmol L⁻¹","B":"1.25 mmol L⁻¹","C":"1.0625 mmol L⁻¹","D":"0.625 mmol L⁻¹"},"correct_answer":"D","correct_answer_value":"0.625 mmol L⁻¹","solution":"140 minutes corresponds to 7 half-lives. The remaining concentration is [A]₀(1/2)ⁿ=80/2^7=0.625 mmol L⁻¹.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.625 mmol L⁻¹, not -2.375 mmol L⁻¹.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.625 mmol L⁻¹, not 1.25 mmol L⁻¹.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.625 mmol L⁻¹, not 1.0625 mmol L⁻¹.","D":"Correct. The worked calculation gives 0.625 mmol L⁻¹."},"concepts_required":["first-order kinetics","half-life","exponential decay"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_first_order_halflife","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_003483","subject":"physics","topic":"mechanics","subtopic":"projectile_range","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A projectile is launched from level ground at 76 m s⁻¹ at 70° to the horizontal. Take g=10 m s⁻². Find its horizontal range.","options":{"A":"-371.27412 m","B":"278.45559 m","C":"445.52895 m","D":"371.27412 m"},"correct_answer":"D","correct_answer_value":"371.27412 m","solution":"Range R=u²sin(2θ)/g. Substituting u=76, θ=70° and g=10 gives R=(76)²sin(140°)/10=371.27412 m.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 371.27412 m, not -371.27412 m.","B":"Incorrect. Substituting the stated values into the governing relation gives 371.27412 m, not 278.45559 m.","C":"Incorrect. Substituting the stated values into the governing relation gives 371.27412 m, not 445.52895 m.","D":"Correct. The worked calculation gives 371.27412 m."},"concepts_required":["projectile motion","range equation","trigonometry"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_projectile_range","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_002626","subject":"mathematics","topic":"linear_algebra","subtopic":"2x2_determinant","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Find the determinant of the matrix [[7, 5], [8, 8]].","options":{"A":"17","B":"-16","C":"16","D":"15"},"correct_answer":"C","correct_answer_value":"16","solution":"For a 2×2 matrix [[a,b],[c,d]], det=ad−bc. Here it is (7)(8)−(5)(8)=16.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 16, not 17.","B":"Incorrect. Substituting the stated values into the governing relation gives 16, not -16.","C":"Correct. The worked calculation gives 16.","D":"Incorrect. Substituting the stated values into the governing relation gives 16, not 15."},"concepts_required":["determinants","matrix operations"],"reasoning_steps":2,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_determinant_2x2","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_008226","subject":"chemistry","topic":"physical_chemistry","subtopic":"faradays_law_copper","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"An aqueous Cu²⁺ solution is electrolysed using a current of 2 A for 2119 s. Assuming 100% current efficiency, calculate the mass of copper deposited. Use M(Cu)=63.5 g mol⁻¹ and F=96500 C mol⁻¹.","options":{"A":"1.04578 g","B":"3.39437 g","C":"1.39437 g","D":"2.09155 g"},"correct_answer":"C","correct_answer_value":"1.39437 g","solution":"Cu²⁺+2e⁻→Cu, so z=2. By Faraday's law m=MIt/(zF)=63.5×2×2119/(2×96500)=1.39437 g.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 1.39437 g, not 1.04578 g.","B":"Incorrect. Substituting the stated values into the governing relation gives 1.39437 g, not 3.39437 g.","C":"Correct. The worked calculation gives 1.39437 g.","D":"Incorrect. Substituting the stated values into the governing relation gives 1.39437 g, not 2.09155 g."},"concepts_required":["electrolysis","Faraday's laws","redox stoichiometry"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_faraday_copper","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_001702","subject":"mathematics","topic":"combinatorics","subtopic":"binomial_coefficient","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"How many distinct subsets containing exactly 12 elements can be selected from a set of 54 distinct elements?","options":{"A":"343006888770","B":"686013777540","C":"34300688878","D":"411608266524"},"correct_answer":"A","correct_answer_value":"343006888770","solution":"Order does not matter, so the count is C(54,12)=54!/[12!(42)!]=343006888770.","option_explanations":{"A":"Correct. The worked calculation gives 343006888770.","B":"Incorrect. Substituting the stated values into the governing relation gives 343006888770, not 686013777540.","C":"Incorrect. Substituting the stated values into the governing relation gives 343006888770, not 34300688878.","D":"Incorrect. Substituting the stated values into the governing relation gives 343006888770, not 411608266524."},"concepts_required":["combinations","factorials"],"reasoning_steps":2,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_combinations","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_007359","subject":"chemistry","topic":"physical_chemistry","subtopic":"strong_acid_base_mixture","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"72 mL of 0.10 M HCl is mixed with 67 mL of 0.10 M NaOH at 25°C. Assuming complete dissociation and additive volumes, find the final pH.","options":{"A":"2.44404","B":"0.44404","C":"1.22202","D":"1.44404"},"correct_answer":"A","correct_answer_value":"2.44404","solution":"Excess H⁺ moles=0.10×(72−67)/1000=0.000500 mol. Total volume=0.139 L, so [H⁺]=0.003597 M and pH=−log₁₀[H⁺]=2.44404.","option_explanations":{"A":"Correct. The worked calculation gives 2.44404.","B":"Incorrect. Substituting the stated values into the governing relation gives 2.44404, not 0.44404.","C":"Incorrect. Substituting the stated values into the governing relation gives 2.44404, not 1.22202.","D":"Incorrect. Substituting the stated values into the governing relation gives 2.44404, not 1.44404."},"concepts_required":["strong acid–strong base neutralization","molarity","pH and pOH"],"reasoning_steps":5,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_acid_base_pH","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_005186","subject":"physics","topic":"electrostatics","subtopic":"coulombs_law","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Two point charges of +8 μC and +8 μC are separated by 11 cm in vacuum. Using k=9×10⁹ N m² C⁻², find the magnitude of the electrostatic force.","options":{"A":"38.08264 N","B":"48.60331 N","C":"45.60331 N","D":"47.60331 N"},"correct_answer":"D","correct_answer_value":"47.60331 N","solution":"Convert charges to coulombs and distance to metres. F=k|q₁q₂|/r²=9×10⁹(8×10⁻⁶)(8×10⁻⁶)/(11×10⁻²)²=47.60331 N.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 47.60331 N, not 38.08264 N.","B":"Incorrect. Substituting the stated values into the governing relation gives 47.60331 N, not 48.60331 N.","C":"Incorrect. Substituting the stated values into the governing relation gives 47.60331 N, not 45.60331 N.","D":"Correct. The worked calculation gives 47.60331 N."},"concepts_required":["Coulomb's law","SI unit conversion","inverse-square law"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_coulomb","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_004384","subject":"physics","topic":"rotational_mechanics","subtopic":"disc_angular_acceleration","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A uniform solid disc of mass 13 kg and radius 2 m is acted on by a constant torque of 60 N m about its central axis. Find its angular acceleration.","options":{"A":"3.30769 rad s⁻²","B":"0.30769 rad s⁻²","C":"2.30769 rad s⁻²","D":"1.30769 rad s⁻²"},"correct_answer":"C","correct_answer_value":"2.30769 rad s⁻²","solution":"For a solid disc, I=½MR²=½(13)(2)²=26 kg m². Using τ=Iα, α=60/26=2.30769 rad s⁻².","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 2.30769 rad s⁻², not 3.30769 rad s⁻².","B":"Incorrect. Substituting the stated values into the governing relation gives 2.30769 rad s⁻², not 0.30769 rad s⁻².","C":"Correct. The worked calculation gives 2.30769 rad s⁻².","D":"Incorrect. Substituting the stated values into the governing relation gives 2.30769 rad s⁻², not 1.30769 rad s⁻²."},"concepts_required":["moment of inertia","rotational dynamics","torque"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_disc_torque","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_005753","subject":"physics","topic":"thermodynamics","subtopic":"ideal_gas_isochoric_pressure","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A fixed amount of ideal gas is kept in a rigid container. Its pressure is 4 atm at 325 K. What is its pressure at 740 K?","options":{"A":"12.10769 atm","B":"8.10769 atm","C":"10.10769 atm","D":"9.10769 atm"},"correct_answer":"D","correct_answer_value":"9.10769 atm","solution":"At constant volume, P/T is constant. P₂=P₁T₂/T₁=(4)(740)/(325)=9.10769 atm.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 9.10769 atm, not 12.10769 atm.","B":"Incorrect. Substituting the stated values into the governing relation gives 9.10769 atm, not 8.10769 atm.","C":"Incorrect. Substituting the stated values into the governing relation gives 9.10769 atm, not 10.10769 atm.","D":"Correct. The worked calculation gives 9.10769 atm."},"concepts_required":["ideal gas law","isochoric process","absolute temperature"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_isochoric_gas","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_003494","subject":"physics","topic":"mechanics","subtopic":"projectile_range","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A projectile is launched from level ground at 115 m s⁻¹ at 40° to the horizontal. Take g=10 m s⁻². Find its horizontal range.","options":{"A":"1301.40825 m","B":"2604.81651 m","C":"1302.40825 m","D":"131.24083 m"},"correct_answer":"C","correct_answer_value":"1302.40825 m","solution":"Range R=u²sin(2θ)/g. Substituting u=115, θ=40° and g=10 gives R=(115)²sin(80°)/10=1302.40825 m.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 1302.40825 m, not 1301.40825 m.","B":"Incorrect. Substituting the stated values into the governing relation gives 1302.40825 m, not 2604.81651 m.","C":"Correct. The worked calculation gives 1302.40825 m.","D":"Incorrect. Substituting the stated values into the governing relation gives 1302.40825 m, not 131.24083 m."},"concepts_required":["projectile motion","range equation","trigonometry"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_projectile_range","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_008847","subject":"chemistry","topic":"physical_chemistry","subtopic":"equilibrium_degree_dissociation","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"For A(g) ⇌ 2B(g), the initial concentration of A is 1.96 mol L⁻¹ and initially B is absent. At equilibrium, 57% of A has dissociated at constant volume. Find K_c.","options":{"A":"-5.92376","B":"5.92376","C":"3.92376","D":"2.96188"},"correct_answer":"B","correct_answer_value":"5.92376","solution":"Let α=0.57. At equilibrium [A]=c₀(1−α) and [B]=2c₀α. Thus K_c=[B]²/[A]=4c₀α²/(1−α)=4(1.96)(0.57)²/(1−0.57)=5.92376.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 5.92376, not -5.92376.","B":"Correct. The worked calculation gives 5.92376.","C":"Incorrect. Substituting the stated values into the governing relation gives 5.92376, not 3.92376.","D":"Incorrect. Substituting the stated values into the governing relation gives 5.92376, not 2.96188."},"concepts_required":["chemical equilibrium","equilibrium constant","degree of dissociation"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_equilibrium_alpha","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_002039","subject":"mathematics","topic":"probability","subtopic":"same_colour_without_replacement","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A bag contains 5 red, 7 blue and 5 green balls. Two balls are drawn uniformly at random without replacement. What is the probability that both have the same colour?","options":{"A":"0.30147","B":"1.30147","C":"0.60294","D":"3.30147"},"correct_answer":"A","correct_answer_value":"0.30147","solution":"Favourable pairs = C(5,2)+C(7,2)+C(5,2)=41. Total pairs=C(17,2)=136. Therefore P=41/136=0.30147.","option_explanations":{"A":"Correct. The worked calculation gives 0.30147.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.30147, not 1.30147.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.30147, not 0.60294.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.30147, not 3.30147."},"concepts_required":["classical probability","combinations","without replacement"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_probability_colours","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_003681","subject":"physics","topic":"mechanics","subtopic":"work_energy_theorem","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A 4 kg block initially moves at 0 m s⁻¹. A constant net force does 20 N of work over 2 m in the direction of motion. Find its final kinetic energy.","options":{"A":"20 J","B":"39 J","C":"80 J","D":"40 J"},"correct_answer":"D","correct_answer_value":"40 J","solution":"By the work–energy theorem, K_f=K_i+W=½mu²+Fs=½(4)(0)²+(20)(2)=40 J.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 40 J, not 20 J.","B":"Incorrect. Substituting the stated values into the governing relation gives 40 J, not 39 J.","C":"Incorrect. Substituting the stated values into the governing relation gives 40 J, not 80 J.","D":"Correct. The worked calculation gives 40 J."},"concepts_required":["work–energy theorem","kinetic energy","work by constant force"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_work_energy","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_006182","subject":"physics","topic":"ray_optics","subtopic":"convex_lens_magnification","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A real object is placed 48 cm in front of a thin convex lens of focal length 19 cm. Using the Cartesian sign convention, what is the magnitude of the linear magnification?","options":{"A":"1.65517","B":"0.65517","C":"-0.34483","D":"1.06552"},"correct_answer":"B","correct_answer_value":"0.65517","solution":"Using distances as positive magnitudes, 1/f=1/v+1/u gives v=uf/(u−f)=(48)(19)/(48−19)=31.44828 cm. Thus |m|=v/u=31.44828/48=0.65517.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.65517, not 1.65517.","B":"Correct. The worked calculation gives 0.65517.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.65517, not -0.34483.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.65517, not 1.06552."},"concepts_required":["thin lens equation","linear magnification","real image"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_lens_magnification","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_005127","subject":"physics","topic":"electrostatics","subtopic":"coulombs_law","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Two point charges of +4 μC and +1 μC are separated by 12 cm in vacuum. Using k=9×10⁹ N m² C⁻², find the magnitude of the electrostatic force.","options":{"A":"1.5 N","B":"-0.5 N","C":"5 N","D":"2.5 N"},"correct_answer":"D","correct_answer_value":"2.5 N","solution":"Convert charges to coulombs and distance to metres. F=k|q₁q₂|/r²=9×10⁹(4×10⁻⁶)(1×10⁻⁶)/(12×10⁻²)²=2.5 N.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 2.5 N, not 1.5 N.","B":"Incorrect. Substituting the stated values into the governing relation gives 2.5 N, not -0.5 N.","C":"Incorrect. Substituting the stated values into the governing relation gives 2.5 N, not 5 N.","D":"Correct. The worked calculation gives 2.5 N."},"concepts_required":["Coulomb's law","SI unit conversion","inverse-square law"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_coulomb","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_006839","subject":"chemistry","topic":"stoichiometry","subtopic":"limiting_reagent_water","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Hydrogen reacts with oxygen according to 2H₂ + O₂ → 2H₂O. If 10 mol H₂ and 6 mol O₂ are mixed and the reaction goes to completion, how many moles of H₂O form?","options":{"A":"9 mol","B":"5 mol","C":"10 mol","D":"7.5 mol"},"correct_answer":"C","correct_answer_value":"10 mol","solution":"From 2 mol H₂ and 1 mol O₂, 2 mol H₂O form. H₂ can produce 10 mol H₂O; O₂ can produce 12 mol H₂O. The limiting condition is H₂; product=10 mol.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 10 mol, not 9 mol.","B":"Incorrect. Substituting the stated values into the governing relation gives 10 mol, not 5 mol.","C":"Correct. The worked calculation gives 10 mol.","D":"Incorrect. Substituting the stated values into the governing relation gives 10 mol, not 7.5 mol."},"concepts_required":["balanced chemical equations","limiting reagent","mole ratios"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_limiting_reagent","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_000078","subject":"mathematics","topic":"algebra","subtopic":"quadratic_roots","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"For the quadratic equation 2x^2 -9x -4=0, let its roots be α and β. What is α²+β²?","options":{"A":"12.125","B":"23.25","C":"24.25","D":"29.1"},"correct_answer":"C","correct_answer_value":"24.25","solution":"By Vieta's relations, α+β = 9/2 and αβ = -4/2. Then α²+β²=(α+β)²−2αβ = (9/2)²−2(-4/2) = 24.25.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 24.25, not 12.125.","B":"Incorrect. Substituting the stated values into the governing relation gives 24.25, not 23.25.","C":"Correct. The worked calculation gives 24.25.","D":"Incorrect. Substituting the stated values into the governing relation gives 24.25, not 29.1."},"concepts_required":["Vieta's formulas","symmetric polynomial expressions"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_quadratic_vieta","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_006781","subject":"chemistry","topic":"stoichiometry","subtopic":"limiting_reagent_water","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Hydrogen reacts with oxygen according to 2H₂ + O₂ → 2H₂O. If 1 mol H₂ and 9 mol O₂ are mixed and the reaction goes to completion, how many moles of H₂O form?","options":{"A":"4 mol","B":"1 mol","C":"3 mol","D":"1.5 mol"},"correct_answer":"B","correct_answer_value":"1 mol","solution":"From 2 mol H₂ and 1 mol O₂, 2 mol H₂O form. H₂ can produce 1 mol H₂O; O₂ can produce 18 mol H₂O. The limiting condition is H₂; product=1 mol.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 1 mol, not 4 mol.","B":"Correct. The worked calculation gives 1 mol.","C":"Incorrect. Substituting the stated values into the governing relation gives 1 mol, not 3 mol.","D":"Incorrect. Substituting the stated values into the governing relation gives 1 mol, not 1.5 mol."},"concepts_required":["balanced chemical equations","limiting reagent","mole ratios"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_limiting_reagent","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_001466","subject":"mathematics","topic":"calculus","subtopic":"definite_integral_polynomial","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Evaluate the definite integral ∫₀^7 (-4x² +5x -3) dx.","options":{"A":"-356.83333","B":"-177.91667","C":"-355.83333","D":"-533.75"},"correct_answer":"C","correct_answer_value":"-355.83333","solution":"An antiderivative is (-4/3)x³+(5/2)x²+(-3)x. Evaluating between 0 and 7 gives (-4/3)(7)³+(5/2)(7)²+(-3)(7)=-355.83333.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives -355.83333, not -356.83333.","B":"Incorrect. Substituting the stated values into the governing relation gives -355.83333, not -177.91667.","C":"Correct. The worked calculation gives -355.83333.","D":"Incorrect. Substituting the stated values into the governing relation gives -355.83333, not -533.75."},"concepts_required":["definite integration","power rule","limits of integration"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_integral_poly","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_008071","subject":"chemistry","topic":"physical_chemistry","subtopic":"faradays_law_copper","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"An aqueous Cu²⁺ solution is electrolysed using a current of 15 A for 12465 s. Assuming 100% current efficiency, calculate the mass of copper deposited. Use M(Cu)=63.5 g mol⁻¹ and F=96500 C mol⁻¹.","options":{"A":"61.51768 g","B":"46.13826 g","C":"63.51768 g","D":"73.82122 g"},"correct_answer":"A","correct_answer_value":"61.51768 g","solution":"Cu²⁺+2e⁻→Cu, so z=2. By Faraday's law m=MIt/(zF)=63.5×15×12465/(2×96500)=61.51768 g.","option_explanations":{"A":"Correct. The worked calculation gives 61.51768 g.","B":"Incorrect. Substituting the stated values into the governing relation gives 61.51768 g, not 46.13826 g.","C":"Incorrect. Substituting the stated values into the governing relation gives 61.51768 g, not 63.51768 g.","D":"Incorrect. Substituting the stated values into the governing relation gives 61.51768 g, not 73.82122 g."},"concepts_required":["electrolysis","Faraday's laws","redox stoichiometry"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_faraday_copper","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_001435","subject":"mathematics","topic":"calculus","subtopic":"definite_integral_polynomial","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Evaluate the definite integral ∫₀^6 (2x² +2x +0) dx.","options":{"A":"181","B":"360","C":"180","D":"144"},"correct_answer":"C","correct_answer_value":"180","solution":"An antiderivative is (2/3)x³+(2/2)x²+(0)x. Evaluating between 0 and 6 gives (2/3)(6)³+(2/2)(6)²+(0)(6)=180.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 180, not 181.","B":"Incorrect. Substituting the stated values into the governing relation gives 180, not 360.","C":"Correct. The worked calculation gives 180.","D":"Incorrect. Substituting the stated values into the governing relation gives 180, not 144."},"concepts_required":["definite integration","power rule","limits of integration"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_integral_poly","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_001856","subject":"mathematics","topic":"combinatorics","subtopic":"binomial_coefficient","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"How many distinct subsets containing exactly 9 elements can be selected from a set of 60 distinct elements?","options":{"A":"14783142660","B":"29566285320","C":"7391571330","D":"14783142662"},"correct_answer":"A","correct_answer_value":"14783142660","solution":"Order does not matter, so the count is C(60,9)=60!/[9!(51)!]=14783142660.","option_explanations":{"A":"Correct. The worked calculation gives 14783142660.","B":"Incorrect. Substituting the stated values into the governing relation gives 14783142660, not 29566285320.","C":"Incorrect. Substituting the stated values into the governing relation gives 14783142660, not 7391571330.","D":"Incorrect. Substituting the stated values into the governing relation gives 14783142660, not 14783142662."},"concepts_required":["combinations","factorials"],"reasoning_steps":2,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_combinations","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_003172","subject":"mathematics","topic":"complex_numbers","subtopic":"real_part_of_complex_quotient","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Let z=(-3-5i)/(9+4i). What is Re(z), the real part of z?","options":{"A":"1.51546","B":"-1.48454","C":"0.51546","D":"-0.48454"},"correct_answer":"D","correct_answer_value":"-0.48454","solution":"Multiply numerator and denominator by the conjugate (9-4i). Then Re(z)=(ac+bd)/(c²+d²)=((-3)(9)+(-5)(4))/(9²+(4)²)=-0.48454.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives -0.48454, not 1.51546.","B":"Incorrect. Substituting the stated values into the governing relation gives -0.48454, not -1.48454.","C":"Incorrect. Substituting the stated values into the governing relation gives -0.48454, not 0.51546.","D":"Correct. The worked calculation gives -0.48454."},"concepts_required":["complex conjugate","complex division","real and imaginary parts"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_complex_quotient","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_009209","subject":"chemistry","topic":"physical_chemistry","subtopic":"nernst_equation_298K","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"At 298 K, a galvanic cell has E°cell=1.05 V, transfers 1 electrons, and has reaction quotient Q=10^4. Using E=E°−(0.0591/n)log₁₀Q, calculate Ecell.","options":{"A":"3.8136 V","B":"-0.8136 V","C":"1.08136 V","D":"0.8136 V"},"correct_answer":"D","correct_answer_value":"0.8136 V","solution":"Apply the Nernst equation: E=1.05−(0.0591/1)(4)=0.8136 V.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.8136 V, not 3.8136 V.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.8136 V, not -0.8136 V.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.8136 V, not 1.08136 V.","D":"Correct. The worked calculation gives 0.8136 V."},"concepts_required":["electrochemical cells","Nernst equation","logarithms"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_nernst","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_003110","subject":"mathematics","topic":"complex_numbers","subtopic":"real_part_of_complex_quotient","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Let z=(1-9i)/(8+7i). What is Re(z), the real part of z?","options":{"A":"-1.48673","B":"0.48673","C":"2.51327","D":"-0.48673"},"correct_answer":"D","correct_answer_value":"-0.48673","solution":"Multiply numerator and denominator by the conjugate (8-7i). Then Re(z)=(ac+bd)/(c²+d²)=((1)(8)+(-9)(7))/(8²+(7)²)=-0.48673.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives -0.48673, not -1.48673.","B":"Incorrect. Substituting the stated values into the governing relation gives -0.48673, not 0.48673.","C":"Incorrect. Substituting the stated values into the governing relation gives -0.48673, not 2.51327.","D":"Correct. The worked calculation gives -0.48673."},"concepts_required":["complex conjugate","complex division","real and imaginary parts"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_complex_quotient","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_006047","subject":"physics","topic":"ray_optics","subtopic":"convex_lens_magnification","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A real object is placed 99 cm in front of a thin convex lens of focal length 21 cm. Using the Cartesian sign convention, what is the magnitude of the linear magnification?","options":{"A":"1.26923","B":"-0.26923","C":"-0.73077","D":"0.26923"},"correct_answer":"D","correct_answer_value":"0.26923","solution":"Using distances as positive magnitudes, 1/f=1/v+1/u gives v=uf/(u−f)=(99)(21)/(99−21)=26.65385 cm. Thus |m|=v/u=26.65385/99=0.26923.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.26923, not 1.26923.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.26923, not -0.26923.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.26923, not -0.73077.","D":"Correct. The worked calculation gives 0.26923."},"concepts_required":["thin lens equation","linear magnification","real image"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_lens_magnification","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_009491","subject":"chemistry","topic":"organic_chemistry","subtopic":"degree_of_unsaturation","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Assuming a neutral closed-shell formula, calculate the degree of unsaturation (DBE) for C_4H_8N_3X_3. Treat X as the total number of monovalent halogen atoms.","options":{"A":"-2","B":"-1","C":"1","D":"4"},"correct_answer":"C","correct_answer_value":"1","solution":"Use DBE=(2C+2+N−H−X)/2. Substitution gives (2×4+2+3−8−3)/2=1.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 1, not -2.","B":"Incorrect. Substituting the stated values into the governing relation gives 1, not -1.","C":"Correct. The worked calculation gives 1.","D":"Incorrect. Substituting the stated values into the governing relation gives 1, not 4."},"concepts_required":["molecular formula","degree of unsaturation","organic structure"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_dbe","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_004617","subject":"physics","topic":"rotational_mechanics","subtopic":"disc_angular_acceleration","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A uniform solid disc of mass 3 kg and radius 6 m is acted on by a constant torque of 68 N m about its central axis. Find its angular acceleration.","options":{"A":"1.12593 rad s⁻²","B":"0.25926 rad s⁻²","C":"1.25926 rad s⁻²","D":"4.25926 rad s⁻²"},"correct_answer":"C","correct_answer_value":"1.25926 rad s⁻²","solution":"For a solid disc, I=½MR²=½(3)(6)²=54 kg m². Using τ=Iα, α=68/54=1.25926 rad s⁻².","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 1.25926 rad s⁻², not 1.12593 rad s⁻².","B":"Incorrect. Substituting the stated values into the governing relation gives 1.25926 rad s⁻², not 0.25926 rad s⁻².","C":"Correct. The worked calculation gives 1.25926 rad s⁻².","D":"Incorrect. Substituting the stated values into the governing relation gives 1.25926 rad s⁻², not 4.25926 rad s⁻²."},"concepts_required":["moment of inertia","rotational dynamics","torque"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_disc_torque","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_001412","subject":"mathematics","topic":"calculus","subtopic":"definite_integral_polynomial","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Evaluate the definite integral ∫₀^1 (3x² -1x +2) dx.","options":{"A":"3.5","B":"2.5","C":"1.25","D":"-2.5"},"correct_answer":"B","correct_answer_value":"2.5","solution":"An antiderivative is (3/3)x³+(-1/2)x²+(2)x. Evaluating between 0 and 1 gives (3/3)(1)³+(-1/2)(1)²+(2)(1)=2.5.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 2.5, not 3.5.","B":"Correct. The worked calculation gives 2.5.","C":"Incorrect. Substituting the stated values into the governing relation gives 2.5, not 1.25.","D":"Incorrect. Substituting the stated values into the governing relation gives 2.5, not -2.5."},"concepts_required":["definite integration","power rule","limits of integration"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_integral_poly","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_002591","subject":"mathematics","topic":"linear_algebra","subtopic":"2x2_determinant","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Find the determinant of the matrix [[1, 7], [1, 4]].","options":{"A":"-1","B":"-3","C":"-1.5","D":"-6"},"correct_answer":"B","correct_answer_value":"-3","solution":"For a 2×2 matrix [[a,b],[c,d]], det=ad−bc. Here it is (1)(4)−(7)(1)=-3.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives -3, not -1.","B":"Correct. The worked calculation gives -3.","C":"Incorrect. Substituting the stated values into the governing relation gives -3, not -1.5.","D":"Incorrect. Substituting the stated values into the governing relation gives -3, not -6."},"concepts_required":["determinants","matrix operations"],"reasoning_steps":2,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_determinant_2x2","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_006181","subject":"physics","topic":"ray_optics","subtopic":"convex_lens_magnification","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A real object is placed 87 cm in front of a thin convex lens of focal length 22 cm. Using the Cartesian sign convention, what is the magnitude of the linear magnification?","options":{"A":"-1.66154","B":"0.50769","C":"1.03385","D":"0.33846"},"correct_answer":"D","correct_answer_value":"0.33846","solution":"Using distances as positive magnitudes, 1/f=1/v+1/u gives v=uf/(u−f)=(87)(22)/(87−22)=29.44615 cm. Thus |m|=v/u=29.44615/87=0.33846.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.33846, not -1.66154.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.33846, not 0.50769.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.33846, not 1.03385.","D":"Correct. The worked calculation gives 0.33846."},"concepts_required":["thin lens equation","linear magnification","real image"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_lens_magnification","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_003783","subject":"physics","topic":"mechanics","subtopic":"work_energy_theorem","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A 10 kg block initially moves at 12 m s⁻¹. A constant net force does 27 N of work over 2 m in the direction of motion. Find its final kinetic energy.","options":{"A":"774 J","B":"78.4 J","C":"772 J","D":"619.2 J"},"correct_answer":"A","correct_answer_value":"774 J","solution":"By the work–energy theorem, K_f=K_i+W=½mu²+Fs=½(10)(12)²+(27)(2)=774 J.","option_explanations":{"A":"Correct. The worked calculation gives 774 J.","B":"Incorrect. Substituting the stated values into the governing relation gives 774 J, not 78.4 J.","C":"Incorrect. Substituting the stated values into the governing relation gives 774 J, not 772 J.","D":"Incorrect. Substituting the stated values into the governing relation gives 774 J, not 619.2 J."},"concepts_required":["work–energy theorem","kinetic energy","work by constant force"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_work_energy","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_000056","subject":"mathematics","topic":"algebra","subtopic":"quadratic_roots","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"For the quadratic equation 6x^2 +17x -8=0, let its roots be α and β. What is α²+β²?","options":{"A":"11.69444","B":"9.69444","C":"10.69444","D":"12.69444"},"correct_answer":"C","correct_answer_value":"10.69444","solution":"By Vieta's relations, α+β = -17/6 and αβ = -8/6. Then α²+β²=(α+β)²−2αβ = (-17/6)²−2(-8/6) = 10.69444.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 10.69444, not 11.69444.","B":"Incorrect. Substituting the stated values into the governing relation gives 10.69444, not 9.69444.","C":"Correct. The worked calculation gives 10.69444.","D":"Incorrect. Substituting the stated values into the governing relation gives 10.69444, not 12.69444."},"concepts_required":["Vieta's formulas","symmetric polynomial expressions"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_quadratic_vieta","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_002194","subject":"mathematics","topic":"probability","subtopic":"same_colour_without_replacement","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A bag contains 12 red, 11 blue and 12 green balls. Two balls are drawn uniformly at random without replacement. What is the probability that both have the same colour?","options":{"A":"0.31429","B":"-2.68571","C":"1.03143","D":"1.31429"},"correct_answer":"A","correct_answer_value":"0.31429","solution":"Favourable pairs = C(12,2)+C(11,2)+C(12,2)=187. Total pairs=C(35,2)=595. Therefore P=187/595=0.31429.","option_explanations":{"A":"Correct. The worked calculation gives 0.31429.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.31429, not -2.68571.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.31429, not 1.03143.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.31429, not 1.31429."},"concepts_required":["classical probability","combinations","without replacement"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_probability_colours","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_001685","subject":"mathematics","topic":"combinatorics","subtopic":"binomial_coefficient","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"How many distinct subsets containing exactly 4 elements can be selected from a set of 57 distinct elements?","options":{"A":"395010","B":"395011","C":"-395010","D":"197505"},"correct_answer":"A","correct_answer_value":"395010","solution":"Order does not matter, so the count is C(57,4)=57!/[4!(53)!]=395010.","option_explanations":{"A":"Correct. The worked calculation gives 395010.","B":"Incorrect. Substituting the stated values into the governing relation gives 395010, not 395011.","C":"Incorrect. Substituting the stated values into the governing relation gives 395010, not -395010.","D":"Incorrect. Substituting the stated values into the governing relation gives 395010, not 197505."},"concepts_required":["combinations","factorials"],"reasoning_steps":2,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_combinations","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_000677","subject":"mathematics","topic":"sequences_and_series","subtopic":"geometric_progression","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A geometric progression has first term 5 and common ratio 5. What is the sum of its first 11 terms?","options":{"A":"61035157","B":"91552732.5","C":"61035155","D":"30517577.5"},"correct_answer":"C","correct_answer_value":"61035155","solution":"For r≠1, Sₙ=a(rⁿ−1)/(r−1). Thus Sₙ=5(5^11−1)/(5−1)=61035155.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 61035155, not 61035157.","B":"Incorrect. Substituting the stated values into the governing relation gives 61035155, not 91552732.5.","C":"Correct. The worked calculation gives 61035155.","D":"Incorrect. Substituting the stated values into the governing relation gives 61035155, not 30517577.5."},"concepts_required":["geometric progression","finite geometric sum"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_gp_sum","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_008296","subject":"chemistry","topic":"physical_chemistry","subtopic":"faradays_law_copper","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"An aqueous Cu²⁺ solution is electrolysed using a current of 8 A for 3079 s. Assuming 100% current efficiency, calculate the mass of copper deposited. Use M(Cu)=63.5 g mol⁻¹ and F=96500 C mol⁻¹.","options":{"A":"10.10431 g","B":"1.81043 g","C":"6.07823 g","D":"8.10431 g"},"correct_answer":"D","correct_answer_value":"8.10431 g","solution":"Cu²⁺+2e⁻→Cu, so z=2. By Faraday's law m=MIt/(zF)=63.5×8×3079/(2×96500)=8.10431 g.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 8.10431 g, not 10.10431 g.","B":"Incorrect. Substituting the stated values into the governing relation gives 8.10431 g, not 1.81043 g.","C":"Incorrect. Substituting the stated values into the governing relation gives 8.10431 g, not 6.07823 g.","D":"Correct. The worked calculation gives 8.10431 g."},"concepts_required":["electrolysis","Faraday's laws","redox stoichiometry"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_faraday_copper","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_003355","subject":"physics","topic":"mechanics","subtopic":"projectile_range","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A projectile is launched from level ground at 57 m s⁻¹ at 15° to the horizontal. Take g=10 m s⁻². Find its horizontal range.","options":{"A":"162.45 m","B":"121.8375 m","C":"194.94 m","D":"163.45 m"},"correct_answer":"A","correct_answer_value":"162.45 m","solution":"Range R=u²sin(2θ)/g. Substituting u=57, θ=15° and g=10 gives R=(57)²sin(30°)/10=162.45 m.","option_explanations":{"A":"Correct. The worked calculation gives 162.45 m.","B":"Incorrect. Substituting the stated values into the governing relation gives 162.45 m, not 121.8375 m.","C":"Incorrect. Substituting the stated values into the governing relation gives 162.45 m, not 194.94 m.","D":"Incorrect. Substituting the stated values into the governing relation gives 162.45 m, not 163.45 m."},"concepts_required":["projectile motion","range equation","trigonometry"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_projectile_range","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_005523","subject":"physics","topic":"electrostatics","subtopic":"capacitor_energy","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A 37 μF capacitor is charged to a potential difference of 13 V. Calculate the energy stored, expressed in μJ.","options":{"A":"3126.5 μJ","B":"3751.8 μJ","C":"3127.5 μJ","D":"4689.75 μJ"},"correct_answer":"A","correct_answer_value":"3126.5 μJ","solution":"U=½CV². Since C=37 μF, U in μJ is ½×37×(13)²=3126.5 μJ.","option_explanations":{"A":"Correct. The worked calculation gives 3126.5 μJ.","B":"Incorrect. Substituting the stated values into the governing relation gives 3126.5 μJ, not 3751.8 μJ.","C":"Incorrect. Substituting the stated values into the governing relation gives 3126.5 μJ, not 3127.5 μJ.","D":"Incorrect. Substituting the stated values into the governing relation gives 3126.5 μJ, not 4689.75 μJ."},"concepts_required":["capacitance","energy stored in a capacitor"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_capacitor_energy","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_007992","subject":"chemistry","topic":"physical_chemistry","subtopic":"first_order_half_life","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A first-order reaction has a half-life of 20 minutes. If the initial reactant concentration is 90 mmol L⁻¹, what concentration remains after 120 minutes?","options":{"A":"1.40625 mmol L⁻¹","B":"0.40625 mmol L⁻¹","C":"2.40625 mmol L��¹","D":"4.40625 mmol L⁻¹"},"correct_answer":"A","correct_answer_value":"1.40625 mmol L⁻¹","solution":"120 minutes corresponds to 6 half-lives. The remaining concentration is [A]₀(1/2)ⁿ=90/2^6=1.40625 mmol L⁻¹.","option_explanations":{"A":"Correct. The worked calculation gives 1.40625 mmol L⁻¹.","B":"Incorrect. Substituting the stated values into the governing relation gives 1.40625 mmol L⁻¹, not 0.40625 mmol L⁻¹.","C":"Incorrect. Substituting the stated values into the governing relation gives 1.40625 mmol L⁻¹, not 2.40625 mmol L⁻¹.","D":"Incorrect. Substituting the stated values into the governing relation gives 1.40625 mmol L⁻¹, not 4.40625 mmol L⁻¹."},"concepts_required":["first-order kinetics","half-life","exponential decay"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_first_order_halflife","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_001508","subject":"mathematics","topic":"calculus","subtopic":"definite_integral_polynomial","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Evaluate the definite integral ∫₀^7 (-4x² +3x +1) dx.","options":{"A":"-376.83333","B":"-565.25","C":"-301.46667","D":"-753.66667"},"correct_answer":"A","correct_answer_value":"-376.83333","solution":"An antiderivative is (-4/3)x³+(3/2)x²+(1)x. Evaluating between 0 and 7 gives (-4/3)(7)³+(3/2)(7)²+(1)(7)=-376.83333.","option_explanations":{"A":"Correct. The worked calculation gives -376.83333.","B":"Incorrect. Substituting the stated values into the governing relation gives -376.83333, not -565.25.","C":"Incorrect. Substituting the stated values into the governing relation gives -376.83333, not -301.46667.","D":"Incorrect. Substituting the stated values into the governing relation gives -376.83333, not -753.66667."},"concepts_required":["definite integration","power rule","limits of integration"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_integral_poly","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_004533","subject":"physics","topic":"rotational_mechanics","subtopic":"disc_angular_acceleration","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A uniform solid disc of mass 7 kg and radius 2 m is acted on by a constant torque of 74 N m about its central axis. Find its angular acceleration.","options":{"A":"1.52857 rad s⁻²","B":"5.28571 rad s⁻²","C":"2.64286 rad s⁻²","D":"3.96429 rad s⁻²"},"correct_answer":"B","correct_answer_value":"5.28571 rad s⁻²","solution":"For a solid disc, I=½MR²=½(7)(2)²=14 kg m². Using τ=Iα, α=74/14=5.28571 rad s⁻².","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 5.28571 rad s⁻², not 1.52857 rad s⁻².","B":"Correct. The worked calculation gives 5.28571 rad s⁻².","C":"Incorrect. Substituting the stated values into the governing relation gives 5.28571 rad s⁻², not 2.64286 rad s⁻².","D":"Incorrect. Substituting the stated values into the governing relation gives 5.28571 rad s⁻², not 3.96429 rad s⁻²."},"concepts_required":["moment of inertia","rotational dynamics","torque"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_disc_torque","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_006555","subject":"physics","topic":"oscillations","subtopic":"mass_spring_period_over_pi","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A mass of 24 kg is attached to an ideal spring of force constant 1131 N m⁻¹ and oscillates horizontally without damping. What is the time period divided by π?","options":{"A":"1.02913 s","B":"-2.70866 s","C":"0.29134 s","D":"-1.70866 s"},"correct_answer":"C","correct_answer_value":"0.29134 s","solution":"For a mass–spring oscillator, T=2π√(m/k). Therefore T/π=2√(24/1131)=0.29134 s.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.29134 s, not 1.02913 s.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.29134 s, not -2.70866 s.","C":"Correct. The worked calculation gives 0.29134 s.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.29134 s, not -1.70866 s."},"concepts_required":["simple harmonic motion","mass–spring oscillator","time period"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_shm_period","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_000394","subject":"mathematics","topic":"sequences_and_series","subtopic":"arithmetic_progression","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"An arithmetic progression has first term -12 and common difference 9. Find the sum of its first 28 terms.","options":{"A":"3066","B":"3064","C":"3067","D":"2452.8"},"correct_answer":"A","correct_answer_value":"3066","solution":"Use Sₙ = n/2[2a+(n−1)d]. Substituting a=-12, d=9, n=28: Sₙ=28/2[2(-12)+(28−1)(9)]=3066.","option_explanations":{"A":"Correct. The worked calculation gives 3066.","B":"Incorrect. Substituting the stated values into the governing relation gives 3066, not 3064.","C":"Incorrect. Substituting the stated values into the governing relation gives 3066, not 3067.","D":"Incorrect. Substituting the stated values into the governing relation gives 3066, not 2452.8."},"concepts_required":["arithmetic progression","finite series"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_ap_sum","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_009512","subject":"chemistry","topic":"organic_chemistry","subtopic":"degree_of_unsaturation","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Assuming a neutral closed-shell formula, calculate the degree of unsaturation (DBE) for C_8H_9X_1. Treat X as the total number of monovalent halogen atoms.","options":{"A":"7","B":"4","C":"1","D":"2"},"correct_answer":"B","correct_answer_value":"4","solution":"Use DBE=(2C+2+N−H−X)/2. Substitution gives (2×8+2+0−9−1)/2=4.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 4, not 7.","B":"Correct. The worked calculation gives 4.","C":"Incorrect. Substituting the stated values into the governing relation gives 4, not 1.","D":"Incorrect. Substituting the stated values into the governing relation gives 4, not 2."},"concepts_required":["molecular formula","degree of unsaturation","organic structure"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_dbe","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_007737","subject":"chemistry","topic":"physical_chemistry","subtopic":"first_order_half_life","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A first-order reaction has a half-life of 20 minutes. If the initial reactant concentration is 74 mmol L⁻¹, what concentration remains after 160 minutes?","options":{"A":"0.28906 mmol L⁻¹","B":"-0.28906 mmol L⁻¹","C":"-0.71094 mmol L⁻¹","D":"0.14453 mmol L⁻¹"},"correct_answer":"A","correct_answer_value":"0.28906 mmol L⁻¹","solution":"160 minutes corresponds to 8 half-lives. The remaining concentration is [A]₀(1/2)ⁿ=74/2^8=0.28906 mmol L⁻¹.","option_explanations":{"A":"Correct. The worked calculation gives 0.28906 mmol L⁻¹.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.28906 mmol L⁻¹, not -0.28906 mmol L⁻¹.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.28906 mmol L⁻¹, not -0.71094 mmol L⁻¹.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.28906 mmol L⁻¹, not 0.14453 mmol L⁻¹."},"concepts_required":["first-order kinetics","half-life","exponential decay"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_first_order_halflife","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_007290","subject":"chemistry","topic":"physical_chemistry","subtopic":"ideal_gas_volume","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Assuming ideal behaviour, what volume is occupied by 20 mol of gas at 208 K and 1.32 atm? Use R=0.082057 L atm mol⁻¹ K⁻¹.","options":{"A":"310.32465 L","B":"258.60388 L","C":"260.60388 L","D":"387.90582 L"},"correct_answer":"B","correct_answer_value":"258.60388 L","solution":"PV=nRT, so V=nRT/P=(20)(0.082057)(208)/(1.32)=258.60388 L.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 258.60388 L, not 310.32465 L.","B":"Correct. The worked calculation gives 258.60388 L.","C":"Incorrect. Substituting the stated values into the governing relation gives 258.60388 L, not 260.60388 L.","D":"Incorrect. Substituting the stated values into the governing relation gives 258.60388 L, not 387.90582 L."},"concepts_required":["ideal gas equation","unit consistency"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_ideal_gas","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_004521","subject":"physics","topic":"rotational_mechanics","subtopic":"disc_angular_acceleration","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A uniform solid disc of mass 11 kg and radius 4 m is acted on by a constant torque of 51 N m about its central axis. Find its angular acceleration.","options":{"A":"-0.57955 rad s⁻²","B":"2.57955 rad s⁻²","C":"0.57955 rad s⁻²","D":"0.28977 rad s⁻²"},"correct_answer":"C","correct_answer_value":"0.57955 rad s⁻²","solution":"For a solid disc, I=½MR²=½(11)(4)²=88 kg m². Using τ=Iα, α=51/88=0.57955 rad s⁻².","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.57955 rad s⁻², not -0.57955 rad s⁻².","B":"Incorrect. Substituting the stated values into the governing relation gives 0.57955 rad s⁻², not 2.57955 rad s⁻².","C":"Correct. The worked calculation gives 0.57955 rad s⁻².","D":"Incorrect. Substituting the stated values into the governing relation gives 0.57955 rad s⁻², not 0.28977 rad s⁻²."},"concepts_required":["moment of inertia","rotational dynamics","torque"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_disc_torque","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_004478","subject":"physics","topic":"rotational_mechanics","subtopic":"disc_angular_acceleration","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A uniform solid disc of mass 9 kg and radius 2 m is acted on by a constant torque of 39 N m about its central axis. Find its angular acceleration.","options":{"A":"1.08333 rad s⁻²","B":"-0.83333 rad s⁻²","C":"5.16667 rad s⁻²","D":"2.16667 rad s⁻²"},"correct_answer":"D","correct_answer_value":"2.16667 rad s⁻²","solution":"For a solid disc, I=½MR²=½(9)(2)²=18 kg m². Using τ=Iα, α=39/18=2.16667 rad s⁻².","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 2.16667 rad s⁻², not 1.08333 rad s⁻².","B":"Incorrect. Substituting the stated values into the governing relation gives 2.16667 rad s⁻², not -0.83333 rad s⁻².","C":"Incorrect. Substituting the stated values into the governing relation gives 2.16667 rad s⁻², not 5.16667 rad s⁻².","D":"Correct. The worked calculation gives 2.16667 rad s⁻²."},"concepts_required":["moment of inertia","rotational dynamics","torque"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_disc_torque","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_002802","subject":"mathematics","topic":"vectors","subtopic":"dot_product","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Given vectors u=(6, -7, -3) and v=(-5, 7, 3), calculate u·v.","options":{"A":"-89","B":"-176","C":"-86","D":"-88"},"correct_answer":"D","correct_answer_value":"-88","solution":"The dot product is the sum of corresponding component products: (6)(-5)+(-7)(7)+(-3)(3)=-88.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives -88, not -89.","B":"Incorrect. Substituting the stated values into the governing relation gives -88, not -176.","C":"Incorrect. Substituting the stated values into the governing relation gives -88, not -86.","D":"Correct. The worked calculation gives -88."},"concepts_required":["three-dimensional vectors","scalar product"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_vector_dot","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_002677","subject":"mathematics","topic":"vectors","subtopic":"dot_product","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Given vectors u=(6, 2, -2) and v=(-3, -4, -6), calculate u·v.","options":{"A":"-14","B":"-17","C":"-12","D":"-13"},"correct_answer":"A","correct_answer_value":"-14","solution":"The dot product is the sum of corresponding component products: (6)(-3)+(2)(-4)+(-2)(-6)=-14.","option_explanations":{"A":"Correct. The worked calculation gives -14.","B":"Incorrect. Substituting the stated values into the governing relation gives -14, not -17.","C":"Incorrect. Substituting the stated values into the governing relation gives -14, not -12.","D":"Incorrect. Substituting the stated values into the governing relation gives -14, not -13."},"concepts_required":["three-dimensional vectors","scalar product"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_vector_dot","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_001448","subject":"mathematics","topic":"calculus","subtopic":"definite_integral_polynomial","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Evaluate the definite integral ∫₀^8 (-4x² +5x +4) dx.","options":{"A":"-489.66667","B":"-736","C":"-48.06667","D":"-490.66667"},"correct_answer":"D","correct_answer_value":"-490.66667","solution":"An antiderivative is (-4/3)x³+(5/2)x²+(4)x. Evaluating between 0 and 8 gives (-4/3)(8)³+(5/2)(8)²+(4)(8)=-490.66667.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives -490.66667, not -489.66667.","B":"Incorrect. Substituting the stated values into the governing relation gives -490.66667, not -736.","C":"Incorrect. Substituting the stated values into the governing relation gives -490.66667, not -48.06667.","D":"Correct. The worked calculation gives -490.66667."},"concepts_required":["definite integration","power rule","limits of integration"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_integral_poly","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_003649","subject":"physics","topic":"mechanics","subtopic":"projectile_range","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A projectile is launched from level ground at 43 m s⁻¹ at 18° to the horizontal. Take g=10 m s⁻². Find its horizontal range.","options":{"A":"110.68149 m","B":"86.94519 m","C":"11.86815 m","D":"108.68149 m"},"correct_answer":"D","correct_answer_value":"108.68149 m","solution":"Range R=u²sin(2θ)/g. Substituting u=43, θ=18° and g=10 gives R=(43)²sin(36°)/10=108.68149 m.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 108.68149 m, not 110.68149 m.","B":"Incorrect. Substituting the stated values into the governing relation gives 108.68149 m, not 86.94519 m.","C":"Incorrect. Substituting the stated values into the governing relation gives 108.68149 m, not 11.86815 m.","D":"Correct. The worked calculation gives 108.68149 m."},"concepts_required":["projectile motion","range equation","trigonometry"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_projectile_range","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_005816","subject":"physics","topic":"thermodynamics","subtopic":"ideal_gas_isochoric_pressure","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A fixed amount of ideal gas is kept in a rigid container. Its pressure is 4 atm at 181 K. What is its pressure at 763 K?","options":{"A":"8.43094 atm","B":"17.86188 atm","C":"16.86188 atm","D":"20.23425 atm"},"correct_answer":"C","correct_answer_value":"16.86188 atm","solution":"At constant volume, P/T is constant. P₂=P₁T₂/T₁=(4)(763)/(181)=16.86188 atm.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 16.86188 atm, not 8.43094 atm.","B":"Incorrect. Substituting the stated values into the governing relation gives 16.86188 atm, not 17.86188 atm.","C":"Correct. The worked calculation gives 16.86188 atm.","D":"Incorrect. Substituting the stated values into the governing relation gives 16.86188 atm, not 20.23425 atm."},"concepts_required":["ideal gas law","isochoric process","absolute temperature"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_isochoric_gas","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_002680","subject":"mathematics","topic":"vectors","subtopic":"dot_product","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Given vectors u=(-4, 5, -6) and v=(-4, -4, -3), calculate u·v.","options":{"A":"14","B":"16","C":"13","D":"12"},"correct_answer":"A","correct_answer_value":"14","solution":"The dot product is the sum of corresponding component products: (-4)(-4)+(5)(-4)+(-6)(-3)=14.","option_explanations":{"A":"Correct. The worked calculation gives 14.","B":"Incorrect. Substituting the stated values into the governing relation gives 14, not 16.","C":"Incorrect. Substituting the stated values into the governing relation gives 14, not 13.","D":"Incorrect. Substituting the stated values into the governing relation gives 14, not 12."},"concepts_required":["three-dimensional vectors","scalar product"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_vector_dot","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_001843","subject":"mathematics","topic":"combinatorics","subtopic":"binomial_coefficient","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"How many distinct subsets containing exactly 9 elements can be selected from a set of 27 distinct elements?","options":{"A":"4686824","B":"4686823","C":"3749460","D":"4686825"},"correct_answer":"D","correct_answer_value":"4686825","solution":"Order does not matter, so the count is C(27,9)=27!/[9!(18)!]=4686825.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 4686825, not 4686824.","B":"Incorrect. Substituting the stated values into the governing relation gives 4686825, not 4686823.","C":"Incorrect. Substituting the stated values into the governing relation gives 4686825, not 3749460.","D":"Correct. The worked calculation gives 4686825."},"concepts_required":["combinations","factorials"],"reasoning_steps":2,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_combinations","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_002914","subject":"mathematics","topic":"vectors","subtopic":"dot_product","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Given vectors u=(1, 3, -1) and v=(-2, 1, 4), calculate u·v.","options":{"A":"-6","B":"0","C":"-3","D":"-5"},"correct_answer":"C","correct_answer_value":"-3","solution":"The dot product is the sum of corresponding component products: (1)(-2)+(3)(1)+(-1)(4)=-3.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives -3, not -6.","B":"Incorrect. Substituting the stated values into the governing relation gives -3, not 0.","C":"Correct. The worked calculation gives -3.","D":"Incorrect. Substituting the stated values into the governing relation gives -3, not -5."},"concepts_required":["three-dimensional vectors","scalar product"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_vector_dot","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_005603","subject":"physics","topic":"electrostatics","subtopic":"capacitor_energy","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A 76 μF capacitor is charged to a potential difference of 16 V. Calculate the energy stored, expressed in μJ.","options":{"A":"973.8 μJ","B":"9730 μJ","C":"9728 μJ","D":"7782.4 μJ"},"correct_answer":"C","correct_answer_value":"9728 μJ","solution":"U=½CV². Since C=76 μF, U in μJ is ½×76×(16)²=9728 μJ.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 9728 μJ, not 973.8 μJ.","B":"Incorrect. Substituting the stated values into the governing relation gives 9728 μJ, not 9730 μJ.","C":"Correct. The worked calculation gives 9728 μJ.","D":"Incorrect. Substituting the stated values into the governing relation gives 9728 μJ, not 7782.4 μJ."},"concepts_required":["capacitance","energy stored in a capacitor"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_capacitor_energy","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_000361","subject":"mathematics","topic":"sequences_and_series","subtopic":"arithmetic_progression","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"An arithmetic progression has first term -3 and common difference 13. Find the sum of its first 9 terms.","options":{"A":"442","B":"45.1","C":"441","D":"330.75"},"correct_answer":"C","correct_answer_value":"441","solution":"Use Sₙ = n/2[2a+(n−1)d]. Substituting a=-3, d=13, n=9: Sₙ=9/2[2(-3)+(9−1)(13)]=441.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 441, not 442.","B":"Incorrect. Substituting the stated values into the governing relation gives 441, not 45.1.","C":"Correct. The worked calculation gives 441.","D":"Incorrect. Substituting the stated values into the governing relation gives 441, not 330.75."},"concepts_required":["arithmetic progression","finite series"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_ap_sum","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_001630","subject":"mathematics","topic":"calculus","subtopic":"definite_integral_polynomial","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Evaluate the definite integral ∫₀^7 (-1x² -4x +4) dx.","options":{"A":"-138.25","B":"-184.33333","C":"-221.2","D":"-185.33333"},"correct_answer":"B","correct_answer_value":"-184.33333","solution":"An antiderivative is (-1/3)x³+(-4/2)x²+(4)x. Evaluating between 0 and 7 gives (-1/3)(7)³+(-4/2)(7)²+(4)(7)=-184.33333.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives -184.33333, not -138.25.","B":"Correct. The worked calculation gives -184.33333.","C":"Incorrect. Substituting the stated values into the governing relation gives -184.33333, not -221.2.","D":"Incorrect. Substituting the stated values into the governing relation gives -184.33333, not -185.33333."},"concepts_required":["definite integration","power rule","limits of integration"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_integral_poly","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_008216","subject":"chemistry","topic":"physical_chemistry","subtopic":"faradays_law_copper","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"An aqueous Cu²⁺ solution is electrolysed using a current of 14 A for 20142 s. Assuming 100% current efficiency, calculate the mass of copper deposited. Use M(Cu)=63.5 g mol⁻¹ and F=96500 C mol⁻¹.","options":{"A":"185.55687 g","B":"-92.77844 g","C":"92.77844 g","D":"91.77844 g"},"correct_answer":"C","correct_answer_value":"92.77844 g","solution":"Cu²⁺+2e⁻→Cu, so z=2. By Faraday's law m=MIt/(zF)=63.5×14×20142/(2×96500)=92.77844 g.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 92.77844 g, not 185.55687 g.","B":"Incorrect. Substituting the stated values into the governing relation gives 92.77844 g, not -92.77844 g.","C":"Correct. The worked calculation gives 92.77844 g.","D":"Incorrect. Substituting the stated values into the governing relation gives 92.77844 g, not 91.77844 g."},"concepts_required":["electrolysis","Faraday's laws","redox stoichiometry"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_faraday_copper","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_005814","subject":"physics","topic":"thermodynamics","subtopic":"ideal_gas_isochoric_pressure","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A fixed amount of ideal gas is kept in a rigid container. Its pressure is 1.5 atm at 449 K. What is its pressure at 477 K?","options":{"A":"1.15935 atm","B":"-1.59354 atm","C":"1.59354 atm","D":"3.18708 atm"},"correct_answer":"C","correct_answer_value":"1.59354 atm","solution":"At constant volume, P/T is constant. P₂=P₁T₂/T₁=(1.5)(477)/(449)=1.59354 atm.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 1.59354 atm, not 1.15935 atm.","B":"Incorrect. Substituting the stated values into the governing relation gives 1.59354 atm, not -1.59354 atm.","C":"Correct. The worked calculation gives 1.59354 atm.","D":"Incorrect. Substituting the stated values into the governing relation gives 1.59354 atm, not 3.18708 atm."},"concepts_required":["ideal gas law","isochoric process","absolute temperature"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_isochoric_gas","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_008893","subject":"chemistry","topic":"physical_chemistry","subtopic":"equilibrium_degree_dissociation","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"For A(g) ⇌ 2B(g), the initial concentration of A is 0.68 mol L⁻¹ and initially B is absent. At equilibrium, 76% of A has dissociated at constant volume. Find K_c.","options":{"A":"6.54613","B":"3.54613","C":"8.54613","D":"4.9096"},"correct_answer":"A","correct_answer_value":"6.54613","solution":"Let α=0.76. At equilibrium [A]=c₀(1−α) and [B]=2c₀α. Thus K_c=[B]²/[A]=4c₀α²/(1−α)=4(0.68)(0.76)²/(1−0.76)=6.54613.","option_explanations":{"A":"Correct. The worked calculation gives 6.54613.","B":"Incorrect. Substituting the stated values into the governing relation gives 6.54613, not 3.54613.","C":"Incorrect. Substituting the stated values into the governing relation gives 6.54613, not 8.54613.","D":"Incorrect. Substituting the stated values into the governing relation gives 6.54613, not 4.9096."},"concepts_required":["chemical equilibrium","equilibrium constant","degree of dissociation"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_equilibrium_alpha","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_004317","subject":"physics","topic":"mechanics","subtopic":"inclined_plane_with_friction","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A 5 kg block slides down a 37° rough incline. The coefficient of kinetic friction is 0.12. Taking g=10 m s⁻², find the net force down the plane. Use sin37°=0.6 and cos37°=0.8.","options":{"A":"30.24 N","B":"37.8 N","C":"24.2 N","D":"25.2 N"},"correct_answer":"D","correct_answer_value":"25.2 N","solution":"Net force=mg(sinθ−μcosθ)=5×10×(0.6−0.12×0.8)=25.2 N.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 25.2 N, not 30.24 N.","B":"Incorrect. Substituting the stated values into the governing relation gives 25.2 N, not 37.8 N.","C":"Incorrect. Substituting the stated values into the governing relation gives 25.2 N, not 24.2 N.","D":"Correct. The worked calculation gives 25.2 N."},"concepts_required":["Newton's laws","inclined plane","kinetic friction"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_incline_friction","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_004769","subject":"physics","topic":"electricity","subtopic":"series_parallel_circuit","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Resistors 10 Ω and 7 Ω are connected in series; this combination is connected in parallel with a 13 Ω resistor across a 9 V ideal battery. Find the total current supplied by the battery.","options":{"A":"0.91629 A","B":"1.22172 A","C":"1.12217 A","D":"4.22172 A"},"correct_answer":"B","correct_answer_value":"1.22172 A","solution":"Series branch resistance=10+7=17 Ω. Parallel equivalent R_eq=(17×13)/(17+13)=7.36667 Ω. Total current I=V/R_eq=9/7.36667=1.22172 A.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 1.22172 A, not 0.91629 A.","B":"Correct. The worked calculation gives 1.22172 A.","C":"Incorrect. Substituting the stated values into the governing relation gives 1.22172 A, not 1.12217 A.","D":"Incorrect. Substituting the stated values into the governing relation gives 1.22172 A, not 4.22172 A."},"concepts_required":["series resistors","parallel resistors","Ohm's law"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_series_parallel","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_005235","subject":"physics","topic":"electrostatics","subtopic":"coulombs_law","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Two point charges of +8 μC and +9 μC are separated by 8 cm in vacuum. Using k=9×10⁹ N m² C⁻², find the magnitude of the electrostatic force.","options":{"A":"100.25 N","B":"151.875 N","C":"75.9375 N","D":"101.25 N"},"correct_answer":"D","correct_answer_value":"101.25 N","solution":"Convert charges to coulombs and distance to metres. F=k|q₁q₂|/r²=9×10⁹(8×10⁻⁶)(9×10⁻⁶)/(8×10⁻²)²=101.25 N.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 101.25 N, not 100.25 N.","B":"Incorrect. Substituting the stated values into the governing relation gives 101.25 N, not 151.875 N.","C":"Incorrect. Substituting the stated values into the governing relation gives 101.25 N, not 75.9375 N.","D":"Correct. The worked calculation gives 101.25 N."},"concepts_required":["Coulomb's law","SI unit conversion","inverse-square law"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_coulomb","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_008657","subject":"chemistry","topic":"physical_chemistry","subtopic":"boiling_point_elevation","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A non-volatile solute is dissolved in water to produce a solution of molality 1.05 mol kg⁻¹. If its van't Hoff factor is 1 and K_b for water is 1.0 K kg mol⁻¹, find the boiling-point elevation.","options":{"A":"4.05 K","B":"-1.95 K","C":"1.05 K","D":"0.7875 K"},"correct_answer":"C","correct_answer_value":"1.05 K","solution":"ΔT_b=iK_bm. Substitution gives ΔT_b=1×1.0×1.05=1.05 K.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 1.05 K, not 4.05 K.","B":"Incorrect. Substituting the stated values into the governing relation gives 1.05 K, not -1.95 K.","C":"Correct. The worked calculation gives 1.05 K.","D":"Incorrect. Substituting the stated values into the governing relation gives 1.05 K, not 0.7875 K."},"concepts_required":["colligative properties","van't Hoff factor","molality"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_boiling_elevation","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_009469","subject":"chemistry","topic":"organic_chemistry","subtopic":"degree_of_unsaturation","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Assuming a neutral closed-shell formula, calculate the degree of unsaturation (DBE) for C_14H_10N_3X_1. Treat X as the total number of monovalent halogen atoms.","options":{"A":"5.5","B":"13","C":"12","D":"11"},"correct_answer":"D","correct_answer_value":"11","solution":"Use DBE=(2C+2+N−H−X)/2. Substitution gives (2×14+2+3−10−1)/2=11.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 11, not 5.5.","B":"Incorrect. Substituting the stated values into the governing relation gives 11, not 13.","C":"Incorrect. Substituting the stated values into the governing relation gives 11, not 12.","D":"Correct. The worked calculation gives 11."},"concepts_required":["molecular formula","degree of unsaturation","organic structure"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_dbe","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_003056","subject":"mathematics","topic":"complex_numbers","subtopic":"real_part_of_complex_quotient","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Let z=(13-8i)/(2+4i). What is Re(z), the real part of z?","options":{"A":"0.7","B":"0.3","C":"-0.3","D":"-0.45"},"correct_answer":"C","correct_answer_value":"-0.3","solution":"Multiply numerator and denominator by the conjugate (2-4i). Then Re(z)=(ac+bd)/(c²+d²)=((13)(2)+(-8)(4))/(2²+(4)²)=-0.3.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives -0.3, not 0.7.","B":"Incorrect. Substituting the stated values into the governing relation gives -0.3, not 0.3.","C":"Correct. The worked calculation gives -0.3.","D":"Incorrect. Substituting the stated values into the governing relation gives -0.3, not -0.45."},"concepts_required":["complex conjugate","complex division","real and imaginary parts"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_complex_quotient","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_000387","subject":"mathematics","topic":"sequences_and_series","subtopic":"arithmetic_progression","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"An arithmetic progression has first term -5 and common difference 4. Find the sum of its first 23 terms.","options":{"A":"1345.5","B":"899","C":"897","D":"717.6"},"correct_answer":"C","correct_answer_value":"897","solution":"Use Sₙ = n/2[2a+(n−1)d]. Substituting a=-5, d=4, n=23: Sₙ=23/2[2(-5)+(23−1)(4)]=897.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 897, not 1345.5.","B":"Incorrect. Substituting the stated values into the governing relation gives 897, not 899.","C":"Correct. The worked calculation gives 897.","D":"Incorrect. Substituting the stated values into the governing relation gives 897, not 717.6."},"concepts_required":["arithmetic progression","finite series"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_ap_sum","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_002395","subject":"mathematics","topic":"linear_algebra","subtopic":"2x2_determinant","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Find the determinant of the matrix [[5, 5], [6, -5]].","options":{"A":"-44","B":"-27.5","C":"-55","D":"-54"},"correct_answer":"C","correct_answer_value":"-55","solution":"For a 2×2 matrix [[a,b],[c,d]], det=ad−bc. Here it is (5)(-5)−(5)(6)=-55.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives -55, not -44.","B":"Incorrect. Substituting the stated values into the governing relation gives -55, not -27.5.","C":"Correct. The worked calculation gives -55.","D":"Incorrect. Substituting the stated values into the governing relation gives -55, not -54."},"concepts_required":["determinants","matrix operations"],"reasoning_steps":2,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_determinant_2x2","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_007538","subject":"chemistry","topic":"physical_chemistry","subtopic":"strong_acid_base_mixture","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"61 mL of 0.10 M HCl is mixed with 42 mL of 0.10 M NaOH at 25°C. Assuming complete dissociation and additive volumes, find the final pH.","options":{"A":"1.17341","B":"1.73408","C":"2.60113","D":"-1.73408"},"correct_answer":"B","correct_answer_value":"1.73408","solution":"Excess H⁺ moles=0.10×(61−42)/1000=0.001900 mol. Total volume=0.103 L, so [H⁺]=0.018447 M and pH=−log₁₀[H⁺]=1.73408.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 1.73408, not 1.17341.","B":"Correct. The worked calculation gives 1.73408.","C":"Incorrect. Substituting the stated values into the governing relation gives 1.73408, not 2.60113.","D":"Incorrect. Substituting the stated values into the governing relation gives 1.73408, not -1.73408."},"concepts_required":["strong acid–strong base neutralization","molarity","pH and pOH"],"reasoning_steps":5,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_acid_base_pH","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_002925","subject":"mathematics","topic":"vectors","subtopic":"dot_product","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Given vectors u=(1, -6, 4) and v=(4, 0, 4), calculate u·v.","options":{"A":"20","B":"16","C":"22","D":"40"},"correct_answer":"A","correct_answer_value":"20","solution":"The dot product is the sum of corresponding component products: (1)(4)+(-6)(0)+(4)(4)=20.","option_explanations":{"A":"Correct. The worked calculation gives 20.","B":"Incorrect. Substituting the stated values into the governing relation gives 20, not 16.","C":"Incorrect. Substituting the stated values into the governing relation gives 20, not 22.","D":"Incorrect. Substituting the stated values into the governing relation gives 20, not 40."},"concepts_required":["three-dimensional vectors","scalar product"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_vector_dot","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_008140","subject":"chemistry","topic":"physical_chemistry","subtopic":"faradays_law_copper","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"An aqueous Cu²⁺ solution is electrolysed using a current of 10 A for 618 s. Assuming 100% current efficiency, calculate the mass of copper deposited. Use M(Cu)=63.5 g mol⁻¹ and F=96500 C mol⁻¹.","options":{"A":"0.03332 g","B":"2.03332 g","C":"3.04997 g","D":"-0.96668 g"},"correct_answer":"B","correct_answer_value":"2.03332 g","solution":"Cu²⁺+2e⁻→Cu, so z=2. By Faraday's law m=MIt/(zF)=63.5×10×618/(2×96500)=2.03332 g.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 2.03332 g, not 0.03332 g.","B":"Correct. The worked calculation gives 2.03332 g.","C":"Incorrect. Substituting the stated values into the governing relation gives 2.03332 g, not 3.04997 g.","D":"Incorrect. Substituting the stated values into the governing relation gives 2.03332 g, not -0.96668 g."},"concepts_required":["electrolysis","Faraday's laws","redox stoichiometry"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_faraday_copper","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_003931","subject":"physics","topic":"mechanics","subtopic":"work_energy_theorem","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A 14 kg block initially moves at 2 m s⁻¹. A constant net force does 26 N of work over 2 m in the direction of motion. Find its final kinetic energy.","options":{"A":"78 J","B":"80 J","C":"-80 J","D":"160 J"},"correct_answer":"B","correct_answer_value":"80 J","solution":"By the work–energy theorem, K_f=K_i+W=½mu²+Fs=½(14)(2)²+(26)(2)=80 J.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 80 J, not 78 J.","B":"Correct. The worked calculation gives 80 J.","C":"Incorrect. Substituting the stated values into the governing relation gives 80 J, not -80 J.","D":"Incorrect. Substituting the stated values into the governing relation gives 80 J, not 160 J."},"concepts_required":["work–energy theorem","kinetic energy","work by constant force"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_work_energy","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_007654","subject":"chemistry","topic":"physical_chemistry","subtopic":"strong_acid_base_mixture","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"32 mL of 0.10 M HCl is mixed with 27 mL of 0.10 M NaOH at 25°C. Assuming complete dissociation and additive volumes, find the final pH.","options":{"A":"1.55391","B":"-0.92812","C":"4.07188","D":"2.07188"},"correct_answer":"D","correct_answer_value":"2.07188","solution":"Excess H⁺ moles=0.10×(32−27)/1000=0.000500 mol. Total volume=0.059 L, so [H⁺]=0.008475 M and pH=−log₁₀[H⁺]=2.07188.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 2.07188, not 1.55391.","B":"Incorrect. Substituting the stated values into the governing relation gives 2.07188, not -0.92812.","C":"Incorrect. Substituting the stated values into the governing relation gives 2.07188, not 4.07188.","D":"Correct. The worked calculation gives 2.07188."},"concepts_required":["strong acid–strong base neutralization","molarity","pH and pOH"],"reasoning_steps":5,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_acid_base_pH","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_008708","subject":"chemistry","topic":"physical_chemistry","subtopic":"equilibrium_degree_dissociation","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"For A(g) ⇌ 2B(g), the initial concentration of A is 1.96 mol L⁻¹ and initially B is absent. At equilibrium, 53% of A has dissociated at constant volume. Find K_c.","options":{"A":"5.68565","B":"2.34283","C":"9.3713","D":"4.68565"},"correct_answer":"D","correct_answer_value":"4.68565","solution":"Let α=0.53. At equilibrium [A]=c₀(1−α) and [B]=2c₀α. Thus K_c=[B]²/[A]=4c₀α²/(1−α)=4(1.96)(0.53)²/(1−0.53)=4.68565.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 4.68565, not 5.68565.","B":"Incorrect. Substituting the stated values into the governing relation gives 4.68565, not 2.34283.","C":"Incorrect. Substituting the stated values into the governing relation gives 4.68565, not 9.3713.","D":"Correct. The worked calculation gives 4.68565."},"concepts_required":["chemical equilibrium","equilibrium constant","degree of dissociation"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_equilibrium_alpha","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_007165","subject":"chemistry","topic":"physical_chemistry","subtopic":"ideal_gas_volume","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Assuming ideal behaviour, what volume is occupied by 18 mol of gas at 575 K and 1.39 atm? Use R=0.082057 L atm mol⁻¹ K⁻¹.","options":{"A":"610.99996 L","B":"1221.99993 L","C":"916.49995 L","D":"609.99996 L"},"correct_answer":"A","correct_answer_value":"610.99996 L","solution":"PV=nRT, so V=nRT/P=(18)(0.082057)(575)/(1.39)=610.99996 L.","option_explanations":{"A":"Correct. The worked calculation gives 610.99996 L.","B":"Incorrect. Substituting the stated values into the governing relation gives 610.99996 L, not 1221.99993 L.","C":"Incorrect. Substituting the stated values into the governing relation gives 610.99996 L, not 916.49995 L.","D":"Incorrect. Substituting the stated values into the governing relation gives 610.99996 L, not 609.99996 L."},"concepts_required":["ideal gas equation","unit consistency"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_ideal_gas","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_008053","subject":"chemistry","topic":"physical_chemistry","subtopic":"faradays_law_copper","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"An aqueous Cu²⁺ solution is electrolysed using a current of 5 A for 762 s. Assuming 100% current efficiency, calculate the mass of copper deposited. Use M(Cu)=63.5 g mol⁻¹ and F=96500 C mol⁻¹.","options":{"A":"2.5071 g","B":"2.25355 g","C":"1.25355 g","D":"4.25355 g"},"correct_answer":"C","correct_answer_value":"1.25355 g","solution":"Cu²⁺+2e⁻→Cu, so z=2. By Faraday's law m=MIt/(zF)=63.5×5×762/(2×96500)=1.25355 g.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 1.25355 g, not 2.5071 g.","B":"Incorrect. Substituting the stated values into the governing relation gives 1.25355 g, not 2.25355 g.","C":"Correct. The worked calculation gives 1.25355 g.","D":"Incorrect. Substituting the stated values into the governing relation gives 1.25355 g, not 4.25355 g."},"concepts_required":["electrolysis","Faraday's laws","redox stoichiometry"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_faraday_copper","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_008721","subject":"chemistry","topic":"physical_chemistry","subtopic":"equilibrium_degree_dissociation","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"For A(g) ⇌ 2B(g), the initial concentration of A is 0.76 mol L⁻¹ and initially B is absent. At equilibrium, 30% of A has dissociated at constant volume. Find K_c.","options":{"A":"0.78171","B":"-0.60914","C":"0.39086","D":"0.29314"},"correct_answer":"C","correct_answer_value":"0.39086","solution":"Let α=0.3. At equilibrium [A]=c₀(1−α) and [B]=2c₀α. Thus K_c=[B]²/[A]=4c₀α²/(1−α)=4(0.76)(0.3)²/(1−0.3)=0.39086.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.39086, not 0.78171.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.39086, not -0.60914.","C":"Correct. The worked calculation gives 0.39086.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.39086, not 0.29314."},"concepts_required":["chemical equilibrium","equilibrium constant","degree of dissociation"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_equilibrium_alpha","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_000708","subject":"mathematics","topic":"sequences_and_series","subtopic":"geometric_progression","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A geometric progression has first term 8 and common ratio 3. What is the sum of its first 7 terms?","options":{"A":"6558","B":"8742","C":"8744","D":"-8744"},"correct_answer":"C","correct_answer_value":"8744","solution":"For r≠1, Sₙ=a(rⁿ−1)/(r−1). Thus Sₙ=8(3^7−1)/(3−1)=8744.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 8744, not 6558.","B":"Incorrect. Substituting the stated values into the governing relation gives 8744, not 8742.","C":"Correct. The worked calculation gives 8744.","D":"Incorrect. Substituting the stated values into the governing relation gives 8744, not -8744."},"concepts_required":["geometric progression","finite geometric sum"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_gp_sum","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_002391","subject":"mathematics","topic":"linear_algebra","subtopic":"2x2_determinant","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Find the determinant of the matrix [[7, -7], [1, -5]].","options":{"A":"-30","B":"-22.4","C":"-21","D":"-28"},"correct_answer":"D","correct_answer_value":"-28","solution":"For a 2×2 matrix [[a,b],[c,d]], det=ad−bc. Here it is (7)(-5)−(-7)(1)=-28.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives -28, not -30.","B":"Incorrect. Substituting the stated values into the governing relation gives -28, not -22.4.","C":"Incorrect. Substituting the stated values into the governing relation gives -28, not -21.","D":"Correct. The worked calculation gives -28."},"concepts_required":["determinants","matrix operations"],"reasoning_steps":2,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_determinant_2x2","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_006299","subject":"physics","topic":"ray_optics","subtopic":"convex_lens_magnification","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A real object is placed 39 cm in front of a thin convex lens of focal length 10 cm. Using the Cartesian sign convention, what is the magnitude of the linear magnification?","options":{"A":"0.17241","B":"0.34483","C":"0.68966","D":"-2.65517"},"correct_answer":"B","correct_answer_value":"0.34483","solution":"Using distances as positive magnitudes, 1/f=1/v+1/u gives v=uf/(u−f)=(39)(10)/(39−10)=13.44828 cm. Thus |m|=v/u=13.44828/39=0.34483.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.34483, not 0.17241.","B":"Correct. The worked calculation gives 0.34483.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.34483, not 0.68966.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.34483, not -2.65517."},"concepts_required":["thin lens equation","linear magnification","real image"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_lens_magnification","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_004768","subject":"physics","topic":"electricity","subtopic":"series_parallel_circuit","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Resistors 2 Ω and 15 Ω are connected in series; this combination is connected in parallel with a 10 Ω resistor across a 36 V ideal battery. Find the total current supplied by the battery.","options":{"A":"3.71765 A","B":"2.71765 A","C":"5.71765 A","D":"4.28824 A"},"correct_answer":"C","correct_answer_value":"5.71765 A","solution":"Series branch resistance=2+15=17 Ω. Parallel equivalent R_eq=(17×10)/(17+10)=6.2963 Ω. Total current I=V/R_eq=36/6.2963=5.71765 A.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 5.71765 A, not 3.71765 A.","B":"Incorrect. Substituting the stated values into the governing relation gives 5.71765 A, not 2.71765 A.","C":"Correct. The worked calculation gives 5.71765 A.","D":"Incorrect. Substituting the stated values into the governing relation gives 5.71765 A, not 4.28824 A."},"concepts_required":["series resistors","parallel resistors","Ohm's law"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_series_parallel","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_001181","subject":"mathematics","topic":"calculus","subtopic":"polynomial_differentiation","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Let f(x)=-2x³ -7x² +10x+7. Evaluate f′(-4).","options":{"A":"-30","B":"-45","C":"30","D":"-28"},"correct_answer":"A","correct_answer_value":"-30","solution":"Differentiate: f′(x)=3(-2)x²+2(-7)x+(10). At x=-4, f′(-4)=3(-2)(-4)²+2(-7)(-4)+(10)=-30.","option_explanations":{"A":"Correct. The worked calculation gives -30.","B":"Incorrect. Substituting the stated values into the governing relation gives -30, not -45.","C":"Incorrect. Substituting the stated values into the governing relation gives -30, not 30.","D":"Incorrect. Substituting the stated values into the governing relation gives -30, not -28."},"concepts_required":["power rule","polynomial derivative"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_derivative_cubic","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_004234","subject":"physics","topic":"mechanics","subtopic":"inclined_plane_with_friction","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A 43 kg block slides down a 37° rough incline. The coefficient of kinetic friction is 0.18. Taking g=10 m s⁻², find the net force down the plane. Use sin37°=0.6 and cos37°=0.8.","options":{"A":"147.06 N","B":"196.08 N","C":"197.08 N","D":"20.608 N"},"correct_answer":"B","correct_answer_value":"196.08 N","solution":"Net force=mg(sinθ−μcosθ)=43×10×(0.6−0.18×0.8)=196.08 N.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 196.08 N, not 147.06 N.","B":"Correct. The worked calculation gives 196.08 N.","C":"Incorrect. Substituting the stated values into the governing relation gives 196.08 N, not 197.08 N.","D":"Incorrect. Substituting the stated values into the governing relation gives 196.08 N, not 20.608 N."},"concepts_required":["Newton's laws","inclined plane","kinetic friction"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_incline_friction","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_001633","subject":"mathematics","topic":"calculus","subtopic":"definite_integral_polynomial","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Evaluate the definite integral ∫₀^8 (3x² -4x +0) dx.","options":{"A":"385","B":"39.4","C":"768","D":"384"},"correct_answer":"D","correct_answer_value":"384","solution":"An antiderivative is (3/3)x³+(-4/2)x²+(0)x. Evaluating between 0 and 8 gives (3/3)(8)³+(-4/2)(8)²+(0)(8)=384.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 384, not 385.","B":"Incorrect. Substituting the stated values into the governing relation gives 384, not 39.4.","C":"Incorrect. Substituting the stated values into the governing relation gives 384, not 768.","D":"Correct. The worked calculation gives 384."},"concepts_required":["definite integration","power rule","limits of integration"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_integral_poly","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_001881","subject":"mathematics","topic":"combinatorics","subtopic":"binomial_coefficient","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"How many distinct subsets containing exactly 5 elements can be selected from a set of 60 distinct elements?","options":{"A":"-5461512","B":"5461512","C":"5461510","D":"5461513"},"correct_answer":"B","correct_answer_value":"5461512","solution":"Order does not matter, so the count is C(60,5)=60!/[5!(55)!]=5461512.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 5461512, not -5461512.","B":"Correct. The worked calculation gives 5461512.","C":"Incorrect. Substituting the stated values into the governing relation gives 5461512, not 5461510.","D":"Incorrect. Substituting the stated values into the governing relation gives 5461512, not 5461513."},"concepts_required":["combinations","factorials"],"reasoning_steps":2,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_combinations","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_004619","subject":"physics","topic":"rotational_mechanics","subtopic":"disc_angular_acceleration","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A uniform solid disc of mass 5 kg and radius 1 m is acted on by a constant torque of 66 N m about its central axis. Find its angular acceleration.","options":{"A":"26.4 rad s⁻²","B":"39.6 rad s⁻²","C":"25.4 rad s⁻²","D":"13.2 rad s⁻²"},"correct_answer":"A","correct_answer_value":"26.4 rad s⁻²","solution":"For a solid disc, I=½MR²=½(5)(1)²=2.5 kg m². Using τ=Iα, α=66/2.5=26.4 rad s⁻².","option_explanations":{"A":"Correct. The worked calculation gives 26.4 rad s⁻².","B":"Incorrect. Substituting the stated values into the governing relation gives 26.4 rad s⁻², not 39.6 rad s⁻².","C":"Incorrect. Substituting the stated values into the governing relation gives 26.4 rad s⁻², not 25.4 rad s⁻².","D":"Incorrect. Substituting the stated values into the governing relation gives 26.4 rad s⁻², not 13.2 rad s⁻²."},"concepts_required":["moment of inertia","rotational dynamics","torque"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_disc_torque","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_007855","subject":"chemistry","topic":"physical_chemistry","subtopic":"first_order_half_life","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A first-order reaction has a half-life of 20 minutes. If the initial reactant concentration is 10 mmol L⁻¹, what concentration remains after 100 minutes?","options":{"A":"0.46875 mmol L⁻¹","B":"0.15625 mmol L⁻¹","C":"-0.6875 mmol L⁻¹","D":"0.3125 mmol L⁻¹"},"correct_answer":"D","correct_answer_value":"0.3125 mmol L⁻¹","solution":"100 minutes corresponds to 5 half-lives. The remaining concentration is [A]₀(1/2)ⁿ=10/2^5=0.3125 mmol L⁻¹.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.3125 mmol L⁻¹, not 0.46875 mmol L⁻¹.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.3125 mmol L⁻¹, not 0.15625 mmol L⁻¹.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.3125 mmol L⁻¹, not -0.6875 mmol L⁻¹.","D":"Correct. The worked calculation gives 0.3125 mmol L⁻¹."},"concepts_required":["first-order kinetics","half-life","exponential decay"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_first_order_halflife","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_004909","subject":"physics","topic":"electricity","subtopic":"series_parallel_circuit","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Resistors 13 Ω and 8 Ω are connected in series; this combination is connected in parallel with a 17 Ω resistor across a 24 V ideal battery. Find the total current supplied by the battery.","options":{"A":"2.55462 A","B":"1.27731 A","C":"-2.55462 A","D":"5.55462 A"},"correct_answer":"A","correct_answer_value":"2.55462 A","solution":"Series branch resistance=13+8=21 Ω. Parallel equivalent R_eq=(21×17)/(21+17)=9.39474 Ω. Total current I=V/R_eq=24/9.39474=2.55462 A.","option_explanations":{"A":"Correct. The worked calculation gives 2.55462 A.","B":"Incorrect. Substituting the stated values into the governing relation gives 2.55462 A, not 1.27731 A.","C":"Incorrect. Substituting the stated values into the governing relation gives 2.55462 A, not -2.55462 A.","D":"Incorrect. Substituting the stated values into the governing relation gives 2.55462 A, not 5.55462 A."},"concepts_required":["series resistors","parallel resistors","Ohm's law"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_series_parallel","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_009766","subject":"chemistry","topic":"inorganic_chemistry","subtopic":"complex_ion_oxidation_state","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"For a hypothetical coordination ion [Cu(CN)_5]^+0, each CN ligand has charge −1. Determine the formal oxidation state of Cu.","options":{"A":"2","B":"-5","C":"5","D":"7.5"},"correct_answer":"C","correct_answer_value":"5","solution":"Let the formal oxidation state of Cu be x. Charge balance gives x+(5)(−1)=+0; hence x=0+5=+5.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 5, not 2.","B":"Incorrect. Substituting the stated values into the governing relation gives 5, not -5.","C":"Correct. The worked calculation gives 5.","D":"Incorrect. Substituting the stated values into the governing relation gives 5, not 7.5."},"concepts_required":["coordination compounds","oxidation state","charge balance"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_complex_oxidation","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_008148","subject":"chemistry","topic":"physical_chemistry","subtopic":"faradays_law_copper","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"An aqueous Cu²⁺ solution is electrolysed using a current of 2 A for 22657 s. Assuming 100% current efficiency, calculate the mass of copper deposited. Use M(Cu)=63.5 g mol⁻¹ and F=96500 C mol⁻¹.","options":{"A":"13.90901 g","B":"14.90901 g","C":"11.18176 g","D":"11.90901 g"},"correct_answer":"B","correct_answer_value":"14.90901 g","solution":"Cu²⁺+2e⁻→Cu, so z=2. By Faraday's law m=MIt/(zF)=63.5×2×22657/(2×96500)=14.90901 g.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 14.90901 g, not 13.90901 g.","B":"Correct. The worked calculation gives 14.90901 g.","C":"Incorrect. Substituting the stated values into the governing relation gives 14.90901 g, not 11.18176 g.","D":"Incorrect. Substituting the stated values into the governing relation gives 14.90901 g, not 11.90901 g."},"concepts_required":["electrolysis","Faraday's laws","redox stoichiometry"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_faraday_copper","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_008582","subject":"chemistry","topic":"physical_chemistry","subtopic":"boiling_point_elevation","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A non-volatile solute is dissolved in water to produce a solution of molality 0.67 mol kg⁻¹. If its van't Hoff factor is 1 and K_b for water is 0.512 K kg mol⁻¹, find the boiling-point elevation.","options":{"A":"1.0343 K","B":"0.34304 K","C":"2.34304 K","D":"-2.65696 K"},"correct_answer":"B","correct_answer_value":"0.34304 K","solution":"ΔT_b=iK_bm. Substitution gives ΔT_b=1×0.512×0.67=0.34304 K.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.34304 K, not 1.0343 K.","B":"Correct. The worked calculation gives 0.34304 K.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.34304 K, not 2.34304 K.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.34304 K, not -2.65696 K."},"concepts_required":["colligative properties","van't Hoff factor","molality"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_boiling_elevation","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_007638","subject":"chemistry","topic":"physical_chemistry","subtopic":"strong_acid_base_mixture","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"92 mL of 0.10 M HCl is mixed with 20 mL of 0.10 M NaOH at 25°C. Assuming complete dissociation and additive volumes, find the final pH.","options":{"A":"0.19189","B":"1.19189","C":"1.11919","D":"-1.80811"},"correct_answer":"B","correct_answer_value":"1.19189","solution":"Excess H⁺ moles=0.10×(92−20)/1000=0.007200 mol. Total volume=0.112 L, so [H⁺]=0.064286 M and pH=−log₁₀[H⁺]=1.19189.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 1.19189, not 0.19189.","B":"Correct. The worked calculation gives 1.19189.","C":"Incorrect. Substituting the stated values into the governing relation gives 1.19189, not 1.11919.","D":"Incorrect. Substituting the stated values into the governing relation gives 1.19189, not -1.80811."},"concepts_required":["strong acid–strong base neutralization","molarity","pH and pOH"],"reasoning_steps":5,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_acid_base_pH","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_003075","subject":"mathematics","topic":"complex_numbers","subtopic":"real_part_of_complex_quotient","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Let z=(-12-12i)/(8+1i). What is Re(z), the real part of z?","options":{"A":"1.66154","B":"-1.24615","C":"-2.66154","D":"-1.66154"},"correct_answer":"D","correct_answer_value":"-1.66154","solution":"Multiply numerator and denominator by the conjugate (8-1i). Then Re(z)=(ac+bd)/(c²+d²)=((-12)(8)+(-12)(1))/(8²+(1)²)=-1.66154.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives -1.66154, not 1.66154.","B":"Incorrect. Substituting the stated values into the governing relation gives -1.66154, not -1.24615.","C":"Incorrect. Substituting the stated values into the governing relation gives -1.66154, not -2.66154.","D":"Correct. The worked calculation gives -1.66154."},"concepts_required":["complex conjugate","complex division","real and imaginary parts"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_complex_quotient","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_006976","subject":"chemistry","topic":"stoichiometry","subtopic":"limiting_reagent_water","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Hydrogen reacts with oxygen according to 2H₂ + O₂ → 2H₂O. If 2 mol H₂ and 29 mol O₂ are mixed and the reaction goes to completion, how many moles of H₂O form?","options":{"A":"2 mol","B":"1.5 mol","C":"1 mol","D":"4 mol"},"correct_answer":"A","correct_answer_value":"2 mol","solution":"From 2 mol H₂ and 1 mol O₂, 2 mol H₂O form. H₂ can produce 2 mol H₂O; O₂ can produce 58 mol H₂O. The limiting condition is H₂; product=2 mol.","option_explanations":{"A":"Correct. The worked calculation gives 2 mol.","B":"Incorrect. Substituting the stated values into the governing relation gives 2 mol, not 1.5 mol.","C":"Incorrect. Substituting the stated values into the governing relation gives 2 mol, not 1 mol.","D":"Incorrect. Substituting the stated values into the governing relation gives 2 mol, not 4 mol."},"concepts_required":["balanced chemical equations","limiting reagent","mole ratios"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_limiting_reagent","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_002941","subject":"mathematics","topic":"vectors","subtopic":"dot_product","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Given vectors u=(7, 8, -2) and v=(2, 5, 0), calculate u·v.","options":{"A":"108","B":"54","C":"27","D":"6.4"},"correct_answer":"B","correct_answer_value":"54","solution":"The dot product is the sum of corresponding component products: (7)(2)+(8)(5)+(-2)(0)=54.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 54, not 108.","B":"Correct. The worked calculation gives 54.","C":"Incorrect. Substituting the stated values into the governing relation gives 54, not 27.","D":"Incorrect. Substituting the stated values into the governing relation gives 54, not 6.4."},"concepts_required":["three-dimensional vectors","scalar product"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_vector_dot","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_008011","subject":"chemistry","topic":"physical_chemistry","subtopic":"faradays_law_copper","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"An aqueous Cu²⁺ solution is electrolysed using a current of 8 A for 21243 s. Assuming 100% current efficiency, calculate the mass of copper deposited. Use M(Cu)=63.5 g mol⁻¹ and F=96500 C mol⁻¹.","options":{"A":"57.91422 g","B":"55.91422 g","C":"54.91422 g","D":"41.93566 g"},"correct_answer":"B","correct_answer_value":"55.91422 g","solution":"Cu²⁺+2e⁻→Cu, so z=2. By Faraday's law m=MIt/(zF)=63.5×8×21243/(2×96500)=55.91422 g.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 55.91422 g, not 57.91422 g.","B":"Correct. The worked calculation gives 55.91422 g.","C":"Incorrect. Substituting the stated values into the governing relation gives 55.91422 g, not 54.91422 g.","D":"Incorrect. Substituting the stated values into the governing relation gives 55.91422 g, not 41.93566 g."},"concepts_required":["electrolysis","Faraday's laws","redox stoichiometry"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_faraday_copper","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_008977","subject":"chemistry","topic":"physical_chemistry","subtopic":"equilibrium_degree_dissociation","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"For A(g) ⇌ 2B(g), the initial concentration of A is 0.56 mol L⁻¹ and initially B is absent. At equilibrium, 48% of A has dissociated at constant volume. Find K_c.","options":{"A":"1.48874","B":"2.99249","C":"-0.00751","D":"0.99249"},"correct_answer":"D","correct_answer_value":"0.99249","solution":"Let α=0.48. At equilibrium [A]=c₀(1−α) and [B]=2c₀α. Thus K_c=[B]²/[A]=4c₀α²/(1−α)=4(0.56)(0.48)²/(1−0.48)=0.99249.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.99249, not 1.48874.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.99249, not 2.99249.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.99249, not -0.00751.","D":"Correct. The worked calculation gives 0.99249."},"concepts_required":["chemical equilibrium","equilibrium constant","degree of dissociation"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_equilibrium_alpha","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_009235","subject":"chemistry","topic":"physical_chemistry","subtopic":"nernst_equation_298K","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"At 298 K, a galvanic cell has E°cell=1.32 V, transfers 1 electrons, and has reaction quotient Q=10^5. Using E=E°−(0.0591/n)log₁₀Q, calculate Ecell.","options":{"A":"-0.9755 V","B":"-1.9755 V","C":"1.0245 V","D":"3.0245 V"},"correct_answer":"C","correct_answer_value":"1.0245 V","solution":"Apply the Nernst equation: E=1.32−(0.0591/1)(5)=1.0245 V.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 1.0245 V, not -0.9755 V.","B":"Incorrect. Substituting the stated values into the governing relation gives 1.0245 V, not -1.9755 V.","C":"Correct. The worked calculation gives 1.0245 V.","D":"Incorrect. Substituting the stated values into the governing relation gives 1.0245 V, not 3.0245 V."},"concepts_required":["electrochemical cells","Nernst equation","logarithms"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_nernst","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_004038","subject":"physics","topic":"mechanics","subtopic":"inclined_plane_with_friction","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A 20 kg block slides down a 37° rough incline. The coefficient of kinetic friction is 0.15. Taking g=10 m s⁻², find the net force down the plane. Use sin37°=0.6 and cos37°=0.8.","options":{"A":"48 N","B":"144 N","C":"95 N","D":"96 N"},"correct_answer":"D","correct_answer_value":"96 N","solution":"Net force=mg(sinθ−μcosθ)=20×10×(0.6−0.15×0.8)=96 N.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 96 N, not 48 N.","B":"Incorrect. Substituting the stated values into the governing relation gives 96 N, not 144 N.","C":"Incorrect. Substituting the stated values into the governing relation gives 96 N, not 95 N.","D":"Correct. The worked calculation gives 96 N."},"concepts_required":["Newton's laws","inclined plane","kinetic friction"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_incline_friction","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_004838","subject":"physics","topic":"electricity","subtopic":"series_parallel_circuit","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Resistors 15 Ω and 12 Ω are connected in series; this combination is connected in parallel with a 16 Ω resistor across a 12 V ideal battery. Find the total current supplied by the battery.","options":{"A":"1.19444 A","B":"4.19444 A","C":"-1.19444 A","D":"0.59722 A"},"correct_answer":"A","correct_answer_value":"1.19444 A","solution":"Series branch resistance=15+12=27 Ω. Parallel equivalent R_eq=(27×16)/(27+16)=10.04651 Ω. Total current I=V/R_eq=12/10.04651=1.19444 A.","option_explanations":{"A":"Correct. The worked calculation gives 1.19444 A.","B":"Incorrect. Substituting the stated values into the governing relation gives 1.19444 A, not 4.19444 A.","C":"Incorrect. Substituting the stated values into the governing relation gives 1.19444 A, not -1.19444 A.","D":"Incorrect. Substituting the stated values into the governing relation gives 1.19444 A, not 0.59722 A."},"concepts_required":["series resistors","parallel resistors","Ohm's law"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_series_parallel","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_005291","subject":"physics","topic":"electrostatics","subtopic":"coulombs_law","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Two point charges of +7 μC and +3 μC are separated by 6 cm in vacuum. Using k=9×10⁹ N m² C⁻², find the magnitude of the electrostatic force.","options":{"A":"26.25 N","B":"52.5 N","C":"51.5 N","D":"78.75 N"},"correct_answer":"B","correct_answer_value":"52.5 N","solution":"Convert charges to coulombs and distance to metres. F=k|q₁q₂|/r²=9×10⁹(7×10⁻⁶)(3×10⁻⁶)/(6×10⁻²)²=52.5 N.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 52.5 N, not 26.25 N.","B":"Correct. The worked calculation gives 52.5 N.","C":"Incorrect. Substituting the stated values into the governing relation gives 52.5 N, not 51.5 N.","D":"Incorrect. Substituting the stated values into the governing relation gives 52.5 N, not 78.75 N."},"concepts_required":["Coulomb's law","SI unit conversion","inverse-square law"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_coulomb","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_000316","subject":"mathematics","topic":"algebra","subtopic":"quadratic_roots","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"For the quadratic equation 3x^2 +5x -6=0, let its roots be α and β. What is α²+β²?","options":{"A":"6.77778","B":"10.16667","C":"-6.77778","D":"9.77778"},"correct_answer":"A","correct_answer_value":"6.77778","solution":"By Vieta's relations, α+β = -5/3 and αβ = -6/3. Then α²+β²=(α+β)²−2αβ = (-5/3)²−2(-6/3) = 6.77778.","option_explanations":{"A":"Correct. The worked calculation gives 6.77778.","B":"Incorrect. Substituting the stated values into the governing relation gives 6.77778, not 10.16667.","C":"Incorrect. Substituting the stated values into the governing relation gives 6.77778, not -6.77778.","D":"Incorrect. Substituting the stated values into the governing relation gives 6.77778, not 9.77778."},"concepts_required":["Vieta's formulas","symmetric polynomial expressions"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_quadratic_vieta","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_006246","subject":"physics","topic":"ray_optics","subtopic":"convex_lens_magnification","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A real object is placed 26 cm in front of a thin convex lens of focal length 6 cm. Using the Cartesian sign convention, what is the magnitude of the linear magnification?","options":{"A":"0.3","B":"-0.7","C":"-0.3","D":"-2.7"},"correct_answer":"A","correct_answer_value":"0.3","solution":"Using distances as positive magnitudes, 1/f=1/v+1/u gives v=uf/(u−f)=(26)(6)/(26−6)=7.8 cm. Thus |m|=v/u=7.8/26=0.3.","option_explanations":{"A":"Correct. The worked calculation gives 0.3.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.3, not -0.7.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.3, not -0.3.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.3, not -2.7."},"concepts_required":["thin lens equation","linear magnification","real image"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_lens_magnification","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_001399","subject":"mathematics","topic":"calculus","subtopic":"definite_integral_polynomial","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Evaluate the definite integral ∫₀^2 (-3x² -8x -4) dx.","options":{"A":"-48","B":"-30","C":"-64","D":"-32"},"correct_answer":"D","correct_answer_value":"-32","solution":"An antiderivative is (-3/3)x³+(-8/2)x²+(-4)x. Evaluating between 0 and 2 gives (-3/3)(2)³+(-8/2)(2)²+(-4)(2)=-32.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives -32, not -48.","B":"Incorrect. Substituting the stated values into the governing relation gives -32, not -30.","C":"Incorrect. Substituting the stated values into the governing relation gives -32, not -64.","D":"Correct. The worked calculation gives -32."},"concepts_required":["definite integration","power rule","limits of integration"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_integral_poly","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_004947","subject":"physics","topic":"electricity","subtopic":"series_parallel_circuit","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Resistors 7 Ω and 12 Ω are connected in series; this combination is connected in parallel with a 7 Ω resistor across a 9 V ideal battery. Find the total current supplied by the battery.","options":{"A":"3.7594 A","B":"1.7594 A","C":"-0.2406 A","D":"0.7594 A"},"correct_answer":"B","correct_answer_value":"1.7594 A","solution":"Series branch resistance=7+12=19 Ω. Parallel equivalent R_eq=(19×7)/(19+7)=5.11538 Ω. Total current I=V/R_eq=9/5.11538=1.7594 A.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 1.7594 A, not 3.7594 A.","B":"Correct. The worked calculation gives 1.7594 A.","C":"Incorrect. Substituting the stated values into the governing relation gives 1.7594 A, not -0.2406 A.","D":"Incorrect. Substituting the stated values into the governing relation gives 1.7594 A, not 0.7594 A."},"concepts_required":["series resistors","parallel resistors","Ohm's law"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_series_parallel","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_008004","subject":"chemistry","topic":"physical_chemistry","subtopic":"faradays_law_copper","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"An aqueous Cu²⁺ solution is electrolysed using a current of 9 A for 27290 s. Assuming 100% current efficiency, calculate the mass of copper deposited. Use M(Cu)=63.5 g mol⁻¹ and F=96500 C mol⁻¹.","options":{"A":"80.80951 g","B":"9.08095 g","C":"161.61902 g","D":"121.21426 g"},"correct_answer":"A","correct_answer_value":"80.80951 g","solution":"Cu²⁺+2e⁻→Cu, so z=2. By Faraday's law m=MIt/(zF)=63.5×9×27290/(2×96500)=80.80951 g.","option_explanations":{"A":"Correct. The worked calculation gives 80.80951 g.","B":"Incorrect. Substituting the stated values into the governing relation gives 80.80951 g, not 9.08095 g.","C":"Incorrect. Substituting the stated values into the governing relation gives 80.80951 g, not 161.61902 g.","D":"Incorrect. Substituting the stated values into the governing relation gives 80.80951 g, not 121.21426 g."},"concepts_required":["electrolysis","Faraday's laws","redox stoichiometry"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_faraday_copper","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_007044","subject":"chemistry","topic":"physical_chemistry","subtopic":"ideal_gas_volume","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Assuming ideal behaviour, what volume is occupied by 14 mol of gas at 282 K and 0.76 atm? Use R=0.082057 L atm mol⁻¹ K⁻¹.","options":{"A":"426.26452 L","B":"213.13226 L","C":"43.62645 L","D":"852.52904 L"},"correct_answer":"A","correct_answer_value":"426.26452 L","solution":"PV=nRT, so V=nRT/P=(14)(0.082057)(282)/(0.76)=426.26452 L.","option_explanations":{"A":"Correct. The worked calculation gives 426.26452 L.","B":"Incorrect. Substituting the stated values into the governing relation gives 426.26452 L, not 213.13226 L.","C":"Incorrect. Substituting the stated values into the governing relation gives 426.26452 L, not 43.62645 L.","D":"Incorrect. Substituting the stated values into the governing relation gives 426.26452 L, not 852.52904 L."},"concepts_required":["ideal gas equation","unit consistency"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_ideal_gas","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_004576","subject":"physics","topic":"rotational_mechanics","subtopic":"disc_angular_acceleration","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A uniform solid disc of mass 12 kg and radius 5 m is acted on by a constant torque of 28 N m about its central axis. Find its angular acceleration.","options":{"A":"3.18667 rad s⁻²","B":"0.09333 rad s⁻²","C":"0.18667 rad s⁻²","D":"1.18667 rad s⁻²"},"correct_answer":"C","correct_answer_value":"0.18667 rad s⁻²","solution":"For a solid disc, I=½MR²=½(12)(5)²=150 kg m². Using τ=Iα, α=28/150=0.18667 rad s⁻².","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.18667 rad s⁻², not 3.18667 rad s⁻².","B":"Incorrect. Substituting the stated values into the governing relation gives 0.18667 rad s⁻², not 0.09333 rad s⁻².","C":"Correct. The worked calculation gives 0.18667 rad s⁻².","D":"Incorrect. Substituting the stated values into the governing relation gives 0.18667 rad s⁻², not 1.18667 rad s⁻²."},"concepts_required":["moment of inertia","rotational dynamics","torque"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_disc_torque","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_000210","subject":"mathematics","topic":"algebra","subtopic":"quadratic_roots","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"For the quadratic equation 7x^2 +16x -5=0, let its roots be α and β. What is α²+β²?","options":{"A":"6.65306","B":"5.65306","C":"13.30612","D":"9.65306"},"correct_answer":"A","correct_answer_value":"6.65306","solution":"By Vieta's relations, α+β = -16/7 and αβ = -5/7. Then α²+β²=(α+β)²−2αβ = (-16/7)²−2(-5/7) = 6.65306.","option_explanations":{"A":"Correct. The worked calculation gives 6.65306.","B":"Incorrect. Substituting the stated values into the governing relation gives 6.65306, not 5.65306.","C":"Incorrect. Substituting the stated values into the governing relation gives 6.65306, not 13.30612.","D":"Incorrect. Substituting the stated values into the governing relation gives 6.65306, not 9.65306."},"concepts_required":["Vieta's formulas","symmetric polynomial expressions"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_quadratic_vieta","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_005895","subject":"physics","topic":"thermodynamics","subtopic":"ideal_gas_isochoric_pressure","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A fixed amount of ideal gas is kept in a rigid container. Its pressure is 3.3 atm at 282 K. What is its pressure at 550 K?","options":{"A":"1.64362 atm","B":"7.43617 atm","C":"9.65426 atm","D":"6.43617 atm"},"correct_answer":"D","correct_answer_value":"6.43617 atm","solution":"At constant volume, P/T is constant. P₂=P₁T₂/T₁=(3.3)(550)/(282)=6.43617 atm.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 6.43617 atm, not 1.64362 atm.","B":"Incorrect. Substituting the stated values into the governing relation gives 6.43617 atm, not 7.43617 atm.","C":"Incorrect. Substituting the stated values into the governing relation gives 6.43617 atm, not 9.65426 atm.","D":"Correct. The worked calculation gives 6.43617 atm."},"concepts_required":["ideal gas law","isochoric process","absolute temperature"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_isochoric_gas","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_008480","subject":"chemistry","topic":"physical_chemistry","subtopic":"boiling_point_elevation","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A non-volatile solute is dissolved in water to produce a solution of molality 1.81 mol kg⁻¹. If its van't Hoff factor is 4 and K_b for water is 1.0 K kg mol⁻¹, find the boiling-point elevation.","options":{"A":"7.24 K","B":"1.724 K","C":"5.24 K","D":"3.62 K"},"correct_answer":"A","correct_answer_value":"7.24 K","solution":"ΔT_b=iK_bm. Substitution gives ΔT_b=4×1.0×1.81=7.24 K.","option_explanations":{"A":"Correct. The worked calculation gives 7.24 K.","B":"Incorrect. Substituting the stated values into the governing relation gives 7.24 K, not 1.724 K.","C":"Incorrect. Substituting the stated values into the governing relation gives 7.24 K, not 5.24 K.","D":"Incorrect. Substituting the stated values into the governing relation gives 7.24 K, not 3.62 K."},"concepts_required":["colligative properties","van't Hoff factor","molality"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_boiling_elevation","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_007218","subject":"chemistry","topic":"physical_chemistry","subtopic":"ideal_gas_volume","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Assuming ideal behaviour, what volume is occupied by 13 mol of gas at 496 K and 1.24 atm? Use R=0.082057 L atm mol⁻¹ K⁻¹.","options":{"A":"213.3482 L","B":"512.03568 L","C":"426.6964 L","D":"341.35712 L"},"correct_answer":"C","correct_answer_value":"426.6964 L","solution":"PV=nRT, so V=nRT/P=(13)(0.082057)(496)/(1.24)=426.6964 L.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 426.6964 L, not 213.3482 L.","B":"Incorrect. Substituting the stated values into the governing relation gives 426.6964 L, not 512.03568 L.","C":"Correct. The worked calculation gives 426.6964 L.","D":"Incorrect. Substituting the stated values into the governing relation gives 426.6964 L, not 341.35712 L."},"concepts_required":["ideal gas equation","unit consistency"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_ideal_gas","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_003614","subject":"physics","topic":"mechanics","subtopic":"projectile_range","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A projectile is launched from level ground at 49 m s⁻¹ at 35° to the horizontal. Take g=10 m s⁻². Find its horizontal range.","options":{"A":"169.21515 m","B":"225.6202 m","C":"226.6202 m","D":"227.6202 m"},"correct_answer":"B","correct_answer_value":"225.6202 m","solution":"Range R=u²sin(2θ)/g. Substituting u=49, θ=35° and g=10 gives R=(49)²sin(70°)/10=225.6202 m.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 225.6202 m, not 169.21515 m.","B":"Correct. The worked calculation gives 225.6202 m.","C":"Incorrect. Substituting the stated values into the governing relation gives 225.6202 m, not 226.6202 m.","D":"Incorrect. Substituting the stated values into the governing relation gives 225.6202 m, not 227.6202 m."},"concepts_required":["projectile motion","range equation","trigonometry"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_projectile_range","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_003485","subject":"physics","topic":"mechanics","subtopic":"projectile_range","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A projectile is launched from level ground at 21 m s⁻¹ at 25° to the horizontal. Take g=10 m s⁻². Find its horizontal range.","options":{"A":"-33.78256 m","B":"35.78256 m","C":"33.78256 m","D":"31.78256 m"},"correct_answer":"C","correct_answer_value":"33.78256 m","solution":"Range R=u²sin(2θ)/g. Substituting u=21, θ=25° and g=10 gives R=(21)²sin(50°)/10=33.78256 m.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 33.78256 m, not -33.78256 m.","B":"Incorrect. Substituting the stated values into the governing relation gives 33.78256 m, not 35.78256 m.","C":"Correct. The worked calculation gives 33.78256 m.","D":"Incorrect. Substituting the stated values into the governing relation gives 33.78256 m, not 31.78256 m."},"concepts_required":["projectile motion","range equation","trigonometry"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_projectile_range","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_003280","subject":"mathematics","topic":"complex_numbers","subtopic":"real_part_of_complex_quotient","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Let z=(12+13i)/(12+7i). What is Re(z), the real part of z?","options":{"A":"0.91321","B":"1.21762","C":"1.82642","D":"3.21762"},"correct_answer":"B","correct_answer_value":"1.21762","solution":"Multiply numerator and denominator by the conjugate (12-7i). Then Re(z)=(ac+bd)/(c��+d²)=((12)(12)+(13)(7))/(12²+(7)²)=1.21762.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 1.21762, not 0.91321.","B":"Correct. The worked calculation gives 1.21762.","C":"Incorrect. Substituting the stated values into the governing relation gives 1.21762, not 1.82642.","D":"Incorrect. Substituting the stated values into the governing relation gives 1.21762, not 3.21762."},"concepts_required":["complex conjugate","complex division","real and imaginary parts"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_complex_quotient","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_002104","subject":"mathematics","topic":"probability","subtopic":"same_colour_without_replacement","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A bag contains 8 red, 9 blue and 10 green balls. Two balls are drawn uniformly at random without replacement. What is the probability that both have the same colour?","options":{"A":"-0.31054","B":"1.31054","C":"2.31054","D":"0.31054"},"correct_answer":"D","correct_answer_value":"0.31054","solution":"Favourable pairs = C(8,2)+C(9,2)+C(10,2)=109. Total pairs=C(27,2)=351. Therefore P=109/351=0.31054.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.31054, not -0.31054.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.31054, not 1.31054.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.31054, not 2.31054.","D":"Correct. The worked calculation gives 0.31054."},"concepts_required":["classical probability","combinations","without replacement"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_probability_colours","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_008116","subject":"chemistry","topic":"physical_chemistry","subtopic":"faradays_law_copper","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"An aqueous Cu²⁺ solution is electrolysed using a current of 2 A for 7575 s. Assuming 100% current efficiency, calculate the mass of copper deposited. Use M(Cu)=63.5 g mol⁻¹ and F=96500 C mol⁻¹.","options":{"A":"1.49846 g","B":"2.49229 g","C":"6.98459 g","D":"4.98459 g"},"correct_answer":"D","correct_answer_value":"4.98459 g","solution":"Cu²⁺+2e⁻→Cu, so z=2. By Faraday's law m=MIt/(zF)=63.5×2×7575/(2×96500)=4.98459 g.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 4.98459 g, not 1.49846 g.","B":"Incorrect. Substituting the stated values into the governing relation gives 4.98459 g, not 2.49229 g.","C":"Incorrect. Substituting the stated values into the governing relation gives 4.98459 g, not 6.98459 g.","D":"Correct. The worked calculation gives 4.98459 g."},"concepts_required":["electrolysis","Faraday's laws","redox stoichiometry"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_faraday_copper","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_003916","subject":"physics","topic":"mechanics","subtopic":"work_energy_theorem","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A 10 kg block initially moves at 11 m s⁻¹. A constant net force does 22 N of work over 12 m in the direction of motion. Find its final kinetic energy.","options":{"A":"1303.5 J","B":"869 J","C":"1042.8 J","D":"867 J"},"correct_answer":"B","correct_answer_value":"869 J","solution":"By the work–energy theorem, K_f=K_i+W=½mu²+Fs=½(10)(11)²+(22)(12)=869 J.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 869 J, not 1303.5 J.","B":"Correct. The worked calculation gives 869 J.","C":"Incorrect. Substituting the stated values into the governing relation gives 869 J, not 1042.8 J.","D":"Incorrect. Substituting the stated values into the governing relation gives 869 J, not 867 J."},"concepts_required":["work–energy theorem","kinetic energy","work by constant force"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_work_energy","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_006164","subject":"physics","topic":"ray_optics","subtopic":"convex_lens_magnification","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A real object is placed 23 cm in front of a thin convex lens of focal length 6 cm. Using the Cartesian sign convention, what is the magnitude of the linear magnification?","options":{"A":"-0.64706","B":"2.35294","C":"0.52941","D":"0.35294"},"correct_answer":"D","correct_answer_value":"0.35294","solution":"Using distances as positive magnitudes, 1/f=1/v+1/u gives v=uf/(u−f)=(23)(6)/(23−6)=8.11765 cm. Thus |m|=v/u=8.11765/23=0.35294.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.35294, not -0.64706.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.35294, not 2.35294.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.35294, not 0.52941.","D":"Correct. The worked calculation gives 0.35294."},"concepts_required":["thin lens equation","linear magnification","real image"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_lens_magnification","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_009864","subject":"chemistry","topic":"inorganic_chemistry","subtopic":"complex_ion_oxidation_state","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"For a hypothetical coordination ion [Cu(CN)_3]^-1, each CN ligand has charge −1. Determine the formal oxidation state of Cu.","options":{"A":"2","B":"1","C":"3","D":"4"},"correct_answer":"A","correct_answer_value":"2","solution":"Let the formal oxidation state of Cu be x. Charge balance gives x+(3)(−1)=-1; hence x=-1+3=+2.","option_explanations":{"A":"Correct. The worked calculation gives 2.","B":"Incorrect. Substituting the stated values into the governing relation gives 2, not 1.","C":"Incorrect. Substituting the stated values into the governing relation gives 2, not 3.","D":"Incorrect. Substituting the stated values into the governing relation gives 2, not 4."},"concepts_required":["coordination compounds","oxidation state","charge balance"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_complex_oxidation","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_000648","subject":"mathematics","topic":"sequences_and_series","subtopic":"arithmetic_progression","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"An arithmetic progression has first term 6 and common difference 15. Find the sum of its first 16 terms.","options":{"A":"1895","B":"1422","C":"948","D":"1896"},"correct_answer":"D","correct_answer_value":"1896","solution":"Use Sₙ = n/2[2a+(n−1)d]. Substituting a=6, d=15, n=16: Sₙ=16/2[2(6)+(16−1)(15)]=1896.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 1896, not 1895.","B":"Incorrect. Substituting the stated values into the governing relation gives 1896, not 1422.","C":"Incorrect. Substituting the stated values into the governing relation gives 1896, not 948.","D":"Correct. The worked calculation gives 1896."},"concepts_required":["arithmetic progression","finite series"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_ap_sum","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_001584","subject":"mathematics","topic":"calculus","subtopic":"definite_integral_polynomial","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Evaluate the definite integral ∫₀^7 (0x² -3x -3) dx.","options":{"A":"-94.5","B":"-75.6","C":"-189","D":"-93.5"},"correct_answer":"A","correct_answer_value":"-94.5","solution":"An antiderivative is (0/3)x³+(-3/2)x²+(-3)x. Evaluating between 0 and 7 gives (0/3)(7)³+(-3/2)(7)²+(-3)(7)=-94.5.","option_explanations":{"A":"Correct. The worked calculation gives -94.5.","B":"Incorrect. Substituting the stated values into the governing relation gives -94.5, not -75.6.","C":"Incorrect. Substituting the stated values into the governing relation gives -94.5, not -189.","D":"Incorrect. Substituting the stated values into the governing relation gives -94.5, not -93.5."},"concepts_required":["definite integration","power rule","limits of integration"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_integral_poly","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_003746","subject":"physics","topic":"mechanics","subtopic":"work_energy_theorem","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A 14 kg block initially moves at 10 m s⁻¹. A constant net force does 18 N of work over 4 m in the direction of motion. Find its final kinetic energy.","options":{"A":"772 J","B":"579 J","C":"774 J","D":"386 J"},"correct_answer":"A","correct_answer_value":"772 J","solution":"By the work–energy theorem, K_f=K_i+W=½mu²+Fs=½(14)(10)²+(18)(4)=772 J.","option_explanations":{"A":"Correct. The worked calculation gives 772 J.","B":"Incorrect. Substituting the stated values into the governing relation gives 772 J, not 579 J.","C":"Incorrect. Substituting the stated values into the governing relation gives 772 J, not 774 J.","D":"Incorrect. Substituting the stated values into the governing relation gives 772 J, not 386 J."},"concepts_required":["work–energy theorem","kinetic energy","work by constant force"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_work_energy","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_005568","subject":"physics","topic":"electrostatics","subtopic":"capacitor_energy","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A 31 μF capacitor is charged to a potential difference of 23 V. Calculate the energy stored, expressed in μJ.","options":{"A":"8201.5 μJ","B":"8198.5 μJ","C":"8199.5 μJ","D":"16399 μJ"},"correct_answer":"C","correct_answer_value":"8199.5 μJ","solution":"U=½CV². Since C=31 μF, U in μJ is ½×31×(23)²=8199.5 μJ.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 8199.5 μJ, not 8201.5 μJ.","B":"Incorrect. Substituting the stated values into the governing relation gives 8199.5 μJ, not 8198.5 μJ.","C":"Correct. The worked calculation gives 8199.5 μJ.","D":"Incorrect. Substituting the stated values into the governing relation gives 8199.5 μJ, not 16399 μJ."},"concepts_required":["capacitance","energy stored in a capacitor"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_capacitor_energy","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_008932","subject":"chemistry","topic":"physical_chemistry","subtopic":"equilibrium_degree_dissociation","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"For A(g) ⇌ 2B(g), the initial concentration of A is 1.24 mol L⁻¹ and initially B is absent. At equilibrium, 28% of A has dissociated at constant volume. Find K_c.","options":{"A":"0.54009","B":"3.54009","C":"1.05401","D":"0.81013"},"correct_answer":"A","correct_answer_value":"0.54009","solution":"Let α=0.28. At equilibrium [A]=c₀(1−α) and [B]=2c₀α. Thus K_c=[B]²/[A]=4c₀α²/(1−α)=4(1.24)(0.28)²/(1−0.28)=0.54009.","option_explanations":{"A":"Correct. The worked calculation gives 0.54009.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.54009, not 3.54009.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.54009, not 1.05401.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.54009, not 0.81013."},"concepts_required":["chemical equilibrium","equilibrium constant","degree of dissociation"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_equilibrium_alpha","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_007576","subject":"chemistry","topic":"physical_chemistry","subtopic":"strong_acid_base_mixture","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"28 mL of 0.10 M HCl is mixed with 88 mL of 0.10 M NaOH at 25°C. Assuming complete dissociation and additive volumes, find the final pH.","options":{"A":"14.71369","B":"9.71369","C":"12.71369","D":"2.27137"},"correct_answer":"C","correct_answer_value":"12.71369","solution":"Excess OH⁻ moles=0.10×(88−28)/1000=0.006000 mol. Total volume=0.116 L, so [OH⁻]=0.051724 M, pOH=−log₁₀[OH⁻], and pH=14−pOH=12.71369.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 12.71369, not 14.71369.","B":"Incorrect. Substituting the stated values into the governing relation gives 12.71369, not 9.71369.","C":"Correct. The worked calculation gives 12.71369.","D":"Incorrect. Substituting the stated values into the governing relation gives 12.71369, not 2.27137."},"concepts_required":["strong acid–strong base neutralization","molarity","pH and pOH"],"reasoning_steps":5,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_acid_base_pH","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_001442","subject":"mathematics","topic":"calculus","subtopic":"definite_integral_polynomial","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Evaluate the definite integral ∫₀^2 (3x² +4x +5) dx.","options":{"A":"-26","B":"26","C":"13","D":"24"},"correct_answer":"B","correct_answer_value":"26","solution":"An antiderivative is (3/3)x³+(4/2)x²+(5)x. Evaluating between 0 and 2 gives (3/3)(2)³+(4/2)(2)²+(5)(2)=26.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 26, not -26.","B":"Correct. The worked calculation gives 26.","C":"Incorrect. Substituting the stated values into the governing relation gives 26, not 13.","D":"Incorrect. Substituting the stated values into the governing relation gives 26, not 24."},"concepts_required":["definite integration","power rule","limits of integration"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_integral_poly","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_004508","subject":"physics","topic":"rotational_mechanics","subtopic":"disc_angular_acceleration","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A uniform solid disc of mass 7 kg and radius 4 m is acted on by a constant torque of 35 N m about its central axis. Find its angular acceleration.","options":{"A":"0.46875 rad s⁻²","B":"0.625 rad s⁻²","C":"1.625 rad s⁻²","D":"-0.375 rad s⁻²"},"correct_answer":"B","correct_answer_value":"0.625 rad s⁻²","solution":"For a solid disc, I=½MR²=½(7)(4)²=56 kg m². Using τ=Iα, α=35/56=0.625 rad s⁻².","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.625 rad s⁻², not 0.46875 rad s⁻².","B":"Correct. The worked calculation gives 0.625 rad s⁻².","C":"Incorrect. Substituting the stated values into the governing relation gives 0.625 rad s⁻², not 1.625 rad s⁻².","D":"Incorrect. Substituting the stated values into the governing relation gives 0.625 rad s⁻², not -0.375 rad s⁻²."},"concepts_required":["moment of inertia","rotational dynamics","torque"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_disc_torque","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_008968","subject":"chemistry","topic":"physical_chemistry","subtopic":"equilibrium_degree_dissociation","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"For A(g) ⇌ 2B(g), the initial concentration of A is 1.81 mol L⁻¹ and initially B is absent. At equilibrium, 65% of A has dissociated at constant volume. Find K_c.","options":{"A":"17.47943","B":"6.55479","C":"8.73971","D":"7.73971"},"correct_answer":"C","correct_answer_value":"8.73971","solution":"Let α=0.65. At equilibrium [A]=c₀(1−α) and [B]=2c₀α. Thus K_c=[B]²/[A]=4c₀α²/(1−α)=4(1.81)(0.65)²/(1−0.65)=8.73971.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 8.73971, not 17.47943.","B":"Incorrect. Substituting the stated values into the governing relation gives 8.73971, not 6.55479.","C":"Correct. The worked calculation gives 8.73971.","D":"Incorrect. Substituting the stated values into the governing relation gives 8.73971, not 7.73971."},"concepts_required":["chemical equilibrium","equilibrium constant","degree of dissociation"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_equilibrium_alpha","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_000417","subject":"mathematics","topic":"sequences_and_series","subtopic":"arithmetic_progression","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"An arithmetic progression has first term 28 and common difference 2. Find the sum of its first 10 terms.","options":{"A":"38","B":"-370","C":"371","D":"370"},"correct_answer":"D","correct_answer_value":"370","solution":"Use Sₙ = n/2[2a+(n−1)d]. Substituting a=28, d=2, n=10: Sₙ=10/2[2(28)+(10−1)(2)]=370.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 370, not 38.","B":"Incorrect. Substituting the stated values into the governing relation gives 370, not -370.","C":"Incorrect. Substituting the stated values into the governing relation gives 370, not 371.","D":"Correct. The worked calculation gives 370."},"concepts_required":["arithmetic progression","finite series"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_ap_sum","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_001367","subject":"mathematics","topic":"calculus","subtopic":"definite_integral_polynomial","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Evaluate the definite integral ∫₀^2 (-1x² -1x -1) dx.","options":{"A":"-5.66667","B":"-13.33333","C":"-6.66667","D":"6.66667"},"correct_answer":"C","correct_answer_value":"-6.66667","solution":"An antiderivative is (-1/3)x³+(-1/2)x²+(-1)x. Evaluating between 0 and 2 gives (-1/3)(2)³+(-1/2)(2)²+(-1)(2)=-6.66667.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives -6.66667, not -5.66667.","B":"Incorrect. Substituting the stated values into the governing relation gives -6.66667, not -13.33333.","C":"Correct. The worked calculation gives -6.66667.","D":"Incorrect. Substituting the stated values into the governing relation gives -6.66667, not 6.66667."},"concepts_required":["definite integration","power rule","limits of integration"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_integral_poly","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_009213","subject":"chemistry","topic":"physical_chemistry","subtopic":"nernst_equation_298K","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"At 298 K, a galvanic cell has E°cell=1.41 V, transfers 3 electrons, and has reaction quotient Q=10^-6. Using E=E°−(0.0591/n)log₁₀Q, calculate Ecell.","options":{"A":"-1.4718 V","B":"0.5282 V","C":"-1.5282 V","D":"1.5282 V"},"correct_answer":"D","correct_answer_value":"1.5282 V","solution":"Apply the Nernst equation: E=1.41−(0.0591/3)(-6)=1.5282 V.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 1.5282 V, not -1.4718 V.","B":"Incorrect. Substituting the stated values into the governing relation gives 1.5282 V, not 0.5282 V.","C":"Incorrect. Substituting the stated values into the governing relation gives 1.5282 V, not -1.5282 V.","D":"Correct. The worked calculation gives 1.5282 V."},"concepts_required":["electrochemical cells","Nernst equation","logarithms"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_nernst","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_006726","subject":"chemistry","topic":"stoichiometry","subtopic":"limiting_reagent_water","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Hydrogen reacts with oxygen according to 2H₂ + O₂ → 2H₂O. If 33 mol H₂ and 34 mol O₂ are mixed and the reaction goes to completion, how many moles of H₂O form?","options":{"A":"35 mol","B":"31 mol","C":"33 mol","D":"39.6 mol"},"correct_answer":"C","correct_answer_value":"33 mol","solution":"From 2 mol H₂ and 1 mol O₂, 2 mol H₂O form. H₂ can produce 33 mol H₂O; O₂ can produce 68 mol H₂O. The limiting condition is H₂; product=33 mol.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 33 mol, not 35 mol.","B":"Incorrect. Substituting the stated values into the governing relation gives 33 mol, not 31 mol.","C":"Correct. The worked calculation gives 33 mol.","D":"Incorrect. Substituting the stated values into the governing relation gives 33 mol, not 39.6 mol."},"concepts_required":["balanced chemical equations","limiting reagent","mole ratios"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_limiting_reagent","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_003884","subject":"physics","topic":"mechanics","subtopic":"work_energy_theorem","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A 14 kg block initially moves at 12 m s⁻¹. A constant net force does 20 N of work over 8 m in the direction of motion. Find its final kinetic energy.","options":{"A":"2336 J","B":"1168 J","C":"1170 J","D":"1401.6 J"},"correct_answer":"B","correct_answer_value":"1168 J","solution":"By the work–energy theorem, K_f=K_i+W=½mu²+Fs=½(14)(12)²+(20)(8)=1168 J.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 1168 J, not 2336 J.","B":"Correct. The worked calculation gives 1168 J.","C":"Incorrect. Substituting the stated values into the governing relation gives 1168 J, not 1170 J.","D":"Incorrect. Substituting the stated values into the governing relation gives 1168 J, not 1401.6 J."},"concepts_required":["work–energy theorem","kinetic energy","work by constant force"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_work_energy","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_007240","subject":"chemistry","topic":"physical_chemistry","subtopic":"ideal_gas_volume","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Assuming ideal behaviour, what volume is occupied by 20 mol of gas at 264 K and 1.51 atm? Use R=0.082057 L atm mol⁻¹ K⁻¹.","options":{"A":"344.31335 L","B":"286.92779 L","C":"29.69278 L","D":"143.46389 L"},"correct_answer":"B","correct_answer_value":"286.92779 L","solution":"PV=nRT, so V=nRT/P=(20)(0.082057)(264)/(1.51)=286.92779 L.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 286.92779 L, not 344.31335 L.","B":"Correct. The worked calculation gives 286.92779 L.","C":"Incorrect. Substituting the stated values into the governing relation gives 286.92779 L, not 29.69278 L.","D":"Incorrect. Substituting the stated values into the governing relation gives 286.92779 L, not 143.46389 L."},"concepts_required":["ideal gas equation","unit consistency"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_ideal_gas","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_001228","subject":"mathematics","topic":"calculus","subtopic":"polynomial_differentiation","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Let f(x)=-2x³ +5x² -3x+7. Evaluate f′(-1).","options":{"A":"-15.2","B":"-19","C":"-38","D":"-20"},"correct_answer":"B","correct_answer_value":"-19","solution":"Differentiate: f′(x)=3(-2)x²+2(5)x+(-3). At x=-1, f′(-1)=3(-2)(-1)²+2(5)(-1)+(-3)=-19.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives -19, not -15.2.","B":"Correct. The worked calculation gives -19.","C":"Incorrect. Substituting the stated values into the governing relation gives -19, not -38.","D":"Incorrect. Substituting the stated values into the governing relation gives -19, not -20."},"concepts_required":["power rule","polynomial derivative"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_derivative_cubic","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_007261","subject":"chemistry","topic":"physical_chemistry","subtopic":"ideal_gas_volume","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Assuming ideal behaviour, what volume is occupied by 3 mol of gas at 604 K and 0.11 atm? Use R=0.082057 L atm mol⁻¹ K⁻¹.","options":{"A":"2703.40516 L","B":"-1351.70258 L","C":"1081.36207 L","D":"1351.70258 L"},"correct_answer":"D","correct_answer_value":"1351.70258 L","solution":"PV=nRT, so V=nRT/P=(3)(0.082057)(604)/(0.11)=1351.70258 L.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 1351.70258 L, not 2703.40516 L.","B":"Incorrect. Substituting the stated values into the governing relation gives 1351.70258 L, not -1351.70258 L.","C":"Incorrect. Substituting the stated values into the governing relation gives 1351.70258 L, not 1081.36207 L.","D":"Correct. The worked calculation gives 1351.70258 L."},"concepts_required":["ideal gas equation","unit consistency"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_ideal_gas","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_007517","subject":"chemistry","topic":"physical_chemistry","subtopic":"strong_acid_base_mixture","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"82 mL of 0.10 M HCl is mixed with 62 mL of 0.10 M NaOH at 25°C. Assuming complete dissociation and additive volumes, find the final pH.","options":{"A":"1.18573","B":"-1.14267","C":"1.85733","D":"2.786"},"correct_answer":"C","correct_answer_value":"1.85733","solution":"Excess H⁺ moles=0.10×(82−62)/1000=0.002000 mol. Total volume=0.144 L, so [H⁺]=0.013889 M and pH=−log₁₀[H⁺]=1.85733.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 1.85733, not 1.18573.","B":"Incorrect. Substituting the stated values into the governing relation gives 1.85733, not -1.14267.","C":"Correct. The worked calculation gives 1.85733.","D":"Incorrect. Substituting the stated values into the governing relation gives 1.85733, not 2.786."},"concepts_required":["strong acid–strong base neutralization","molarity","pH and pOH"],"reasoning_steps":5,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_acid_base_pH","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_006102","subject":"physics","topic":"ray_optics","subtopic":"convex_lens_magnification","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A real object is placed 16 cm in front of a thin convex lens of focal length 6 cm. Using the Cartesian sign convention, what is the magnitude of the linear magnification?","options":{"A":"-0.6","B":"-1.4","C":"0.6","D":"1.06"},"correct_answer":"C","correct_answer_value":"0.6","solution":"Using distances as positive magnitudes, 1/f=1/v+1/u gives v=uf/(u−f)=(16)(6)/(16−6)=9.6 cm. Thus |m|=v/u=9.6/16=0.6.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.6, not -0.6.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.6, not -1.4.","C":"Correct. The worked calculation gives 0.6.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.6, not 1.06."},"concepts_required":["thin lens equation","linear magnification","real image"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_lens_magnification","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_000608","subject":"mathematics","topic":"sequences_and_series","subtopic":"arithmetic_progression","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"An arithmetic progression has first term 19 and common difference 14. Find the sum of its first 33 terms.","options":{"A":"-8019","B":"8018","C":"8019","D":"9622.8"},"correct_answer":"C","correct_answer_value":"8019","solution":"Use Sₙ = n/2[2a+(n−1)d]. Substituting a=19, d=14, n=33: Sₙ=33/2[2(19)+(33−1)(14)]=8019.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 8019, not -8019.","B":"Incorrect. Substituting the stated values into the governing relation gives 8019, not 8018.","C":"Correct. The worked calculation gives 8019.","D":"Incorrect. Substituting the stated values into the governing relation gives 8019, not 9622.8."},"concepts_required":["arithmetic progression","finite series"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_ap_sum","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_006461","subject":"physics","topic":"oscillations","subtopic":"mass_spring_period_over_pi","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A mass of 8 kg is attached to an ideal spring of force constant 311 N m⁻¹ and oscillates horizontally without damping. What is the time period divided by π?","options":{"A":"0.48116 s","B":"0.32077 s","C":"-0.67923 s","D":"2.32077 s"},"correct_answer":"B","correct_answer_value":"0.32077 s","solution":"For a mass–spring oscillator, T=2π√(m/k). Therefore T/π=2√(8/311)=0.32077 s.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.32077 s, not 0.48116 s.","B":"Correct. The worked calculation gives 0.32077 s.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.32077 s, not -0.67923 s.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.32077 s, not 2.32077 s."},"concepts_required":["simple harmonic motion","mass–spring oscillator","time period"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_shm_period","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_007519","subject":"chemistry","topic":"physical_chemistry","subtopic":"strong_acid_base_mixture","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"70 mL of 0.10 M HCl is mixed with 21 mL of 0.10 M NaOH at 25°C. Assuming complete dissociation and additive volumes, find the final pH.","options":{"A":"1.26885","B":"-1.26885","C":"1.90327","D":"3.26885"},"correct_answer":"A","correct_answer_value":"1.26885","solution":"Excess H⁺ moles=0.10×(70−21)/1000=0.004900 mol. Total volume=0.091 L, so [H⁺]=0.053846 M and pH=−log₁₀[H⁺]=1.26885.","option_explanations":{"A":"Correct. The worked calculation gives 1.26885.","B":"Incorrect. Substituting the stated values into the governing relation gives 1.26885, not -1.26885.","C":"Incorrect. Substituting the stated values into the governing relation gives 1.26885, not 1.90327.","D":"Incorrect. Substituting the stated values into the governing relation gives 1.26885, not 3.26885."},"concepts_required":["strong acid–strong base neutralization","molarity","pH and pOH"],"reasoning_steps":5,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_acid_base_pH","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_000096","subject":"mathematics","topic":"algebra","subtopic":"quadratic_roots","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"For the quadratic equation 6x^2 +17x +1=0, let its roots be α and β. What is α²+β²?","options":{"A":"8.69444","B":"7.69444","C":"15.38889","D":"10.69444"},"correct_answer":"B","correct_answer_value":"7.69444","solution":"By Vieta's relations, α+β = -17/6 and αβ = 1/6. Then α²+β²=(α+β)²−2αβ = (-17/6)²−2(1/6) = 7.69444.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 7.69444, not 8.69444.","B":"Correct. The worked calculation gives 7.69444.","C":"Incorrect. Substituting the stated values into the governing relation gives 7.69444, not 15.38889.","D":"Incorrect. Substituting the stated values into the governing relation gives 7.69444, not 10.69444."},"concepts_required":["Vieta's formulas","symmetric polynomial expressions"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_quadratic_vieta","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_001007","subject":"mathematics","topic":"calculus","subtopic":"polynomial_differentiation","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Let f(x)=1x³ +9x² +4x+7. Evaluate f′(-3).","options":{"A":"-24","B":"-27.6","C":"-17.25","D":"-23"},"correct_answer":"D","correct_answer_value":"-23","solution":"Differentiate: f′(x)=3(1)x²+2(9)x+(4). At x=-3, f′(-3)=3(1)(-3)²+2(9)(-3)+(4)=-23.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives -23, not -24.","B":"Incorrect. Substituting the stated values into the governing relation gives -23, not -27.6.","C":"Incorrect. Substituting the stated values into the governing relation gives -23, not -17.25.","D":"Correct. The worked calculation gives -23."},"concepts_required":["power rule","polynomial derivative"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_derivative_cubic","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_008851","subject":"chemistry","topic":"physical_chemistry","subtopic":"equilibrium_degree_dissociation","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"For A(g) ⇌ 2B(g), the initial concentration of A is 1.36 mol L⁻¹ and initially B is absent. At equilibrium, 80% of A has dissociated at constant volume. Find K_c.","options":{"A":"13.056","B":"-17.408","C":"17.408","D":"20.8896"},"correct_answer":"C","correct_answer_value":"17.408","solution":"Let α=0.8. At equilibrium [A]=c₀(1−α) and [B]=2c₀α. Thus K_c=[B]²/[A]=4c₀α²/(1−α)=4(1.36)(0.8)²/(1−0.8)=17.408.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 17.408, not 13.056.","B":"Incorrect. Substituting the stated values into the governing relation gives 17.408, not -17.408.","C":"Correct. The worked calculation gives 17.408.","D":"Incorrect. Substituting the stated values into the governing relation gives 17.408, not 20.8896."},"concepts_required":["chemical equilibrium","equilibrium constant","degree of dissociation"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_equilibrium_alpha","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_004981","subject":"physics","topic":"electricity","subtopic":"series_parallel_circuit","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Resistors 4 Ω and 7 Ω are connected in series; this combination is connected in parallel with a 13 Ω resistor across a 24 V ideal battery. Find the total current supplied by the battery.","options":{"A":"6.04196 A","B":"5.02797 A","C":"1.4028 A","D":"4.02797 A"},"correct_answer":"D","correct_answer_value":"4.02797 A","solution":"Series branch resistance=4+7=11 Ω. Parallel equivalent R_eq=(11×13)/(11+13)=5.95833 Ω. Total current I=V/R_eq=24/5.95833=4.02797 A.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 4.02797 A, not 6.04196 A.","B":"Incorrect. Substituting the stated values into the governing relation gives 4.02797 A, not 5.02797 A.","C":"Incorrect. Substituting the stated values into the governing relation gives 4.02797 A, not 1.4028 A.","D":"Correct. The worked calculation gives 4.02797 A."},"concepts_required":["series resistors","parallel resistors","Ohm's law"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_series_parallel","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_007838","subject":"chemistry","topic":"physical_chemistry","subtopic":"first_order_half_life","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A first-order reaction has a half-life of 20 minutes. If the initial reactant concentration is 39 mmol L⁻¹, what concentration remains after 80 minutes?","options":{"A":"-2.4375 mmol L⁻¹","B":"2.4375 mmol L⁻¹","C":"1.82812 mmol L⁻¹","D":"-0.5625 mmol L⁻¹"},"correct_answer":"B","correct_answer_value":"2.4375 mmol L⁻¹","solution":"80 minutes corresponds to 4 half-lives. The remaining concentration is [A]₀(1/2)ⁿ=39/2^4=2.4375 mmol L⁻¹.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 2.4375 mmol L⁻¹, not -2.4375 mmol L⁻¹.","B":"Correct. The worked calculation gives 2.4375 mmol L⁻¹.","C":"Incorrect. Substituting the stated values into the governing relation gives 2.4375 mmol L⁻¹, not 1.82812 mmol L⁻¹.","D":"Incorrect. Substituting the stated values into the governing relation gives 2.4375 mmol L⁻¹, not -0.5625 mmol L⁻¹."},"concepts_required":["first-order kinetics","half-life","exponential decay"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_first_order_halflife","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_009008","subject":"chemistry","topic":"physical_chemistry","subtopic":"nernst_equation_298K","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"At 298 K, a galvanic cell has E°cell=0.61 V, transfers 3 electrons, and has reaction quotient Q=10^1. Using E=E°−(0.0591/n)log₁₀Q, calculate Ecell.","options":{"A":"0.29515 V","B":"0.44272 V","C":"3.5903 V","D":"0.5903 V"},"correct_answer":"D","correct_answer_value":"0.5903 V","solution":"Apply the Nernst equation: E=0.61−(0.0591/3)(1)=0.5903 V.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.5903 V, not 0.29515 V.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.5903 V, not 0.44272 V.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.5903 V, not 3.5903 V.","D":"Correct. The worked calculation gives 0.5903 V."},"concepts_required":["electrochemical cells","Nernst equation","logarithms"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_nernst","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_002398","subject":"mathematics","topic":"linear_algebra","subtopic":"2x2_determinant","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Find the determinant of the matrix [[8, 3], [7, -2]].","options":{"A":"-29.6","B":"-37","C":"-35","D":"-18.5"},"correct_answer":"B","correct_answer_value":"-37","solution":"For a 2×2 matrix [[a,b],[c,d]], det=ad−bc. Here it is (8)(-2)−(3)(7)=-37.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives -37, not -29.6.","B":"Correct. The worked calculation gives -37.","C":"Incorrect. Substituting the stated values into the governing relation gives -37, not -35.","D":"Incorrect. Substituting the stated values into the governing relation gives -37, not -18.5."},"concepts_required":["determinants","matrix operations"],"reasoning_steps":2,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_determinant_2x2","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_000508","subject":"mathematics","topic":"sequences_and_series","subtopic":"arithmetic_progression","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"An arithmetic progression has first term 26 and common difference 2. Find the sum of its first 35 terms.","options":{"A":"2100","B":"211","C":"2099","D":"2101"},"correct_answer":"A","correct_answer_value":"2100","solution":"Use Sₙ = n/2[2a+(n−1)d]. Substituting a=26, d=2, n=35: Sₙ=35/2[2(26)+(35−1)(2)]=2100.","option_explanations":{"A":"Correct. The worked calculation gives 2100.","B":"Incorrect. Substituting the stated values into the governing relation gives 2100, not 211.","C":"Incorrect. Substituting the stated values into the governing relation gives 2100, not 2099.","D":"Incorrect. Substituting the stated values into the governing relation gives 2100, not 2101."},"concepts_required":["arithmetic progression","finite series"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_ap_sum","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_001499","subject":"mathematics","topic":"calculus","subtopic":"definite_integral_polynomial","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Evaluate the definite integral ∫₀^6 (1x² +4x +1) dx.","options":{"A":"151","B":"148","C":"152","D":"150"},"correct_answer":"D","correct_answer_value":"150","solution":"An antiderivative is (1/3)x³+(4/2)x²+(1)x. Evaluating between 0 and 6 gives (1/3)(6)³+(4/2)(6)²+(1)(6)=150.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 150, not 151.","B":"Incorrect. Substituting the stated values into the governing relation gives 150, not 148.","C":"Incorrect. Substituting the stated values into the governing relation gives 150, not 152.","D":"Correct. The worked calculation gives 150."},"concepts_required":["definite integration","power rule","limits of integration"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_integral_poly","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_005743","subject":"physics","topic":"thermodynamics","subtopic":"ideal_gas_isochoric_pressure","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A fixed amount of ideal gas is kept in a rigid container. Its pressure is 4.7 atm at 327 K. What is its pressure at 412 K?","options":{"A":"1.59217 atm","B":"5.92171 atm","C":"7.92171 atm","D":"8.88257 atm"},"correct_answer":"B","correct_answer_value":"5.92171 atm","solution":"At constant volume, P/T is constant. P₂=P₁T₂/T₁=(4.7)(412)/(327)=5.92171 atm.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 5.92171 atm, not 1.59217 atm.","B":"Correct. The worked calculation gives 5.92171 atm.","C":"Incorrect. Substituting the stated values into the governing relation gives 5.92171 atm, not 7.92171 atm.","D":"Incorrect. Substituting the stated values into the governing relation gives 5.92171 atm, not 8.88257 atm."},"concepts_required":["ideal gas law","isochoric process","absolute temperature"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_isochoric_gas","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_002495","subject":"mathematics","topic":"linear_algebra","subtopic":"2x2_determinant","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Find the determinant of the matrix [[4, 3], [2, -2]].","options":{"A":"-14","B":"-10.5","C":"14","D":"-12"},"correct_answer":"A","correct_answer_value":"-14","solution":"For a 2×2 matrix [[a,b],[c,d]], det=ad−bc. Here it is (4)(-2)−(3)(2)=-14.","option_explanations":{"A":"Correct. The worked calculation gives -14.","B":"Incorrect. Substituting the stated values into the governing relation gives -14, not -10.5.","C":"Incorrect. Substituting the stated values into the governing relation gives -14, not 14.","D":"Incorrect. Substituting the stated values into the governing relation gives -14, not -12."},"concepts_required":["determinants","matrix operations"],"reasoning_steps":2,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_determinant_2x2","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_005958","subject":"physics","topic":"thermodynamics","subtopic":"ideal_gas_isochoric_pressure","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A fixed amount of ideal gas is kept in a rigid container. Its pressure is 4.3 atm at 432 K. What is its pressure at 709 K?","options":{"A":"7.05718 atm","B":"9.05718 atm","C":"1.70572 atm","D":"5.05718 atm"},"correct_answer":"A","correct_answer_value":"7.05718 atm","solution":"At constant volume, P/T is constant. P₂=P₁T₂/T₁=(4.3)(709)/(432)=7.05718 atm.","option_explanations":{"A":"Correct. The worked calculation gives 7.05718 atm.","B":"Incorrect. Substituting the stated values into the governing relation gives 7.05718 atm, not 9.05718 atm.","C":"Incorrect. Substituting the stated values into the governing relation gives 7.05718 atm, not 1.70572 atm.","D":"Incorrect. Substituting the stated values into the governing relation gives 7.05718 atm, not 5.05718 atm."},"concepts_required":["ideal gas law","isochoric process","absolute temperature"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_isochoric_gas","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_002089","subject":"mathematics","topic":"probability","subtopic":"same_colour_without_replacement","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A bag contains 5 red, 9 blue and 6 green balls. Two balls are drawn uniformly at random without replacement. What is the probability that both have the same colour?","options":{"A":"0.16053","B":"0.32105","C":"-0.32105","D":"1.32105"},"correct_answer":"B","correct_answer_value":"0.32105","solution":"Favourable pairs = C(5,2)+C(9,2)+C(6,2)=61. Total pairs=C(20,2)=190. Therefore P=61/190=0.32105.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.32105, not 0.16053.","B":"Correct. The worked calculation gives 0.32105.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.32105, not -0.32105.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.32105, not 1.32105."},"concepts_required":["classical probability","combinations","without replacement"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_probability_colours","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_003342","subject":"physics","topic":"mechanics","subtopic":"projectile_range","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A projectile is launched from level ground at 117 m s⁻¹ at 40° to the horizontal. Take g=10 m s⁻². Find its horizontal range.","options":{"A":"1348.10333 m","B":"1347.10333 m","C":"1078.48267 m","D":"135.81033 m"},"correct_answer":"A","correct_answer_value":"1348.10333 m","solution":"Range R=u²sin(2θ)/g. Substituting u=117, θ=40° and g=10 gives R=(117)²sin(80°)/10=1348.10333 m.","option_explanations":{"A":"Correct. The worked calculation gives 1348.10333 m.","B":"Incorrect. Substituting the stated values into the governing relation gives 1348.10333 m, not 1347.10333 m.","C":"Incorrect. Substituting the stated values into the governing relation gives 1348.10333 m, not 1078.48267 m.","D":"Incorrect. Substituting the stated values into the governing relation gives 1348.10333 m, not 135.81033 m."},"concepts_required":["projectile motion","range equation","trigonometry"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_projectile_range","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_003171","subject":"mathematics","topic":"complex_numbers","subtopic":"real_part_of_complex_quotient","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Let z=(-13+13i)/(11+8i). What is Re(z), the real part of z?","options":{"A":"-0.21081","B":"2.78919","C":"-3.21081","D":"-1.21081"},"correct_answer":"A","correct_answer_value":"-0.21081","solution":"Multiply numerator and denominator by the conjugate (11-8i). Then Re(z)=(ac+bd)/(c²+d²)=((-13)(11)+(13)(8))/(11²+(8)²)=-0.21081.","option_explanations":{"A":"Correct. The worked calculation gives -0.21081.","B":"Incorrect. Substituting the stated values into the governing relation gives -0.21081, not 2.78919.","C":"Incorrect. Substituting the stated values into the governing relation gives -0.21081, not -3.21081.","D":"Incorrect. Substituting the stated values into the governing relation gives -0.21081, not -1.21081."},"concepts_required":["complex conjugate","complex division","real and imaginary parts"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_complex_quotient","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_009639","subject":"chemistry","topic":"organic_chemistry","subtopic":"degree_of_unsaturation","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Assuming a neutral closed-shell formula, calculate the degree of unsaturation (DBE) for C_17H_29N_1X_2. Treat X as the total number of monovalent halogen atoms.","options":{"A":"6","B":"4.5","C":"3","D":"2"},"correct_answer":"C","correct_answer_value":"3","solution":"Use DBE=(2C+2+N−H−X)/2. Substitution gives (2×17+2+1−29−2)/2=3.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 3, not 6.","B":"Incorrect. Substituting the stated values into the governing relation gives 3, not 4.5.","C":"Correct. The worked calculation gives 3.","D":"Incorrect. Substituting the stated values into the governing relation gives 3, not 2."},"concepts_required":["molecular formula","degree of unsaturation","organic structure"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_dbe","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_008125","subject":"chemistry","topic":"physical_chemistry","subtopic":"faradays_law_copper","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"An aqueous Cu²⁺ solution is electrolysed using a current of 3 A for 28891 s. Assuming 100% current efficiency, calculate the mass of copper deposited. Use M(Cu)=63.5 g mol⁻¹ and F=96500 C mol⁻¹.","options":{"A":"28.51676 g","B":"26.51676 g","C":"57.03353 g","D":"29.51676 g"},"correct_answer":"A","correct_answer_value":"28.51676 g","solution":"Cu²⁺+2e⁻→Cu, so z=2. By Faraday's law m=MIt/(zF)=63.5×3×28891/(2×96500)=28.51676 g.","option_explanations":{"A":"Correct. The worked calculation gives 28.51676 g.","B":"Incorrect. Substituting the stated values into the governing relation gives 28.51676 g, not 26.51676 g.","C":"Incorrect. Substituting the stated values into the governing relation gives 28.51676 g, not 57.03353 g.","D":"Incorrect. Substituting the stated values into the governing relation gives 28.51676 g, not 29.51676 g."},"concepts_required":["electrolysis","Faraday's laws","redox stoichiometry"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_faraday_copper","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_005443","subject":"physics","topic":"electrostatics","subtopic":"capacitor_energy","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A 18 μF capacitor is charged to a potential difference of 35 V. Calculate the energy stored, expressed in μJ.","options":{"A":"-11025 μJ","B":"11025 μJ","C":"8268.75 μJ","D":"22050 μJ"},"correct_answer":"B","correct_answer_value":"11025 μJ","solution":"U=½CV². Since C=18 μF, U in μJ is ½×18×(35)²=11025 μJ.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 11025 μJ, not -11025 μJ.","B":"Correct. The worked calculation gives 11025 μJ.","C":"Incorrect. Substituting the stated values into the governing relation gives 11025 μJ, not 8268.75 μJ.","D":"Incorrect. Substituting the stated values into the governing relation gives 11025 μJ, not 22050 μJ."},"concepts_required":["capacitance","energy stored in a capacitor"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_capacitor_energy","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_006840","subject":"chemistry","topic":"stoichiometry","subtopic":"limiting_reagent_water","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Hydrogen reacts with oxygen according to 2H₂ + O₂ → 2H₂O. If 38 mol H₂ and 6 mol O₂ are mixed and the reaction goes to completion, how many moles of H₂O form?","options":{"A":"9 mol","B":"12 mol","C":"10 mol","D":"18 mol"},"correct_answer":"B","correct_answer_value":"12 mol","solution":"From 2 mol H₂ and 1 mol O₂, 2 mol H₂O form. H₂ can produce 38 mol H₂O; O₂ can produce 12 mol H₂O. The limiting condition is O₂; product=12 mol.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 12 mol, not 9 mol.","B":"Correct. The worked calculation gives 12 mol.","C":"Incorrect. Substituting the stated values into the governing relation gives 12 mol, not 10 mol.","D":"Incorrect. Substituting the stated values into the governing relation gives 12 mol, not 18 mol."},"concepts_required":["balanced chemical equations","limiting reagent","mole ratios"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_limiting_reagent","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_006909","subject":"chemistry","topic":"stoichiometry","subtopic":"limiting_reagent_water","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Hydrogen reacts with oxygen according to 2H₂ + O₂ → 2H₂O. If 38 mol H₂ and 25 mol O₂ are mixed and the reaction goes to completion, how many moles of H₂O form?","options":{"A":"28.5 mol","B":"37 mol","C":"38 mol","D":"4.8 mol"},"correct_answer":"C","correct_answer_value":"38 mol","solution":"From 2 mol H₂ and 1 mol O₂, 2 mol H₂O form. H₂ can produce 38 mol H₂O; O₂ can produce 50 mol H₂O. The limiting condition is H₂; product=38 mol.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 38 mol, not 28.5 mol.","B":"Incorrect. Substituting the stated values into the governing relation gives 38 mol, not 37 mol.","C":"Correct. The worked calculation gives 38 mol.","D":"Incorrect. Substituting the stated values into the governing relation gives 38 mol, not 4.8 mol."},"concepts_required":["balanced chemical equations","limiting reagent","mole ratios"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_limiting_reagent","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_006888","subject":"chemistry","topic":"stoichiometry","subtopic":"limiting_reagent_water","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Hydrogen reacts with oxygen according to 2H₂ + O₂ → 2H₂O. If 46 mol H₂ and 2 mol O₂ are mixed and the reaction goes to completion, how many moles of H₂O form?","options":{"A":"2 mol","B":"6 mol","C":"-4 mol","D":"4 mol"},"correct_answer":"D","correct_answer_value":"4 mol","solution":"From 2 mol H₂ and 1 mol O₂, 2 mol H₂O form. H₂ can produce 46 mol H₂O; O₂ can produce 4 mol H₂O. The limiting condition is O₂; product=4 mol.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 4 mol, not 2 mol.","B":"Incorrect. Substituting the stated values into the governing relation gives 4 mol, not 6 mol.","C":"Incorrect. Substituting the stated values into the governing relation gives 4 mol, not -4 mol.","D":"Correct. The worked calculation gives 4 mol."},"concepts_required":["balanced chemical equations","limiting reagent","mole ratios"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_limiting_reagent","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_004102","subject":"physics","topic":"mechanics","subtopic":"inclined_plane_with_friction","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A 27 kg block slides down a 37° rough incline. The coefficient of kinetic friction is 0.45. Taking g=10 m s⁻², find the net force down the plane. Use sin37°=0.6 and cos37°=0.8.","options":{"A":"-64.8 N","B":"65.8 N","C":"64.8 N","D":"51.84 N"},"correct_answer":"C","correct_answer_value":"64.8 N","solution":"Net force=mg(sinθ−μcosθ)=27×10×(0.6−0.45×0.8)=64.8 N.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 64.8 N, not -64.8 N.","B":"Incorrect. Substituting the stated values into the governing relation gives 64.8 N, not 65.8 N.","C":"Correct. The worked calculation gives 64.8 N.","D":"Incorrect. Substituting the stated values into the governing relation gives 64.8 N, not 51.84 N."},"concepts_required":["Newton's laws","inclined plane","kinetic friction"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_incline_friction","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_002661","subject":"mathematics","topic":"linear_algebra","subtopic":"2x2_determinant","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Find the determinant of the matrix [[1, -1], [7, 0]].","options":{"A":"4","B":"10","C":"6","D":"7"},"correct_answer":"D","correct_answer_value":"7","solution":"For a 2×2 matrix [[a,b],[c,d]], det=ad−bc. Here it is (1)(0)−(-1)(7)=7.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 7, not 4.","B":"Incorrect. Substituting the stated values into the governing relation gives 7, not 10.","C":"Incorrect. Substituting the stated values into the governing relation gives 7, not 6.","D":"Correct. The worked calculation gives 7."},"concepts_required":["determinants","matrix operations"],"reasoning_steps":2,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_determinant_2x2","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_002500","subject":"mathematics","topic":"linear_algebra","subtopic":"2x2_determinant","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Find the determinant of the matrix [[9, 4], [3, 7]].","options":{"A":"53","B":"51","C":"40.8","D":"61.2"},"correct_answer":"B","correct_answer_value":"51","solution":"For a 2×2 matrix [[a,b],[c,d]], det=ad−bc. Here it is (9)(7)−(4)(3)=51.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 51, not 53.","B":"Correct. The worked calculation gives 51.","C":"Incorrect. Substituting the stated values into the governing relation gives 51, not 40.8.","D":"Incorrect. Substituting the stated values into the governing relation gives 51, not 61.2."},"concepts_required":["determinants","matrix operations"],"reasoning_steps":2,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_determinant_2x2","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_003421","subject":"physics","topic":"mechanics","subtopic":"projectile_range","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A projectile is launched from level ground at 19 m s⁻¹ at 65° to the horizontal. Take g=10 m s⁻². Find its horizontal range.","options":{"A":"25.6542 m","B":"20.74065 m","C":"27.6542 m","D":"29.6542 m"},"correct_answer":"C","correct_answer_value":"27.6542 m","solution":"Range R=u²sin(2θ)/g. Substituting u=19, θ=65° and g=10 gives R=(19)²sin(130°)/10=27.6542 m.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 27.6542 m, not 25.6542 m.","B":"Incorrect. Substituting the stated values into the governing relation gives 27.6542 m, not 20.74065 m.","C":"Correct. The worked calculation gives 27.6542 m.","D":"Incorrect. Substituting the stated values into the governing relation gives 27.6542 m, not 29.6542 m."},"concepts_required":["projectile motion","range equation","trigonometry"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_projectile_range","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_009071","subject":"chemistry","topic":"physical_chemistry","subtopic":"nernst_equation_298K","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"At 298 K, a galvanic cell has E°cell=1.43 V, transfers 1 electrons, and has reaction quotient Q=10^-3. Using E=E°−(0.0591/n)log₁₀Q, calculate Ecell.","options":{"A":"1.6073 V","B":"3.6073 V","C":"4.6073 V","D":"-1.6073 V"},"correct_answer":"A","correct_answer_value":"1.6073 V","solution":"Apply the Nernst equation: E=1.43−(0.0591/1)(-3)=1.6073 V.","option_explanations":{"A":"Correct. The worked calculation gives 1.6073 V.","B":"Incorrect. Substituting the stated values into the governing relation gives 1.6073 V, not 3.6073 V.","C":"Incorrect. Substituting the stated values into the governing relation gives 1.6073 V, not 4.6073 V.","D":"Incorrect. Substituting the stated values into the governing relation gives 1.6073 V, not -1.6073 V."},"concepts_required":["electrochemical cells","Nernst equation","logarithms"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_nernst","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_003969","subject":"physics","topic":"mechanics","subtopic":"work_energy_theorem","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A 6 kg block initially moves at 10 m s⁻¹. A constant net force does 7 N of work over 18 m in the direction of motion. Find its final kinetic energy.","options":{"A":"639 J","B":"426 J","C":"-426 J","D":"852 J"},"correct_answer":"B","correct_answer_value":"426 J","solution":"By the work–energy theorem, K_f=K_i+W=½mu²+Fs=½(6)(10)²+(7)(18)=426 J.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 426 J, not 639 J.","B":"Correct. The worked calculation gives 426 J.","C":"Incorrect. Substituting the stated values into the governing relation gives 426 J, not -426 J.","D":"Incorrect. Substituting the stated values into the governing relation gives 426 J, not 852 J."},"concepts_required":["work–energy theorem","kinetic energy","work by constant force"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_work_energy","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_006975","subject":"chemistry","topic":"stoichiometry","subtopic":"limiting_reagent_water","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Hydrogen reacts with oxygen according to 2H₂ + O₂ → 2H₂O. If 34 mol H₂ and 9 mol O₂ are mixed and the reaction goes to completion, how many moles of H₂O form?","options":{"A":"21.6 mol","B":"19 mol","C":"18 mol","D":"9 mol"},"correct_answer":"C","correct_answer_value":"18 mol","solution":"From 2 mol H₂ and 1 mol O₂, 2 mol H₂O form. H₂ can produce 34 mol H₂O; O₂ can produce 18 mol H₂O. The limiting condition is O₂; product=18 mol.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 18 mol, not 21.6 mol.","B":"Incorrect. Substituting the stated values into the governing relation gives 18 mol, not 19 mol.","C":"Correct. The worked calculation gives 18 mol.","D":"Incorrect. Substituting the stated values into the governing relation gives 18 mol, not 9 mol."},"concepts_required":["balanced chemical equations","limiting reagent","mole ratios"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_limiting_reagent","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_001672","subject":"mathematics","topic":"combinatorics","subtopic":"binomial_coefficient","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"How many distinct subsets containing exactly 2 elements can be selected from a set of 18 distinct elements?","options":{"A":"-153","B":"16.3","C":"153","D":"183.6"},"correct_answer":"C","correct_answer_value":"153","solution":"Order does not matter, so the count is C(18,2)=18!/[2!(16)!]=153.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 153, not -153.","B":"Incorrect. Substituting the stated values into the governing relation gives 153, not 16.3.","C":"Correct. The worked calculation gives 153.","D":"Incorrect. Substituting the stated values into the governing relation gives 153, not 183.6."},"concepts_required":["combinations","factorials"],"reasoning_steps":2,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_combinations","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_000010","subject":"mathematics","topic":"algebra","subtopic":"quadratic_roots","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"For the quadratic equation 5x^2 -17x +7=0, let its roots be α and β. What is α²+β²?","options":{"A":"11.76","B":"1.876","C":"8.76","D":"17.52"},"correct_answer":"C","correct_answer_value":"8.76","solution":"By Vieta's relations, α+β = 17/5 and αβ = 7/5. Then α²+β²=(α+β)²−2αβ = (17/5)²−2(7/5) = 8.76.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 8.76, not 11.76.","B":"Incorrect. Substituting the stated values into the governing relation gives 8.76, not 1.876.","C":"Correct. The worked calculation gives 8.76.","D":"Incorrect. Substituting the stated values into the governing relation gives 8.76, not 17.52."},"concepts_required":["Vieta's formulas","symmetric polynomial expressions"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_quadratic_vieta","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_000246","subject":"mathematics","topic":"algebra","subtopic":"quadratic_roots","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"For the quadratic equation 3x^2 -13x -3=0, let its roots be α and β. What is α²+β²?","options":{"A":"41.55556","B":"21.77778","C":"20.77778","D":"18.77778"},"correct_answer":"C","correct_answer_value":"20.77778","solution":"By Vieta's relations, α+β = 13/3 and αβ = -3/3. Then α²+β²=(α+β)²−2αβ = (13/3)²−2(-3/3) = 20.77778.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 20.77778, not 41.55556.","B":"Incorrect. Substituting the stated values into the governing relation gives 20.77778, not 21.77778.","C":"Correct. The worked calculation gives 20.77778.","D":"Incorrect. Substituting the stated values into the governing relation gives 20.77778, not 18.77778."},"concepts_required":["Vieta's formulas","symmetric polynomial expressions"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_quadratic_vieta","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_007136","subject":"chemistry","topic":"physical_chemistry","subtopic":"ideal_gas_volume","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Assuming ideal behaviour, what volume is occupied by 8 mol of gas at 489 K and 2.26 atm? Use R=0.082057 L atm mol⁻¹ K⁻¹.","options":{"A":"144.03849 L","B":"-142.03849 L","C":"71.01924 L","D":"142.03849 L"},"correct_answer":"D","correct_answer_value":"142.03849 L","solution":"PV=nRT, so V=nRT/P=(8)(0.082057)(489)/(2.26)=142.03849 L.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 142.03849 L, not 144.03849 L.","B":"Incorrect. Substituting the stated values into the governing relation gives 142.03849 L, not -142.03849 L.","C":"Incorrect. Substituting the stated values into the governing relation gives 142.03849 L, not 71.01924 L.","D":"Correct. The worked calculation gives 142.03849 L."},"concepts_required":["ideal gas equation","unit consistency"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_ideal_gas","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_002672","subject":"mathematics","topic":"linear_algebra","subtopic":"2x2_determinant","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Find the determinant of the matrix [[7, 0], [1, -6]].","options":{"A":"-42","B":"-43","C":"-63","D":"-41"},"correct_answer":"A","correct_answer_value":"-42","solution":"For a 2×2 matrix [[a,b],[c,d]], det=ad−bc. Here it is (7)(-6)−(0)(1)=-42.","option_explanations":{"A":"Correct. The worked calculation gives -42.","B":"Incorrect. Substituting the stated values into the governing relation gives -42, not -43.","C":"Incorrect. Substituting the stated values into the governing relation gives -42, not -63.","D":"Incorrect. Substituting the stated values into the governing relation gives -42, not -41."},"concepts_required":["determinants","matrix operations"],"reasoning_steps":2,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_determinant_2x2","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_000479","subject":"mathematics","topic":"sequences_and_series","subtopic":"arithmetic_progression","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"An arithmetic progression has first term 18 and common difference 13. Find the sum of its first 28 terms.","options":{"A":"542.8","B":"2709","C":"-5418","D":"5418"},"correct_answer":"D","correct_answer_value":"5418","solution":"Use Sₙ = n/2[2a+(n−1)d]. Substituting a=18, d=13, n=28: Sₙ=28/2[2(18)+(28−1)(13)]=5418.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 5418, not 542.8.","B":"Incorrect. Substituting the stated values into the governing relation gives 5418, not 2709.","C":"Incorrect. Substituting the stated values into the governing relation gives 5418, not -5418.","D":"Correct. The worked calculation gives 5418."},"concepts_required":["arithmetic progression","finite series"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_ap_sum","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_006657","subject":"physics","topic":"oscillations","subtopic":"mass_spring_period_over_pi","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A mass of 20 kg is attached to an ideal spring of force constant 33 N m⁻¹ and oscillates horizontally without damping. What is the time period divided by π?","options":{"A":"-0.443 s","B":"1.557 s","C":"2.3355 s","D":"1.1557 s"},"correct_answer":"B","correct_answer_value":"1.557 s","solution":"For a mass–spring oscillator, T=2π√(m/k). Therefore T/π=2√(20/33)=1.557 s.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 1.557 s, not -0.443 s.","B":"Correct. The worked calculation gives 1.557 s.","C":"Incorrect. Substituting the stated values into the governing relation gives 1.557 s, not 2.3355 s.","D":"Incorrect. Substituting the stated values into the governing relation gives 1.557 s, not 1.1557 s."},"concepts_required":["simple harmonic motion","mass–spring oscillator","time period"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_shm_period","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_001022","subject":"mathematics","topic":"calculus","subtopic":"polynomial_differentiation","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Let f(x)=-5x³ +2x² +11x+7. Evaluate f′(2).","options":{"A":"-41","B":"-82","C":"-49.2","D":"-39"},"correct_answer":"A","correct_answer_value":"-41","solution":"Differentiate: f′(x)=3(-5)x²+2(2)x+(11). At x=2, f′(2)=3(-5)(2)²+2(2)(2)+(11)=-41.","option_explanations":{"A":"Correct. The worked calculation gives -41.","B":"Incorrect. Substituting the stated values into the governing relation gives -41, not -82.","C":"Incorrect. Substituting the stated values into the governing relation gives -41, not -49.2.","D":"Incorrect. Substituting the stated values into the governing relation gives -41, not -39."},"concepts_required":["power rule","polynomial derivative"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_derivative_cubic","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_006889","subject":"chemistry","topic":"stoichiometry","subtopic":"limiting_reagent_water","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Hydrogen reacts with oxygen according to 2H₂ + O₂ → 2H₂O. If 40 mol H₂ and 10 mol O₂ are mixed and the reaction goes to completion, how many moles of H₂O form?","options":{"A":"20 mol","B":"16 mol","C":"-20 mol","D":"30 mol"},"correct_answer":"A","correct_answer_value":"20 mol","solution":"From 2 mol H₂ and 1 mol O₂, 2 mol H₂O form. H₂ can produce 40 mol H₂O; O₂ can produce 20 mol H₂O. The limiting condition is O₂; product=20 mol.","option_explanations":{"A":"Correct. The worked calculation gives 20 mol.","B":"Incorrect. Substituting the stated values into the governing relation gives 20 mol, not 16 mol.","C":"Incorrect. Substituting the stated values into the governing relation gives 20 mol, not -20 mol.","D":"Incorrect. Substituting the stated values into the governing relation gives 20 mol, not 30 mol."},"concepts_required":["balanced chemical equations","limiting reagent","mole ratios"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_limiting_reagent","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_004555","subject":"physics","topic":"rotational_mechanics","subtopic":"disc_angular_acceleration","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A uniform solid disc of mass 10 kg and radius 3 m is acted on by a constant torque of 22 N m about its central axis. Find its angular acceleration.","options":{"A":"0.48889 rad s⁻²","B":"3.48889 rad s⁻²","C":"-2.51111 rad s⁻²","D":"1.04889 rad s⁻²"},"correct_answer":"A","correct_answer_value":"0.48889 rad s⁻²","solution":"For a solid disc, I=½MR²=½(10)(3)²=45 kg m². Using τ=Iα, α=22/45=0.48889 rad s⁻².","option_explanations":{"A":"Correct. The worked calculation gives 0.48889 rad s⁻².","B":"Incorrect. Substituting the stated values into the governing relation gives 0.48889 rad s⁻², not 3.48889 rad s⁻².","C":"Incorrect. Substituting the stated values into the governing relation gives 0.48889 rad s⁻², not -2.51111 rad s⁻².","D":"Incorrect. Substituting the stated values into the governing relation gives 0.48889 rad s⁻², not 1.04889 rad s⁻²."},"concepts_required":["moment of inertia","rotational dynamics","torque"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_disc_torque","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_009750","subject":"chemistry","topic":"inorganic_chemistry","subtopic":"complex_ion_oxidation_state","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"For a hypothetical coordination ion [Pt(Cl)_4]^+3, each Cl ligand has charge −1. Determine the formal oxidation state of Pt.","options":{"A":"8","B":"4","C":"7","D":"5"},"correct_answer":"C","correct_answer_value":"7","solution":"Let the formal oxidation state of Pt be x. Charge balance gives x+(4)(−1)=+3; hence x=3+4=+7.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 7, not 8.","B":"Incorrect. Substituting the stated values into the governing relation gives 7, not 4.","C":"Correct. The worked calculation gives 7.","D":"Incorrect. Substituting the stated values into the governing relation gives 7, not 5."},"concepts_required":["coordination compounds","oxidation state","charge balance"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_complex_oxidation","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_004166","subject":"physics","topic":"mechanics","subtopic":"inclined_plane_with_friction","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A 45 kg block slides down a 37° rough incline. The coefficient of kinetic friction is 0.45. Taking g=10 m s⁻², find the net force down the plane. Use sin37°=0.6 and cos37°=0.8.","options":{"A":"162 N","B":"107 N","C":"216 N","D":"108 N"},"correct_answer":"D","correct_answer_value":"108 N","solution":"Net force=mg(sinθ−μcosθ)=45×10×(0.6−0.45×0.8)=108 N.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 108 N, not 162 N.","B":"Incorrect. Substituting the stated values into the governing relation gives 108 N, not 107 N.","C":"Incorrect. Substituting the stated values into the governing relation gives 108 N, not 216 N.","D":"Correct. The worked calculation gives 108 N."},"concepts_required":["Newton's laws","inclined plane","kinetic friction"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_incline_friction","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_005292","subject":"physics","topic":"electrostatics","subtopic":"coulombs_law","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Two point charges of +3 μC and +9 μC are separated by 12 cm in vacuum. Using k=9×10⁹ N m² C⁻², find the magnitude of the electrostatic force.","options":{"A":"16.875 N","B":"-16.875 N","C":"15.875 N","D":"25.3125 N"},"correct_answer":"A","correct_answer_value":"16.875 N","solution":"Convert charges to coulombs and distance to metres. F=k|q₁q₂|/r²=9×10⁹(3×10⁻⁶)(9×10⁻⁶)/(12×10⁻²)²=16.875 N.","option_explanations":{"A":"Correct. The worked calculation gives 16.875 N.","B":"Incorrect. Substituting the stated values into the governing relation gives 16.875 N, not -16.875 N.","C":"Incorrect. Substituting the stated values into the governing relation gives 16.875 N, not 15.875 N.","D":"Incorrect. Substituting the stated values into the governing relation gives 16.875 N, not 25.3125 N."},"concepts_required":["Coulomb's law","SI unit conversion","inverse-square law"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_coulomb","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_000193","subject":"mathematics","topic":"algebra","subtopic":"quadratic_roots","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"For the quadratic equation 5x^2 +18x +1=0, let its roots be α and β. What is α²+β²?","options":{"A":"-12.56","B":"13.56","C":"9.56","D":"12.56"},"correct_answer":"D","correct_answer_value":"12.56","solution":"By Vieta's relations, α+β = -18/5 and αβ = 1/5. Then α²+β²=(α+β)²−2αβ = (-18/5)²−2(1/5) = 12.56.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 12.56, not -12.56.","B":"Incorrect. Substituting the stated values into the governing relation gives 12.56, not 13.56.","C":"Incorrect. Substituting the stated values into the governing relation gives 12.56, not 9.56.","D":"Correct. The worked calculation gives 12.56."},"concepts_required":["Vieta's formulas","symmetric polynomial expressions"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_quadratic_vieta","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_003375","subject":"physics","topic":"mechanics","subtopic":"projectile_range","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A projectile is launched from level ground at 27 m s⁻¹ at 65° to the horizontal. Take g=10 m s⁻². Find its horizontal range.","options":{"A":"67.01357 m","B":"53.84464 m","C":"55.84464 m","D":"56.84464 m"},"correct_answer":"C","correct_answer_value":"55.84464 m","solution":"Range R=u²sin(2θ)/g. Substituting u=27, θ=65° and g=10 gives R=(27)²sin(130°)/10=55.84464 m.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 55.84464 m, not 67.01357 m.","B":"Incorrect. Substituting the stated values into the governing relation gives 55.84464 m, not 53.84464 m.","C":"Correct. The worked calculation gives 55.84464 m.","D":"Incorrect. Substituting the stated values into the governing relation gives 55.84464 m, not 56.84464 m."},"concepts_required":["projectile motion","range equation","trigonometry"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_projectile_range","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_005224","subject":"physics","topic":"electrostatics","subtopic":"coulombs_law","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Two point charges of +1 μC and +4 μC are separated by 11 cm in vacuum. Using k=9×10⁹ N m² C⁻², find the magnitude of the electrostatic force.","options":{"A":"1.97521 N","B":"-0.02479 N","C":"3.97521 N","D":"2.97521 N"},"correct_answer":"D","correct_answer_value":"2.97521 N","solution":"Convert charges to coulombs and distance to metres. F=k|q₁q₂|/r²=9×10⁹(1×10⁻⁶)(4×10⁻⁶)/(11×10⁻²)²=2.97521 N.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 2.97521 N, not 1.97521 N.","B":"Incorrect. Substituting the stated values into the governing relation gives 2.97521 N, not -0.02479 N.","C":"Incorrect. Substituting the stated values into the governing relation gives 2.97521 N, not 3.97521 N.","D":"Correct. The worked calculation gives 2.97521 N."},"concepts_required":["Coulomb's law","SI unit conversion","inverse-square law"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_coulomb","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_002701","subject":"mathematics","topic":"vectors","subtopic":"dot_product","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Given vectors u=(-6, -3, 9) and v=(-7, -2, -6), calculate u·v.","options":{"A":"-6","B":"-3","C":"-4.5","D":"-9"},"correct_answer":"A","correct_answer_value":"-6","solution":"The dot product is the sum of corresponding component products: (-6)(-7)+(-3)(-2)+(9)(-6)=-6.","option_explanations":{"A":"Correct. The worked calculation gives -6.","B":"Incorrect. Substituting the stated values into the governing relation gives -6, not -3.","C":"Incorrect. Substituting the stated values into the governing relation gives -6, not -4.5.","D":"Incorrect. Substituting the stated values into the governing relation gives -6, not -9."},"concepts_required":["three-dimensional vectors","scalar product"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_vector_dot","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_009611","subject":"chemistry","topic":"organic_chemistry","subtopic":"degree_of_unsaturation","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Assuming a neutral closed-shell formula, calculate the degree of unsaturation (DBE) for C_14H_18N_2X_4. Treat X as the total number of monovalent halogen atoms.","options":{"A":"-5","B":"2","C":"3","D":"5"},"correct_answer":"D","correct_answer_value":"5","solution":"Use DBE=(2C+2+N−H−X)/2. Substitution gives (2×14+2+2−18−4)/2=5.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 5, not -5.","B":"Incorrect. Substituting the stated values into the governing relation gives 5, not 2.","C":"Incorrect. Substituting the stated values into the governing relation gives 5, not 3.","D":"Correct. The worked calculation gives 5."},"concepts_required":["molecular formula","degree of unsaturation","organic structure"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_dbe","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_006327","subject":"physics","topic":"ray_optics","subtopic":"convex_lens_magnification","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A real object is placed 72 cm in front of a thin convex lens of focal length 21 cm. Using the Cartesian sign convention, what is the magnitude of the linear magnification?","options":{"A":"0.41176","B":"1.41176","C":"0.82353","D":"-0.41176"},"correct_answer":"A","correct_answer_value":"0.41176","solution":"Using distances as positive magnitudes, 1/f=1/v+1/u gives v=uf/(u−f)=(72)(21)/(72−21)=29.64706 cm. Thus |m|=v/u=29.64706/72=0.41176.","option_explanations":{"A":"Correct. The worked calculation gives 0.41176.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.41176, not 1.41176.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.41176, not 0.82353.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.41176, not -0.41176."},"concepts_required":["thin lens equation","linear magnification","real image"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_lens_magnification","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_004957","subject":"physics","topic":"electricity","subtopic":"series_parallel_circuit","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Resistors 9 Ω and 3 Ω are connected in series; this combination is connected in parallel with a 5 Ω resistor across a 18 V ideal battery. Find the total current supplied by the battery.","options":{"A":"5.1 A","B":"3.825 A","C":"7.1 A","D":"2.1 A"},"correct_answer":"A","correct_answer_value":"5.1 A","solution":"Series branch resistance=9+3=12 Ω. Parallel equivalent R_eq=(12×5)/(12+5)=3.52941 Ω. Total current I=V/R_eq=18/3.52941=5.1 A.","option_explanations":{"A":"Correct. The worked calculation gives 5.1 A.","B":"Incorrect. Substituting the stated values into the governing relation gives 5.1 A, not 3.825 A.","C":"Incorrect. Substituting the stated values into the governing relation gives 5.1 A, not 7.1 A.","D":"Incorrect. Substituting the stated values into the governing relation gives 5.1 A, not 2.1 A."},"concepts_required":["series resistors","parallel resistors","Ohm's law"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_series_parallel","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_000370","subject":"mathematics","topic":"sequences_and_series","subtopic":"arithmetic_progression","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"An arithmetic progression has first term 9 and common difference 7. Find the sum of its first 33 terms.","options":{"A":"3993","B":"2994.75","C":"5989.5","D":"3194.4"},"correct_answer":"A","correct_answer_value":"3993","solution":"Use Sₙ = n/2[2a+(n−1)d]. Substituting a=9, d=7, n=33: Sₙ=33/2[2(9)+(33−1)(7)]=3993.","option_explanations":{"A":"Correct. The worked calculation gives 3993.","B":"Incorrect. Substituting the stated values into the governing relation gives 3993, not 2994.75.","C":"Incorrect. Substituting the stated values into the governing relation gives 3993, not 5989.5.","D":"Incorrect. Substituting the stated values into the governing relation gives 3993, not 3194.4."},"concepts_required":["arithmetic progression","finite series"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_ap_sum","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_007687","subject":"chemistry","topic":"physical_chemistry","subtopic":"first_order_half_life","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A first-order reaction has a half-life of 20 minutes. If the initial reactant concentration is 70 mmol L⁻¹, what concentration remains after 60 minutes?","options":{"A":"6.5625 mmol L⁻¹","B":"8.75 mmol L⁻¹","C":"-8.75 mmol L⁻¹","D":"7.75 mmol L⁻¹"},"correct_answer":"B","correct_answer_value":"8.75 mmol L⁻¹","solution":"60 minutes corresponds to 3 half-lives. The remaining concentration is [A]₀(1/2)ⁿ=70/2^3=8.75 mmol L⁻¹.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 8.75 mmol L⁻¹, not 6.5625 mmol L⁻¹.","B":"Correct. The worked calculation gives 8.75 mmol L⁻¹.","C":"Incorrect. Substituting the stated values into the governing relation gives 8.75 mmol L⁻¹, not -8.75 mmol L⁻¹.","D":"Incorrect. Substituting the stated values into the governing relation gives 8.75 mmol L⁻¹, not 7.75 mmol L⁻¹."},"concepts_required":["first-order kinetics","half-life","exponential decay"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_first_order_halflife","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_000516","subject":"mathematics","topic":"sequences_and_series","subtopic":"arithmetic_progression","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"An arithmetic progression has first term 4 and common difference 13. Find the sum of its first 35 terms.","options":{"A":"-7875","B":"7875","C":"15750","D":"7876"},"correct_answer":"B","correct_answer_value":"7875","solution":"Use Sₙ = n/2[2a+(n−1)d]. Substituting a=4, d=13, n=35: Sₙ=35/2[2(4)+(35−1)(13)]=7875.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 7875, not -7875.","B":"Correct. The worked calculation gives 7875.","C":"Incorrect. Substituting the stated values into the governing relation gives 7875, not 15750.","D":"Incorrect. Substituting the stated values into the governing relation gives 7875, not 7876."},"concepts_required":["arithmetic progression","finite series"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_ap_sum","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_005729","subject":"physics","topic":"thermodynamics","subtopic":"ideal_gas_isochoric_pressure","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A fixed amount of ideal gas is kept in a rigid container. Its pressure is 5 atm at 320 K. What is its pressure at 784 K?","options":{"A":"9.1875 atm","B":"6.125 atm","C":"12.25 atm","D":"18.375 atm"},"correct_answer":"C","correct_answer_value":"12.25 atm","solution":"At constant volume, P/T is constant. P₂=P₁T₂/T₁=(5)(784)/(320)=12.25 atm.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 12.25 atm, not 9.1875 atm.","B":"Incorrect. Substituting the stated values into the governing relation gives 12.25 atm, not 6.125 atm.","C":"Correct. The worked calculation gives 12.25 atm.","D":"Incorrect. Substituting the stated values into the governing relation gives 12.25 atm, not 18.375 atm."},"concepts_required":["ideal gas law","isochoric process","absolute temperature"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_isochoric_gas","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_007302","subject":"chemistry","topic":"physical_chemistry","subtopic":"ideal_gas_volume","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Assuming ideal behaviour, what volume is occupied by 8 mol of gas at 592 K and 1.74 atm? Use R=0.082057 L atm mol⁻¹ K⁻¹.","options":{"A":"111.67297 L","B":"224.34595 L","C":"178.67676 L","D":"223.34595 L"},"correct_answer":"D","correct_answer_value":"223.34595 L","solution":"PV=nRT, so V=nRT/P=(8)(0.082057)(592)/(1.74)=223.34595 L.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 223.34595 L, not 111.67297 L.","B":"Incorrect. Substituting the stated values into the governing relation gives 223.34595 L, not 224.34595 L.","C":"Incorrect. Substituting the stated values into the governing relation gives 223.34595 L, not 178.67676 L.","D":"Correct. The worked calculation gives 223.34595 L."},"concepts_required":["ideal gas equation","unit consistency"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_ideal_gas","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_004902","subject":"physics","topic":"electricity","subtopic":"series_parallel_circuit","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Resistors 10 Ω and 6 Ω are connected in series; this combination is connected in parallel with a 13 Ω resistor across a 6 V ideal battery. Find the total current supplied by the battery.","options":{"A":"0.83654 A","B":"1.83654 A","C":"1.67308 A","D":"0.41827 A"},"correct_answer":"A","correct_answer_value":"0.83654 A","solution":"Series branch resistance=10+6=16 Ω. Parallel equivalent R_eq=(16×13)/(16+13)=7.17241 Ω. Total current I=V/R_eq=6/7.17241=0.83654 A.","option_explanations":{"A":"Correct. The worked calculation gives 0.83654 A.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.83654 A, not 1.83654 A.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.83654 A, not 1.67308 A.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.83654 A, not 0.41827 A."},"concepts_required":["series resistors","parallel resistors","Ohm's law"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_series_parallel","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_003325","subject":"mathematics","topic":"complex_numbers","subtopic":"real_part_of_complex_quotient","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Let z=(-8-6i)/(9+10i). What is Re(z), the real part of z?","options":{"A":"-1.09392","B":"0.92707","C":"-1.72928","D":"-0.72928"},"correct_answer":"D","correct_answer_value":"-0.72928","solution":"Multiply numerator and denominator by the conjugate (9-10i). Then Re(z)=(ac+bd)/(c²+d²)=((-8)(9)+(-6)(10))/(9²+(10)²)=-0.72928.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives -0.72928, not -1.09392.","B":"Incorrect. Substituting the stated values into the governing relation gives -0.72928, not 0.92707.","C":"Incorrect. Substituting the stated values into the governing relation gives -0.72928, not -1.72928.","D":"Correct. The worked calculation gives -0.72928."},"concepts_required":["complex conjugate","complex division","real and imaginary parts"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_complex_quotient","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_000391","subject":"mathematics","topic":"sequences_and_series","subtopic":"arithmetic_progression","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"An arithmetic progression has first term -6 and common difference 9. Find the sum of its first 15 terms.","options":{"A":"86.5","B":"1710","C":"855","D":"1026"},"correct_answer":"C","correct_answer_value":"855","solution":"Use Sₙ = n/2[2a+(n−1)d]. Substituting a=-6, d=9, n=15: Sₙ=15/2[2(-6)+(15−1)(9)]=855.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 855, not 86.5.","B":"Incorrect. Substituting the stated values into the governing relation gives 855, not 1710.","C":"Correct. The worked calculation gives 855.","D":"Incorrect. Substituting the stated values into the governing relation gives 855, not 1026."},"concepts_required":["arithmetic progression","finite series"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_ap_sum","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_001906","subject":"mathematics","topic":"combinatorics","subtopic":"binomial_coefficient","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"How many distinct subsets containing exactly 4 elements can be selected from a set of 28 distinct elements?","options":{"A":"20476","B":"40950","C":"20473","D":"20475"},"correct_answer":"D","correct_answer_value":"20475","solution":"Order does not matter, so the count is C(28,4)=28!/[4!(24)!]=20475.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 20475, not 20476.","B":"Incorrect. Substituting the stated values into the governing relation gives 20475, not 40950.","C":"Incorrect. Substituting the stated values into the governing relation gives 20475, not 20473.","D":"Correct. The worked calculation gives 20475."},"concepts_required":["combinations","factorials"],"reasoning_steps":2,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_combinations","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_004517","subject":"physics","topic":"rotational_mechanics","subtopic":"disc_angular_acceleration","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A uniform solid disc of mass 7 kg and radius 2 m is acted on by a constant torque of 12 N m about its central axis. Find its angular acceleration.","options":{"A":"0.64286 rad s⁻²","B":"2.85714 rad s⁻²","C":"-2.14286 rad s⁻²","D":"0.85714 rad s⁻²"},"correct_answer":"D","correct_answer_value":"0.85714 rad s⁻²","solution":"For a solid disc, I=½MR²=½(7)(2)²=14 kg m². Using τ=Iα, α=12/14=0.85714 rad s⁻².","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.85714 rad s⁻², not 0.64286 rad s⁻².","B":"Incorrect. Substituting the stated values into the governing relation gives 0.85714 rad s⁻², not 2.85714 rad s⁻².","C":"Incorrect. Substituting the stated values into the governing relation gives 0.85714 rad s⁻², not -2.14286 rad s⁻².","D":"Correct. The worked calculation gives 0.85714 rad s⁻²."},"concepts_required":["moment of inertia","rotational dynamics","torque"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_disc_torque","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_002407","subject":"mathematics","topic":"linear_algebra","subtopic":"2x2_determinant","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Find the determinant of the matrix [[9, -1], [3, -2]].","options":{"A":"-15","B":"-16","C":"-30","D":"-18"},"correct_answer":"A","correct_answer_value":"-15","solution":"For a 2×2 matrix [[a,b],[c,d]], det=ad−bc. Here it is (9)(-2)−(-1)(3)=-15.","option_explanations":{"A":"Correct. The worked calculation gives -15.","B":"Incorrect. Substituting the stated values into the governing relation gives -15, not -16.","C":"Incorrect. Substituting the stated values into the governing relation gives -15, not -30.","D":"Incorrect. Substituting the stated values into the governing relation gives -15, not -18."},"concepts_required":["determinants","matrix operations"],"reasoning_steps":2,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_determinant_2x2","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_004105","subject":"physics","topic":"mechanics","subtopic":"inclined_plane_with_friction","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A 21 kg block slides down a 37° rough incline. The coefficient of kinetic friction is 0.31. Taking g=10 m s⁻², find the net force down the plane. Use sin37°=0.6 and cos37°=0.8.","options":{"A":"72.92 N","B":"147.84 N","C":"36.96 N","D":"73.92 N"},"correct_answer":"D","correct_answer_value":"73.92 N","solution":"Net force=mg(sinθ−μcosθ)=21×10×(0.6−0.31×0.8)=73.92 N.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 73.92 N, not 72.92 N.","B":"Incorrect. Substituting the stated values into the governing relation gives 73.92 N, not 147.84 N.","C":"Incorrect. Substituting the stated values into the governing relation gives 73.92 N, not 36.96 N.","D":"Correct. The worked calculation gives 73.92 N."},"concepts_required":["Newton's laws","inclined plane","kinetic friction"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_incline_friction","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_005165","subject":"physics","topic":"electrostatics","subtopic":"coulombs_law","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Two point charges of +4 μC and +7 μC are separated by 8 cm in vacuum. Using k=9×10⁹ N m² C⁻², find the magnitude of the electrostatic force.","options":{"A":"39.375 N","B":"47.25 N","C":"59.0625 N","D":"19.6875 N"},"correct_answer":"A","correct_answer_value":"39.375 N","solution":"Convert charges to coulombs and distance to metres. F=k|q₁q₂|/r²=9×10⁹(4×10⁻⁶)(7×10⁻⁶)/(8×10⁻²)²=39.375 N.","option_explanations":{"A":"Correct. The worked calculation gives 39.375 N.","B":"Incorrect. Substituting the stated values into the governing relation gives 39.375 N, not 47.25 N.","C":"Incorrect. Substituting the stated values into the governing relation gives 39.375 N, not 59.0625 N.","D":"Incorrect. Substituting the stated values into the governing relation gives 39.375 N, not 19.6875 N."},"concepts_required":["Coulomb's law","SI unit conversion","inverse-square law"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_coulomb","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_003928","subject":"physics","topic":"mechanics","subtopic":"work_energy_theorem","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A 2 kg block initially moves at 12 m s⁻¹. A constant net force does 30 N of work over 9 m in the direction of motion. Find its final kinetic energy.","options":{"A":"331.2 J","B":"415 J","C":"414 J","D":"412 J"},"correct_answer":"C","correct_answer_value":"414 J","solution":"By the work–energy theorem, K_f=K_i+W=½mu²+Fs=½(2)(12)²+(30)(9)=414 J.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 414 J, not 331.2 J.","B":"Incorrect. Substituting the stated values into the governing relation gives 414 J, not 415 J.","C":"Correct. The worked calculation gives 414 J.","D":"Incorrect. Substituting the stated values into the governing relation gives 414 J, not 412 J."},"concepts_required":["work–energy theorem","kinetic energy","work by constant force"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_work_energy","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_005444","subject":"physics","topic":"electrostatics","subtopic":"capacitor_energy","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A 63 μF capacitor is charged to a potential difference of 36 V. Calculate the energy stored, expressed in μJ.","options":{"A":"32659.2 μJ","B":"4083.4 μJ","C":"40824 μJ","D":"40822 μJ"},"correct_answer":"C","correct_answer_value":"40824 μJ","solution":"U=½CV². Since C=63 μF, U in μJ is ½×63×(36)²=40824 μJ.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 40824 μJ, not 32659.2 μJ.","B":"Incorrect. Substituting the stated values into the governing relation gives 40824 μJ, not 4083.4 μJ.","C":"Correct. The worked calculation gives 40824 μJ.","D":"Incorrect. Substituting the stated values into the governing relation gives 40824 μJ, not 40822 μJ."},"concepts_required":["capacitance","energy stored in a capacitor"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_capacitor_energy","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_007105","subject":"chemistry","topic":"physical_chemistry","subtopic":"ideal_gas_volume","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Assuming ideal behaviour, what volume is occupied by 1 mol of gas at 352 K and 0.26 atm? Use R=0.082057 L atm mol⁻¹ K⁻¹.","options":{"A":"112.09255 L","B":"111.09255 L","C":"166.63883 L","D":"12.10926 L"},"correct_answer":"B","correct_answer_value":"111.09255 L","solution":"PV=nRT, so V=nRT/P=(1)(0.082057)(352)/(0.26)=111.09255 L.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 111.09255 L, not 112.09255 L.","B":"Correct. The worked calculation gives 111.09255 L.","C":"Incorrect. Substituting the stated values into the governing relation gives 111.09255 L, not 166.63883 L.","D":"Incorrect. Substituting the stated values into the governing relation gives 111.09255 L, not 12.10926 L."},"concepts_required":["ideal gas equation","unit consistency"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_ideal_gas","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_009255","subject":"chemistry","topic":"physical_chemistry","subtopic":"nernst_equation_298K","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"At 298 K, a galvanic cell has E°cell=0.63 V, transfers 3 electrons, and has reaction quotient Q=10^-6. Using E=E°−(0.0591/n)log₁₀Q, calculate Ecell.","options":{"A":"-1.2518 V","B":"-0.7482 V","C":"0.56115 V","D":"0.7482 V"},"correct_answer":"D","correct_answer_value":"0.7482 V","solution":"Apply the Nernst equation: E=0.63−(0.0591/3)(-6)=0.7482 V.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.7482 V, not -1.2518 V.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.7482 V, not -0.7482 V.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.7482 V, not 0.56115 V.","D":"Correct. The worked calculation gives 0.7482 V."},"concepts_required":["electrochemical cells","Nernst equation","logarithms"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_nernst","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_000722","subject":"mathematics","topic":"sequences_and_series","subtopic":"geometric_progression","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A geometric progression has first term 10 and common ratio 3. What is the sum of its first 6 terms?","options":{"A":"3641","B":"5460","C":"2912","D":"3640"},"correct_answer":"D","correct_answer_value":"3640","solution":"For r≠1, Sₙ=a(rⁿ−1)/(r−1). Thus Sₙ=10(3^6−1)/(3−1)=3640.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 3640, not 3641.","B":"Incorrect. Substituting the stated values into the governing relation gives 3640, not 5460.","C":"Incorrect. Substituting the stated values into the governing relation gives 3640, not 2912.","D":"Correct. The worked calculation gives 3640."},"concepts_required":["geometric progression","finite geometric sum"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_gp_sum","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_009645","subject":"chemistry","topic":"organic_chemistry","subtopic":"degree_of_unsaturation","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Assuming a neutral closed-shell formula, calculate the degree of unsaturation (DBE) for C_16H_27N_3X_2. Treat X as the total number of monovalent halogen atoms.","options":{"A":"6","B":"4","C":"-4","D":"8"},"correct_answer":"B","correct_answer_value":"4","solution":"Use DBE=(2C+2+N−H−X)/2. Substitution gives (2×16+2+3−27−2)/2=4.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 4, not 6.","B":"Correct. The worked calculation gives 4.","C":"Incorrect. Substituting the stated values into the governing relation gives 4, not -4.","D":"Incorrect. Substituting the stated values into the governing relation gives 4, not 8."},"concepts_required":["molecular formula","degree of unsaturation","organic structure"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_dbe","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_008074","subject":"chemistry","topic":"physical_chemistry","subtopic":"faradays_law_copper","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"An aqueous Cu²⁺ solution is electrolysed using a current of 15 A for 953 s. Assuming 100% current efficiency, calculate the mass of copper deposited. Use M(Cu)=63.5 g mol⁻¹ and F=96500 C mol⁻¹.","options":{"A":"7.70328 g","B":"2.70328 g","C":"4.70328 g","D":"5.70328 g"},"correct_answer":"C","correct_answer_value":"4.70328 g","solution":"Cu²⁺+2e⁻→Cu, so z=2. By Faraday's law m=MIt/(zF)=63.5×15×953/(2×96500)=4.70328 g.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 4.70328 g, not 7.70328 g.","B":"Incorrect. Substituting the stated values into the governing relation gives 4.70328 g, not 2.70328 g.","C":"Correct. The worked calculation gives 4.70328 g.","D":"Incorrect. Substituting the stated values into the governing relation gives 4.70328 g, not 5.70328 g."},"concepts_required":["electrolysis","Faraday's laws","redox stoichiometry"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_faraday_copper","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_008420","subject":"chemistry","topic":"physical_chemistry","subtopic":"boiling_point_elevation","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A non-volatile solute is dissolved in water to produce a solution of molality 1.2 mol kg⁻¹. If its van't Hoff factor is 1 and K_b for water is 0.512 K kg mol⁻¹, find the boiling-point elevation.","options":{"A":"1.06144 K","B":"-2.3856 K","C":"0.9216 K","D":"0.6144 K"},"correct_answer":"D","correct_answer_value":"0.6144 K","solution":"ΔT_b=iK_bm. Substitution gives ΔT_b=1×0.512×1.2=0.6144 K.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.6144 K, not 1.06144 K.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.6144 K, not -2.3856 K.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.6144 K, not 0.9216 K.","D":"Correct. The worked calculation gives 0.6144 K."},"concepts_required":["colligative properties","van't Hoff factor","molality"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_boiling_elevation","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_002774","subject":"mathematics","topic":"vectors","subtopic":"dot_product","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Given vectors u=(5, 4, -1) and v=(-2, -7, -5), calculate u·v.","options":{"A":"-24.75","B":"-26.4","C":"-16.5","D":"-33"},"correct_answer":"D","correct_answer_value":"-33","solution":"The dot product is the sum of corresponding component products: (5)(-2)+(4)(-7)+(-1)(-5)=-33.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives -33, not -24.75.","B":"Incorrect. Substituting the stated values into the governing relation gives -33, not -26.4.","C":"Incorrect. Substituting the stated values into the governing relation gives -33, not -16.5.","D":"Correct. The worked calculation gives -33."},"concepts_required":["three-dimensional vectors","scalar product"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_vector_dot","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_008327","subject":"chemistry","topic":"physical_chemistry","subtopic":"faradays_law_copper","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"An aqueous Cu²⁺ solution is electrolysed using a current of 5 A for 6065 s. Assuming 100% current efficiency, calculate the mass of copper deposited. Use M(Cu)=63.5 g mol⁻¹ and F=96500 C mol⁻¹.","options":{"A":"14.96609 g","B":"9.9774 g","C":"19.95479 g","D":"11.9774 g"},"correct_answer":"B","correct_answer_value":"9.9774 g","solution":"Cu²⁺+2e⁻→Cu, so z=2. By Faraday's law m=MIt/(zF)=63.5×5×6065/(2×96500)=9.9774 g.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 9.9774 g, not 14.96609 g.","B":"Correct. The worked calculation gives 9.9774 g.","C":"Incorrect. Substituting the stated values into the governing relation gives 9.9774 g, not 19.95479 g.","D":"Incorrect. Substituting the stated values into the governing relation gives 9.9774 g, not 11.9774 g."},"concepts_required":["electrolysis","Faraday's laws","redox stoichiometry"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_faraday_copper","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_007387","subject":"chemistry","topic":"physical_chemistry","subtopic":"strong_acid_base_mixture","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"55 mL of 0.10 M HCl is mixed with 53 mL of 0.10 M NaOH at 25°C. Assuming complete dissociation and additive volumes, find the final pH.","options":{"A":"1.27324","B":"4.09859","C":"5.73239","D":"2.73239"},"correct_answer":"D","correct_answer_value":"2.73239","solution":"Excess H⁺ moles=0.10×(55−53)/1000=0.000200 mol. Total volume=0.108 L, so [H⁺]=0.001852 M and pH=−log₁₀[H⁺]=2.73239.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 2.73239, not 1.27324.","B":"Incorrect. Substituting the stated values into the governing relation gives 2.73239, not 4.09859.","C":"Incorrect. Substituting the stated values into the governing relation gives 2.73239, not 5.73239.","D":"Correct. The worked calculation gives 2.73239."},"concepts_required":["strong acid–strong base neutralization","molarity","pH and pOH"],"reasoning_steps":5,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_acid_base_pH","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_009894","subject":"chemistry","topic":"inorganic_chemistry","subtopic":"complex_ion_oxidation_state","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"For a hypothetical coordination ion [Mn(OH)_6]^+1, each OH ligand has charge −1. Determine the formal oxidation state of Mn.","options":{"A":"10","B":"7","C":"9","D":"5.25"},"correct_answer":"B","correct_answer_value":"7","solution":"Let the formal oxidation state of Mn be x. Charge balance gives x+(6)(−1)=+1; hence x=1+6=+7.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 7, not 10.","B":"Correct. The worked calculation gives 7.","C":"Incorrect. Substituting the stated values into the governing relation gives 7, not 9.","D":"Incorrect. Substituting the stated values into the governing relation gives 7, not 5.25."},"concepts_required":["coordination compounds","oxidation state","charge balance"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_complex_oxidation","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_003623","subject":"physics","topic":"mechanics","subtopic":"projectile_range","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A projectile is launched from level ground at 18 m s⁻¹ at 70° to the horizontal. Take g=10 m s⁻². Find its horizontal range.","options":{"A":"19.82632 m","B":"22.82632 m","C":"18.82632 m","D":"20.82632 m"},"correct_answer":"D","correct_answer_value":"20.82632 m","solution":"Range R=u²sin(2θ)/g. Substituting u=18, θ=70° and g=10 gives R=(18)²sin(140°)/10=20.82632 m.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 20.82632 m, not 19.82632 m.","B":"Incorrect. Substituting the stated values into the governing relation gives 20.82632 m, not 22.82632 m.","C":"Incorrect. Substituting the stated values into the governing relation gives 20.82632 m, not 18.82632 m.","D":"Correct. The worked calculation gives 20.82632 m."},"concepts_required":["projectile motion","range equation","trigonometry"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_projectile_range","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_006301","subject":"physics","topic":"ray_optics","subtopic":"convex_lens_magnification","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A real object is placed 103 cm in front of a thin convex lens of focal length 29 cm. Using the Cartesian sign convention, what is the magnitude of the linear magnification?","options":{"A":"2.39189","B":"0.39189","C":"3.39189","D":"0.29392"},"correct_answer":"B","correct_answer_value":"0.39189","solution":"Using distances as positive magnitudes, 1/f=1/v+1/u gives v=uf/(u−f)=(103)(29)/(103−29)=40.36486 cm. Thus |m|=v/u=40.36486/103=0.39189.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.39189, not 2.39189.","B":"Correct. The worked calculation gives 0.39189.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.39189, not 3.39189.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.39189, not 0.29392."},"concepts_required":["thin lens equation","linear magnification","real image"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_lens_magnification","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_005925","subject":"physics","topic":"thermodynamics","subtopic":"ideal_gas_isochoric_pressure","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A fixed amount of ideal gas is kept in a rigid container. Its pressure is 1.7 atm at 311 K. What is its pressure at 485 K?","options":{"A":"2.65113 atm","B":"0.65113 atm","C":"1.65113 atm","D":"4.65113 atm"},"correct_answer":"A","correct_answer_value":"2.65113 atm","solution":"At constant volume, P/T is constant. P₂=P₁T₂/T₁=(1.7)(485)/(311)=2.65113 atm.","option_explanations":{"A":"Correct. The worked calculation gives 2.65113 atm.","B":"Incorrect. Substituting the stated values into the governing relation gives 2.65113 atm, not 0.65113 atm.","C":"Incorrect. Substituting the stated values into the governing relation gives 2.65113 atm, not 1.65113 atm.","D":"Incorrect. Substituting the stated values into the governing relation gives 2.65113 atm, not 4.65113 atm."},"concepts_required":["ideal gas law","isochoric process","absolute temperature"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_isochoric_gas","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_000257","subject":"mathematics","topic":"algebra","subtopic":"quadratic_roots","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"For the quadratic equation 6x^2 +1x -6=0, let its roots be α and β. What is α²+β²?","options":{"A":"1.01389","B":"1.52083","C":"4.02778","D":"2.02778"},"correct_answer":"D","correct_answer_value":"2.02778","solution":"By Vieta's relations, α+β = -1/6 and αβ = -6/6. Then α²+β²=(α+β)²−2αβ = (-1/6)²−2(-6/6) = 2.02778.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 2.02778, not 1.01389.","B":"Incorrect. Substituting the stated values into the governing relation gives 2.02778, not 1.52083.","C":"Incorrect. Substituting the stated values into the governing relation gives 2.02778, not 4.02778.","D":"Correct. The worked calculation gives 2.02778."},"concepts_required":["Vieta's formulas","symmetric polynomial expressions"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_quadratic_vieta","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_006238","subject":"physics","topic":"ray_optics","subtopic":"convex_lens_magnification","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A real object is placed 45 cm in front of a thin convex lens of focal length 9 cm. Using the Cartesian sign convention, what is the magnitude of the linear magnification?","options":{"A":"0.375","B":"2.25","C":"0.25","D":"0.1875"},"correct_answer":"C","correct_answer_value":"0.25","solution":"Using distances as positive magnitudes, 1/f=1/v+1/u gives v=uf/(u−f)=(45)(9)/(45−9)=11.25 cm. Thus |m|=v/u=11.25/45=0.25.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.25, not 0.375.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.25, not 2.25.","C":"Correct. The worked calculation gives 0.25.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.25, not 0.1875."},"concepts_required":["thin lens equation","linear magnification","real image"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_lens_magnification","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_004837","subject":"physics","topic":"electricity","subtopic":"series_parallel_circuit","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Resistors 3 Ω and 15 Ω are connected in series; this combination is connected in parallel with a 8 Ω resistor across a 24 V ideal battery. Find the total current supplied by the battery.","options":{"A":"5.33333 A","B":"4.33333 A","C":"7.33333 A","D":"2.33333 A"},"correct_answer":"B","correct_answer_value":"4.33333 A","solution":"Series branch resistance=3+15=18 Ω. Parallel equivalent R_eq=(18×8)/(18+8)=5.53846 Ω. Total current I=V/R_eq=24/5.53846=4.33333 A.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 4.33333 A, not 5.33333 A.","B":"Correct. The worked calculation gives 4.33333 A.","C":"Incorrect. Substituting the stated values into the governing relation gives 4.33333 A, not 7.33333 A.","D":"Incorrect. Substituting the stated values into the governing relation gives 4.33333 A, not 2.33333 A."},"concepts_required":["series resistors","parallel resistors","Ohm's law"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_series_parallel","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_003007","subject":"mathematics","topic":"complex_numbers","subtopic":"real_part_of_complex_quotient","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Let z=(11+9i)/(11+4i). What is Re(z), the real part of z?","options":{"A":"0.14599","B":"1.14599","C":"0.57299","D":"-0.85401"},"correct_answer":"B","correct_answer_value":"1.14599","solution":"Multiply numerator and denominator by the conjugate (11-4i). Then Re(z)=(ac+bd)/(c²+d²)=((11)(11)+(9)(4))/(11²+(4)²)=1.14599.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 1.14599, not 0.14599.","B":"Correct. The worked calculation gives 1.14599.","C":"Incorrect. Substituting the stated values into the governing relation gives 1.14599, not 0.57299.","D":"Incorrect. Substituting the stated values into the governing relation gives 1.14599, not -0.85401."},"concepts_required":["complex conjugate","complex division","real and imaginary parts"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_complex_quotient","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_007407","subject":"chemistry","topic":"physical_chemistry","subtopic":"strong_acid_base_mixture","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"66 mL of 0.10 M HCl is mixed with 66 mL of 0.10 M NaOH at 25°C. Assuming complete dissociation and additive volumes, find the final pH.","options":{"A":"5","B":"14","C":"10.5","D":"7"},"correct_answer":"D","correct_answer_value":"7","solution":"Equal moles of strong acid and base are present. They neutralize completely, so at 25°C pH=7.00.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 7, not 5.","B":"Incorrect. Substituting the stated values into the governing relation gives 7, not 14.","C":"Incorrect. Substituting the stated values into the governing relation gives 7, not 10.5.","D":"Correct. The worked calculation gives 7."},"concepts_required":["strong acid–strong base neutralization","molarity","pH and pOH"],"reasoning_steps":5,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_acid_base_pH","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_002308","subject":"mathematics","topic":"probability","subtopic":"same_colour_without_replacement","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A bag contains 3 red, 11 blue and 11 green balls. Two balls are drawn uniformly at random without replacement. What is the probability that both have the same colour?","options":{"A":"0.37667","B":"0.565","C":"-0.37667","D":"0.75333"},"correct_answer":"A","correct_answer_value":"0.37667","solution":"Favourable pairs = C(3,2)+C(11,2)+C(11,2)=113. Total pairs=C(25,2)=300. Therefore P=113/300=0.37667.","option_explanations":{"A":"Correct. The worked calculation gives 0.37667.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.37667, not 0.565.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.37667, not -0.37667.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.37667, not 0.75333."},"concepts_required":["classical probability","combinations","without replacement"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_probability_colours","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_003190","subject":"mathematics","topic":"complex_numbers","subtopic":"real_part_of_complex_quotient","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Let z=(-6+9i)/(4+7i). What is Re(z), the real part of z?","options":{"A":"0.9","B":"0.3","C":"0.6","D":"-1.4"},"correct_answer":"C","correct_answer_value":"0.6","solution":"Multiply numerator and denominator by the conjugate (4-7i). Then Re(z)=(ac+bd)/(c²+d²)=((-6)(4)+(9)(7))/(4²+(7)²)=0.6.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.6, not 0.9.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.6, not 0.3.","C":"Correct. The worked calculation gives 0.6.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.6, not -1.4."},"concepts_required":["complex conjugate","complex division","real and imaginary parts"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_complex_quotient","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_001773","subject":"mathematics","topic":"combinatorics","subtopic":"binomial_coefficient","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"How many distinct subsets containing exactly 8 elements can be selected from a set of 55 distinct elements?","options":{"A":"974053080","B":"1461079620","C":"1217566352","D":"1217566350"},"correct_answer":"D","correct_answer_value":"1217566350","solution":"Order does not matter, so the count is C(55,8)=55!/[8!(47)!]=1217566350.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 1217566350, not 974053080.","B":"Incorrect. Substituting the stated values into the governing relation gives 1217566350, not 1461079620.","C":"Incorrect. Substituting the stated values into the governing relation gives 1217566350, not 1217566352.","D":"Correct. The worked calculation gives 1217566350."},"concepts_required":["combinations","factorials"],"reasoning_steps":2,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_combinations","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_008800","subject":"chemistry","topic":"physical_chemistry","subtopic":"equilibrium_degree_dissociation","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"For A(g) ⇌ 2B(g), the initial concentration of A is 1.33 mol L⁻¹ and initially B is absent. At equilibrium, 31% of A has dissociated at constant volume. Find K_c.","options":{"A":"0.74094","B":"-0.25906","C":"1.11142","D":"3.74094"},"correct_answer":"A","correct_answer_value":"0.74094","solution":"Let α=0.31. At equilibrium [A]=c₀(1−α) and [B]=2c₀α. Thus K_c=[B]²/[A]=4c₀α²/(1−α)=4(1.33)(0.31)²/(1−0.31)=0.74094.","option_explanations":{"A":"Correct. The worked calculation gives 0.74094.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.74094, not -0.25906.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.74094, not 1.11142.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.74094, not 3.74094."},"concepts_required":["chemical equilibrium","equilibrium constant","degree of dissociation"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_equilibrium_alpha","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_000556","subject":"mathematics","topic":"sequences_and_series","subtopic":"arithmetic_progression","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"An arithmetic progression has first term 18 and common difference 14. Find the sum of its first 25 terms.","options":{"A":"4650","B":"4651","C":"466","D":"4648"},"correct_answer":"A","correct_answer_value":"4650","solution":"Use Sₙ = n/2[2a+(n−1)d]. Substituting a=18, d=14, n=25: Sₙ=25/2[2(18)+(25−1)(14)]=4650.","option_explanations":{"A":"Correct. The worked calculation gives 4650.","B":"Incorrect. Substituting the stated values into the governing relation gives 4650, not 4651.","C":"Incorrect. Substituting the stated values into the governing relation gives 4650, not 466.","D":"Incorrect. Substituting the stated values into the governing relation gives 4650, not 4648."},"concepts_required":["arithmetic progression","finite series"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_ap_sum","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_008748","subject":"chemistry","topic":"physical_chemistry","subtopic":"equilibrium_degree_dissociation","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"For A(g) ⇌ 2B(g), the initial concentration of A is 1.97 mol L⁻¹ and initially B is absent. At equilibrium, 59% of A has dissociated at constant volume. Find K_c.","options":{"A":"4.69031","B":"6.69031","C":"10.03547","D":"5.01773"},"correct_answer":"B","correct_answer_value":"6.69031","solution":"Let α=0.59. At equilibrium [A]=c₀(1−α) and [B]=2c₀α. Thus K_c=[B]²/[A]=4c₀α²/(1−α)=4(1.97)(0.59)²/(1−0.59)=6.69031.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 6.69031, not 4.69031.","B":"Correct. The worked calculation gives 6.69031.","C":"Incorrect. Substituting the stated values into the governing relation gives 6.69031, not 10.03547.","D":"Incorrect. Substituting the stated values into the governing relation gives 6.69031, not 5.01773."},"concepts_required":["chemical equilibrium","equilibrium constant","degree of dissociation"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_equilibrium_alpha","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_005642","subject":"physics","topic":"electrostatics","subtopic":"capacitor_energy","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A 2 μF capacitor is charged to a potential difference of 6 V. Calculate the energy stored, expressed in μJ.","options":{"A":"18 μJ","B":"36 μJ","C":"43.2 μJ","D":"34 μJ"},"correct_answer":"B","correct_answer_value":"36 μJ","solution":"U=½CV². Since C=2 μF, U in μJ is ½×2×(6)²=36 μJ.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 36 μJ, not 18 μJ.","B":"Correct. The worked calculation gives 36 μJ.","C":"Incorrect. Substituting the stated values into the governing relation gives 36 μJ, not 43.2 μJ.","D":"Incorrect. Substituting the stated values into the governing relation gives 36 μJ, not 34 μJ."},"concepts_required":["capacitance","energy stored in a capacitor"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_capacitor_energy","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_004279","subject":"physics","topic":"mechanics","subtopic":"inclined_plane_with_friction","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A 40 kg block slides down a 37° rough incline. The coefficient of kinetic friction is 0.34. Taking g=10 m s⁻², find the net force down the plane. Use sin37°=0.6 and cos37°=0.8.","options":{"A":"104.96 N","B":"131.2 N","C":"129.2 N","D":"65.6 N"},"correct_answer":"B","correct_answer_value":"131.2 N","solution":"Net force=mg(sinθ−μcosθ)=40×10×(0.6−0.34×0.8)=131.2 N.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 131.2 N, not 104.96 N.","B":"Correct. The worked calculation gives 131.2 N.","C":"Incorrect. Substituting the stated values into the governing relation gives 131.2 N, not 129.2 N.","D":"Incorrect. Substituting the stated values into the governing relation gives 131.2 N, not 65.6 N."},"concepts_required":["Newton's laws","inclined plane","kinetic friction"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_incline_friction","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_006766","subject":"chemistry","topic":"stoichiometry","subtopic":"limiting_reagent_water","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Hydrogen reacts with oxygen according to 2H₂ + O₂ → 2H₂O. If 16 mol H₂ and 23 mol O₂ are mixed and the reaction goes to completion, how many moles of H₂O form?","options":{"A":"17 mol","B":"12 mol","C":"12.8 mol","D":"16 mol"},"correct_answer":"D","correct_answer_value":"16 mol","solution":"From 2 mol H₂ and 1 mol O₂, 2 mol H₂O form. H₂ can produce 16 mol H₂O; O₂ can produce 46 mol H₂O. The limiting condition is H₂; product=16 mol.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 16 mol, not 17 mol.","B":"Incorrect. Substituting the stated values into the governing relation gives 16 mol, not 12 mol.","C":"Incorrect. Substituting the stated values into the governing relation gives 16 mol, not 12.8 mol.","D":"Correct. The worked calculation gives 16 mol."},"concepts_required":["balanced chemical equations","limiting reagent","mole ratios"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_limiting_reagent","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_002169","subject":"mathematics","topic":"probability","subtopic":"same_colour_without_replacement","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A bag contains 9 red, 9 blue and 9 green balls. Two balls are drawn uniformly at random without replacement. What is the probability that both have the same colour?","options":{"A":"-0.69231","B":"0.30769","C":"0.61538","D":"-0.30769"},"correct_answer":"B","correct_answer_value":"0.30769","solution":"Favourable pairs = C(9,2)+C(9,2)+C(9,2)=108. Total pairs=C(27,2)=351. Therefore P=108/351=0.30769.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.30769, not -0.69231.","B":"Correct. The worked calculation gives 0.30769.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.30769, not 0.61538.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.30769, not -0.30769."},"concepts_required":["classical probability","combinations","without replacement"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_probability_colours","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_005420","subject":"physics","topic":"electrostatics","subtopic":"capacitor_energy","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A 28 μF capacitor is charged to a potential difference of 30 V. Calculate the energy stored, expressed in μJ.","options":{"A":"9450 μJ","B":"6300 μJ","C":"12600 μJ","D":"12599 μJ"},"correct_answer":"C","correct_answer_value":"12600 μJ","solution":"U=½CV². Since C=28 μF, U in μJ is ½×28×(30)²=12600 μJ.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 12600 μJ, not 9450 μJ.","B":"Incorrect. Substituting the stated values into the governing relation gives 12600 μJ, not 6300 μJ.","C":"Correct. The worked calculation gives 12600 μJ.","D":"Incorrect. Substituting the stated values into the governing relation gives 12600 μJ, not 12599 μJ."},"concepts_required":["capacitance","energy stored in a capacitor"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_capacitor_energy","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_008276","subject":"chemistry","topic":"physical_chemistry","subtopic":"faradays_law_copper","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"An aqueous Cu²⁺ solution is electrolysed using a current of 6 A for 3801 s. Assuming 100% current efficiency, calculate the mass of copper deposited. Use M(Cu)=63.5 g mol⁻¹ and F=96500 C mol⁻¹.","options":{"A":"7.50353 g","B":"11.25529 g","C":"1.75035 g","D":"15.00706 g"},"correct_answer":"A","correct_answer_value":"7.50353 g","solution":"Cu²⁺+2e⁻→Cu, so z=2. By Faraday's law m=MIt/(zF)=63.5×6×3801/(2×96500)=7.50353 g.","option_explanations":{"A":"Correct. The worked calculation gives 7.50353 g.","B":"Incorrect. Substituting the stated values into the governing relation gives 7.50353 g, not 11.25529 g.","C":"Incorrect. Substituting the stated values into the governing relation gives 7.50353 g, not 1.75035 g.","D":"Incorrect. Substituting the stated values into the governing relation gives 7.50353 g, not 15.00706 g."},"concepts_required":["electrolysis","Faraday's laws","redox stoichiometry"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_faraday_copper","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_009267","subject":"chemistry","topic":"physical_chemistry","subtopic":"nernst_equation_298K","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"At 298 K, a galvanic cell has E°cell=0.57 V, transfers 2 electrons, and has reaction quotient Q=10^-6. Using E=E°−(0.0591/n)log₁₀Q, calculate Ecell.","options":{"A":"-0.7473 V","B":"-0.2527 V","C":"0.7473 V","D":"-2.2527 V"},"correct_answer":"C","correct_answer_value":"0.7473 V","solution":"Apply the Nernst equation: E=0.57−(0.0591/2)(-6)=0.7473 V.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.7473 V, not -0.7473 V.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.7473 V, not -0.2527 V.","C":"Correct. The worked calculation gives 0.7473 V.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.7473 V, not -2.2527 V."},"concepts_required":["electrochemical cells","Nernst equation","logarithms"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_nernst","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_000301","subject":"mathematics","topic":"algebra","subtopic":"quadratic_roots","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"For the quadratic equation 7x^2 -14x +8=0, let its roots be α and β. What is α²+β²?","options":{"A":"1.71429","B":"2.71429","C":"-1.28571","D":"1.17143"},"correct_answer":"A","correct_answer_value":"1.71429","solution":"By Vieta's relations, α+β = 14/7 and αβ = 8/7. Then α²+β²=(α+β)²−2αβ = (14/7)²−2(8/7) = 1.71429.","option_explanations":{"A":"Correct. The worked calculation gives 1.71429.","B":"Incorrect. Substituting the stated values into the governing relation gives 1.71429, not 2.71429.","C":"Incorrect. Substituting the stated values into the governing relation gives 1.71429, not -1.28571.","D":"Incorrect. Substituting the stated values into the governing relation gives 1.71429, not 1.17143."},"concepts_required":["Vieta's formulas","symmetric polynomial expressions"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_quadratic_vieta","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_007556","subject":"chemistry","topic":"physical_chemistry","subtopic":"strong_acid_base_mixture","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"14 mL of 0.10 M HCl is mixed with 57 mL of 0.10 M NaOH at 25°C. Assuming complete dissociation and additive volumes, find the final pH.","options":{"A":"25.56442","B":"-12.78221","C":"9.58666","D":"12.78221"},"correct_answer":"D","correct_answer_value":"12.78221","solution":"Excess OH⁻ moles=0.10×(57−14)/1000=0.004300 mol. Total volume=0.071 L, so [OH⁻]=0.060563 M, pOH=−log₁₀[OH⁻], and pH=14−pOH=12.78221.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 12.78221, not 25.56442.","B":"Incorrect. Substituting the stated values into the governing relation gives 12.78221, not -12.78221.","C":"Incorrect. Substituting the stated values into the governing relation gives 12.78221, not 9.58666.","D":"Correct. The worked calculation gives 12.78221."},"concepts_required":["strong acid–strong base neutralization","molarity","pH and pOH"],"reasoning_steps":5,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_acid_base_pH","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_005460","subject":"physics","topic":"electrostatics","subtopic":"capacitor_energy","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A 57 μF capacitor is charged to a potential difference of 14 V. Calculate the energy stored, expressed in μJ.","options":{"A":"5586 μJ","B":"2793 μJ","C":"8379 μJ","D":"5585 μJ"},"correct_answer":"A","correct_answer_value":"5586 μJ","solution":"U=½CV². Since C=57 μF, U in μJ is ½×57×(14)²=5586 μJ.","option_explanations":{"A":"Correct. The worked calculation gives 5586 μJ.","B":"Incorrect. Substituting the stated values into the governing relation gives 5586 μJ, not 2793 μJ.","C":"Incorrect. Substituting the stated values into the governing relation gives 5586 μJ, not 8379 μJ.","D":"Incorrect. Substituting the stated values into the governing relation gives 5586 μJ, not 5585 μJ."},"concepts_required":["capacitance","energy stored in a capacitor"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_capacitor_energy","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_005228","subject":"physics","topic":"electrostatics","subtopic":"coulombs_law","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Two point charges of +3 μC and +1 μC are separated by 10 cm in vacuum. Using k=9×10⁹ N m² C⁻², find the magnitude of the electrostatic force.","options":{"A":"1.35 N","B":"2.7 N","C":"1.7 N","D":"5.7 N"},"correct_answer":"B","correct_answer_value":"2.7 N","solution":"Convert charges to coulombs and distance to metres. F=k|q₁q₂|/r²=9×10⁹(3×10⁻⁶)(1×10⁻⁶)/(10×10⁻²)²=2.7 N.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 2.7 N, not 1.35 N.","B":"Correct. The worked calculation gives 2.7 N.","C":"Incorrect. Substituting the stated values into the governing relation gives 2.7 N, not 1.7 N.","D":"Incorrect. Substituting the stated values into the governing relation gives 2.7 N, not 5.7 N."},"concepts_required":["Coulomb's law","SI unit conversion","inverse-square law"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_coulomb","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_001648","subject":"mathematics","topic":"calculus","subtopic":"definite_integral_polynomial","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Evaluate the definite integral ∫₀^8 (-2x² -6x +4) dx.","options":{"A":"-501.33333","B":"-500.33333","C":"-499.33333","D":"-401.06667"},"correct_answer":"A","correct_answer_value":"-501.33333","solution":"An antiderivative is (-2/3)x³+(-6/2)x²+(4)x. Evaluating between 0 and 8 gives (-2/3)(8)³+(-6/2)(8)²+(4)(8)=-501.33333.","option_explanations":{"A":"Correct. The worked calculation gives -501.33333.","B":"Incorrect. Substituting the stated values into the governing relation gives -501.33333, not -500.33333.","C":"Incorrect. Substituting the stated values into the governing relation gives -501.33333, not -499.33333.","D":"Incorrect. Substituting the stated values into the governing relation gives -501.33333, not -401.06667."},"concepts_required":["definite integration","power rule","limits of integration"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_integral_poly","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_008017","subject":"chemistry","topic":"physical_chemistry","subtopic":"faradays_law_copper","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"An aqueous Cu²⁺ solution is electrolysed using a current of 3 A for 19665 s. Assuming 100% current efficiency, calculate the mass of copper deposited. Use M(Cu)=63.5 g mol⁻¹ and F=96500 C mol⁻¹.","options":{"A":"19.41027 g","B":"2.94103 g","C":"9.70514 g","D":"21.41027 g"},"correct_answer":"A","correct_answer_value":"19.41027 g","solution":"Cu²⁺+2e⁻→Cu, so z=2. By Faraday's law m=MIt/(zF)=63.5×3×19665/(2×96500)=19.41027 g.","option_explanations":{"A":"Correct. The worked calculation gives 19.41027 g.","B":"Incorrect. Substituting the stated values into the governing relation gives 19.41027 g, not 2.94103 g.","C":"Incorrect. Substituting the stated values into the governing relation gives 19.41027 g, not 9.70514 g.","D":"Incorrect. Substituting the stated values into the governing relation gives 19.41027 g, not 21.41027 g."},"concepts_required":["electrolysis","Faraday's laws","redox stoichiometry"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_faraday_copper","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_005980","subject":"physics","topic":"thermodynamics","subtopic":"ideal_gas_isochoric_pressure","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A fixed amount of ideal gas is kept in a rigid container. Its pressure is 4.1 atm at 442 K. What is its pressure at 647 K?","options":{"A":"3.00158 atm","B":"3.00079 atm","C":"8.00158 atm","D":"6.00158 atm"},"correct_answer":"D","correct_answer_value":"6.00158 atm","solution":"At constant volume, P/T is constant. P₂=P₁T₂/T₁=(4.1)(647)/(442)=6.00158 atm.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 6.00158 atm, not 3.00158 atm.","B":"Incorrect. Substituting the stated values into the governing relation gives 6.00158 atm, not 3.00079 atm.","C":"Incorrect. Substituting the stated values into the governing relation gives 6.00158 atm, not 8.00158 atm.","D":"Correct. The worked calculation gives 6.00158 atm."},"concepts_required":["ideal gas law","isochoric process","absolute temperature"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_isochoric_gas","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_002592","subject":"mathematics","topic":"linear_algebra","subtopic":"2x2_determinant","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Find the determinant of the matrix [[8, -1], [7, -2]].","options":{"A":"-9","B":"9","C":"0.1","D":"-18"},"correct_answer":"A","correct_answer_value":"-9","solution":"For a 2×2 matrix [[a,b],[c,d]], det=ad−bc. Here it is (8)(-2)−(-1)(7)=-9.","option_explanations":{"A":"Correct. The worked calculation gives -9.","B":"Incorrect. Substituting the stated values into the governing relation gives -9, not 9.","C":"Incorrect. Substituting the stated values into the governing relation gives -9, not 0.1.","D":"Incorrect. Substituting the stated values into the governing relation gives -9, not -18."},"concepts_required":["determinants","matrix operations"],"reasoning_steps":2,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_determinant_2x2","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_002160","subject":"mathematics","topic":"probability","subtopic":"same_colour_without_replacement","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A bag contains 8 red, 4 blue and 12 green balls. Two balls are drawn uniformly at random without replacement. What is the probability that both have the same colour?","options":{"A":"0.18116","B":"0.36232","C":"2.36232","D":"-0.63768"},"correct_answer":"B","correct_answer_value":"0.36232","solution":"Favourable pairs = C(8,2)+C(4,2)+C(12,2)=100. Total pairs=C(24,2)=276. Therefore P=100/276=0.36232.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.36232, not 0.18116.","B":"Correct. The worked calculation gives 0.36232.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.36232, not 2.36232.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.36232, not -0.63768."},"concepts_required":["classical probability","combinations","without replacement"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_probability_colours","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_001506","subject":"mathematics","topic":"calculus","subtopic":"definite_integral_polynomial","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Evaluate the definite integral ∫₀^7 (3x² +2x +4) dx.","options":{"A":"210","B":"420","C":"315","D":"840"},"correct_answer":"B","correct_answer_value":"420","solution":"An antiderivative is (3/3)x³+(2/2)x²+(4)x. Evaluating between 0 and 7 gives (3/3)(7)³+(2/2)(7)²+(4)(7)=420.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 420, not 210.","B":"Correct. The worked calculation gives 420.","C":"Incorrect. Substituting the stated values into the governing relation gives 420, not 315.","D":"Incorrect. Substituting the stated values into the governing relation gives 420, not 840."},"concepts_required":["definite integration","power rule","limits of integration"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_integral_poly","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_003509","subject":"physics","topic":"mechanics","subtopic":"projectile_range","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A projectile is launched from level ground at 107 m s⁻¹ at 70° to the horizontal. Take g=10 m s⁻². Find its horizontal range.","options":{"A":"1103.8913 m","B":"1471.85507 m","C":"735.92753 m","D":"588.74203 m"},"correct_answer":"C","correct_answer_value":"735.92753 m","solution":"Range R=u²sin(2θ)/g. Substituting u=107, θ=70° and g=10 gives R=(107)²sin(140°)/10=735.92753 m.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 735.92753 m, not 1103.8913 m.","B":"Incorrect. Substituting the stated values into the governing relation gives 735.92753 m, not 1471.85507 m.","C":"Correct. The worked calculation gives 735.92753 m.","D":"Incorrect. Substituting the stated values into the governing relation gives 735.92753 m, not 588.74203 m."},"concepts_required":["projectile motion","range equation","trigonometry"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_projectile_range","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_009460","subject":"chemistry","topic":"organic_chemistry","subtopic":"degree_of_unsaturation","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Assuming a neutral closed-shell formula, calculate the degree of unsaturation (DBE) for C_5H_9N_1X_4. Treat X as the total number of monovalent halogen atoms.","options":{"A":"2","B":"-3","C":"0","D":"1"},"correct_answer":"C","correct_answer_value":"0","solution":"Use DBE=(2C+2+N−H−X)/2. Substitution gives (2×5+2+1−9−4)/2=0.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0, not 2.","B":"Incorrect. Substituting the stated values into the governing relation gives 0, not -3.","C":"Correct. The worked calculation gives 0.","D":"Incorrect. Substituting the stated values into the governing relation gives 0, not 1."},"concepts_required":["molecular formula","degree of unsaturation","organic structure"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_dbe","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_006663","subject":"physics","topic":"oscillations","subtopic":"mass_spring_period_over_pi","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A mass of 3 kg is attached to an ideal spring of force constant 960 N m⁻¹ and oscillates horizontally without damping. What is the time period divided by π?","options":{"A":"1.01118 s","B":"0.1118 s","C":"1.1118 s","D":"-1.8882 s"},"correct_answer":"B","correct_answer_value":"0.1118 s","solution":"For a mass–spring oscillator, T=2π√(m/k). Therefore T/π=2√(3/960)=0.1118 s.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.1118 s, not 1.01118 s.","B":"Correct. The worked calculation gives 0.1118 s.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.1118 s, not 1.1118 s.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.1118 s, not -1.8882 s."},"concepts_required":["simple harmonic motion","mass–spring oscillator","time period"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_shm_period","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_007232","subject":"chemistry","topic":"physical_chemistry","subtopic":"ideal_gas_volume","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Assuming ideal behaviour, what volume is occupied by 5 mol of gas at 506 K and 1.89 atm? Use R=0.082057 L atm mol⁻¹ K⁻¹.","options":{"A":"109.8435 L","B":"87.8748 L","C":"54.92175 L","D":"219.68699 L"},"correct_answer":"A","correct_answer_value":"109.8435 L","solution":"PV=nRT, so V=nRT/P=(5)(0.082057)(506)/(1.89)=109.8435 L.","option_explanations":{"A":"Correct. The worked calculation gives 109.8435 L.","B":"Incorrect. Substituting the stated values into the governing relation gives 109.8435 L, not 87.8748 L.","C":"Incorrect. Substituting the stated values into the governing relation gives 109.8435 L, not 54.92175 L.","D":"Incorrect. Substituting the stated values into the governing relation gives 109.8435 L, not 219.68699 L."},"concepts_required":["ideal gas equation","unit consistency"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_ideal_gas","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_004008","subject":"physics","topic":"mechanics","subtopic":"inclined_plane_with_friction","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A 14 kg block slides down a 37° rough incline. The coefficient of kinetic friction is 0.11. Taking g=10 m s⁻², find the net force down the plane. Use sin37°=0.6 and cos37°=0.8.","options":{"A":"86.016 N","B":"71.68 N","C":"70.68 N","D":"8.168 N"},"correct_answer":"B","correct_answer_value":"71.68 N","solution":"Net force=mg(sinθ−μcosθ)=14×10×(0.6−0.11×0.8)=71.68 N.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 71.68 N, not 86.016 N.","B":"Correct. The worked calculation gives 71.68 N.","C":"Incorrect. Substituting the stated values into the governing relation gives 71.68 N, not 70.68 N.","D":"Incorrect. Substituting the stated values into the governing relation gives 71.68 N, not 8.168 N."},"concepts_required":["Newton's laws","inclined plane","kinetic friction"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_incline_friction","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_000295","subject":"mathematics","topic":"algebra","subtopic":"quadratic_roots","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"For the quadratic equation 5x^2 +12x -4=0, let its roots be α and β. What is α²+β²?","options":{"A":"5.52","B":"5.36","C":"7.36","D":"11.04"},"correct_answer":"C","correct_answer_value":"7.36","solution":"By Vieta's relations, α+β = -12/5 and αβ = -4/5. Then α²+β²=(α+β)²−2αβ = (-12/5)²−2(-4/5) = 7.36.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 7.36, not 5.52.","B":"Incorrect. Substituting the stated values into the governing relation gives 7.36, not 5.36.","C":"Correct. The worked calculation gives 7.36.","D":"Incorrect. Substituting the stated values into the governing relation gives 7.36, not 11.04."},"concepts_required":["Vieta's formulas","symmetric polynomial expressions"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_quadratic_vieta","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_004425","subject":"physics","topic":"rotational_mechanics","subtopic":"disc_angular_acceleration","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A uniform solid disc of mass 15 kg and radius 4 m is acted on by a constant torque of 71 N m about its central axis. Find its angular acceleration.","options":{"A":"0.29583 rad s⁻²","B":"1.18333 rad s⁻²","C":"0.59167 rad s⁻²","D":"-0.40833 rad s⁻²"},"correct_answer":"C","correct_answer_value":"0.59167 rad s⁻²","solution":"For a solid disc, I=½MR²=½(15)(4)²=120 kg m². Using τ=Iα, α=71/120=0.59167 rad s⁻².","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.59167 rad s⁻², not 0.29583 rad s⁻².","B":"Incorrect. Substituting the stated values into the governing relation gives 0.59167 rad s⁻², not 1.18333 rad s⁻².","C":"Correct. The worked calculation gives 0.59167 rad s⁻².","D":"Incorrect. Substituting the stated values into the governing relation gives 0.59167 rad s⁻², not -0.40833 rad s⁻²."},"concepts_required":["moment of inertia","rotational dynamics","torque"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_disc_torque","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_005480","subject":"physics","topic":"electrostatics","subtopic":"capacitor_energy","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A 75 μF capacitor is charged to a potential difference of 18 V. Calculate the energy stored, expressed in μJ.","options":{"A":"12150 μJ","B":"9720 μJ","C":"12152 μJ","D":"-12150 μJ"},"correct_answer":"A","correct_answer_value":"12150 μJ","solution":"U=½CV². Since C=75 μF, U in μJ is ½×75×(18)²=12150 μJ.","option_explanations":{"A":"Correct. The worked calculation gives 12150 μJ.","B":"Incorrect. Substituting the stated values into the governing relation gives 12150 μJ, not 9720 μJ.","C":"Incorrect. Substituting the stated values into the governing relation gives 12150 μJ, not 12152 μJ.","D":"Incorrect. Substituting the stated values into the governing relation gives 12150 μJ, not -12150 μJ."},"concepts_required":["capacitance","energy stored in a capacitor"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_capacitor_energy","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_000703","subject":"mathematics","topic":"sequences_and_series","subtopic":"geometric_progression","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A geometric progression has first term 11 and common ratio 2. What is the sum of its first 10 terms?","options":{"A":"22506","B":"5626.5","C":"11251","D":"11253"},"correct_answer":"D","correct_answer_value":"11253","solution":"For r≠1, Sₙ=a(rⁿ−1)/(r−1). Thus Sₙ=11(2^10−1)/(2−1)=11253.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 11253, not 22506.","B":"Incorrect. Substituting the stated values into the governing relation gives 11253, not 5626.5.","C":"Incorrect. Substituting the stated values into the governing relation gives 11253, not 11251.","D":"Correct. The worked calculation gives 11253."},"concepts_required":["geometric progression","finite geometric sum"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_gp_sum","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_007800","subject":"chemistry","topic":"physical_chemistry","subtopic":"first_order_half_life","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A first-order reaction has a half-life of 20 minutes. If the initial reactant concentration is 69 mmol L⁻¹, what concentration remains after 100 minutes?","options":{"A":"3.23438 mmol L⁻¹","B":"0.15625 mmol L⁻¹","C":"1.15625 mmol L⁻¹","D":"2.15625 mmol L⁻¹"},"correct_answer":"D","correct_answer_value":"2.15625 mmol L⁻¹","solution":"100 minutes corresponds to 5 half-lives. The remaining concentration is [A]₀(1/2)ⁿ=69/2^5=2.15625 mmol L⁻¹.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 2.15625 mmol L⁻¹, not 3.23438 mmol L⁻¹.","B":"Incorrect. Substituting the stated values into the governing relation gives 2.15625 mmol L⁻¹, not 0.15625 mmol L⁻¹.","C":"Incorrect. Substituting the stated values into the governing relation gives 2.15625 mmol L⁻¹, not 1.15625 mmol L⁻¹.","D":"Correct. The worked calculation gives 2.15625 mmol L⁻¹."},"concepts_required":["first-order kinetics","half-life","exponential decay"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_first_order_halflife","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_002586","subject":"mathematics","topic":"linear_algebra","subtopic":"2x2_determinant","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Find the determinant of the matrix [[1, 4], [2, 4]].","options":{"A":"-6","B":"-4","C":"-3","D":"4"},"correct_answer":"B","correct_answer_value":"-4","solution":"For a 2×2 matrix [[a,b],[c,d]], det=ad−bc. Here it is (1)(4)−(4)(2)=-4.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives -4, not -6.","B":"Correct. The worked calculation gives -4.","C":"Incorrect. Substituting the stated values into the governing relation gives -4, not -3.","D":"Incorrect. Substituting the stated values into the governing relation gives -4, not 4."},"concepts_required":["determinants","matrix operations"],"reasoning_steps":2,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_determinant_2x2","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_002081","subject":"mathematics","topic":"probability","subtopic":"same_colour_without_replacement","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A bag contains 4 red, 7 blue and 11 green balls. Two balls are drawn uniformly at random without replacement. What is the probability that both have the same colour?","options":{"A":"0.35498","B":"-0.64502","C":"2.35498","D":"0.26623"},"correct_answer":"A","correct_answer_value":"0.35498","solution":"Favourable pairs = C(4,2)+C(7,2)+C(11,2)=82. Total pairs=C(22,2)=231. Therefore P=82/231=0.35498.","option_explanations":{"A":"Correct. The worked calculation gives 0.35498.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.35498, not -0.64502.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.35498, not 2.35498.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.35498, not 0.26623."},"concepts_required":["classical probability","combinations","without replacement"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_probability_colours","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_008873","subject":"chemistry","topic":"physical_chemistry","subtopic":"equilibrium_degree_dissociation","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"For A(g) ⇌ 2B(g), the initial concentration of A is 1.18 mol L⁻¹ and initially B is absent. At equilibrium, 28% of A has dissociated at constant volume. Find K_c.","options":{"A":"0.38547","B":"0.51396","C":"3.51396","D":"1.02791"},"correct_answer":"B","correct_answer_value":"0.51396","solution":"Let α=0.28. At equilibrium [A]=c₀(1−α) and [B]=2c₀α. Thus K_c=[B]²/[A]=4c₀α²/(1−α)=4(1.18)(0.28)²/(1−0.28)=0.51396.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.51396, not 0.38547.","B":"Correct. The worked calculation gives 0.51396.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.51396, not 3.51396.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.51396, not 1.02791."},"concepts_required":["chemical equilibrium","equilibrium constant","degree of dissociation"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_equilibrium_alpha","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_002067","subject":"mathematics","topic":"probability","subtopic":"same_colour_without_replacement","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A bag contains 7 red, 10 blue and 4 green balls. Two balls are drawn uniformly at random without replacement. What is the probability that both have the same colour?","options":{"A":"-2.65714","B":"1.03429","C":"-0.34286","D":"0.34286"},"correct_answer":"D","correct_answer_value":"0.34286","solution":"Favourable pairs = C(7,2)+C(10,2)+C(4,2)=72. Total pairs=C(21,2)=210. Therefore P=72/210=0.34286.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.34286, not -2.65714.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.34286, not 1.03429.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.34286, not -0.34286.","D":"Correct. The worked calculation gives 0.34286."},"concepts_required":["classical probability","combinations","without replacement"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_probability_colours","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_009395","subject":"chemistry","topic":"organic_chemistry","subtopic":"degree_of_unsaturation","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Assuming a neutral closed-shell formula, calculate the degree of unsaturation (DBE) for C_10H_8X_2. Treat X as the total number of monovalent halogen atoms.","options":{"A":"6","B":"12","C":"7","D":"-6"},"correct_answer":"A","correct_answer_value":"6","solution":"Use DBE=(2C+2+N−H−X)/2. Substitution gives (2×10+2+0−8−2)/2=6.","option_explanations":{"A":"Correct. The worked calculation gives 6.","B":"Incorrect. Substituting the stated values into the governing relation gives 6, not 12.","C":"Incorrect. Substituting the stated values into the governing relation gives 6, not 7.","D":"Incorrect. Substituting the stated values into the governing relation gives 6, not -6."},"concepts_required":["molecular formula","degree of unsaturation","organic structure"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_dbe","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_005432","subject":"physics","topic":"electrostatics","subtopic":"capacitor_energy","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A 3 μF capacitor is charged to a potential difference of 22 V. Calculate the energy stored, expressed in μJ.","options":{"A":"725 μJ","B":"728 μJ","C":"726 μJ","D":"871.2 μJ"},"correct_answer":"C","correct_answer_value":"726 μJ","solution":"U=½CV². Since C=3 μF, U in μJ is ½×3×(22)²=726 μJ.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 726 μJ, not 725 μJ.","B":"Incorrect. Substituting the stated values into the governing relation gives 726 μJ, not 728 μJ.","C":"Correct. The worked calculation gives 726 μJ.","D":"Incorrect. Substituting the stated values into the governing relation gives 726 μJ, not 871.2 μJ."},"concepts_required":["capacitance","energy stored in a capacitor"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_capacitor_energy","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_008093","subject":"chemistry","topic":"physical_chemistry","subtopic":"faradays_law_copper","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"An aqueous Cu²⁺ solution is electrolysed using a current of 10 A for 18326 s. Assuming 100% current efficiency, calculate the mass of copper deposited. Use M(Cu)=63.5 g mol⁻¹ and F=96500 C mol⁻¹.","options":{"A":"60.29539 g","B":"59.29539 g","C":"-60.29539 g","D":"61.29539 g"},"correct_answer":"A","correct_answer_value":"60.29539 g","solution":"Cu²⁺+2e⁻→Cu, so z=2. By Faraday's law m=MIt/(zF)=63.5×10×18326/(2×96500)=60.29539 g.","option_explanations":{"A":"Correct. The worked calculation gives 60.29539 g.","B":"Incorrect. Substituting the stated values into the governing relation gives 60.29539 g, not 59.29539 g.","C":"Incorrect. Substituting the stated values into the governing relation gives 60.29539 g, not -60.29539 g.","D":"Incorrect. Substituting the stated values into the governing relation gives 60.29539 g, not 61.29539 g."},"concepts_required":["electrolysis","Faraday's laws","redox stoichiometry"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_faraday_copper","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_004506","subject":"physics","topic":"rotational_mechanics","subtopic":"disc_angular_acceleration","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A uniform solid disc of mass 5 kg and radius 2 m is acted on by a constant torque of 88 N m about its central axis. Find its angular acceleration.","options":{"A":"7.8 rad s⁻²","B":"8.8 rad s⁻²","C":"5.8 rad s⁻²","D":"-8.8 rad s⁻²"},"correct_answer":"B","correct_answer_value":"8.8 rad s⁻²","solution":"For a solid disc, I=½MR²=½(5)(2)²=10 kg m². Using τ=Iα, α=88/10=8.8 rad s⁻².","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 8.8 rad s⁻², not 7.8 rad s⁻².","B":"Correct. The worked calculation gives 8.8 rad s⁻².","C":"Incorrect. Substituting the stated values into the governing relation gives 8.8 rad s⁻², not 5.8 rad s⁻².","D":"Incorrect. Substituting the stated values into the governing relation gives 8.8 rad s⁻², not -8.8 rad s⁻²."},"concepts_required":["moment of inertia","rotational dynamics","torque"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_disc_torque","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_000239","subject":"mathematics","topic":"algebra","subtopic":"quadratic_roots","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"For the quadratic equation 3x^2 +6x -10=0, let its roots be α and β. What is α²+β²?","options":{"A":"11.66667","B":"10.66667","C":"9.66667","D":"2.06667"},"correct_answer":"B","correct_answer_value":"10.66667","solution":"By Vieta's relations, α+β = -6/3 and αβ = -10/3. Then α²+β²=(α+β)²−2αβ = (-6/3)²−2(-10/3) = 10.66667.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 10.66667, not 11.66667.","B":"Correct. The worked calculation gives 10.66667.","C":"Incorrect. Substituting the stated values into the governing relation gives 10.66667, not 9.66667.","D":"Incorrect. Substituting the stated values into the governing relation gives 10.66667, not 2.06667."},"concepts_required":["Vieta's formulas","symmetric polynomial expressions"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_quadratic_vieta","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_005263","subject":"physics","topic":"electrostatics","subtopic":"coulombs_law","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Two point charges of +2 μC and +8 μC are separated by 3 cm in vacuum. Using k=9×10⁹ N m² C⁻², find the magnitude of the electrostatic force.","options":{"A":"161 N","B":"192 N","C":"80 N","D":"160 N"},"correct_answer":"D","correct_answer_value":"160 N","solution":"Convert charges to coulombs and distance to metres. F=k|q₁q₂|/r²=9×10⁹(2×10⁻⁶)(8×10⁻⁶)/(3×10⁻²)²=160 N.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 160 N, not 161 N.","B":"Incorrect. Substituting the stated values into the governing relation gives 160 N, not 192 N.","C":"Incorrect. Substituting the stated values into the governing relation gives 160 N, not 80 N.","D":"Correct. The worked calculation gives 160 N."},"concepts_required":["Coulomb's law","SI unit conversion","inverse-square law"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_coulomb","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_002125","subject":"mathematics","topic":"probability","subtopic":"same_colour_without_replacement","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A bag contains 4 red, 7 blue and 5 green balls. Two balls are drawn uniformly at random without replacement. What is the probability that both have the same colour?","options":{"A":"1.30833","B":"0.30833","C":"1.03083","D":"2.30833"},"correct_answer":"B","correct_answer_value":"0.30833","solution":"Favourable pairs = C(4,2)+C(7,2)+C(5,2)=37. Total pairs=C(16,2)=120. Therefore P=37/120=0.30833.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.30833, not 1.30833.","B":"Correct. The worked calculation gives 0.30833.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.30833, not 1.03083.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.30833, not 2.30833."},"concepts_required":["classical probability","combinations","without replacement"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_probability_colours","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_002787","subject":"mathematics","topic":"vectors","subtopic":"dot_product","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Given vectors u=(7, 6, 3) and v=(8, 6, 9), calculate u·v.","options":{"A":"118","B":"59.5","C":"119","D":"120"},"correct_answer":"C","correct_answer_value":"119","solution":"The dot product is the sum of corresponding component products: (7)(8)+(6)(6)+(3)(9)=119.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 119, not 118.","B":"Incorrect. Substituting the stated values into the governing relation gives 119, not 59.5.","C":"Correct. The worked calculation gives 119.","D":"Incorrect. Substituting the stated values into the governing relation gives 119, not 120."},"concepts_required":["three-dimensional vectors","scalar product"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_vector_dot","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_004428","subject":"physics","topic":"rotational_mechanics","subtopic":"disc_angular_acceleration","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A uniform solid disc of mass 7 kg and radius 1 m is acted on by a constant torque of 56 N m about its central axis. Find its angular acceleration.","options":{"A":"16 rad s⁻²","B":"15 rad s⁻²","C":"12 rad s⁻²","D":"24 rad s⁻²"},"correct_answer":"A","correct_answer_value":"16 rad s⁻²","solution":"For a solid disc, I=½MR²=½(7)(1)²=3.5 kg m². Using τ=Iα, α=56/3.5=16 rad s⁻².","option_explanations":{"A":"Correct. The worked calculation gives 16 rad s⁻².","B":"Incorrect. Substituting the stated values into the governing relation gives 16 rad s⁻², not 15 rad s⁻².","C":"Incorrect. Substituting the stated values into the governing relation gives 16 rad s⁻², not 12 rad s⁻².","D":"Incorrect. Substituting the stated values into the governing relation gives 16 rad s⁻², not 24 rad s⁻²."},"concepts_required":["moment of inertia","rotational dynamics","torque"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_disc_torque","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_001269","subject":"mathematics","topic":"calculus","subtopic":"polynomial_differentiation","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Let f(x)=3x³ +8x² +8x+7. Evaluate f′(3).","options":{"A":"205.5","B":"137","C":"164.4","D":"109.6"},"correct_answer":"B","correct_answer_value":"137","solution":"Differentiate: f′(x)=3(3)x²+2(8)x+(8). At x=3, f′(3)=3(3)(3)²+2(8)(3)+(8)=137.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 137, not 205.5.","B":"Correct. The worked calculation gives 137.","C":"Incorrect. Substituting the stated values into the governing relation gives 137, not 164.4.","D":"Incorrect. Substituting the stated values into the governing relation gives 137, not 109.6."},"concepts_required":["power rule","polynomial derivative"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_derivative_cubic","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_004560","subject":"physics","topic":"rotational_mechanics","subtopic":"disc_angular_acceleration","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A uniform solid disc of mass 15 kg and radius 3 m is acted on by a constant torque of 100 N m about its central axis. Find its angular acceleration.","options":{"A":"-1.48148 rad s⁻²","B":"2.48148 rad s⁻²","C":"1.48148 rad s⁻²","D":"-0.51852 rad s⁻²"},"correct_answer":"C","correct_answer_value":"1.48148 rad s⁻²","solution":"For a solid disc, I=½MR²=½(15)(3)²=67.5 kg m². Using τ=Iα, α=100/67.5=1.48148 rad s⁻².","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 1.48148 rad s⁻², not -1.48148 rad s⁻².","B":"Incorrect. Substituting the stated values into the governing relation gives 1.48148 rad s⁻², not 2.48148 rad s⁻².","C":"Correct. The worked calculation gives 1.48148 rad s⁻².","D":"Incorrect. Substituting the stated values into the governing relation gives 1.48148 rad s⁻², not -0.51852 rad s⁻²."},"concepts_required":["moment of inertia","rotational dynamics","torque"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_disc_torque","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_007424","subject":"chemistry","topic":"physical_chemistry","subtopic":"strong_acid_base_mixture","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"35 mL of 0.10 M HCl is mixed with 49 mL of 0.10 M NaOH at 25°C. Assuming complete dissociation and additive volumes, find the final pH.","options":{"A":"9.16639","B":"13.22185","C":"14.22185","D":"12.22185"},"correct_answer":"D","correct_answer_value":"12.22185","solution":"Excess OH⁻ moles=0.10×(49−35)/1000=0.001400 mol. Total volume=0.084 L, so [OH⁻]=0.016667 M, pOH=−log₁₀[OH⁻], and pH=14−pOH=12.22185.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 12.22185, not 9.16639.","B":"Incorrect. Substituting the stated values into the governing relation gives 12.22185, not 13.22185.","C":"Incorrect. Substituting the stated values into the governing relation gives 12.22185, not 14.22185.","D":"Correct. The worked calculation gives 12.22185."},"concepts_required":["strong acid–strong base neutralization","molarity","pH and pOH"],"reasoning_steps":5,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_acid_base_pH","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_009203","subject":"chemistry","topic":"physical_chemistry","subtopic":"nernst_equation_298K","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"At 298 K, a galvanic cell has E°cell=0.68 V, transfers 3 electrons, and has reaction quotient Q=10^5. Using E=E°−(0.0591/n)log₁₀Q, calculate Ecell.","options":{"A":"-0.4185 V","B":"-2.4185 V","C":"0.5815 V","D":"-1.4185 V"},"correct_answer":"C","correct_answer_value":"0.5815 V","solution":"Apply the Nernst equation: E=0.68−(0.0591/3)(5)=0.5815 V.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.5815 V, not -0.4185 V.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.5815 V, not -2.4185 V.","C":"Correct. The worked calculation gives 0.5815 V.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.5815 V, not -1.4185 V."},"concepts_required":["electrochemical cells","Nernst equation","logarithms"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_nernst","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_005619","subject":"physics","topic":"electrostatics","subtopic":"capacitor_energy","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A 39 μF capacitor is charged to a potential difference of 19 V. Calculate the energy stored, expressed in μJ.","options":{"A":"3519.75 μJ","B":"7039.5 μJ","C":"14079 μJ","D":"7037.5 μJ"},"correct_answer":"B","correct_answer_value":"7039.5 μJ","solution":"U=½CV². Since C=39 μF, U in μJ is ½×39×(19)²=7039.5 μJ.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 7039.5 μJ, not 3519.75 μJ.","B":"Correct. The worked calculation gives 7039.5 μJ.","C":"Incorrect. Substituting the stated values into the governing relation gives 7039.5 μJ, not 14079 μJ.","D":"Incorrect. Substituting the stated values into the governing relation gives 7039.5 μJ, not 7037.5 μJ."},"concepts_required":["capacitance","energy stored in a capacitor"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_capacitor_energy","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_000925","subject":"mathematics","topic":"sequences_and_series","subtopic":"geometric_progression","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A geometric progression has first term 6 and common ratio 4. What is the sum of its first 6 terms?","options":{"A":"8191","B":"8190","C":"8192","D":"8188"},"correct_answer":"B","correct_answer_value":"8190","solution":"For r≠1, Sₙ=a(rⁿ−1)/(r−1). Thus Sₙ=6(4^6−1)/(4−1)=8190.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 8190, not 8191.","B":"Correct. The worked calculation gives 8190.","C":"Incorrect. Substituting the stated values into the governing relation gives 8190, not 8192.","D":"Incorrect. Substituting the stated values into the governing relation gives 8190, not 8188."},"concepts_required":["geometric progression","finite geometric sum"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_gp_sum","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_003613","subject":"physics","topic":"mechanics","subtopic":"projectile_range","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A projectile is launched from level ground at 93 m s⁻¹ at 15° to the horizontal. Take g=10 m s⁻². Find its horizontal range.","options":{"A":"432.45 m","B":"-432.45 m","C":"44.245 m","D":"324.3375 m"},"correct_answer":"A","correct_answer_value":"432.45 m","solution":"Range R=u²sin(2θ)/g. Substituting u=93, θ=15° and g=10 gives R=(93)²sin(30°)/10=432.45 m.","option_explanations":{"A":"Correct. The worked calculation gives 432.45 m.","B":"Incorrect. Substituting the stated values into the governing relation gives 432.45 m, not -432.45 m.","C":"Incorrect. Substituting the stated values into the governing relation gives 432.45 m, not 44.245 m.","D":"Incorrect. Substituting the stated values into the governing relation gives 432.45 m, not 324.3375 m."},"concepts_required":["projectile motion","range equation","trigonometry"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_projectile_range","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_009804","subject":"chemistry","topic":"inorganic_chemistry","subtopic":"complex_ion_oxidation_state","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"For a hypothetical coordination ion [Pd(CN)_6]^+0, each CN ligand has charge −1. Determine the formal oxidation state of Pd.","options":{"A":"1.6","B":"3","C":"4.5","D":"6"},"correct_answer":"D","correct_answer_value":"6","solution":"Let the formal oxidation state of Pd be x. Charge balance gives x+(6)(−1)=+0; hence x=0+6=+6.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 6, not 1.6.","B":"Incorrect. Substituting the stated values into the governing relation gives 6, not 3.","C":"Incorrect. Substituting the stated values into the governing relation gives 6, not 4.5.","D":"Correct. The worked calculation gives 6."},"concepts_required":["coordination compounds","oxidation state","charge balance"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_complex_oxidation","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_008428","subject":"chemistry","topic":"physical_chemistry","subtopic":"boiling_point_elevation","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A non-volatile solute is dissolved in water to produce a solution of molality 0.93 mol kg⁻¹. If its van't Hoff factor is 4 and K_b for water is 0.6 K kg mol⁻¹, find the boiling-point elevation.","options":{"A":"2.232 K","B":"3.232 K","C":"-2.232 K","D":"1.232 K"},"correct_answer":"A","correct_answer_value":"2.232 K","solution":"ΔT_b=iK_bm. Substitution gives ΔT_b=4×0.6×0.93=2.232 K.","option_explanations":{"A":"Correct. The worked calculation gives 2.232 K.","B":"Incorrect. Substituting the stated values into the governing relation gives 2.232 K, not 3.232 K.","C":"Incorrect. Substituting the stated values into the governing relation gives 2.232 K, not -2.232 K.","D":"Incorrect. Substituting the stated values into the governing relation gives 2.232 K, not 1.232 K."},"concepts_required":["colligative properties","van't Hoff factor","molality"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_boiling_elevation","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_000440","subject":"mathematics","topic":"sequences_and_series","subtopic":"arithmetic_progression","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"An arithmetic progression has first term -19 and common difference 14. Find the sum of its first 32 terms.","options":{"A":"634.6","B":"6336","C":"9504","D":"6334"},"correct_answer":"B","correct_answer_value":"6336","solution":"Use Sₙ = n/2[2a+(n−1)d]. Substituting a=-19, d=14, n=32: Sₙ=32/2[2(-19)+(32−1)(14)]=6336.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 6336, not 634.6.","B":"Correct. The worked calculation gives 6336.","C":"Incorrect. Substituting the stated values into the governing relation gives 6336, not 9504.","D":"Incorrect. Substituting the stated values into the governing relation gives 6336, not 6334."},"concepts_required":["arithmetic progression","finite series"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_ap_sum","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_009696","subject":"chemistry","topic":"inorganic_chemistry","subtopic":"complex_ion_oxidation_state","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"For a hypothetical coordination ion [Fe(Cl)_3]^-1, each Cl ligand has charge −1. Determine the formal oxidation state of Fe.","options":{"A":"4","B":"2","C":"1","D":"3"},"correct_answer":"B","correct_answer_value":"2","solution":"Let the formal oxidation state of Fe be x. Charge balance gives x+(3)(−1)=-1; hence x=-1+3=+2.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 2, not 4.","B":"Correct. The worked calculation gives 2.","C":"Incorrect. Substituting the stated values into the governing relation gives 2, not 1.","D":"Incorrect. Substituting the stated values into the governing relation gives 2, not 3."},"concepts_required":["coordination compounds","oxidation state","charge balance"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_complex_oxidation","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_005908","subject":"physics","topic":"thermodynamics","subtopic":"ideal_gas_isochoric_pressure","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A fixed amount of ideal gas is kept in a rigid container. Its pressure is 0.1 atm at 381 K. What is its pressure at 624 K?","options":{"A":"2.16378 atm","B":"0.32756 atm","C":"0.16378 atm","D":"-0.16378 atm"},"correct_answer":"C","correct_answer_value":"0.16378 atm","solution":"At constant volume, P/T is constant. P₂=P₁T₂/T₁=(0.1)(624)/(381)=0.16378 atm.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.16378 atm, not 2.16378 atm.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.16378 atm, not 0.32756 atm.","C":"Correct. The worked calculation gives 0.16378 atm.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.16378 atm, not -0.16378 atm."},"concepts_required":["ideal gas law","isochoric process","absolute temperature"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_isochoric_gas","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_007726","subject":"chemistry","topic":"physical_chemistry","subtopic":"first_order_half_life","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A first-order reaction has a half-life of 20 minutes. If the initial reactant concentration is 83 mmol L⁻¹, what concentration remains after 20 minutes?","options":{"A":"31.125 mmol L⁻¹","B":"39.5 mmol L⁻¹","C":"42.5 mmol L⁻¹","D":"41.5 mmol L⁻¹"},"correct_answer":"D","correct_answer_value":"41.5 mmol L⁻¹","solution":"20 minutes corresponds to 1 half-lives. The remaining concentration is [A]₀(1/2)ⁿ=83/2^1=41.5 mmol L⁻¹.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 41.5 mmol L⁻¹, not 31.125 mmol L⁻¹.","B":"Incorrect. Substituting the stated values into the governing relation gives 41.5 mmol L⁻¹, not 39.5 mmol L⁻¹.","C":"Incorrect. Substituting the stated values into the governing relation gives 41.5 mmol L⁻¹, not 42.5 mmol L⁻¹.","D":"Correct. The worked calculation gives 41.5 mmol L⁻¹."},"concepts_required":["first-order kinetics","half-life","exponential decay"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_first_order_halflife","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_002178","subject":"mathematics","topic":"probability","subtopic":"same_colour_without_replacement","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A bag contains 12 red, 7 blue and 8 green balls. Two balls are drawn uniformly at random without replacement. What is the probability that both have the same colour?","options":{"A":"3.32764","B":"-0.32764","C":"0.32764","D":"0.16382"},"correct_answer":"C","correct_answer_value":"0.32764","solution":"Favourable pairs = C(12,2)+C(7,2)+C(8,2)=115. Total pairs=C(27,2)=351. Therefore P=115/351=0.32764.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.32764, not 3.32764.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.32764, not -0.32764.","C":"Correct. The worked calculation gives 0.32764.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.32764, not 0.16382."},"concepts_required":["classical probability","combinations","without replacement"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_probability_colours","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_005941","subject":"physics","topic":"thermodynamics","subtopic":"ideal_gas_isochoric_pressure","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A fixed amount of ideal gas is kept in a rigid container. Its pressure is 1.5 atm at 202 K. What is its pressure at 331 K?","options":{"A":"4.45792 atm","B":"0.45792 atm","C":"-2.45792 atm","D":"2.45792 atm"},"correct_answer":"D","correct_answer_value":"2.45792 atm","solution":"At constant volume, P/T is constant. P₂=P₁T₂/T₁=(1.5)(331)/(202)=2.45792 atm.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 2.45792 atm, not 4.45792 atm.","B":"Incorrect. Substituting the stated values into the governing relation gives 2.45792 atm, not 0.45792 atm.","C":"Incorrect. Substituting the stated values into the governing relation gives 2.45792 atm, not -2.45792 atm.","D":"Correct. The worked calculation gives 2.45792 atm."},"concepts_required":["ideal gas law","isochoric process","absolute temperature"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_isochoric_gas","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_005732","subject":"physics","topic":"thermodynamics","subtopic":"ideal_gas_isochoric_pressure","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A fixed amount of ideal gas is kept in a rigid container. Its pressure is 1.4 atm at 393 K. What is its pressure at 459 K?","options":{"A":"3.63511 atm","B":"-1.36489 atm","C":"-0.36489 atm","D":"1.63511 atm"},"correct_answer":"D","correct_answer_value":"1.63511 atm","solution":"At constant volume, P/T is constant. P₂=P₁T₂/T₁=(1.4)(459)/(393)=1.63511 atm.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 1.63511 atm, not 3.63511 atm.","B":"Incorrect. Substituting the stated values into the governing relation gives 1.63511 atm, not -1.36489 atm.","C":"Incorrect. Substituting the stated values into the governing relation gives 1.63511 atm, not -0.36489 atm.","D":"Correct. The worked calculation gives 1.63511 atm."},"concepts_required":["ideal gas law","isochoric process","absolute temperature"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_isochoric_gas","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_009068","subject":"chemistry","topic":"physical_chemistry","subtopic":"nernst_equation_298K","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"At 298 K, a galvanic cell has E°cell=0.72 V, transfers 1 electrons, and has reaction quotient Q=10^-5. Using E=E°−(0.0591/n)log₁₀Q, calculate Ecell.","options":{"A":"-1.0155 V","B":"1.0155 V","C":"0.76162 V","D":"2.0155 V"},"correct_answer":"B","correct_answer_value":"1.0155 V","solution":"Apply the Nernst equation: E=0.72−(0.0591/1)(-5)=1.0155 V.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 1.0155 V, not -1.0155 V.","B":"Correct. The worked calculation gives 1.0155 V.","C":"Incorrect. Substituting the stated values into the governing relation gives 1.0155 V, not 0.76162 V.","D":"Incorrect. Substituting the stated values into the governing relation gives 1.0155 V, not 2.0155 V."},"concepts_required":["electrochemical cells","Nernst equation","logarithms"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_nernst","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_003736","subject":"physics","topic":"mechanics","subtopic":"work_energy_theorem","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A 12 kg block initially moves at 9 m s⁻¹. A constant net force does 14 N of work over 3 m in the direction of motion. Find its final kinetic energy.","options":{"A":"-528 J","B":"526 J","C":"264 J","D":"528 J"},"correct_answer":"D","correct_answer_value":"528 J","solution":"By the work–energy theorem, K_f=K_i+W=½mu²+Fs=½(12)(9)²+(14)(3)=528 J.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 528 J, not -528 J.","B":"Incorrect. Substituting the stated values into the governing relation gives 528 J, not 526 J.","C":"Incorrect. Substituting the stated values into the governing relation gives 528 J, not 264 J.","D":"Correct. The worked calculation gives 528 J."},"concepts_required":["work–energy theorem","kinetic energy","work by constant force"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_work_energy","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_000127","subject":"mathematics","topic":"algebra","subtopic":"quadratic_roots","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"For the quadratic equation 4x^2 +0x -2=0, let its roots be α and β. What is α²+β²?","options":{"A":"0.5","B":"0.75","C":"1","D":"-1"},"correct_answer":"C","correct_answer_value":"1","solution":"By Vieta's relations, α+β = 0/4 and αβ = -2/4. Then α²+β²=(α+β)²−2αβ = (0/4)²−2(-2/4) = 1.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 1, not 0.5.","B":"Incorrect. Substituting the stated values into the governing relation gives 1, not 0.75.","C":"Correct. The worked calculation gives 1.","D":"Incorrect. Substituting the stated values into the governing relation gives 1, not -1."},"concepts_required":["Vieta's formulas","symmetric polynomial expressions"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_quadratic_vieta","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_002478","subject":"mathematics","topic":"linear_algebra","subtopic":"2x2_determinant","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Find the determinant of the matrix [[1, 8], [5, 4]].","options":{"A":"-36","B":"-28.8","C":"-34","D":"36"},"correct_answer":"A","correct_answer_value":"-36","solution":"For a 2×2 matrix [[a,b],[c,d]], det=ad−bc. Here it is (1)(4)−(8)(5)=-36.","option_explanations":{"A":"Correct. The worked calculation gives -36.","B":"Incorrect. Substituting the stated values into the governing relation gives -36, not -28.8.","C":"Incorrect. Substituting the stated values into the governing relation gives -36, not -34.","D":"Incorrect. Substituting the stated values into the governing relation gives -36, not 36."},"concepts_required":["determinants","matrix operations"],"reasoning_steps":2,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_determinant_2x2","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_002839","subject":"mathematics","topic":"vectors","subtopic":"dot_product","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Given vectors u=(-6, -3, 7) and v=(0, -2, -3), calculate u·v.","options":{"A":"-0.5","B":"-30","C":"15","D":"-15"},"correct_answer":"D","correct_answer_value":"-15","solution":"The dot product is the sum of corresponding component products: (-6)(0)+(-3)(-2)+(7)(-3)=-15.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives -15, not -0.5.","B":"Incorrect. Substituting the stated values into the governing relation gives -15, not -30.","C":"Incorrect. Substituting the stated values into the governing relation gives -15, not 15.","D":"Correct. The worked calculation gives -15."},"concepts_required":["three-dimensional vectors","scalar product"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_vector_dot","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_000202","subject":"mathematics","topic":"algebra","subtopic":"quadratic_roots","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"For the quadratic equation 2x^2 +18x +9=0, let its roots be α and β. What is α²+β²?","options":{"A":"72","B":"54","C":"74","D":"86.4"},"correct_answer":"A","correct_answer_value":"72","solution":"By Vieta's relations, α+β = -18/2 and αβ = 9/2. Then α²+β²=(α+β)²−2αβ = (-18/2)²−2(9/2) = 72.","option_explanations":{"A":"Correct. The worked calculation gives 72.","B":"Incorrect. Substituting the stated values into the governing relation gives 72, not 54.","C":"Incorrect. Substituting the stated values into the governing relation gives 72, not 74.","D":"Incorrect. Substituting the stated values into the governing relation gives 72, not 86.4."},"concepts_required":["Vieta's formulas","symmetric polynomial expressions"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_quadratic_vieta","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_003503","subject":"physics","topic":"mechanics","subtopic":"projectile_range","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A projectile is launched from level ground at 40 m s⁻¹ at 45° to the horizontal. Take g=10 m s⁻². Find its horizontal range.","options":{"A":"159 m","B":"162 m","C":"128 m","D":"160 m"},"correct_answer":"D","correct_answer_value":"160 m","solution":"Range R=u²sin(2θ)/g. Substituting u=40, θ=45° and g=10 gives R=(40)²sin(90°)/10=160 m.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 160 m, not 159 m.","B":"Incorrect. Substituting the stated values into the governing relation gives 160 m, not 162 m.","C":"Incorrect. Substituting the stated values into the governing relation gives 160 m, not 128 m.","D":"Correct. The worked calculation gives 160 m."},"concepts_required":["projectile motion","range equation","trigonometry"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_projectile_range","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_006749","subject":"chemistry","topic":"stoichiometry","subtopic":"limiting_reagent_water","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Hydrogen reacts with oxygen according to 2H₂ + O₂ → 2H₂O. If 23 mol H₂ and 9 mol O₂ are mixed and the reaction goes to completion, how many moles of H₂O form?","options":{"A":"19 mol","B":"36 mol","C":"17 mol","D":"18 mol"},"correct_answer":"D","correct_answer_value":"18 mol","solution":"From 2 mol H₂ and 1 mol O₂, 2 mol H₂O form. H₂ can produce 23 mol H₂O; O₂ can produce 18 mol H₂O. The limiting condition is O₂; product=18 mol.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 18 mol, not 19 mol.","B":"Incorrect. Substituting the stated values into the governing relation gives 18 mol, not 36 mol.","C":"Incorrect. Substituting the stated values into the governing relation gives 18 mol, not 17 mol.","D":"Correct. The worked calculation gives 18 mol."},"concepts_required":["balanced chemical equations","limiting reagent","mole ratios"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_limiting_reagent","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_001048","subject":"mathematics","topic":"calculus","subtopic":"polynomial_differentiation","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Let f(x)=5x³ +10x² -5x+7. Evaluate f′(0).","options":{"A":"-6","B":"5","C":"-3","D":"-5"},"correct_answer":"D","correct_answer_value":"-5","solution":"Differentiate: f′(x)=3(5)x²+2(10)x+(-5). At x=0, f′(0)=3(5)(0)²+2(10)(0)+(-5)=-5.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives -5, not -6.","B":"Incorrect. Substituting the stated values into the governing relation gives -5, not 5.","C":"Incorrect. Substituting the stated values into the governing relation gives -5, not -3.","D":"Correct. The worked calculation gives -5."},"concepts_required":["power rule","polynomial derivative"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_derivative_cubic","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_007573","subject":"chemistry","topic":"physical_chemistry","subtopic":"strong_acid_base_mixture","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"42 mL of 0.10 M HCl is mixed with 34 mL of 0.10 M NaOH at 25°C. Assuming complete dissociation and additive volumes, find the final pH.","options":{"A":"2.97772","B":"0.97772","C":"-1.97772","D":"1.97772"},"correct_answer":"D","correct_answer_value":"1.97772","solution":"Excess H⁺ moles=0.10×(42−34)/1000=0.000800 mol. Total volume=0.076 L, so [H⁺]=0.010526 M and pH=−log₁₀[H⁺]=1.97772.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 1.97772, not 2.97772.","B":"Incorrect. Substituting the stated values into the governing relation gives 1.97772, not 0.97772.","C":"Incorrect. Substituting the stated values into the governing relation gives 1.97772, not -1.97772.","D":"Correct. The worked calculation gives 1.97772."},"concepts_required":["strong acid–strong base neutralization","molarity","pH and pOH"],"reasoning_steps":5,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_acid_base_pH","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_007006","subject":"chemistry","topic":"physical_chemistry","subtopic":"ideal_gas_volume","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Assuming ideal behaviour, what volume is occupied by 20 mol of gas at 249 K and 0.93 atm? Use R=0.082057 L atm mol⁻¹ K⁻¹.","options":{"A":"329.5515 L","B":"659.103 L","C":"439.402 L","D":"-439.402 L"},"correct_answer":"C","correct_answer_value":"439.402 L","solution":"PV=nRT, so V=nRT/P=(20)(0.082057)(249)/(0.93)=439.402 L.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 439.402 L, not 329.5515 L.","B":"Incorrect. Substituting the stated values into the governing relation gives 439.402 L, not 659.103 L.","C":"Correct. The worked calculation gives 439.402 L.","D":"Incorrect. Substituting the stated values into the governing relation gives 439.402 L, not -439.402 L."},"concepts_required":["ideal gas equation","unit consistency"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_ideal_gas","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_005300","subject":"physics","topic":"electrostatics","subtopic":"coulombs_law","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Two point charges of +2 μC and +5 μC are separated by 11 cm in vacuum. Using k=9×10⁹ N m² C⁻², find the magnitude of the electrostatic force.","options":{"A":"7.43802 N","B":"-7.43802 N","C":"14.87603 N","D":"8.43802 N"},"correct_answer":"A","correct_answer_value":"7.43802 N","solution":"Convert charges to coulombs and distance to metres. F=k|q₁q₂|/r²=9×10⁹(2×10⁻⁶)(5×10⁻⁶)/(11×10⁻²)²=7.43802 N.","option_explanations":{"A":"Correct. The worked calculation gives 7.43802 N.","B":"Incorrect. Substituting the stated values into the governing relation gives 7.43802 N, not -7.43802 N.","C":"Incorrect. Substituting the stated values into the governing relation gives 7.43802 N, not 14.87603 N.","D":"Incorrect. Substituting the stated values into the governing relation gives 7.43802 N, not 8.43802 N."},"concepts_required":["Coulomb's law","SI unit conversion","inverse-square law"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_coulomb","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_007331","subject":"chemistry","topic":"physical_chemistry","subtopic":"ideal_gas_volume","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Assuming ideal behaviour, what volume is occupied by 14 mol of gas at 397 K and 2.08 atm? Use R=0.082057 L atm mol⁻¹ K⁻¹.","options":{"A":"219.26577 L","B":"217.26577 L","C":"109.63289 L","D":"175.41262 L"},"correct_answer":"A","correct_answer_value":"219.26577 L","solution":"PV=nRT, so V=nRT/P=(14)(0.082057)(397)/(2.08)=219.26577 L.","option_explanations":{"A":"Correct. The worked calculation gives 219.26577 L.","B":"Incorrect. Substituting the stated values into the governing relation gives 219.26577 L, not 217.26577 L.","C":"Incorrect. Substituting the stated values into the governing relation gives 219.26577 L, not 109.63289 L.","D":"Incorrect. Substituting the stated values into the governing relation gives 219.26577 L, not 175.41262 L."},"concepts_required":["ideal gas equation","unit consistency"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_ideal_gas","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_007606","subject":"chemistry","topic":"physical_chemistry","subtopic":"strong_acid_base_mixture","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"59 mL of 0.10 M HCl is mixed with 52 mL of 0.10 M NaOH at 25°C. Assuming complete dissociation and additive volumes, find the final pH.","options":{"A":"3.20022","B":"1.22002","C":"2.20022","D":"4.20022"},"correct_answer":"C","correct_answer_value":"2.20022","solution":"Excess H⁺ moles=0.10×(59−52)/1000=0.000700 mol. Total volume=0.111 L, so [H⁺]=0.006306 M and pH=−log₁₀[H⁺]=2.20022.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 2.20022, not 3.20022.","B":"Incorrect. Substituting the stated values into the governing relation gives 2.20022, not 1.22002.","C":"Correct. The worked calculation gives 2.20022.","D":"Incorrect. Substituting the stated values into the governing relation gives 2.20022, not 4.20022."},"concepts_required":["strong acid–strong base neutralization","molarity","pH and pOH"],"reasoning_steps":5,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_acid_base_pH","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_004715","subject":"physics","topic":"electricity","subtopic":"series_parallel_circuit","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Resistors 13 Ω and 12 Ω are connected in series; this combination is connected in parallel with a 9 Ω resistor across a 30 V ideal battery. Find the total current supplied by the battery.","options":{"A":"6.53333 A","B":"9.06667 A","C":"7.53333 A","D":"4.53333 A"},"correct_answer":"D","correct_answer_value":"4.53333 A","solution":"Series branch resistance=13+12=25 Ω. Parallel equivalent R_eq=(25×9)/(25+9)=6.61765 Ω. Total current I=V/R_eq=30/6.61765=4.53333 A.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 4.53333 A, not 6.53333 A.","B":"Incorrect. Substituting the stated values into the governing relation gives 4.53333 A, not 9.06667 A.","C":"Incorrect. Substituting the stated values into the governing relation gives 4.53333 A, not 7.53333 A.","D":"Correct. The worked calculation gives 4.53333 A."},"concepts_required":["series resistors","parallel resistors","Ohm's law"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_series_parallel","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_005199","subject":"physics","topic":"electrostatics","subtopic":"coulombs_law","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Two point charges of +3 μC and +1 μC are separated by 3 cm in vacuum. Using k=9×10⁹ N m² C⁻², find the magnitude of the electrostatic force.","options":{"A":"22.5 N","B":"36 N","C":"30 N","D":"29 N"},"correct_answer":"C","correct_answer_value":"30 N","solution":"Convert charges to coulombs and distance to metres. F=k|q₁q₂|/r²=9×10⁹(3×10⁻⁶)(1×10⁻⁶)/(3×10⁻²)²=30 N.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 30 N, not 22.5 N.","B":"Incorrect. Substituting the stated values into the governing relation gives 30 N, not 36 N.","C":"Correct. The worked calculation gives 30 N.","D":"Incorrect. Substituting the stated values into the governing relation gives 30 N, not 29 N."},"concepts_required":["Coulomb's law","SI unit conversion","inverse-square law"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_coulomb","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_002239","subject":"mathematics","topic":"probability","subtopic":"same_colour_without_replacement","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A bag contains 8 red, 7 blue and 6 green balls. Two balls are drawn uniformly at random without replacement. What is the probability that both have the same colour?","options":{"A":"0.30476","B":"1.03048","C":"-2.69524","D":"1.30476"},"correct_answer":"A","correct_answer_value":"0.30476","solution":"Favourable pairs = C(8,2)+C(7,2)+C(6,2)=64. Total pairs=C(21,2)=210. Therefore P=64/210=0.30476.","option_explanations":{"A":"Correct. The worked calculation gives 0.30476.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.30476, not 1.03048.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.30476, not -2.69524.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.30476, not 1.30476."},"concepts_required":["classical probability","combinations","without replacement"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_probability_colours","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_001942","subject":"mathematics","topic":"combinatorics","subtopic":"binomial_coefficient","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"How many distinct subsets containing exactly 3 elements can be selected from a set of 57 distinct elements?","options":{"A":"21945","B":"2927","C":"58520","D":"29260"},"correct_answer":"D","correct_answer_value":"29260","solution":"Order does not matter, so the count is C(57,3)=57!/[3!(54)!]=29260.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 29260, not 21945.","B":"Incorrect. Substituting the stated values into the governing relation gives 29260, not 2927.","C":"Incorrect. Substituting the stated values into the governing relation gives 29260, not 58520.","D":"Correct. The worked calculation gives 29260."},"concepts_required":["combinations","factorials"],"reasoning_steps":2,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_combinations","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_000071","subject":"mathematics","topic":"algebra","subtopic":"quadratic_roots","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"For the quadratic equation 1x^2 -5x -3=0, let its roots be α and β. What is α²+β²?","options":{"A":"23.25","B":"24.8","C":"31","D":"-31"},"correct_answer":"C","correct_answer_value":"31","solution":"By Vieta's relations, α+β = 5/1 and αβ = -3/1. Then α²+β²=(α+β)²−2αβ = (5/1)²−2(-3/1) = 31.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 31, not 23.25.","B":"Incorrect. Substituting the stated values into the governing relation gives 31, not 24.8.","C":"Correct. The worked calculation gives 31.","D":"Incorrect. Substituting the stated values into the governing relation gives 31, not -31."},"concepts_required":["Vieta's formulas","symmetric polynomial expressions"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_quadratic_vieta","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_001136","subject":"mathematics","topic":"calculus","subtopic":"polynomial_differentiation","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Let f(x)=7x³ -1x² +9x+7. Evaluate f′(-1).","options":{"A":"30","B":"48","C":"25.6","D":"32"},"correct_answer":"D","correct_answer_value":"32","solution":"Differentiate: f′(x)=3(7)x²+2(-1)x+(9). At x=-1, f′(-1)=3(7)(-1)²+2(-1)(-1)+(9)=32.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 32, not 30.","B":"Incorrect. Substituting the stated values into the governing relation gives 32, not 48.","C":"Incorrect. Substituting the stated values into the governing relation gives 32, not 25.6.","D":"Correct. The worked calculation gives 32."},"concepts_required":["power rule","polynomial derivative"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_derivative_cubic","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_002930","subject":"mathematics","topic":"vectors","subtopic":"dot_product","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Given vectors u=(8, 6, 6) and v=(9, 5, -4), calculate u·v.","options":{"A":"117","B":"78","C":"77","D":"93.6"},"correct_answer":"B","correct_answer_value":"78","solution":"The dot product is the sum of corresponding component products: (8)(9)+(6)(5)+(6)(-4)=78.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 78, not 117.","B":"Correct. The worked calculation gives 78.","C":"Incorrect. Substituting the stated values into the governing relation gives 78, not 77.","D":"Incorrect. Substituting the stated values into the governing relation gives 78, not 93.6."},"concepts_required":["three-dimensional vectors","scalar product"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_vector_dot","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_001860","subject":"mathematics","topic":"combinatorics","subtopic":"binomial_coefficient","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"How many distinct subsets containing exactly 5 elements can be selected from a set of 13 distinct elements?","options":{"A":"965.25","B":"1287","C":"1544.4","D":"2574"},"correct_answer":"B","correct_answer_value":"1287","solution":"Order does not matter, so the count is C(13,5)=13!/[5!(8)!]=1287.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 1287, not 965.25.","B":"Correct. The worked calculation gives 1287.","C":"Incorrect. Substituting the stated values into the governing relation gives 1287, not 1544.4.","D":"Incorrect. Substituting the stated values into the governing relation gives 1287, not 2574."},"concepts_required":["combinations","factorials"],"reasoning_steps":2,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_combinations","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_007371","subject":"chemistry","topic":"physical_chemistry","subtopic":"strong_acid_base_mixture","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"61 mL of 0.10 M HCl is mixed with 48 mL of 0.10 M NaOH at 25°C. Assuming complete dissociation and additive volumes, find the final pH.","options":{"A":"1.92348","B":"3.84697","C":"2.88522","D":"0.92348"},"correct_answer":"A","correct_answer_value":"1.92348","solution":"Excess H⁺ moles=0.10×(61−48)/1000=0.001300 mol. Total volume=0.109 L, so [H⁺]=0.011927 M and pH=−log₁₀[H⁺]=1.92348.","option_explanations":{"A":"Correct. The worked calculation gives 1.92348.","B":"Incorrect. Substituting the stated values into the governing relation gives 1.92348, not 3.84697.","C":"Incorrect. Substituting the stated values into the governing relation gives 1.92348, not 2.88522.","D":"Incorrect. Substituting the stated values into the governing relation gives 1.92348, not 0.92348."},"concepts_required":["strong acid–strong base neutralization","molarity","pH and pOH"],"reasoning_steps":5,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_acid_base_pH","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_009315","subject":"chemistry","topic":"physical_chemistry","subtopic":"nernst_equation_298K","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"At 298 K, a galvanic cell has E°cell=0.65 V, transfers 1 electrons, and has reaction quotient Q=10^-6. Using E=E°−(0.0591/n)log₁₀Q, calculate Ecell.","options":{"A":"1.10046 V","B":"1.0046 V","C":"0.75345 V","D":"2.0092 V"},"correct_answer":"B","correct_answer_value":"1.0046 V","solution":"Apply the Nernst equation: E=0.65−(0.0591/1)(-6)=1.0046 V.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 1.0046 V, not 1.10046 V.","B":"Correct. The worked calculation gives 1.0046 V.","C":"Incorrect. Substituting the stated values into the governing relation gives 1.0046 V, not 0.75345 V.","D":"Incorrect. Substituting the stated values into the governing relation gives 1.0046 V, not 2.0092 V."},"concepts_required":["electrochemical cells","Nernst equation","logarithms"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_nernst","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_008301","subject":"chemistry","topic":"physical_chemistry","subtopic":"faradays_law_copper","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"An aqueous Cu²⁺ solution is electrolysed using a current of 7 A for 14815 s. Assuming 100% current efficiency, calculate the mass of copper deposited. Use M(Cu)=63.5 g mol⁻¹ and F=96500 C mol⁻¹.","options":{"A":"34.12056 g","B":"40.94467 g","C":"35.12056 g","D":"51.18084 g"},"correct_answer":"A","correct_answer_value":"34.12056 g","solution":"Cu²⁺+2e⁻→Cu, so z=2. By Faraday's law m=MIt/(zF)=63.5×7×14815/(2×96500)=34.12056 g.","option_explanations":{"A":"Correct. The worked calculation gives 34.12056 g.","B":"Incorrect. Substituting the stated values into the governing relation gives 34.12056 g, not 40.94467 g.","C":"Incorrect. Substituting the stated values into the governing relation gives 34.12056 g, not 35.12056 g.","D":"Incorrect. Substituting the stated values into the governing relation gives 34.12056 g, not 51.18084 g."},"concepts_required":["electrolysis","Faraday's laws","redox stoichiometry"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_faraday_copper","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_002770","subject":"mathematics","topic":"vectors","subtopic":"dot_product","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Given vectors u=(8, 9, -3) and v=(8, 7, 3), calculate u·v.","options":{"A":"118","B":"116","C":"141.6","D":"119"},"correct_answer":"A","correct_answer_value":"118","solution":"The dot product is the sum of corresponding component products: (8)(8)+(9)(7)+(-3)(3)=118.","option_explanations":{"A":"Correct. The worked calculation gives 118.","B":"Incorrect. Substituting the stated values into the governing relation gives 118, not 116.","C":"Incorrect. Substituting the stated values into the governing relation gives 118, not 141.6.","D":"Incorrect. Substituting the stated values into the governing relation gives 118, not 119."},"concepts_required":["three-dimensional vectors","scalar product"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_vector_dot","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_006668","subject":"physics","topic":"oscillations","subtopic":"mass_spring_period_over_pi","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A mass of 22 kg is attached to an ideal spring of force constant 407 N m⁻¹ and oscillates horizontally without damping. What is the time period divided by π?","options":{"A":"-0.53501 s","B":"0.92998 s","C":"0.46499 s","D":"3.46499 s"},"correct_answer":"C","correct_answer_value":"0.46499 s","solution":"For a mass–spring oscillator, T=2π√(m/k). Therefore T/π=2√(22/407)=0.46499 s.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.46499 s, not -0.53501 s.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.46499 s, not 0.92998 s.","C":"Correct. The worked calculation gives 0.46499 s.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.46499 s, not 3.46499 s."},"concepts_required":["simple harmonic motion","mass–spring oscillator","time period"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_shm_period","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_007636","subject":"chemistry","topic":"physical_chemistry","subtopic":"strong_acid_base_mixture","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"10 mL of 0.10 M HCl is mixed with 41 mL of 0.10 M NaOH at 25°C. Assuming complete dissociation and additive volumes, find the final pH.","options":{"A":"12.78379","B":"19.17569","C":"-12.78379","D":"13.78379"},"correct_answer":"A","correct_answer_value":"12.78379","solution":"Excess OH⁻ moles=0.10×(41−10)/1000=0.003100 mol. Total volume=0.051 L, so [OH⁻]=0.060784 M, pOH=−log₁₀[OH⁻], and pH=14−pOH=12.78379.","option_explanations":{"A":"Correct. The worked calculation gives 12.78379.","B":"Incorrect. Substituting the stated values into the governing relation gives 12.78379, not 19.17569.","C":"Incorrect. Substituting the stated values into the governing relation gives 12.78379, not -12.78379.","D":"Incorrect. Substituting the stated values into the governing relation gives 12.78379, not 13.78379."},"concepts_required":["strong acid–strong base neutralization","molarity","pH and pOH"],"reasoning_steps":5,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_acid_base_pH","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_005196","subject":"physics","topic":"electrostatics","subtopic":"coulombs_law","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Two point charges of +6 μC and +6 μC are separated by 12 cm in vacuum. Using k=9×10⁹ N m² C⁻², find the magnitude of the electrostatic force.","options":{"A":"22.5 N","B":"16.875 N","C":"33.75 N","D":"3.25 N"},"correct_answer":"A","correct_answer_value":"22.5 N","solution":"Convert charges to coulombs and distance to metres. F=k|q₁q₂|/r²=9×10⁹(6×10⁻⁶)(6×10⁻⁶)/(12×10⁻²)²=22.5 N.","option_explanations":{"A":"Correct. The worked calculation gives 22.5 N.","B":"Incorrect. Substituting the stated values into the governing relation gives 22.5 N, not 16.875 N.","C":"Incorrect. Substituting the stated values into the governing relation gives 22.5 N, not 33.75 N.","D":"Incorrect. Substituting the stated values into the governing relation gives 22.5 N, not 3.25 N."},"concepts_required":["Coulomb's law","SI unit conversion","inverse-square law"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_coulomb","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_007874","subject":"chemistry","topic":"physical_chemistry","subtopic":"first_order_half_life","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A first-order reaction has a half-life of 20 minutes. If the initial reactant concentration is 69 mmol L⁻¹, what concentration remains after 40 minutes?","options":{"A":"2.725 mmol L⁻¹","B":"19.25 mmol L⁻¹","C":"17.25 mmol L⁻¹","D":"8.625 mmol L⁻¹"},"correct_answer":"C","correct_answer_value":"17.25 mmol L⁻¹","solution":"40 minutes corresponds to 2 half-lives. The remaining concentration is [A]₀(1/2)ⁿ=69/2^2=17.25 mmol L⁻¹.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 17.25 mmol L⁻¹, not 2.725 mmol L⁻¹.","B":"Incorrect. Substituting the stated values into the governing relation gives 17.25 mmol L⁻¹, not 19.25 mmol L⁻¹.","C":"Correct. The worked calculation gives 17.25 mmol L⁻¹.","D":"Incorrect. Substituting the stated values into the governing relation gives 17.25 mmol L⁻¹, not 8.625 mmol L⁻¹."},"concepts_required":["first-order kinetics","half-life","exponential decay"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_first_order_halflife","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_009178","subject":"chemistry","topic":"physical_chemistry","subtopic":"nernst_equation_298K","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"At 298 K, a galvanic cell has E°cell=0.95 V, transfers 2 electrons, and has reaction quotient Q=10^-4. Using E=E°−(0.0591/n)log₁₀Q, calculate Ecell.","options":{"A":"-1.0682 V","B":"1.0682 V","C":"3.0682 V","D":"0.80115 V"},"correct_answer":"B","correct_answer_value":"1.0682 V","solution":"Apply the Nernst equation: E=0.95−(0.0591/2)(-4)=1.0682 V.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 1.0682 V, not -1.0682 V.","B":"Correct. The worked calculation gives 1.0682 V.","C":"Incorrect. Substituting the stated values into the governing relation gives 1.0682 V, not 3.0682 V.","D":"Incorrect. Substituting the stated values into the governing relation gives 1.0682 V, not 0.80115 V."},"concepts_required":["electrochemical cells","Nernst equation","logarithms"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_nernst","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_001212","subject":"mathematics","topic":"calculus","subtopic":"polynomial_differentiation","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Let f(x)=-3x³ +7x² -3x+7. Evaluate f′(3).","options":{"A":"-44","B":"-33.6","C":"-21","D":"-42"},"correct_answer":"D","correct_answer_value":"-42","solution":"Differentiate: f′(x)=3(-3)x²+2(7)x+(-3). At x=3, f′(3)=3(-3)(3)²+2(7)(3)+(-3)=-42.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives -42, not -44.","B":"Incorrect. Substituting the stated values into the governing relation gives -42, not -33.6.","C":"Incorrect. Substituting the stated values into the governing relation gives -42, not -21.","D":"Correct. The worked calculation gives -42."},"concepts_required":["power rule","polynomial derivative"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_derivative_cubic","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_009334","subject":"chemistry","topic":"physical_chemistry","subtopic":"nernst_equation_298K","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"At 298 K, a galvanic cell has E°cell=1.22 V, transfers 2 electrons, and has reaction quotient Q=10^1. Using E=E°−(0.0591/n)log₁₀Q, calculate Ecell.","options":{"A":"1.78567 V","B":"1.19045 V","C":"-1.19045 V","D":"0.59523 V"},"correct_answer":"B","correct_answer_value":"1.19045 V","solution":"Apply the Nernst equation: E=1.22−(0.0591/2)(1)=1.19045 V.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 1.19045 V, not 1.78567 V.","B":"Correct. The worked calculation gives 1.19045 V.","C":"Incorrect. Substituting the stated values into the governing relation gives 1.19045 V, not -1.19045 V.","D":"Incorrect. Substituting the stated values into the governing relation gives 1.19045 V, not 0.59523 V."},"concepts_required":["electrochemical cells","Nernst equation","logarithms"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_nernst","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_007516","subject":"chemistry","topic":"physical_chemistry","subtopic":"strong_acid_base_mixture","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"25 mL of 0.10 M HCl is mixed with 68 mL of 0.10 M NaOH at 25°C. Assuming complete dissociation and additive volumes, find the final pH.","options":{"A":"18.99748","B":"9.66499","C":"6.33249","D":"12.66499"},"correct_answer":"D","correct_answer_value":"12.66499","solution":"Excess OH⁻ moles=0.10×(68−25)/1000=0.004300 mol. Total volume=0.093 L, so [OH⁻]=0.046237 M, pOH=−log₁₀[OH⁻], and pH=14−pOH=12.66499.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 12.66499, not 18.99748.","B":"Incorrect. Substituting the stated values into the governing relation gives 12.66499, not 9.66499.","C":"Incorrect. Substituting the stated values into the governing relation gives 12.66499, not 6.33249.","D":"Correct. The worked calculation gives 12.66499."},"concepts_required":["strong acid–strong base neutralization","molarity","pH and pOH"],"reasoning_steps":5,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_acid_base_pH","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_000768","subject":"mathematics","topic":"sequences_and_series","subtopic":"geometric_progression","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A geometric progression has first term 3 and common ratio 5. What is the sum of its first 5 terms?","options":{"A":"2343","B":"2345","C":"2342","D":"2341"},"correct_answer":"A","correct_answer_value":"2343","solution":"For r≠1, Sₙ=a(rⁿ−1)/(r−1). Thus Sₙ=3(5^5−1)/(5−1)=2343.","option_explanations":{"A":"Correct. The worked calculation gives 2343.","B":"Incorrect. Substituting the stated values into the governing relation gives 2343, not 2345.","C":"Incorrect. Substituting the stated values into the governing relation gives 2343, not 2342.","D":"Incorrect. Substituting the stated values into the governing relation gives 2343, not 2341."},"concepts_required":["geometric progression","finite geometric sum"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_gp_sum","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_007158","subject":"chemistry","topic":"physical_chemistry","subtopic":"ideal_gas_volume","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Assuming ideal behaviour, what volume is occupied by 8 mol of gas at 229 K and 0.26 atm? Use R=0.082057 L atm mol⁻¹ K⁻¹.","options":{"A":"580.18625 L","B":"462.549 L","C":"577.18625 L","D":"578.18625 L"},"correct_answer":"D","correct_answer_value":"578.18625 L","solution":"PV=nRT, so V=nRT/P=(8)(0.082057)(229)/(0.26)=578.18625 L.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 578.18625 L, not 580.18625 L.","B":"Incorrect. Substituting the stated values into the governing relation gives 578.18625 L, not 462.549 L.","C":"Incorrect. Substituting the stated values into the governing relation gives 578.18625 L, not 577.18625 L.","D":"Correct. The worked calculation gives 578.18625 L."},"concepts_required":["ideal gas equation","unit consistency"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_ideal_gas","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_002882","subject":"mathematics","topic":"vectors","subtopic":"dot_product","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Given vectors u=(6, 4, -7) and v=(6, 0, -5), calculate u·v.","options":{"A":"70","B":"85.2","C":"56.8","D":"71"},"correct_answer":"D","correct_answer_value":"71","solution":"The dot product is the sum of corresponding component products: (6)(6)+(4)(0)+(-7)(-5)=71.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 71, not 70.","B":"Incorrect. Substituting the stated values into the governing relation gives 71, not 85.2.","C":"Incorrect. Substituting the stated values into the governing relation gives 71, not 56.8.","D":"Correct. The worked calculation gives 71."},"concepts_required":["three-dimensional vectors","scalar product"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_vector_dot","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_008598","subject":"chemistry","topic":"physical_chemistry","subtopic":"boiling_point_elevation","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A non-volatile solute is dissolved in water to produce a solution of molality 1.5 mol kg⁻¹. If its van't Hoff factor is 4 and K_b for water is 1.0 K kg mol⁻¹, find the boiling-point elevation.","options":{"A":"7 K","B":"6 K","C":"1.6 K","D":"12 K"},"correct_answer":"B","correct_answer_value":"6 K","solution":"ΔT_b=iK_bm. Substitution gives ΔT_b=4×1.0×1.5=6 K.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 6 K, not 7 K.","B":"Correct. The worked calculation gives 6 K.","C":"Incorrect. Substituting the stated values into the governing relation gives 6 K, not 1.6 K.","D":"Incorrect. Substituting the stated values into the governing relation gives 6 K, not 12 K."},"concepts_required":["colligative properties","van't Hoff factor","molality"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_boiling_elevation","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_007678","subject":"chemistry","topic":"physical_chemistry","subtopic":"first_order_half_life","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A first-order reaction has a half-life of 20 minutes. If the initial reactant concentration is 60 mmol L⁻¹, what concentration remains after 120 minutes?","options":{"A":"-0.0625 mmol L⁻¹","B":"1.09375 mmol L⁻¹","C":"-0.9375 mmol L⁻¹","D":"0.9375 mmol L⁻¹"},"correct_answer":"D","correct_answer_value":"0.9375 mmol L⁻¹","solution":"120 minutes corresponds to 6 half-lives. The remaining concentration is [A]₀(1/2)ⁿ=60/2^6=0.9375 mmol L⁻¹.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.9375 mmol L⁻¹, not -0.0625 mmol L⁻¹.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.9375 mmol L⁻¹, not 1.09375 mmol L⁻¹.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.9375 mmol L⁻¹, not -0.9375 mmol L⁻¹.","D":"Correct. The worked calculation gives 0.9375 mmol L⁻¹."},"concepts_required":["first-order kinetics","half-life","exponential decay"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_first_order_halflife","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_005473","subject":"physics","topic":"electrostatics","subtopic":"capacitor_energy","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A 21 μF capacitor is charged to a potential difference of 21 V. Calculate the energy stored, expressed in μJ.","options":{"A":"3704.4 μJ","B":"9261 μJ","C":"4630.5 μJ","D":"3472.875 μJ"},"correct_answer":"C","correct_answer_value":"4630.5 μJ","solution":"U=½CV². Since C=21 μF, U in μJ is ½×21×(21)²=4630.5 μJ.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 4630.5 μJ, not 3704.4 μJ.","B":"Incorrect. Substituting the stated values into the governing relation gives 4630.5 μJ, not 9261 μJ.","C":"Correct. The worked calculation gives 4630.5 μJ.","D":"Incorrect. Substituting the stated values into the governing relation gives 4630.5 μJ, not 3472.875 μJ."},"concepts_required":["capacitance","energy stored in a capacitor"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_capacitor_energy","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_005987","subject":"physics","topic":"thermodynamics","subtopic":"ideal_gas_isochoric_pressure","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A fixed amount of ideal gas is kept in a rigid container. Its pressure is 5 atm at 249 K. What is its pressure at 311 K?","options":{"A":"3.24498 atm","B":"6.24498 atm","C":"9.36747 atm","D":"4.24498 atm"},"correct_answer":"B","correct_answer_value":"6.24498 atm","solution":"At constant volume, P/T is constant. P₂=P₁T₂/T₁=(5)(311)/(249)=6.24498 atm.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 6.24498 atm, not 3.24498 atm.","B":"Correct. The worked calculation gives 6.24498 atm.","C":"Incorrect. Substituting the stated values into the governing relation gives 6.24498 atm, not 9.36747 atm.","D":"Incorrect. Substituting the stated values into the governing relation gives 6.24498 atm, not 4.24498 atm."},"concepts_required":["ideal gas law","isochoric process","absolute temperature"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_isochoric_gas","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_000242","subject":"mathematics","topic":"algebra","subtopic":"quadratic_roots","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"For the quadratic equation 5x^2 +4x +6=0, let its roots be α and β. What is α²+β²?","options":{"A":"-2.64","B":"-4.76","C":"-1.76","D":"-3.52"},"correct_answer":"C","correct_answer_value":"-1.76","solution":"By Vieta's relations, α+β = -4/5 and αβ = 6/5. Then α²+β²=(α+β)²−2αβ = (-4/5)²−2(6/5) = -1.76.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives -1.76, not -2.64.","B":"Incorrect. Substituting the stated values into the governing relation gives -1.76, not -4.76.","C":"Correct. The worked calculation gives -1.76.","D":"Incorrect. Substituting the stated values into the governing relation gives -1.76, not -3.52."},"concepts_required":["Vieta's formulas","symmetric polynomial expressions"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_quadratic_vieta","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_001935","subject":"mathematics","topic":"combinatorics","subtopic":"binomial_coefficient","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"How many distinct subsets containing exactly 3 elements can be selected from a set of 61 distinct elements?","options":{"A":"35990","B":"28792","C":"53985","D":"35989"},"correct_answer":"A","correct_answer_value":"35990","solution":"Order does not matter, so the count is C(61,3)=61!/[3!(58)!]=35990.","option_explanations":{"A":"Correct. The worked calculation gives 35990.","B":"Incorrect. Substituting the stated values into the governing relation gives 35990, not 28792.","C":"Incorrect. Substituting the stated values into the governing relation gives 35990, not 53985.","D":"Incorrect. Substituting the stated values into the governing relation gives 35990, not 35989."},"concepts_required":["combinations","factorials"],"reasoning_steps":2,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_combinations","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_005893","subject":"physics","topic":"thermodynamics","subtopic":"ideal_gas_isochoric_pressure","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A fixed amount of ideal gas is kept in a rigid container. Its pressure is 2.3 atm at 401 K. What is its pressure at 667 K?","options":{"A":"1.82569 atm","B":"4.82569 atm","C":"2.86926 atm","D":"3.82569 atm"},"correct_answer":"D","correct_answer_value":"3.82569 atm","solution":"At constant volume, P/T is constant. P₂=P₁T₂/T₁=(2.3)(667)/(401)=3.82569 atm.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 3.82569 atm, not 1.82569 atm.","B":"Incorrect. Substituting the stated values into the governing relation gives 3.82569 atm, not 4.82569 atm.","C":"Incorrect. Substituting the stated values into the governing relation gives 3.82569 atm, not 2.86926 atm.","D":"Correct. The worked calculation gives 3.82569 atm."},"concepts_required":["ideal gas law","isochoric process","absolute temperature"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_isochoric_gas","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_006743","subject":"chemistry","topic":"stoichiometry","subtopic":"limiting_reagent_water","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Hydrogen reacts with oxygen according to 2H₂ + O₂ → 2H₂O. If 35 mol H₂ and 25 mol O₂ are mixed and the reaction goes to completion, how many moles of H₂O form?","options":{"A":"33 mol","B":"36 mol","C":"35 mol","D":"70 mol"},"correct_answer":"C","correct_answer_value":"35 mol","solution":"From 2 mol H₂ and 1 mol O₂, 2 mol H₂O form. H₂ can produce 35 mol H₂O; O₂ can produce 50 mol H₂O. The limiting condition is H₂; product=35 mol.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 35 mol, not 33 mol.","B":"Incorrect. Substituting the stated values into the governing relation gives 35 mol, not 36 mol.","C":"Correct. The worked calculation gives 35 mol.","D":"Incorrect. Substituting the stated values into the governing relation gives 35 mol, not 70 mol."},"concepts_required":["balanced chemical equations","limiting reagent","mole ratios"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_limiting_reagent","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_003706","subject":"physics","topic":"mechanics","subtopic":"work_energy_theorem","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A 4 kg block initially moves at 3 m s⁻¹. A constant net force does 12 N of work over 14 m in the direction of motion. Find its final kinetic energy.","options":{"A":"19.6 J","B":"93 J","C":"186 J","D":"187 J"},"correct_answer":"C","correct_answer_value":"186 J","solution":"By the work–energy theorem, K_f=K_i+W=½mu²+Fs=½(4)(3)²+(12)(14)=186 J.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 186 J, not 19.6 J.","B":"Incorrect. Substituting the stated values into the governing relation gives 186 J, not 93 J.","C":"Correct. The worked calculation gives 186 J.","D":"Incorrect. Substituting the stated values into the governing relation gives 186 J, not 187 J."},"concepts_required":["work–energy theorem","kinetic energy","work by constant force"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_work_energy","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_007506","subject":"chemistry","topic":"physical_chemistry","subtopic":"strong_acid_base_mixture","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"88 mL of 0.10 M HCl is mixed with 35 mL of 0.10 M NaOH at 25°C. Assuming complete dissociation and additive volumes, find the final pH.","options":{"A":"0.68281","B":"2.04844","C":"0.36563","D":"1.36563"},"correct_answer":"D","correct_answer_value":"1.36563","solution":"Excess H⁺ moles=0.10×(88−35)/1000=0.005300 mol. Total volume=0.123 L, so [H⁺]=0.043089 M and pH=−log₁₀[H⁺]=1.36563.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 1.36563, not 0.68281.","B":"Incorrect. Substituting the stated values into the governing relation gives 1.36563, not 2.04844.","C":"Incorrect. Substituting the stated values into the governing relation gives 1.36563, not 0.36563.","D":"Correct. The worked calculation gives 1.36563."},"concepts_required":["strong acid–strong base neutralization","molarity","pH and pOH"],"reasoning_steps":5,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_acid_base_pH","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_007468","subject":"chemistry","topic":"physical_chemistry","subtopic":"strong_acid_base_mixture","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"24 mL of 0.10 M HCl is mixed with 60 mL of 0.10 M NaOH at 25°C. Assuming complete dissociation and additive volumes, find the final pH.","options":{"A":"12.63202","B":"-12.63202","C":"6.31601","D":"9.63202"},"correct_answer":"A","correct_answer_value":"12.63202","solution":"Excess OH⁻ moles=0.10×(60−24)/1000=0.003600 mol. Total volume=0.084 L, so [OH⁻]=0.042857 M, pOH=−log₁₀[OH⁻], and pH=14−pOH=12.63202.","option_explanations":{"A":"Correct. The worked calculation gives 12.63202.","B":"Incorrect. Substituting the stated values into the governing relation gives 12.63202, not -12.63202.","C":"Incorrect. Substituting the stated values into the governing relation gives 12.63202, not 6.31601.","D":"Incorrect. Substituting the stated values into the governing relation gives 12.63202, not 9.63202."},"concepts_required":["strong acid–strong base neutralization","molarity","pH and pOH"],"reasoning_steps":5,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_acid_base_pH","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_000962","subject":"mathematics","topic":"sequences_and_series","subtopic":"geometric_progression","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A geometric progression has first term 9 and common ratio -2. What is the sum of its first 5 terms?","options":{"A":"49.5","B":"99","C":"148.5","D":"100"},"correct_answer":"B","correct_answer_value":"99","solution":"For r≠1, Sₙ=a(rⁿ−1)/(r−1). Thus Sₙ=9(-2^5−1)/(-2−1)=99.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 99, not 49.5.","B":"Correct. The worked calculation gives 99.","C":"Incorrect. Substituting the stated values into the governing relation gives 99, not 148.5.","D":"Incorrect. Substituting the stated values into the governing relation gives 99, not 100."},"concepts_required":["geometric progression","finite geometric sum"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_gp_sum","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_006356","subject":"physics","topic":"oscillations","subtopic":"mass_spring_period_over_pi","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A mass of 9 kg is attached to an ideal spring of force constant 1027 N m⁻¹ and oscillates horizontally without damping. What is the time period divided by π?","options":{"A":"0.18723 s","B":"0.09361 s","C":"0.37445 s","D":"-0.18723 s"},"correct_answer":"A","correct_answer_value":"0.18723 s","solution":"For a mass–spring oscillator, T=2π√(m/k). Therefore T/π=2√(9/1027)=0.18723 s.","option_explanations":{"A":"Correct. The worked calculation gives 0.18723 s.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.18723 s, not 0.09361 s.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.18723 s, not 0.37445 s.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.18723 s, not -0.18723 s."},"concepts_required":["simple harmonic motion","mass–spring oscillator","time period"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_shm_period","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_003284","subject":"mathematics","topic":"complex_numbers","subtopic":"real_part_of_complex_quotient","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Let z=(13-7i)/(4+5i). What is Re(z), the real part of z?","options":{"A":"-2.58537","B":"0.41463","C":"1.41463","D":"0.82927"},"correct_answer":"B","correct_answer_value":"0.41463","solution":"Multiply numerator and denominator by the conjugate (4-5i). Then Re(z)=(ac+bd)/(c²+d²)=((13)(4)+(-7)(5))/(4²+(5)²)=0.41463.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.41463, not -2.58537.","B":"Correct. The worked calculation gives 0.41463.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.41463, not 1.41463.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.41463, not 0.82927."},"concepts_required":["complex conjugate","complex division","real and imaginary parts"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_complex_quotient","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_002918","subject":"mathematics","topic":"vectors","subtopic":"dot_product","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Given vectors u=(-5, 3, 2) and v=(3, 4, 7), calculate u·v.","options":{"A":"11","B":"22","C":"-11","D":"12"},"correct_answer":"A","correct_answer_value":"11","solution":"The dot product is the sum of corresponding component products: (-5)(3)+(3)(4)+(2)(7)=11.","option_explanations":{"A":"Correct. The worked calculation gives 11.","B":"Incorrect. Substituting the stated values into the governing relation gives 11, not 22.","C":"Incorrect. Substituting the stated values into the governing relation gives 11, not -11.","D":"Incorrect. Substituting the stated values into the governing relation gives 11, not 12."},"concepts_required":["three-dimensional vectors","scalar product"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_vector_dot","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_000322","subject":"mathematics","topic":"algebra","subtopic":"quadratic_roots","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"For the quadratic equation 2x^2 +13x +12=0, let its roots be α and β. What is α²+β²?","options":{"A":"30.25","B":"45.375","C":"31.25","D":"28.25"},"correct_answer":"A","correct_answer_value":"30.25","solution":"By Vieta's relations, α+β = -13/2 and αβ = 12/2. Then α²+β²=(α+β)²−2αβ = (-13/2)²−2(12/2) = 30.25.","option_explanations":{"A":"Correct. The worked calculation gives 30.25.","B":"Incorrect. Substituting the stated values into the governing relation gives 30.25, not 45.375.","C":"Incorrect. Substituting the stated values into the governing relation gives 30.25, not 31.25.","D":"Incorrect. Substituting the stated values into the governing relation gives 30.25, not 28.25."},"concepts_required":["Vieta's formulas","symmetric polynomial expressions"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_quadratic_vieta","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_004911","subject":"physics","topic":"electricity","subtopic":"series_parallel_circuit","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Resistors 4 Ω and 7 Ω are connected in series; this combination is connected in parallel with a 17 Ω resistor across a 36 V ideal battery. Find the total current supplied by the battery.","options":{"A":"-5.39037 A","B":"5.39037 A","C":"10.78075 A","D":"6.39037 A"},"correct_answer":"B","correct_answer_value":"5.39037 A","solution":"Series branch resistance=4+7=11 Ω. Parallel equivalent R_eq=(11×17)/(11+17)=6.67857 Ω. Total current I=V/R_eq=36/6.67857=5.39037 A.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 5.39037 A, not -5.39037 A.","B":"Correct. The worked calculation gives 5.39037 A.","C":"Incorrect. Substituting the stated values into the governing relation gives 5.39037 A, not 10.78075 A.","D":"Incorrect. Substituting the stated values into the governing relation gives 5.39037 A, not 6.39037 A."},"concepts_required":["series resistors","parallel resistors","Ohm's law"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_series_parallel","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_006914","subject":"chemistry","topic":"stoichiometry","subtopic":"limiting_reagent_water","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Hydrogen reacts with oxygen according to 2H₂ + O₂ → 2H₂O. If 4 mol H₂ and 4 mol O₂ are mixed and the reaction goes to completion, how many moles of H₂O form?","options":{"A":"-4 mol","B":"6 mol","C":"4 mol","D":"1 mol"},"correct_answer":"C","correct_answer_value":"4 mol","solution":"From 2 mol H₂ and 1 mol O₂, 2 mol H₂O form. H₂ can produce 4 mol H₂O; O₂ can produce 8 mol H₂O. The limiting condition is H₂; product=4 mol.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 4 mol, not -4 mol.","B":"Incorrect. Substituting the stated values into the governing relation gives 4 mol, not 6 mol.","C":"Correct. The worked calculation gives 4 mol.","D":"Incorrect. Substituting the stated values into the governing relation gives 4 mol, not 1 mol."},"concepts_required":["balanced chemical equations","limiting reagent","mole ratios"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_limiting_reagent","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_007169","subject":"chemistry","topic":"physical_chemistry","subtopic":"ideal_gas_volume","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Assuming ideal behaviour, what volume is occupied by 17 mol of gas at 315 K and 0.18 atm? Use R=0.082057 L atm mol⁻¹ K⁻¹.","options":{"A":"2441.19575 L","B":"2440.19575 L","C":"2442.19575 L","D":"2439.19575 L"},"correct_answer":"A","correct_answer_value":"2441.19575 L","solution":"PV=nRT, so V=nRT/P=(17)(0.082057)(315)/(0.18)=2441.19575 L.","option_explanations":{"A":"Correct. The worked calculation gives 2441.19575 L.","B":"Incorrect. Substituting the stated values into the governing relation gives 2441.19575 L, not 2440.19575 L.","C":"Incorrect. Substituting the stated values into the governing relation gives 2441.19575 L, not 2442.19575 L.","D":"Incorrect. Substituting the stated values into the governing relation gives 2441.19575 L, not 2439.19575 L."},"concepts_required":["ideal gas equation","unit consistency"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_ideal_gas","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_004363","subject":"physics","topic":"rotational_mechanics","subtopic":"disc_angular_acceleration","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A uniform solid disc of mass 11 kg and radius 4 m is acted on by a constant torque of 58 N m about its central axis. Find its angular acceleration.","options":{"A":"0.65909 rad s⁻²","B":"0.98864 rad s⁻²","C":"1.06591 rad s⁻²","D":"3.65909 rad s⁻²"},"correct_answer":"A","correct_answer_value":"0.65909 rad s⁻²","solution":"For a solid disc, I=½MR²=½(11)(4)²=88 kg m². Using τ=Iα, α=58/88=0.65909 rad s⁻².","option_explanations":{"A":"Correct. The worked calculation gives 0.65909 rad s⁻².","B":"Incorrect. Substituting the stated values into the governing relation gives 0.65909 rad s⁻², not 0.98864 rad s⁻².","C":"Incorrect. Substituting the stated values into the governing relation gives 0.65909 rad s⁻², not 1.06591 rad s⁻².","D":"Incorrect. Substituting the stated values into the governing relation gives 0.65909 rad s⁻², not 3.65909 rad s⁻²."},"concepts_required":["moment of inertia","rotational dynamics","torque"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_disc_torque","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_000506","subject":"mathematics","topic":"sequences_and_series","subtopic":"arithmetic_progression","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"An arithmetic progression has first term 7 and common difference 2. Find the sum of its first 34 terms.","options":{"A":"1361","B":"1360","C":"1020","D":"1359"},"correct_answer":"B","correct_answer_value":"1360","solution":"Use Sₙ = n/2[2a+(n−1)d]. Substituting a=7, d=2, n=34: Sₙ=34/2[2(7)+(34−1)(2)]=1360.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 1360, not 1361.","B":"Correct. The worked calculation gives 1360.","C":"Incorrect. Substituting the stated values into the governing relation gives 1360, not 1020.","D":"Incorrect. Substituting the stated values into the governing relation gives 1360, not 1359."},"concepts_required":["arithmetic progression","finite series"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_ap_sum","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_003944","subject":"physics","topic":"mechanics","subtopic":"work_energy_theorem","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A 15 kg block initially moves at 0 m s⁻¹. A constant net force does 25 N of work over 7 m in the direction of motion. Find its final kinetic energy.","options":{"A":"173 J","B":"174 J","C":"18.5 J","D":"175 J"},"correct_answer":"D","correct_answer_value":"175 J","solution":"By the work–energy theorem, K_f=K_i+W=½mu²+Fs=½(15)(0)²+(25)(7)=175 J.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 175 J, not 173 J.","B":"Incorrect. Substituting the stated values into the governing relation gives 175 J, not 174 J.","C":"Incorrect. Substituting the stated values into the governing relation gives 175 J, not 18.5 J.","D":"Correct. The worked calculation gives 175 J."},"concepts_required":["work–energy theorem","kinetic energy","work by constant force"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_work_energy","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_001586","subject":"mathematics","topic":"calculus","subtopic":"definite_integral_polynomial","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Evaluate the definite integral ∫₀^6 (2x² +7x +1) dx.","options":{"A":"414","B":"275","C":"276","D":"274"},"correct_answer":"C","correct_answer_value":"276","solution":"An antiderivative is (2/3)x³+(7/2)x²+(1)x. Evaluating between 0 and 6 gives (2/3)(6)³+(7/2)(6)²+(1)(6)=276.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 276, not 414.","B":"Incorrect. Substituting the stated values into the governing relation gives 276, not 275.","C":"Correct. The worked calculation gives 276.","D":"Incorrect. Substituting the stated values into the governing relation gives 276, not 274."},"concepts_required":["definite integration","power rule","limits of integration"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_integral_poly","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_005119","subject":"physics","topic":"electrostatics","subtopic":"coulombs_law","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Two point charges of +9 μC and +7 μC are separated by 5 cm in vacuum. Using k=9×10⁹ N m² C⁻², find the magnitude of the electrostatic force.","options":{"A":"272.16 N","B":"226.8 N","C":"453.6 N","D":"224.8 N"},"correct_answer":"B","correct_answer_value":"226.8 N","solution":"Convert charges to coulombs and distance to metres. F=k|q₁q₂|/r²=9×10⁹(9×10⁻⁶)(7×10⁻⁶)/(5×10⁻²)²=226.8 N.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 226.8 N, not 272.16 N.","B":"Correct. The worked calculation gives 226.8 N.","C":"Incorrect. Substituting the stated values into the governing relation gives 226.8 N, not 453.6 N.","D":"Incorrect. Substituting the stated values into the governing relation gives 226.8 N, not 224.8 N."},"concepts_required":["Coulomb's law","SI unit conversion","inverse-square law"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_coulomb","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_008240","subject":"chemistry","topic":"physical_chemistry","subtopic":"faradays_law_copper","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"An aqueous Cu²⁺ solution is electrolysed using a current of 9 A for 16784 s. Assuming 100% current efficiency, calculate the mass of copper deposited. Use M(Cu)=63.5 g mol⁻¹ and F=96500 C mol⁻¹.","options":{"A":"-49.69977 g","B":"24.84989 g","C":"39.75982 g","D":"49.69977 g"},"correct_answer":"D","correct_answer_value":"49.69977 g","solution":"Cu²⁺+2e⁻→Cu, so z=2. By Faraday's law m=MIt/(zF)=63.5×9×16784/(2×96500)=49.69977 g.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 49.69977 g, not -49.69977 g.","B":"Incorrect. Substituting the stated values into the governing relation gives 49.69977 g, not 24.84989 g.","C":"Incorrect. Substituting the stated values into the governing relation gives 49.69977 g, not 39.75982 g.","D":"Correct. The worked calculation gives 49.69977 g."},"concepts_required":["electrolysis","Faraday's laws","redox stoichiometry"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_faraday_copper","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_003742","subject":"physics","topic":"mechanics","subtopic":"work_energy_theorem","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A 8 kg block initially moves at 2 m s⁻¹. A constant net force does 30 N of work over 13 m in the direction of motion. Find its final kinetic energy.","options":{"A":"324.8 J","B":"406 J","C":"304.5 J","D":"408 J"},"correct_answer":"B","correct_answer_value":"406 J","solution":"By the work–energy theorem, K_f=K_i+W=½mu²+Fs=½(8)(2)²+(30)(13)=406 J.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 406 J, not 324.8 J.","B":"Correct. The worked calculation gives 406 J.","C":"Incorrect. Substituting the stated values into the governing relation gives 406 J, not 304.5 J.","D":"Incorrect. Substituting the stated values into the governing relation gives 406 J, not 408 J."},"concepts_required":["work–energy theorem","kinetic energy","work by constant force"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_work_energy","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_007524","subject":"chemistry","topic":"physical_chemistry","subtopic":"strong_acid_base_mixture","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"11 mL of 0.10 M HCl is mixed with 53 mL of 0.10 M NaOH at 25°C. Assuming complete dissociation and additive volumes, find the final pH.","options":{"A":"-12.81707","B":"25.63414","C":"12.81707","D":"6.40853"},"correct_answer":"C","correct_answer_value":"12.81707","solution":"Excess OH⁻ moles=0.10×(53−11)/1000=0.004200 mol. Total volume=0.064 L, so [OH⁻]=0.065625 M, pOH=−log₁₀[OH⁻], and pH=14−pOH=12.81707.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 12.81707, not -12.81707.","B":"Incorrect. Substituting the stated values into the governing relation gives 12.81707, not 25.63414.","C":"Correct. The worked calculation gives 12.81707.","D":"Incorrect. Substituting the stated values into the governing relation gives 12.81707, not 6.40853."},"concepts_required":["strong acid–strong base neutralization","molarity","pH and pOH"],"reasoning_steps":5,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_acid_base_pH","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_008270","subject":"chemistry","topic":"physical_chemistry","subtopic":"faradays_law_copper","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"An aqueous Cu²⁺ solution is electrolysed using a current of 1 A for 2210 s. Assuming 100% current efficiency, calculate the mass of copper deposited. Use M(Cu)=63.5 g mol⁻¹ and F=96500 C mol⁻¹.","options":{"A":"0.72712 g","B":"0.36356 g","C":"-0.72712 g","D":"-1.27288 g"},"correct_answer":"A","correct_answer_value":"0.72712 g","solution":"Cu²⁺+2e⁻→Cu, so z=2. By Faraday's law m=MIt/(zF)=63.5×1×2210/(2×96500)=0.72712 g.","option_explanations":{"A":"Correct. The worked calculation gives 0.72712 g.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.72712 g, not 0.36356 g.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.72712 g, not -0.72712 g.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.72712 g, not -1.27288 g."},"concepts_required":["electrolysis","Faraday's laws","redox stoichiometry"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_faraday_copper","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_002926","subject":"mathematics","topic":"vectors","subtopic":"dot_product","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Given vectors u=(1, -7, 7) and v=(7, -1, -3), calculate u·v.","options":{"A":"-5.25","B":"-7","C":"-4","D":"-6"},"correct_answer":"B","correct_answer_value":"-7","solution":"The dot product is the sum of corresponding component products: (1)(7)+(-7)(-1)+(7)(-3)=-7.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives -7, not -5.25.","B":"Correct. The worked calculation gives -7.","C":"Incorrect. Substituting the stated values into the governing relation gives -7, not -4.","D":"Incorrect. Substituting the stated values into the governing relation gives -7, not -6."},"concepts_required":["three-dimensional vectors","scalar product"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_vector_dot","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_003605","subject":"physics","topic":"mechanics","subtopic":"projectile_range","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A projectile is launched from level ground at 67 m s⁻¹ at 60° to the horizontal. Take g=10 m s⁻². Find its horizontal range.","options":{"A":"388.7588 m","B":"194.3794 m","C":"583.13821 m","D":"777.51761 m"},"correct_answer":"A","correct_answer_value":"388.7588 m","solution":"Range R=u²sin(2θ)/g. Substituting u=67, θ=60° and g=10 gives R=(67)²sin(120°)/10=388.7588 m.","option_explanations":{"A":"Correct. The worked calculation gives 388.7588 m.","B":"Incorrect. Substituting the stated values into the governing relation gives 388.7588 m, not 194.3794 m.","C":"Incorrect. Substituting the stated values into the governing relation gives 388.7588 m, not 583.13821 m.","D":"Incorrect. Substituting the stated values into the governing relation gives 388.7588 m, not 777.51761 m."},"concepts_required":["projectile motion","range equation","trigonometry"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_projectile_range","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_009697","subject":"chemistry","topic":"inorganic_chemistry","subtopic":"complex_ion_oxidation_state","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"For a hypothetical coordination ion [Pd(CN)_5]^+3, each CN ligand has charge −1. Determine the formal oxidation state of Pd.","options":{"A":"8","B":"7","C":"5","D":"-8"},"correct_answer":"A","correct_answer_value":"8","solution":"Let the formal oxidation state of Pd be x. Charge balance gives x+(5)(−1)=+3; hence x=3+5=+8.","option_explanations":{"A":"Correct. The worked calculation gives 8.","B":"Incorrect. Substituting the stated values into the governing relation gives 8, not 7.","C":"Incorrect. Substituting the stated values into the governing relation gives 8, not 5.","D":"Incorrect. Substituting the stated values into the governing relation gives 8, not -8."},"concepts_required":["coordination compounds","oxidation state","charge balance"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_complex_oxidation","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_005268","subject":"physics","topic":"electrostatics","subtopic":"coulombs_law","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Two point charges of +2 μC and +7 μC are separated by 10 cm in vacuum. Using k=9×10⁹ N m² C⁻², find the magnitude of the electrostatic force.","options":{"A":"-12.6 N","B":"18.9 N","C":"12.6 N","D":"6.3 N"},"correct_answer":"C","correct_answer_value":"12.6 N","solution":"Convert charges to coulombs and distance to metres. F=k|q₁q₂|/r²=9×10⁹(2×10⁻⁶)(7×10⁻⁶)/(10×10⁻²)²=12.6 N.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 12.6 N, not -12.6 N.","B":"Incorrect. Substituting the stated values into the governing relation gives 12.6 N, not 18.9 N.","C":"Correct. The worked calculation gives 12.6 N.","D":"Incorrect. Substituting the stated values into the governing relation gives 12.6 N, not 6.3 N."},"concepts_required":["Coulomb's law","SI unit conversion","inverse-square law"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_coulomb","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_005301","subject":"physics","topic":"electrostatics","subtopic":"coulombs_law","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Two point charges of +9 μC and +6 μC are separated by 12 cm in vacuum. Using k=9×10⁹ N m² C⁻², find the magnitude of the electrostatic force.","options":{"A":"27 N","B":"32.75 N","C":"33.75 N","D":"31.75 N"},"correct_answer":"C","correct_answer_value":"33.75 N","solution":"Convert charges to coulombs and distance to metres. F=k|q₁q₂|/r²=9×10⁹(9×10⁻⁶)(6×10⁻⁶)/(12×10⁻²)²=33.75 N.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 33.75 N, not 27 N.","B":"Incorrect. Substituting the stated values into the governing relation gives 33.75 N, not 32.75 N.","C":"Correct. The worked calculation gives 33.75 N.","D":"Incorrect. Substituting the stated values into the governing relation gives 33.75 N, not 31.75 N."},"concepts_required":["Coulomb's law","SI unit conversion","inverse-square law"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_coulomb","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_009174","subject":"chemistry","topic":"physical_chemistry","subtopic":"nernst_equation_298K","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"At 298 K, a galvanic cell has E°cell=1.31 V, transfers 2 electrons, and has reaction quotient Q=10^0. Using E=E°−(0.0591/n)log₁₀Q, calculate Ecell.","options":{"A":"3.31 V","B":"1.131 V","C":"0.9825 V","D":"1.31 V"},"correct_answer":"D","correct_answer_value":"1.31 V","solution":"Apply the Nernst equation: E=1.31−(0.0591/2)(0)=1.31 V.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 1.31 V, not 3.31 V.","B":"Incorrect. Substituting the stated values into the governing relation gives 1.31 V, not 1.131 V.","C":"Incorrect. Substituting the stated values into the governing relation gives 1.31 V, not 0.9825 V.","D":"Correct. The worked calculation gives 1.31 V."},"concepts_required":["electrochemical cells","Nernst equation","logarithms"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_nernst","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_002351","subject":"mathematics","topic":"linear_algebra","subtopic":"2x2_determinant","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Find the determinant of the matrix [[8, 4], [5, -5]].","options":{"A":"-120","B":"-60","C":"60","D":"-72"},"correct_answer":"B","correct_answer_value":"-60","solution":"For a 2×2 matrix [[a,b],[c,d]], det=ad−bc. Here it is (8)(-5)−(4)(5)=-60.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives -60, not -120.","B":"Correct. The worked calculation gives -60.","C":"Incorrect. Substituting the stated values into the governing relation gives -60, not 60.","D":"Incorrect. Substituting the stated values into the governing relation gives -60, not -72."},"concepts_required":["determinants","matrix operations"],"reasoning_steps":2,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_determinant_2x2","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_004114","subject":"physics","topic":"mechanics","subtopic":"inclined_plane_with_friction","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A 16 kg block slides down a 37° rough incline. The coefficient of kinetic friction is 0.16. Taking g=10 m s⁻², find the net force down the plane. Use sin37°=0.6 and cos37°=0.8.","options":{"A":"76.52 N","B":"-75.52 N","C":"74.52 N","D":"75.52 N"},"correct_answer":"D","correct_answer_value":"75.52 N","solution":"Net force=mg(sinθ−μcosθ)=16×10×(0.6−0.16×0.8)=75.52 N.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 75.52 N, not 76.52 N.","B":"Incorrect. Substituting the stated values into the governing relation gives 75.52 N, not -75.52 N.","C":"Incorrect. Substituting the stated values into the governing relation gives 75.52 N, not 74.52 N.","D":"Correct. The worked calculation gives 75.52 N."},"concepts_required":["Newton's laws","inclined plane","kinetic friction"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_incline_friction","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_004375","subject":"physics","topic":"rotational_mechanics","subtopic":"disc_angular_acceleration","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A uniform solid disc of mass 12 kg and radius 2 m is acted on by a constant torque of 55 N m about its central axis. Find its angular acceleration.","options":{"A":"3.4375 rad s⁻²","B":"5.29167 rad s⁻²","C":"1.22917 rad s⁻²","D":"2.29167 rad s⁻²"},"correct_answer":"D","correct_answer_value":"2.29167 rad s⁻²","solution":"For a solid disc, I=½MR²=½(12)(2)²=24 kg m². Using τ=Iα, α=55/24=2.29167 rad s⁻².","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 2.29167 rad s⁻², not 3.4375 rad s⁻².","B":"Incorrect. Substituting the stated values into the governing relation gives 2.29167 rad s⁻², not 5.29167 rad s⁻².","C":"Incorrect. Substituting the stated values into the governing relation gives 2.29167 rad s⁻², not 1.22917 rad s⁻².","D":"Correct. The worked calculation gives 2.29167 rad s⁻²."},"concepts_required":["moment of inertia","rotational dynamics","torque"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_disc_torque","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_000255","subject":"mathematics","topic":"algebra","subtopic":"quadratic_roots","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"For the quadratic equation 6x^2 +11x -1=0, let its roots be α and β. What is α²+β²?","options":{"A":"4.69444","B":"1.36944","C":"3.69444","D":"6.69444"},"correct_answer":"C","correct_answer_value":"3.69444","solution":"By Vieta's relations, α+β = -11/6 and αβ = -1/6. Then α²+β²=(α+β)²−2αβ = (-11/6)²−2(-1/6) = 3.69444.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 3.69444, not 4.69444.","B":"Incorrect. Substituting the stated values into the governing relation gives 3.69444, not 1.36944.","C":"Correct. The worked calculation gives 3.69444.","D":"Incorrect. Substituting the stated values into the governing relation gives 3.69444, not 6.69444."},"concepts_required":["Vieta's formulas","symmetric polynomial expressions"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_quadratic_vieta","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_007364","subject":"chemistry","topic":"physical_chemistry","subtopic":"strong_acid_base_mixture","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"34 mL of 0.10 M HCl is mixed with 35 mL of 0.10 M NaOH at 25°C. Assuming complete dissociation and additive volumes, find the final pH.","options":{"A":"11.16115","B":"10.16115","C":"5.58058","D":"9.16115"},"correct_answer":"A","correct_answer_value":"11.16115","solution":"Excess OH⁻ moles=0.10×(35−34)/1000=0.000100 mol. Total volume=0.069 L, so [OH⁻]=0.001449 M, pOH=−log₁₀[OH⁻], and pH=14−pOH=11.16115.","option_explanations":{"A":"Correct. The worked calculation gives 11.16115.","B":"Incorrect. Substituting the stated values into the governing relation gives 11.16115, not 10.16115.","C":"Incorrect. Substituting the stated values into the governing relation gives 11.16115, not 5.58058.","D":"Incorrect. Substituting the stated values into the governing relation gives 11.16115, not 9.16115."},"concepts_required":["strong acid–strong base neutralization","molarity","pH and pOH"],"reasoning_steps":5,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_acid_base_pH","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_007481","subject":"chemistry","topic":"physical_chemistry","subtopic":"strong_acid_base_mixture","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"54 mL of 0.10 M HCl is mixed with 60 mL of 0.10 M NaOH at 25°C. Assuming complete dissociation and additive volumes, find the final pH.","options":{"A":"8.72125","B":"11.72125","C":"8.79093","D":"-11.72125"},"correct_answer":"B","correct_answer_value":"11.72125","solution":"Excess OH⁻ moles=0.10×(60−54)/1000=0.000600 mol. Total volume=0.114 L, so [OH⁻]=0.005263 M, pOH=−log₁₀[OH⁻], and pH=14−pOH=11.72125.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 11.72125, not 8.72125.","B":"Correct. The worked calculation gives 11.72125.","C":"Incorrect. Substituting the stated values into the governing relation gives 11.72125, not 8.79093.","D":"Incorrect. Substituting the stated values into the governing relation gives 11.72125, not -11.72125."},"concepts_required":["strong acid–strong base neutralization","molarity","pH and pOH"],"reasoning_steps":5,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_acid_base_pH","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_001000","subject":"mathematics","topic":"sequences_and_series","subtopic":"geometric_progression","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A geometric progression has first term 8 and common ratio -2. What is the sum of its first 8 terms?","options":{"A":"680","B":"-1360","C":"-544","D":"-680"},"correct_answer":"D","correct_answer_value":"-680","solution":"For r≠1, Sₙ=a(rⁿ−1)/(r−1). Thus Sₙ=8(-2^8−1)/(-2−1)=-680.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives -680, not 680.","B":"Incorrect. Substituting the stated values into the governing relation gives -680, not -1360.","C":"Incorrect. Substituting the stated values into the governing relation gives -680, not -544.","D":"Correct. The worked calculation gives -680."},"concepts_required":["geometric progression","finite geometric sum"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_gp_sum","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_006259","subject":"physics","topic":"ray_optics","subtopic":"convex_lens_magnification","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A real object is placed 38 cm in front of a thin convex lens of focal length 16 cm. Using the Cartesian sign convention, what is the magnitude of the linear magnification?","options":{"A":"0.72727","B":"1.72727","C":"-1.27273","D":"0.36364"},"correct_answer":"A","correct_answer_value":"0.72727","solution":"Using distances as positive magnitudes, 1/f=1/v+1/u gives v=uf/(u−f)=(38)(16)/(38−16)=27.63636 cm. Thus |m|=v/u=27.63636/38=0.72727.","option_explanations":{"A":"Correct. The worked calculation gives 0.72727.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.72727, not 1.72727.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.72727, not -1.27273.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.72727, not 0.36364."},"concepts_required":["thin lens equation","linear magnification","real image"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_lens_magnification","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_003637","subject":"physics","topic":"mechanics","subtopic":"projectile_range","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A projectile is launched from level ground at 50 m s⁻¹ at 25° to the horizontal. Take g=10 m s⁻². Find its horizontal range.","options":{"A":"229.81333 m","B":"191.51111 m","C":"383.02222 m","D":"287.26667 m"},"correct_answer":"B","correct_answer_value":"191.51111 m","solution":"Range R=u²sin(2θ)/g. Substituting u=50, θ=25° and g=10 gives R=(50)²sin(50°)/10=191.51111 m.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 191.51111 m, not 229.81333 m.","B":"Correct. The worked calculation gives 191.51111 m.","C":"Incorrect. Substituting the stated values into the governing relation gives 191.51111 m, not 383.02222 m.","D":"Incorrect. Substituting the stated values into the governing relation gives 191.51111 m, not 287.26667 m."},"concepts_required":["projectile motion","range equation","trigonometry"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_projectile_range","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_005417","subject":"physics","topic":"electrostatics","subtopic":"capacitor_energy","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A 11 μF capacitor is charged to a potential difference of 20 V. Calculate the energy stored, expressed in μJ.","options":{"A":"1760 μJ","B":"2199 μJ","C":"221 μJ","D":"2200 μJ"},"correct_answer":"D","correct_answer_value":"2200 μJ","solution":"U=½CV². Since C=11 μF, U in μJ is ½×11×(20)²=2200 μJ.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 2200 μJ, not 1760 μJ.","B":"Incorrect. Substituting the stated values into the governing relation gives 2200 μJ, not 2199 μJ.","C":"Incorrect. Substituting the stated values into the governing relation gives 2200 μJ, not 221 μJ.","D":"Correct. The worked calculation gives 2200 μJ."},"concepts_required":["capacitance","energy stored in a capacitor"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_capacitor_energy","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_005331","subject":"physics","topic":"electrostatics","subtopic":"coulombs_law","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Two point charges of +6 μC and +7 μC are separated by 10 cm in vacuum. Using k=9×10⁹ N m² C⁻², find the magnitude of the electrostatic force.","options":{"A":"39.8 N","B":"18.9 N","C":"37.8 N","D":"-37.8 N"},"correct_answer":"C","correct_answer_value":"37.8 N","solution":"Convert charges to coulombs and distance to metres. F=k|q₁q₂|/r²=9×10⁹(6×10⁻⁶)(7×10⁻⁶)/(10×10⁻²)²=37.8 N.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 37.8 N, not 39.8 N.","B":"Incorrect. Substituting the stated values into the governing relation gives 37.8 N, not 18.9 N.","C":"Correct. The worked calculation gives 37.8 N.","D":"Incorrect. Substituting the stated values into the governing relation gives 37.8 N, not -37.8 N."},"concepts_required":["Coulomb's law","SI unit conversion","inverse-square law"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_coulomb","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_007005","subject":"chemistry","topic":"physical_chemistry","subtopic":"ideal_gas_volume","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Assuming ideal behaviour, what volume is occupied by 14 mol of gas at 524 K and 2.18 atm? Use R=0.082057 L atm mol⁻¹ K⁻¹.","options":{"A":"552.26619 L","B":"220.90648 L","C":"276.1331 L","D":"138.06655 L"},"correct_answer":"C","correct_answer_value":"276.1331 L","solution":"PV=nRT, so V=nRT/P=(14)(0.082057)(524)/(2.18)=276.1331 L.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 276.1331 L, not 552.26619 L.","B":"Incorrect. Substituting the stated values into the governing relation gives 276.1331 L, not 220.90648 L.","C":"Correct. The worked calculation gives 276.1331 L.","D":"Incorrect. Substituting the stated values into the governing relation gives 276.1331 L, not 138.06655 L."},"concepts_required":["ideal gas equation","unit consistency"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_ideal_gas","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_001815","subject":"mathematics","topic":"combinatorics","subtopic":"binomial_coefficient","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"How many distinct subsets containing exactly 7 elements can be selected from a set of 38 distinct elements?","options":{"A":"25240512","B":"12620256","C":"12620258","D":"12620257"},"correct_answer":"B","correct_answer_value":"12620256","solution":"Order does not matter, so the count is C(38,7)=38!/[7!(31)!]=12620256.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 12620256, not 25240512.","B":"Correct. The worked calculation gives 12620256.","C":"Incorrect. Substituting the stated values into the governing relation gives 12620256, not 12620258.","D":"Incorrect. Substituting the stated values into the governing relation gives 12620256, not 12620257."},"concepts_required":["combinations","factorials"],"reasoning_steps":2,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_combinations","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_002904","subject":"mathematics","topic":"vectors","subtopic":"dot_product","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Given vectors u=(2, 0, -7) and v=(-5, 3, -1), calculate u·v.","options":{"A":"3","B":"-3","C":"-6","D":"-2"},"correct_answer":"B","correct_answer_value":"-3","solution":"The dot product is the sum of corresponding component products: (2)(-5)+(0)(3)+(-7)(-1)=-3.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives -3, not 3.","B":"Correct. The worked calculation gives -3.","C":"Incorrect. Substituting the stated values into the governing relation gives -3, not -6.","D":"Incorrect. Substituting the stated values into the governing relation gives -3, not -2."},"concepts_required":["three-dimensional vectors","scalar product"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_vector_dot","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_001587","subject":"mathematics","topic":"calculus","subtopic":"definite_integral_polynomial","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Evaluate the definite integral ∫₀^8 (-3x² -1x -2) dx.","options":{"A":"-420","B":"-560","C":"-558","D":"-280"},"correct_answer":"B","correct_answer_value":"-560","solution":"An antiderivative is (-3/3)x³+(-1/2)x²+(-2)x. Evaluating between 0 and 8 gives (-3/3)(8)³+(-1/2)(8)²+(-2)(8)=-560.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives -560, not -420.","B":"Correct. The worked calculation gives -560.","C":"Incorrect. Substituting the stated values into the governing relation gives -560, not -558.","D":"Incorrect. Substituting the stated values into the governing relation gives -560, not -280."},"concepts_required":["definite integration","power rule","limits of integration"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_integral_poly","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_002712","subject":"mathematics","topic":"vectors","subtopic":"dot_product","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Given vectors u=(-5, -7, 8) and v=(2, 1, -4), calculate u·v.","options":{"A":"-39.2","B":"-49","C":"-58.8","D":"-3.9"},"correct_answer":"B","correct_answer_value":"-49","solution":"The dot product is the sum of corresponding component products: (-5)(2)+(-7)(1)+(8)(-4)=-49.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives -49, not -39.2.","B":"Correct. The worked calculation gives -49.","C":"Incorrect. Substituting the stated values into the governing relation gives -49, not -58.8.","D":"Incorrect. Substituting the stated values into the governing relation gives -49, not -3.9."},"concepts_required":["three-dimensional vectors","scalar product"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_vector_dot","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_007151","subject":"chemistry","topic":"physical_chemistry","subtopic":"ideal_gas_volume","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Assuming ideal behaviour, what volume is occupied by 20 mol of gas at 467 K and 0.08 atm? Use R=0.082057 L atm mol⁻¹ K⁻¹.","options":{"A":"-9580.15475 L","B":"959.01548 L","C":"9579.15475 L","D":"9580.15475 L"},"correct_answer":"D","correct_answer_value":"9580.15475 L","solution":"PV=nRT, so V=nRT/P=(20)(0.082057)(467)/(0.08)=9580.15475 L.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 9580.15475 L, not -9580.15475 L.","B":"Incorrect. Substituting the stated values into the governing relation gives 9580.15475 L, not 959.01548 L.","C":"Incorrect. Substituting the stated values into the governing relation gives 9580.15475 L, not 9579.15475 L.","D":"Correct. The worked calculation gives 9580.15475 L."},"concepts_required":["ideal gas equation","unit consistency"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_ideal_gas","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_004281","subject":"physics","topic":"mechanics","subtopic":"inclined_plane_with_friction","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A 21 kg block slides down a 37° rough incline. The coefficient of kinetic friction is 0.43. Taking g=10 m s⁻², find the net force down the plane. Use sin37°=0.6 and cos37°=0.8.","options":{"A":"43.008 N","B":"52.76 N","C":"53.76 N","D":"54.76 N"},"correct_answer":"C","correct_answer_value":"53.76 N","solution":"Net force=mg(sinθ−μcosθ)=21×10×(0.6−0.43×0.8)=53.76 N.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 53.76 N, not 43.008 N.","B":"Incorrect. Substituting the stated values into the governing relation gives 53.76 N, not 52.76 N.","C":"Correct. The worked calculation gives 53.76 N.","D":"Incorrect. Substituting the stated values into the governing relation gives 53.76 N, not 54.76 N."},"concepts_required":["Newton's laws","inclined plane","kinetic friction"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_incline_friction","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_002188","subject":"mathematics","topic":"probability","subtopic":"same_colour_without_replacement","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A bag contains 7 red, 5 blue and 5 green balls. Two balls are drawn uniformly at random without replacement. What is the probability that both have the same colour?","options":{"A":"0.2261","B":"0.45221","C":"2.30147","D":"0.30147"},"correct_answer":"D","correct_answer_value":"0.30147","solution":"Favourable pairs = C(7,2)+C(5,2)+C(5,2)=41. Total pairs=C(17,2)=136. Therefore P=41/136=0.30147.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.30147, not 0.2261.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.30147, not 0.45221.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.30147, not 2.30147.","D":"Correct. The worked calculation gives 0.30147."},"concepts_required":["classical probability","combinations","without replacement"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_probability_colours","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_006838","subject":"chemistry","topic":"stoichiometry","subtopic":"limiting_reagent_water","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Hydrogen reacts with oxygen according to 2H₂ + O₂ → 2H₂O. If 12 mol H₂ and 9 mol O₂ are mixed and the reaction goes to completion, how many moles of H₂O form?","options":{"A":"10 mol","B":"9 mol","C":"11 mol","D":"12 mol"},"correct_answer":"D","correct_answer_value":"12 mol","solution":"From 2 mol H₂ and 1 mol O₂, 2 mol H₂O form. H₂ can produce 12 mol H₂O; O₂ can produce 18 mol H₂O. The limiting condition is H₂; product=12 mol.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 12 mol, not 10 mol.","B":"Incorrect. Substituting the stated values into the governing relation gives 12 mol, not 9 mol.","C":"Incorrect. Substituting the stated values into the governing relation gives 12 mol, not 11 mol.","D":"Correct. The worked calculation gives 12 mol."},"concepts_required":["balanced chemical equations","limiting reagent","mole ratios"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_limiting_reagent","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_006370","subject":"physics","topic":"oscillations","subtopic":"mass_spring_period_over_pi","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A mass of 13 kg is attached to an ideal spring of force constant 1030 N m⁻¹ and oscillates horizontally without damping. What is the time period divided by π?","options":{"A":"0.22469 s","B":"3.22469 s","C":"0.11234 s","D":"0.44938 s"},"correct_answer":"A","correct_answer_value":"0.22469 s","solution":"For a mass–spring oscillator, T=2π√(m/k). Therefore T/π=2√(13/1030)=0.22469 s.","option_explanations":{"A":"Correct. The worked calculation gives 0.22469 s.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.22469 s, not 3.22469 s.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.22469 s, not 0.11234 s.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.22469 s, not 0.44938 s."},"concepts_required":["simple harmonic motion","mass–spring oscillator","time period"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_shm_period","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_006500","subject":"physics","topic":"oscillations","subtopic":"mass_spring_period_over_pi","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A mass of 6 kg is attached to an ideal spring of force constant 310 N m⁻¹ and oscillates horizontally without damping. What is the time period divided by π?","options":{"A":"0.55649 s","B":"2.27824 s","C":"0.41737 s","D":"0.27824 s"},"correct_answer":"D","correct_answer_value":"0.27824 s","solution":"For a mass–spring oscillator, T=2π√(m/k). Therefore T/π=2√(6/310)=0.27824 s.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.27824 s, not 0.55649 s.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.27824 s, not 2.27824 s.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.27824 s, not 0.41737 s.","D":"Correct. The worked calculation gives 0.27824 s."},"concepts_required":["simple harmonic motion","mass–spring oscillator","time period"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_shm_period","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_006428","subject":"physics","topic":"oscillations","subtopic":"mass_spring_period_over_pi","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A mass of 15 kg is attached to an ideal spring of force constant 414 N m⁻¹ and oscillates horizontally without damping. What is the time period divided by π?","options":{"A":"-2.61931 s","B":"-0.61931 s","C":"1.03807 s","D":"0.38069 s"},"correct_answer":"D","correct_answer_value":"0.38069 s","solution":"For a mass–spring oscillator, T=2π√(m/k). Therefore T/π=2√(15/414)=0.38069 s.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.38069 s, not -2.61931 s.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.38069 s, not -0.61931 s.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.38069 s, not 1.03807 s.","D":"Correct. The worked calculation gives 0.38069 s."},"concepts_required":["simple harmonic motion","mass–spring oscillator","time period"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_shm_period","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_000868","subject":"mathematics","topic":"sequences_and_series","subtopic":"geometric_progression","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A geometric progression has first term 6 and common ratio -2. What is the sum of its first 9 terms?","options":{"A":"513","B":"-1026","C":"1026","D":"1231.2"},"correct_answer":"C","correct_answer_value":"1026","solution":"For r≠1, Sₙ=a(rⁿ−1)/(r−1). Thus Sₙ=6(-2^9−1)/(-2−1)=1026.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 1026, not 513.","B":"Incorrect. Substituting the stated values into the governing relation gives 1026, not -1026.","C":"Correct. The worked calculation gives 1026.","D":"Incorrect. Substituting the stated values into the governing relation gives 1026, not 1231.2."},"concepts_required":["geometric progression","finite geometric sum"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_gp_sum","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_002975","subject":"mathematics","topic":"vectors","subtopic":"dot_product","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Given vectors u=(1, 4, 2) and v=(5, 9, -5), calculate u·v.","options":{"A":"31","B":"23.25","C":"29","D":"46.5"},"correct_answer":"A","correct_answer_value":"31","solution":"The dot product is the sum of corresponding component products: (1)(5)+(4)(9)+(2)(-5)=31.","option_explanations":{"A":"Correct. The worked calculation gives 31.","B":"Incorrect. Substituting the stated values into the governing relation gives 31, not 23.25.","C":"Incorrect. Substituting the stated values into the governing relation gives 31, not 29.","D":"Incorrect. Substituting the stated values into the governing relation gives 31, not 46.5."},"concepts_required":["three-dimensional vectors","scalar product"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_vector_dot","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_006120","subject":"physics","topic":"ray_optics","subtopic":"convex_lens_magnification","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A real object is placed 29 cm in front of a thin convex lens of focal length 14 cm. Using the Cartesian sign convention, what is the magnitude of the linear magnification?","options":{"A":"-1.06667","B":"-0.06667","C":"1.09333","D":"0.93333"},"correct_answer":"D","correct_answer_value":"0.93333","solution":"Using distances as positive magnitudes, 1/f=1/v+1/u gives v=uf/(u−f)=(29)(14)/(29−14)=27.06667 cm. Thus |m|=v/u=27.06667/29=0.93333.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.93333, not -1.06667.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.93333, not -0.06667.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.93333, not 1.09333.","D":"Correct. The worked calculation gives 0.93333."},"concepts_required":["thin lens equation","linear magnification","real image"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_lens_magnification","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_006171","subject":"physics","topic":"ray_optics","subtopic":"convex_lens_magnification","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A real object is placed 36 cm in front of a thin convex lens of focal length 10 cm. Using the Cartesian sign convention, what is the magnitude of the linear magnification?","options":{"A":"1.38462","B":"2.38462","C":"-0.61538","D":"0.38462"},"correct_answer":"D","correct_answer_value":"0.38462","solution":"Using distances as positive magnitudes, 1/f=1/v+1/u gives v=uf/(u−f)=(36)(10)/(36−10)=13.84615 cm. Thus |m|=v/u=13.84615/36=0.38462.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.38462, not 1.38462.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.38462, not 2.38462.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.38462, not -0.61538.","D":"Correct. The worked calculation gives 0.38462."},"concepts_required":["thin lens equation","linear magnification","real image"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_lens_magnification","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_000630","subject":"mathematics","topic":"sequences_and_series","subtopic":"arithmetic_progression","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"An arithmetic progression has first term 1 and common difference 9. Find the sum of its first 18 terms.","options":{"A":"1046.25","B":"1395","C":"-1395","D":"1674"},"correct_answer":"B","correct_answer_value":"1395","solution":"Use Sₙ = n/2[2a+(n−1)d]. Substituting a=1, d=9, n=18: Sₙ=18/2[2(1)+(18−1)(9)]=1395.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 1395, not 1046.25.","B":"Correct. The worked calculation gives 1395.","C":"Incorrect. Substituting the stated values into the governing relation gives 1395, not -1395.","D":"Incorrect. Substituting the stated values into the governing relation gives 1395, not 1674."},"concepts_required":["arithmetic progression","finite series"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_ap_sum","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_003426","subject":"physics","topic":"mechanics","subtopic":"projectile_range","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A projectile is launched from level ground at 60 m s⁻¹ at 30° to the horizontal. Take g=10 m s⁻². Find its horizontal range.","options":{"A":"249.41532 m","B":"-311.76915 m","C":"311.76915 m","D":"155.88457 m"},"correct_answer":"C","correct_answer_value":"311.76915 m","solution":"Range R=u²sin(2θ)/g. Substituting u=60, θ=30° and g=10 gives R=(60)²sin(60°)/10=311.76915 m.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 311.76915 m, not 249.41532 m.","B":"Incorrect. Substituting the stated values into the governing relation gives 311.76915 m, not -311.76915 m.","C":"Correct. The worked calculation gives 311.76915 m.","D":"Incorrect. Substituting the stated values into the governing relation gives 311.76915 m, not 155.88457 m."},"concepts_required":["projectile motion","range equation","trigonometry"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_projectile_range","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_007444","subject":"chemistry","topic":"physical_chemistry","subtopic":"strong_acid_base_mixture","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"11 mL of 0.10 M HCl is mixed with 54 mL of 0.10 M NaOH at 25°C. Assuming complete dissociation and additive volumes, find the final pH.","options":{"A":"9.61542","B":"11.82056","C":"10.82056","D":"12.82056"},"correct_answer":"D","correct_answer_value":"12.82056","solution":"Excess OH⁻ moles=0.10×(54−11)/1000=0.004300 mol. Total volume=0.065 L, so [OH⁻]=0.066154 M, pOH=−log₁₀[OH⁻], and pH=14−pOH=12.82056.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 12.82056, not 9.61542.","B":"Incorrect. Substituting the stated values into the governing relation gives 12.82056, not 11.82056.","C":"Incorrect. Substituting the stated values into the governing relation gives 12.82056, not 10.82056.","D":"Correct. The worked calculation gives 12.82056."},"concepts_required":["strong acid–strong base neutralization","molarity","pH and pOH"],"reasoning_steps":5,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_acid_base_pH","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_004640","subject":"physics","topic":"rotational_mechanics","subtopic":"disc_angular_acceleration","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A uniform solid disc of mass 13 kg and radius 4 m is acted on by a constant torque of 7 N m about its central axis. Find its angular acceleration.","options":{"A":"0.06731 rad s⁻²","B":"0.10096 rad s⁻²","C":"0.03365 rad s⁻²","D":"-0.06731 rad s⁻²"},"correct_answer":"A","correct_answer_value":"0.06731 rad s⁻²","solution":"For a solid disc, I=½MR²=½(13)(4)²=104 kg m². Using τ=Iα, α=7/104=0.06731 rad s⁻².","option_explanations":{"A":"Correct. The worked calculation gives 0.06731 rad s⁻².","B":"Incorrect. Substituting the stated values into the governing relation gives 0.06731 rad s⁻², not 0.10096 rad s⁻².","C":"Incorrect. Substituting the stated values into the governing relation gives 0.06731 rad s⁻², not 0.03365 rad s⁻².","D":"Incorrect. Substituting the stated values into the governing relation gives 0.06731 rad s⁻², not -0.06731 rad s⁻²."},"concepts_required":["moment of inertia","rotational dynamics","torque"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_disc_torque","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_000115","subject":"mathematics","topic":"algebra","subtopic":"quadratic_roots","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"For the quadratic equation 1x^2 -13x -3=0, let its roots be α and β. What is α²+β²?","options":{"A":"175","B":"173","C":"-175","D":"176"},"correct_answer":"A","correct_answer_value":"175","solution":"By Vieta's relations, α+β = 13/1 and αβ = -3/1. Then α²+β²=(α+β)²−2αβ = (13/1)²−2(-3/1) = 175.","option_explanations":{"A":"Correct. The worked calculation gives 175.","B":"Incorrect. Substituting the stated values into the governing relation gives 175, not 173.","C":"Incorrect. Substituting the stated values into the governing relation gives 175, not -175.","D":"Incorrect. Substituting the stated values into the governing relation gives 175, not 176."},"concepts_required":["Vieta's formulas","symmetric polynomial expressions"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_quadratic_vieta","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_004493","subject":"physics","topic":"rotational_mechanics","subtopic":"disc_angular_acceleration","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A uniform solid disc of mass 9 kg and radius 2 m is acted on by a constant torque of 24 N m about its central axis. Find its angular acceleration.","options":{"A":"3.33333 rad s⁻²","B":"1 rad s⁻²","C":"-1.33333 rad s⁻²","D":"1.33333 rad s⁻²"},"correct_answer":"D","correct_answer_value":"1.33333 rad s⁻²","solution":"For a solid disc, I=½MR²=½(9)(2)²=18 kg m². Using τ=Iα, α=24/18=1.33333 rad s⁻².","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 1.33333 rad s⁻², not 3.33333 rad s⁻².","B":"Incorrect. Substituting the stated values into the governing relation gives 1.33333 rad s⁻², not 1 rad s⁻².","C":"Incorrect. Substituting the stated values into the governing relation gives 1.33333 rad s⁻², not -1.33333 rad s⁻².","D":"Correct. The worked calculation gives 1.33333 rad s⁻²."},"concepts_required":["moment of inertia","rotational dynamics","torque"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_disc_torque","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_005487","subject":"physics","topic":"electrostatics","subtopic":"capacitor_energy","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A 61 μF capacitor is charged to a potential difference of 20 V. Calculate the energy stored, expressed in μJ.","options":{"A":"12200 μJ","B":"12199 μJ","C":"9150 μJ","D":"9760 μJ"},"correct_answer":"A","correct_answer_value":"12200 μJ","solution":"U=½CV². Since C=61 μF, U in μJ is ½×61×(20)²=12200 μJ.","option_explanations":{"A":"Correct. The worked calculation gives 12200 μJ.","B":"Incorrect. Substituting the stated values into the governing relation gives 12200 μJ, not 12199 μJ.","C":"Incorrect. Substituting the stated values into the governing relation gives 12200 μJ, not 9150 μJ.","D":"Incorrect. Substituting the stated values into the governing relation gives 12200 μJ, not 9760 μJ."},"concepts_required":["capacitance","energy stored in a capacitor"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_capacitor_energy","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_000093","subject":"mathematics","topic":"algebra","subtopic":"quadratic_roots","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"For the quadratic equation 6x^2 -7x -10=0, let its roots be α and β. What is α²+β²?","options":{"A":"1.46944","B":"4.69444","C":"7.04167","D":"7.69444"},"correct_answer":"B","correct_answer_value":"4.69444","solution":"By Vieta's relations, α+β = 7/6 and αβ = -10/6. Then α²+β²=(α+β)²−2αβ = (7/6)²−2(-10/6) = 4.69444.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 4.69444, not 1.46944.","B":"Correct. The worked calculation gives 4.69444.","C":"Incorrect. Substituting the stated values into the governing relation gives 4.69444, not 7.04167.","D":"Incorrect. Substituting the stated values into the governing relation gives 4.69444, not 7.69444."},"concepts_required":["Vieta's formulas","symmetric polynomial expressions"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_quadratic_vieta","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_002967","subject":"mathematics","topic":"vectors","subtopic":"dot_product","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Given vectors u=(3, -3, 2) and v=(6, -2, -1), calculate u·v.","options":{"A":"16.5","B":"44","C":"22","D":"11"},"correct_answer":"C","correct_answer_value":"22","solution":"The dot product is the sum of corresponding component products: (3)(6)+(-3)(-2)+(2)(-1)=22.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 22, not 16.5.","B":"Incorrect. Substituting the stated values into the governing relation gives 22, not 44.","C":"Correct. The worked calculation gives 22.","D":"Incorrect. Substituting the stated values into the governing relation gives 22, not 11."},"concepts_required":["three-dimensional vectors","scalar product"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_vector_dot","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_001816","subject":"mathematics","topic":"combinatorics","subtopic":"binomial_coefficient","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"How many distinct subsets containing exactly 8 elements can be selected from a set of 74 distinct elements?","options":{"A":"15071474661","B":"15071474660","C":"1507147467.1","D":"15071474659"},"correct_answer":"A","correct_answer_value":"15071474661","solution":"Order does not matter, so the count is C(74,8)=74!/[8!(66)!]=15071474661.","option_explanations":{"A":"Correct. The worked calculation gives 15071474661.","B":"Incorrect. Substituting the stated values into the governing relation gives 15071474661, not 15071474660.","C":"Incorrect. Substituting the stated values into the governing relation gives 15071474661, not 1507147467.1.","D":"Incorrect. Substituting the stated values into the governing relation gives 15071474661, not 15071474659."},"concepts_required":["combinations","factorials"],"reasoning_steps":2,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_combinations","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_008219","subject":"chemistry","topic":"physical_chemistry","subtopic":"faradays_law_copper","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"An aqueous Cu²⁺ solution is electrolysed using a current of 2 A for 16151 s. Assuming 100% current efficiency, calculate the mass of copper deposited. Use M(Cu)=63.5 g mol⁻¹ and F=96500 C mol⁻¹.","options":{"A":"10.62786 g","B":"8.62786 g","C":"7.9709 g","D":"11.62786 g"},"correct_answer":"A","correct_answer_value":"10.62786 g","solution":"Cu²⁺+2e⁻→Cu, so z=2. By Faraday's law m=MIt/(zF)=63.5×2×16151/(2×96500)=10.62786 g.","option_explanations":{"A":"Correct. The worked calculation gives 10.62786 g.","B":"Incorrect. Substituting the stated values into the governing relation gives 10.62786 g, not 8.62786 g.","C":"Incorrect. Substituting the stated values into the governing relation gives 10.62786 g, not 7.9709 g.","D":"Incorrect. Substituting the stated values into the governing relation gives 10.62786 g, not 11.62786 g."},"concepts_required":["electrolysis","Faraday's laws","redox stoichiometry"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_faraday_copper","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_006369","subject":"physics","topic":"oscillations","subtopic":"mass_spring_period_over_pi","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A mass of 4 kg is attached to an ideal spring of force constant 1157 N m⁻¹ and oscillates horizontally without damping. What is the time period divided by π?","options":{"A":"0.0588 s","B":"1.1176 s","C":"2.1176 s","D":"0.1176 s"},"correct_answer":"D","correct_answer_value":"0.1176 s","solution":"For a mass–spring oscillator, T=2π√(m/k). Therefore T/π=2√(4/1157)=0.1176 s.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.1176 s, not 0.0588 s.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.1176 s, not 1.1176 s.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.1176 s, not 2.1176 s.","D":"Correct. The worked calculation gives 0.1176 s."},"concepts_required":["simple harmonic motion","mass–spring oscillator","time period"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_shm_period","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_007763","subject":"chemistry","topic":"physical_chemistry","subtopic":"first_order_half_life","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A first-order reaction has a half-life of 20 minutes. If the initial reactant concentration is 50 mmol L⁻¹, what concentration remains after 20 minutes?","options":{"A":"20 mmol L⁻¹","B":"24 mmol L⁻¹","C":"37.5 mmol L⁻¹","D":"25 mmol L⁻¹"},"correct_answer":"D","correct_answer_value":"25 mmol L⁻¹","solution":"20 minutes corresponds to 1 half-lives. The remaining concentration is [A]₀(1/2)ⁿ=50/2^1=25 mmol L⁻¹.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 25 mmol L⁻¹, not 20 mmol L⁻¹.","B":"Incorrect. Substituting the stated values into the governing relation gives 25 mmol L⁻¹, not 24 mmol L⁻¹.","C":"Incorrect. Substituting the stated values into the governing relation gives 25 mmol L⁻¹, not 37.5 mmol L⁻¹.","D":"Correct. The worked calculation gives 25 mmol L⁻¹."},"concepts_required":["first-order kinetics","half-life","exponential decay"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_first_order_halflife","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_005945","subject":"physics","topic":"thermodynamics","subtopic":"ideal_gas_isochoric_pressure","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A fixed amount of ideal gas is kept in a rigid container. Its pressure is 4.9 atm at 229 K. What is its pressure at 623 K?","options":{"A":"26.66114 atm","B":"19.99585 atm","C":"10.33057 atm","D":"13.33057 atm"},"correct_answer":"D","correct_answer_value":"13.33057 atm","solution":"At constant volume, P/T is constant. P₂=P₁T₂/T₁=(4.9)(623)/(229)=13.33057 atm.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 13.33057 atm, not 26.66114 atm.","B":"Incorrect. Substituting the stated values into the governing relation gives 13.33057 atm, not 19.99585 atm.","C":"Incorrect. Substituting the stated values into the governing relation gives 13.33057 atm, not 10.33057 atm.","D":"Correct. The worked calculation gives 13.33057 atm."},"concepts_required":["ideal gas law","isochoric process","absolute temperature"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_isochoric_gas","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_003457","subject":"physics","topic":"mechanics","subtopic":"projectile_range","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A projectile is launched from level ground at 46 m s⁻¹ at 18° to the horizontal. Take g=10 m s⁻². Find its horizontal range.","options":{"A":"13.43754 m","B":"124.37536 m","C":"126.37536 m","D":"62.18768 m"},"correct_answer":"B","correct_answer_value":"124.37536 m","solution":"Range R=u²sin(2θ)/g. Substituting u=46, θ=18° and g=10 gives R=(46)²sin(36°)/10=124.37536 m.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 124.37536 m, not 13.43754 m.","B":"Correct. The worked calculation gives 124.37536 m.","C":"Incorrect. Substituting the stated values into the governing relation gives 124.37536 m, not 126.37536 m.","D":"Incorrect. Substituting the stated values into the governing relation gives 124.37536 m, not 62.18768 m."},"concepts_required":["projectile motion","range equation","trigonometry"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_projectile_range","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_008938","subject":"chemistry","topic":"physical_chemistry","subtopic":"equilibrium_degree_dissociation","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"For A(g) ⇌ 2B(g), the initial concentration of A is 1.09 mol L⁻¹ and initially B is absent. At equilibrium, 77% of A has dissociated at constant volume. Find K_c.","options":{"A":"9.23932","B":"5.61966","C":"-11.23932","D":"11.23932"},"correct_answer":"D","correct_answer_value":"11.23932","solution":"Let α=0.77. At equilibrium [A]=c₀(1−α) and [B]=2c₀α. Thus K_c=[B]²/[A]=4c₀α²/(1−α)=4(1.09)(0.77)²/(1−0.77)=11.23932.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 11.23932, not 9.23932.","B":"Incorrect. Substituting the stated values into the governing relation gives 11.23932, not 5.61966.","C":"Incorrect. Substituting the stated values into the governing relation gives 11.23932, not -11.23932.","D":"Correct. The worked calculation gives 11.23932."},"concepts_required":["chemical equilibrium","equilibrium constant","degree of dissociation"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_equilibrium_alpha","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_004670","subject":"physics","topic":"rotational_mechanics","subtopic":"disc_angular_acceleration","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A uniform solid disc of mass 2 kg and radius 6 m is acted on by a constant torque of 63 N m about its central axis. Find its angular acceleration.","options":{"A":"0.875 rad s⁻²","B":"2.625 rad s⁻²","C":"1.75 rad s⁻²","D":"-1.75 rad s⁻²"},"correct_answer":"C","correct_answer_value":"1.75 rad s⁻²","solution":"For a solid disc, I=½MR²=½(2)(6)²=36 kg m². Using τ=Iα, α=63/36=1.75 rad s⁻².","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 1.75 rad s⁻², not 0.875 rad s⁻².","B":"Incorrect. Substituting the stated values into the governing relation gives 1.75 rad s⁻², not 2.625 rad s⁻².","C":"Correct. The worked calculation gives 1.75 rad s⁻².","D":"Incorrect. Substituting the stated values into the governing relation gives 1.75 rad s⁻², not -1.75 rad s⁻²."},"concepts_required":["moment of inertia","rotational dynamics","torque"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_disc_torque","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_008461","subject":"chemistry","topic":"physical_chemistry","subtopic":"boiling_point_elevation","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A non-volatile solute is dissolved in water to produce a solution of molality 1.52 mol kg⁻¹. If its van't Hoff factor is 4 and K_b for water is 0.512 K kg mol⁻¹, find the boiling-point elevation.","options":{"A":"5.11296 K","B":"3.11296 K","C":"6.22592 K","D":"0.11296 K"},"correct_answer":"B","correct_answer_value":"3.11296 K","solution":"ΔT_b=iK_bm. Substitution gives ΔT_b=4×0.512×1.52=3.11296 K.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 3.11296 K, not 5.11296 K.","B":"Correct. The worked calculation gives 3.11296 K.","C":"Incorrect. Substituting the stated values into the governing relation gives 3.11296 K, not 6.22592 K.","D":"Incorrect. Substituting the stated values into the governing relation gives 3.11296 K, not 0.11296 K."},"concepts_required":["colligative properties","van't Hoff factor","molality"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_boiling_elevation","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_008513","subject":"chemistry","topic":"physical_chemistry","subtopic":"boiling_point_elevation","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A non-volatile solute is dissolved in water to produce a solution of molality 0.38 mol kg⁻¹. If its van't Hoff factor is 3 and K_b for water is 1.0 K kg mol⁻¹, find the boiling-point elevation.","options":{"A":"2.14 K","B":"0.855 K","C":"3.14 K","D":"1.14 K"},"correct_answer":"D","correct_answer_value":"1.14 K","solution":"ΔT_b=iK_bm. Substitution gives ΔT_b=3×1.0×0.38=1.14 K.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 1.14 K, not 2.14 K.","B":"Incorrect. Substituting the stated values into the governing relation gives 1.14 K, not 0.855 K.","C":"Incorrect. Substituting the stated values into the governing relation gives 1.14 K, not 3.14 K.","D":"Correct. The worked calculation gives 1.14 K."},"concepts_required":["colligative properties","van't Hoff factor","molality"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_boiling_elevation","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_001057","subject":"mathematics","topic":"calculus","subtopic":"polynomial_differentiation","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Let f(x)=4x³ -7x² +2x+7. Evaluate f′(-3).","options":{"A":"182.4","B":"152","C":"76","D":"228"},"correct_answer":"B","correct_answer_value":"152","solution":"Differentiate: f′(x)=3(4)x²+2(-7)x+(2). At x=-3, f′(-3)=3(4)(-3)²+2(-7)(-3)+(2)=152.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 152, not 182.4.","B":"Correct. The worked calculation gives 152.","C":"Incorrect. Substituting the stated values into the governing relation gives 152, not 76.","D":"Incorrect. Substituting the stated values into the governing relation gives 152, not 228."},"concepts_required":["power rule","polynomial derivative"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_derivative_cubic","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_001457","subject":"mathematics","topic":"calculus","subtopic":"definite_integral_polynomial","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Evaluate the definite integral ∫₀^8 (-2x² -2x +7) dx.","options":{"A":"-279.46667","B":"-262","C":"-349.33333","D":"-347.33333"},"correct_answer":"C","correct_answer_value":"-349.33333","solution":"An antiderivative is (-2/3)x³+(-2/2)x²+(7)x. Evaluating between 0 and 8 gives (-2/3)(8)³+(-2/2)(8)²+(7)(8)=-349.33333.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives -349.33333, not -279.46667.","B":"Incorrect. Substituting the stated values into the governing relation gives -349.33333, not -262.","C":"Correct. The worked calculation gives -349.33333.","D":"Incorrect. Substituting the stated values into the governing relation gives -349.33333, not -347.33333."},"concepts_required":["definite integration","power rule","limits of integration"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_integral_poly","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_007225","subject":"chemistry","topic":"physical_chemistry","subtopic":"ideal_gas_volume","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Assuming ideal behaviour, what volume is occupied by 16 mol of gas at 306 K and 0.49 atm? Use R=0.082057 L atm mol⁻¹ K⁻¹.","options":{"A":"820.90015 L","B":"614.92511 L","C":"819.90015 L","D":"1229.85022 L"},"correct_answer":"C","correct_answer_value":"819.90015 L","solution":"PV=nRT, so V=nRT/P=(16)(0.082057)(306)/(0.49)=819.90015 L.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 819.90015 L, not 820.90015 L.","B":"Incorrect. Substituting the stated values into the governing relation gives 819.90015 L, not 614.92511 L.","C":"Correct. The worked calculation gives 819.90015 L.","D":"Incorrect. Substituting the stated values into the governing relation gives 819.90015 L, not 1229.85022 L."},"concepts_required":["ideal gas equation","unit consistency"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_ideal_gas","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_005175","subject":"physics","topic":"electrostatics","subtopic":"coulombs_law","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Two point charges of +8 μC and +3 μC are separated by 12 cm in vacuum. Using k=9×10⁹ N m² C⁻², find the magnitude of the electrostatic force.","options":{"A":"15 N","B":"13 N","C":"14 N","D":"18 N"},"correct_answer":"A","correct_answer_value":"15 N","solution":"Convert charges to coulombs and distance to metres. F=k|q₁q₂|/r²=9×10⁹(8×10⁻⁶)(3×10⁻⁶)/(12×10⁻²)²=15 N.","option_explanations":{"A":"Correct. The worked calculation gives 15 N.","B":"Incorrect. Substituting the stated values into the governing relation gives 15 N, not 13 N.","C":"Incorrect. Substituting the stated values into the governing relation gives 15 N, not 14 N.","D":"Incorrect. Substituting the stated values into the governing relation gives 15 N, not 18 N."},"concepts_required":["Coulomb's law","SI unit conversion","inverse-square law"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_coulomb","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_009477","subject":"chemistry","topic":"organic_chemistry","subtopic":"degree_of_unsaturation","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Assuming a neutral closed-shell formula, calculate the degree of unsaturation (DBE) for C_11H_9X_1. Treat X as the total number of monovalent halogen atoms.","options":{"A":"-7","B":"6","C":"10.5","D":"7"},"correct_answer":"D","correct_answer_value":"7","solution":"Use DBE=(2C+2+N−H−X)/2. Substitution gives (2×11+2+0−9−1)/2=7.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 7, not -7.","B":"Incorrect. Substituting the stated values into the governing relation gives 7, not 6.","C":"Incorrect. Substituting the stated values into the governing relation gives 7, not 10.5.","D":"Correct. The worked calculation gives 7."},"concepts_required":["molecular formula","degree of unsaturation","organic structure"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_dbe","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_005132","subject":"physics","topic":"electrostatics","subtopic":"coulombs_law","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Two point charges of +9 μC and +8 μC are separated by 9 cm in vacuum. Using k=9×10⁹ N m² C⁻², find the magnitude of the electrostatic force.","options":{"A":"80 N","B":"40 N","C":"79 N","D":"96 N"},"correct_answer":"A","correct_answer_value":"80 N","solution":"Convert charges to coulombs and distance to metres. F=k|q₁q₂|/r²=9×10⁹(9×10⁻⁶)(8×10⁻⁶)/(9×10⁻²)²=80 N.","option_explanations":{"A":"Correct. The worked calculation gives 80 N.","B":"Incorrect. Substituting the stated values into the governing relation gives 80 N, not 40 N.","C":"Incorrect. Substituting the stated values into the governing relation gives 80 N, not 79 N.","D":"Incorrect. Substituting the stated values into the governing relation gives 80 N, not 96 N."},"concepts_required":["Coulomb's law","SI unit conversion","inverse-square law"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_coulomb","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_001305","subject":"mathematics","topic":"calculus","subtopic":"polynomial_differentiation","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Let f(x)=-1x³ +1x² -7x+7. Evaluate f′(4).","options":{"A":"-47","B":"-48","C":"-70.5","D":"-23.5"},"correct_answer":"A","correct_answer_value":"-47","solution":"Differentiate: f′(x)=3(-1)x²+2(1)x+(-7). At x=4, f′(4)=3(-1)(4)²+2(1)(4)+(-7)=-47.","option_explanations":{"A":"Correct. The worked calculation gives -47.","B":"Incorrect. Substituting the stated values into the governing relation gives -47, not -48.","C":"Incorrect. Substituting the stated values into the governing relation gives -47, not -70.5.","D":"Incorrect. Substituting the stated values into the governing relation gives -47, not -23.5."},"concepts_required":["power rule","polynomial derivative"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_derivative_cubic","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_009240","subject":"chemistry","topic":"physical_chemistry","subtopic":"nernst_equation_298K","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"At 298 K, a galvanic cell has E°cell=1.27 V, transfers 1 electrons, and has reaction quotient Q=10^-5. Using E=E°−(0.0591/n)log₁₀Q, calculate Ecell.","options":{"A":"1.5655 V","B":"3.131 V","C":"0.78275 V","D":"2.34825 V"},"correct_answer":"A","correct_answer_value":"1.5655 V","solution":"Apply the Nernst equation: E=1.27−(0.0591/1)(-5)=1.5655 V.","option_explanations":{"A":"Correct. The worked calculation gives 1.5655 V.","B":"Incorrect. Substituting the stated values into the governing relation gives 1.5655 V, not 3.131 V.","C":"Incorrect. Substituting the stated values into the governing relation gives 1.5655 V, not 0.78275 V.","D":"Incorrect. Substituting the stated values into the governing relation gives 1.5655 V, not 2.34825 V."},"concepts_required":["electrochemical cells","Nernst equation","logarithms"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_nernst","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_009557","subject":"chemistry","topic":"organic_chemistry","subtopic":"degree_of_unsaturation","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Assuming a neutral closed-shell formula, calculate the degree of unsaturation (DBE) for C_17H_33N_2X_1. Treat X as the total number of monovalent halogen atoms.","options":{"A":"-2","B":"2","C":"3","D":"1"},"correct_answer":"B","correct_answer_value":"2","solution":"Use DBE=(2C+2+N−H−X)/2. Substitution gives (2×17+2+2−33−1)/2=2.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 2, not -2.","B":"Correct. The worked calculation gives 2.","C":"Incorrect. Substituting the stated values into the governing relation gives 2, not 3.","D":"Incorrect. Substituting the stated values into the governing relation gives 2, not 1."},"concepts_required":["molecular formula","degree of unsaturation","organic structure"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_dbe","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_005997","subject":"physics","topic":"thermodynamics","subtopic":"ideal_gas_isochoric_pressure","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A fixed amount of ideal gas is kept in a rigid container. Its pressure is 5 atm at 317 K. What is its pressure at 712 K?","options":{"A":"-11.23028 atm","B":"2.12303 atm","C":"13.23028 atm","D":"11.23028 atm"},"correct_answer":"D","correct_answer_value":"11.23028 atm","solution":"At constant volume, P/T is constant. P₂=P₁T₂/T₁=(5)(712)/(317)=11.23028 atm.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 11.23028 atm, not -11.23028 atm.","B":"Incorrect. Substituting the stated values into the governing relation gives 11.23028 atm, not 2.12303 atm.","C":"Incorrect. Substituting the stated values into the governing relation gives 11.23028 atm, not 13.23028 atm.","D":"Correct. The worked calculation gives 11.23028 atm."},"concepts_required":["ideal gas law","isochoric process","absolute temperature"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_isochoric_gas","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_006951","subject":"chemistry","topic":"stoichiometry","subtopic":"limiting_reagent_water","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Hydrogen reacts with oxygen according to 2H₂ + O₂ → 2H₂O. If 19 mol H₂ and 33 mol O₂ are mixed and the reaction goes to completion, how many moles of H₂O form?","options":{"A":"9.5 mol","B":"19 mol","C":"21 mol","D":"14.25 mol"},"correct_answer":"B","correct_answer_value":"19 mol","solution":"From 2 mol H₂ and 1 mol O₂, 2 mol H₂O form. H₂ can produce 19 mol H₂O; O₂ can produce 66 mol H₂O. The limiting condition is H₂; product=19 mol.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 19 mol, not 9.5 mol.","B":"Correct. The worked calculation gives 19 mol.","C":"Incorrect. Substituting the stated values into the governing relation gives 19 mol, not 21 mol.","D":"Incorrect. Substituting the stated values into the governing relation gives 19 mol, not 14.25 mol."},"concepts_required":["balanced chemical equations","limiting reagent","mole ratios"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_limiting_reagent","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_005130","subject":"physics","topic":"electrostatics","subtopic":"coulombs_law","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Two point charges of +4 μC and +1 μC are separated by 6 cm in vacuum. Using k=9×10⁹ N m² C⁻², find the magnitude of the electrostatic force.","options":{"A":"5 N","B":"8 N","C":"11 N","D":"10 N"},"correct_answer":"D","correct_answer_value":"10 N","solution":"Convert charges to coulombs and distance to metres. F=k|q₁q₂|/r²=9×10⁹(4×10⁻⁶)(1×10⁻⁶)/(6×10⁻²)²=10 N.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 10 N, not 5 N.","B":"Incorrect. Substituting the stated values into the governing relation gives 10 N, not 8 N.","C":"Incorrect. Substituting the stated values into the governing relation gives 10 N, not 11 N.","D":"Correct. The worked calculation gives 10 N."},"concepts_required":["Coulomb's law","SI unit conversion","inverse-square law"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_coulomb","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_008670","subject":"chemistry","topic":"physical_chemistry","subtopic":"equilibrium_degree_dissociation","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"For A(g) ⇌ 2B(g), the initial concentration of A is 1.76 mol L⁻¹ and initially B is absent. At equilibrium, 20% of A has dissociated at constant volume. Find K_c.","options":{"A":"0.528","B":"0.264","C":"0.352","D":"3.352"},"correct_answer":"C","correct_answer_value":"0.352","solution":"Let α=0.2. At equilibrium [A]=c₀(1−α) and [B]=2c₀α. Thus K_c=[B]²/[A]=4c₀α²/(1−α)=4(1.76)(0.2)²/(1−0.2)=0.352.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.352, not 0.528.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.352, not 0.264.","C":"Correct. The worked calculation gives 0.352.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.352, not 3.352."},"concepts_required":["chemical equilibrium","equilibrium constant","degree of dissociation"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_equilibrium_alpha","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_007088","subject":"chemistry","topic":"physical_chemistry","subtopic":"ideal_gas_volume","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Assuming ideal behaviour, what volume is occupied by 16 mol of gas at 300 K and 2.37 atm? Use R=0.082057 L atm mol⁻¹ K⁻¹.","options":{"A":"166.19139 L","B":"167.19139 L","C":"199.42967 L","D":"17.61914 L"},"correct_answer":"A","correct_answer_value":"166.19139 L","solution":"PV=nRT, so V=nRT/P=(16)(0.082057)(300)/(2.37)=166.19139 L.","option_explanations":{"A":"Correct. The worked calculation gives 166.19139 L.","B":"Incorrect. Substituting the stated values into the governing relation gives 166.19139 L, not 167.19139 L.","C":"Incorrect. Substituting the stated values into the governing relation gives 166.19139 L, not 199.42967 L.","D":"Incorrect. Substituting the stated values into the governing relation gives 166.19139 L, not 17.61914 L."},"concepts_required":["ideal gas equation","unit consistency"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_ideal_gas","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_002219","subject":"mathematics","topic":"probability","subtopic":"same_colour_without_replacement","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A bag contains 3 red, 3 blue and 5 green balls. Two balls are drawn uniformly at random without replacement. What is the probability that both have the same colour?","options":{"A":"-2.70909","B":"0.43636","C":"0.14545","D":"0.29091"},"correct_answer":"D","correct_answer_value":"0.29091","solution":"Favourable pairs = C(3,2)+C(3,2)+C(5,2)=16. Total pairs=C(11,2)=55. Therefore P=16/55=0.29091.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.29091, not -2.70909.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.29091, not 0.43636.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.29091, not 0.14545.","D":"Correct. The worked calculation gives 0.29091."},"concepts_required":["classical probability","combinations","without replacement"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_probability_colours","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_007984","subject":"chemistry","topic":"physical_chemistry","subtopic":"first_order_half_life","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A first-order reaction has a half-life of 20 minutes. If the initial reactant concentration is 28 mmol L⁻¹, what concentration remains after 40 minutes?","options":{"A":"10.5 mmol L⁻¹","B":"7 mmol L⁻¹","C":"6 mmol L⁻¹","D":"14 mmol L⁻¹"},"correct_answer":"B","correct_answer_value":"7 mmol L⁻¹","solution":"40 minutes corresponds to 2 half-lives. The remaining concentration is [A]₀(1/2)ⁿ=28/2^2=7 mmol L⁻¹.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 7 mmol L⁻¹, not 10.5 mmol L⁻¹.","B":"Correct. The worked calculation gives 7 mmol L⁻¹.","C":"Incorrect. Substituting the stated values into the governing relation gives 7 mmol L⁻¹, not 6 mmol L⁻¹.","D":"Incorrect. Substituting the stated values into the governing relation gives 7 mmol L⁻¹, not 14 mmol L⁻¹."},"concepts_required":["first-order kinetics","half-life","exponential decay"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_first_order_halflife","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_009285","subject":"chemistry","topic":"physical_chemistry","subtopic":"nernst_equation_298K","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"At 298 K, a galvanic cell has E°cell=1.10 V, transfers 3 electrons, and has reaction quotient Q=10^-2. Using E=E°−(0.0591/n)log₁₀Q, calculate Ecell.","options":{"A":"4.1394 V","B":"3.1394 V","C":"1.1394 V","D":"-1.1394 V"},"correct_answer":"C","correct_answer_value":"1.1394 V","solution":"Apply the Nernst equation: E=1.10−(0.0591/3)(-2)=1.1394 V.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 1.1394 V, not 4.1394 V.","B":"Incorrect. Substituting the stated values into the governing relation gives 1.1394 V, not 3.1394 V.","C":"Correct. The worked calculation gives 1.1394 V.","D":"Incorrect. Substituting the stated values into the governing relation gives 1.1394 V, not -1.1394 V."},"concepts_required":["electrochemical cells","Nernst equation","logarithms"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_nernst","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_008610","subject":"chemistry","topic":"physical_chemistry","subtopic":"boiling_point_elevation","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A non-volatile solute is dissolved in water to produce a solution of molality 0.2 mol kg⁻¹. If its van't Hoff factor is 3 and K_b for water is 0.512 K kg mol⁻¹, find the boiling-point elevation.","options":{"A":"-2.6928 K","B":"0.6144 K","C":"0.3072 K","D":"-1.6928 K"},"correct_answer":"C","correct_answer_value":"0.3072 K","solution":"ΔT_b=iK_bm. Substitution gives ΔT_b=3×0.512×0.2=0.3072 K.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.3072 K, not -2.6928 K.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.3072 K, not 0.6144 K.","C":"Correct. The worked calculation gives 0.3072 K.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.3072 K, not -1.6928 K."},"concepts_required":["colligative properties","van't Hoff factor","molality"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_boiling_elevation","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_000683","subject":"mathematics","topic":"sequences_and_series","subtopic":"geometric_progression","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A geometric progression has first term 12 and common ratio 4. What is the sum of its first 6 terms?","options":{"A":"24570","B":"16380","C":"8190","D":"1639"},"correct_answer":"B","correct_answer_value":"16380","solution":"For r≠1, Sₙ=a(rⁿ−1)/(r−1). Thus Sₙ=12(4^6−1)/(4−1)=16380.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 16380, not 24570.","B":"Correct. The worked calculation gives 16380.","C":"Incorrect. Substituting the stated values into the governing relation gives 16380, not 8190.","D":"Incorrect. Substituting the stated values into the governing relation gives 16380, not 1639."},"concepts_required":["geometric progression","finite geometric sum"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_gp_sum","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_008798","subject":"chemistry","topic":"physical_chemistry","subtopic":"equilibrium_degree_dissociation","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"For A(g) ⇌ 2B(g), the initial concentration of A is 1.29 mol L⁻¹ and initially B is absent. At equilibrium, 23% of A has dissociated at constant volume. Find K_c.","options":{"A":"0.3545","B":"-0.6455","C":"-2.6455","D":"0.53175"},"correct_answer":"A","correct_answer_value":"0.3545","solution":"Let α=0.23. At equilibrium [A]=c₀(1−α) and [B]=2c₀α. Thus K_c=[B]²/[A]=4c₀α²/(1−α)=4(1.29)(0.23)²/(1−0.23)=0.3545.","option_explanations":{"A":"Correct. The worked calculation gives 0.3545.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.3545, not -0.6455.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.3545, not -2.6455.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.3545, not 0.53175."},"concepts_required":["chemical equilibrium","equilibrium constant","degree of dissociation"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_equilibrium_alpha","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_005183","subject":"physics","topic":"electrostatics","subtopic":"coulombs_law","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Two point charges of +2 μC and +6 μC are separated by 6 cm in vacuum. Using k=9×10⁹ N m² C⁻², find the magnitude of the electrostatic force.","options":{"A":"4 N","B":"22.5 N","C":"30 N","D":"32 N"},"correct_answer":"C","correct_answer_value":"30 N","solution":"Convert charges to coulombs and distance to metres. F=k|q₁q₂|/r²=9×10⁹(2×10⁻⁶)(6×10⁻⁶)/(6×10⁻²)²=30 N.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 30 N, not 4 N.","B":"Incorrect. Substituting the stated values into the governing relation gives 30 N, not 22.5 N.","C":"Correct. The worked calculation gives 30 N.","D":"Incorrect. Substituting the stated values into the governing relation gives 30 N, not 32 N."},"concepts_required":["Coulomb's law","SI unit conversion","inverse-square law"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_coulomb","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_006199","subject":"physics","topic":"ray_optics","subtopic":"convex_lens_magnification","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A real object is placed 82 cm in front of a thin convex lens of focal length 17 cm. Using the Cartesian sign convention, what is the magnitude of the linear magnification?","options":{"A":"0.26154","B":"-2.73846","C":"1.02615","D":"0.52308"},"correct_answer":"A","correct_answer_value":"0.26154","solution":"Using distances as positive magnitudes, 1/f=1/v+1/u gives v=uf/(u−f)=(82)(17)/(82−17)=21.44615 cm. Thus |m|=v/u=21.44615/82=0.26154.","option_explanations":{"A":"Correct. The worked calculation gives 0.26154.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.26154, not -2.73846.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.26154, not 1.02615.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.26154, not 0.52308."},"concepts_required":["thin lens equation","linear magnification","real image"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_lens_magnification","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_002523","subject":"mathematics","topic":"linear_algebra","subtopic":"2x2_determinant","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Find the determinant of the matrix [[2, -1], [9, -5]].","options":{"A":"-4","B":"-1","C":"-3","D":"2"},"correct_answer":"B","correct_answer_value":"-1","solution":"For a 2×2 matrix [[a,b],[c,d]], det=ad−bc. Here it is (2)(-5)−(-1)(9)=-1.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives -1, not -4.","B":"Correct. The worked calculation gives -1.","C":"Incorrect. Substituting the stated values into the governing relation gives -1, not -3.","D":"Incorrect. Substituting the stated values into the governing relation gives -1, not 2."},"concepts_required":["determinants","matrix operations"],"reasoning_steps":2,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_determinant_2x2","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_003704","subject":"physics","topic":"mechanics","subtopic":"work_energy_theorem","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A 2 kg block initially moves at 7 m s⁻¹. A constant net force does 9 N of work over 13 m in the direction of motion. Find its final kinetic energy.","options":{"A":"165 J","B":"164 J","C":"132.8 J","D":"166 J"},"correct_answer":"D","correct_answer_value":"166 J","solution":"By the work–energy theorem, K_f=K_i+W=½mu²+Fs=½(2)(7)²+(9)(13)=166 J.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 166 J, not 165 J.","B":"Incorrect. Substituting the stated values into the governing relation gives 166 J, not 164 J.","C":"Incorrect. Substituting the stated values into the governing relation gives 166 J, not 132.8 J.","D":"Correct. The worked calculation gives 166 J."},"concepts_required":["work–energy theorem","kinetic energy","work by constant force"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_work_energy","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_001548","subject":"mathematics","topic":"calculus","subtopic":"definite_integral_polynomial","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Evaluate the definite integral ∫₀^2 (-2x² -3x -1) dx.","options":{"A":"-10.33333","B":"-20","C":"-13.33333","D":"-0.33333"},"correct_answer":"C","correct_answer_value":"-13.33333","solution":"An antiderivative is (-2/3)x³+(-3/2)x²+(-1)x. Evaluating between 0 and 2 gives (-2/3)(2)³+(-3/2)(2)²+(-1)(2)=-13.33333.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives -13.33333, not -10.33333.","B":"Incorrect. Substituting the stated values into the governing relation gives -13.33333, not -20.","C":"Correct. The worked calculation gives -13.33333.","D":"Incorrect. Substituting the stated values into the governing relation gives -13.33333, not -0.33333."},"concepts_required":["definite integration","power rule","limits of integration"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_integral_poly","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_001009","subject":"mathematics","topic":"calculus","subtopic":"polynomial_differentiation","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Let f(x)=-2x³ +3x² -12x+7. Evaluate f′(1).","options":{"A":"-9","B":"-0.2","C":"-13","D":"-12"},"correct_answer":"D","correct_answer_value":"-12","solution":"Differentiate: f′(x)=3(-2)x²+2(3)x+(-12). At x=1, f′(1)=3(-2)(1)²+2(3)(1)+(-12)=-12.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives -12, not -9.","B":"Incorrect. Substituting the stated values into the governing relation gives -12, not -0.2.","C":"Incorrect. Substituting the stated values into the governing relation gives -12, not -13.","D":"Correct. The worked calculation gives -12."},"concepts_required":["power rule","polynomial derivative"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_derivative_cubic","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_007447","subject":"chemistry","topic":"physical_chemistry","subtopic":"strong_acid_base_mixture","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"22 mL of 0.10 M HCl is mixed with 86 mL of 0.10 M NaOH at 25°C. Assuming complete dissociation and additive volumes, find the final pH.","options":{"A":"25.54551","B":"6.38638","C":"-12.77276","D":"12.77276"},"correct_answer":"D","correct_answer_value":"12.77276","solution":"Excess OH⁻ moles=0.10×(86−22)/1000=0.006400 mol. Total volume=0.108 L, so [OH⁻]=0.059259 M, pOH=−log₁₀[OH⁻], and pH=14−pOH=12.77276.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 12.77276, not 25.54551.","B":"Incorrect. Substituting the stated values into the governing relation gives 12.77276, not 6.38638.","C":"Incorrect. Substituting the stated values into the governing relation gives 12.77276, not -12.77276.","D":"Correct. The worked calculation gives 12.77276."},"concepts_required":["strong acid–strong base neutralization","molarity","pH and pOH"],"reasoning_steps":5,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_acid_base_pH","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_003549","subject":"physics","topic":"mechanics","subtopic":"projectile_range","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A projectile is launched from level ground at 32 m s⁻¹ at 75° to the horizontal. Take g=10 m s⁻². Find its horizontal range.","options":{"A":"6.12 m","B":"53.2 m","C":"50.2 m","D":"51.2 m"},"correct_answer":"D","correct_answer_value":"51.2 m","solution":"Range R=u²sin(2θ)/g. Substituting u=32, θ=75° and g=10 gives R=(32)²sin(150°)/10=51.2 m.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 51.2 m, not 6.12 m.","B":"Incorrect. Substituting the stated values into the governing relation gives 51.2 m, not 53.2 m.","C":"Incorrect. Substituting the stated values into the governing relation gives 51.2 m, not 50.2 m.","D":"Correct. The worked calculation gives 51.2 m."},"concepts_required":["projectile motion","range equation","trigonometry"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_projectile_range","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_002143","subject":"mathematics","topic":"probability","subtopic":"same_colour_without_replacement","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A bag contains 10 red, 12 blue and 3 green balls. Two balls are drawn uniformly at random without replacement. What is the probability that both have the same colour?","options":{"A":"0.57","B":"-0.62","C":"-1.62","D":"0.38"},"correct_answer":"D","correct_answer_value":"0.38","solution":"Favourable pairs = C(10,2)+C(12,2)+C(3,2)=114. Total pairs=C(25,2)=300. Therefore P=114/300=0.38.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.38, not 0.57.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.38, not -0.62.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.38, not -1.62.","D":"Correct. The worked calculation gives 0.38."},"concepts_required":["classical probability","combinations","without replacement"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_probability_colours","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_001239","subject":"mathematics","topic":"calculus","subtopic":"polynomial_differentiation","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Let f(x)=-4x³ -9x² +1x+7. Evaluate f′(0).","options":{"A":"1","B":"-1","C":"0","D":"0.5"},"correct_answer":"A","correct_answer_value":"1","solution":"Differentiate: f′(x)=3(-4)x²+2(-9)x+(1). At x=0, f′(0)=3(-4)(0)²+2(-9)(0)+(1)=1.","option_explanations":{"A":"Correct. The worked calculation gives 1.","B":"Incorrect. Substituting the stated values into the governing relation gives 1, not -1.","C":"Incorrect. Substituting the stated values into the governing relation gives 1, not 0.","D":"Incorrect. Substituting the stated values into the governing relation gives 1, not 0.5."},"concepts_required":["power rule","polynomial derivative"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_derivative_cubic","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_001438","subject":"mathematics","topic":"calculus","subtopic":"definite_integral_polynomial","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Evaluate the definite integral ∫₀^3 (3x² +1x +4) dx.","options":{"A":"5.35","B":"34.8","C":"43.5","D":"44.5"},"correct_answer":"C","correct_answer_value":"43.5","solution":"An antiderivative is (3/3)x³+(1/2)x²+(4)x. Evaluating between 0 and 3 gives (3/3)(3)³+(1/2)(3)²+(4)(3)=43.5.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 43.5, not 5.35.","B":"Incorrect. Substituting the stated values into the governing relation gives 43.5, not 34.8.","C":"Correct. The worked calculation gives 43.5.","D":"Incorrect. Substituting the stated values into the governing relation gives 43.5, not 44.5."},"concepts_required":["definite integration","power rule","limits of integration"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_integral_poly","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_004408","subject":"physics","topic":"rotational_mechanics","subtopic":"disc_angular_acceleration","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A uniform solid disc of mass 3 kg and radius 1 m is acted on by a constant torque of 69 N m about its central axis. Find its angular acceleration.","options":{"A":"36.8 rad s⁻²","B":"69 rad s⁻²","C":"5.6 rad s⁻²","D":"46 rad s⁻²"},"correct_answer":"D","correct_answer_value":"46 rad s⁻²","solution":"For a solid disc, I=½MR²=½(3)(1)²=1.5 kg m². Using τ=Iα, α=69/1.5=46 rad s⁻².","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 46 rad s⁻², not 36.8 rad s⁻².","B":"Incorrect. Substituting the stated values into the governing relation gives 46 rad s⁻², not 69 rad s⁻².","C":"Incorrect. Substituting the stated values into the governing relation gives 46 rad s⁻², not 5.6 rad s⁻².","D":"Correct. The worked calculation gives 46 rad s⁻²."},"concepts_required":["moment of inertia","rotational dynamics","torque"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_disc_torque","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_007614","subject":"chemistry","topic":"physical_chemistry","subtopic":"strong_acid_base_mixture","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"64 mL of 0.10 M HCl is mixed with 64 mL of 0.10 M NaOH at 25°C. Assuming complete dissociation and additive volumes, find the final pH.","options":{"A":"6","B":"7","C":"10","D":"5.25"},"correct_answer":"B","correct_answer_value":"7","solution":"Equal moles of strong acid and base are present. They neutralize completely, so at 25°C pH=7.00.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 7, not 6.","B":"Correct. The worked calculation gives 7.","C":"Incorrect. Substituting the stated values into the governing relation gives 7, not 10.","D":"Incorrect. Substituting the stated values into the governing relation gives 7, not 5.25."},"concepts_required":["strong acid–strong base neutralization","molarity","pH and pOH"],"reasoning_steps":5,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_acid_base_pH","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_007342","subject":"chemistry","topic":"physical_chemistry","subtopic":"strong_acid_base_mixture","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"42 mL of 0.10 M HCl is mixed with 92 mL of 0.10 M NaOH at 25°C. Assuming complete dissociation and additive volumes, find the final pH.","options":{"A":"14.57187","B":"12.57187","C":"6.28593","D":"15.57187"},"correct_answer":"B","correct_answer_value":"12.57187","solution":"Excess OH⁻ moles=0.10×(92−42)/1000=0.005000 mol. Total volume=0.134 L, so [OH⁻]=0.037313 M, pOH=−log₁₀[OH⁻], and pH=14−pOH=12.57187.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 12.57187, not 14.57187.","B":"Correct. The worked calculation gives 12.57187.","C":"Incorrect. Substituting the stated values into the governing relation gives 12.57187, not 6.28593.","D":"Incorrect. Substituting the stated values into the governing relation gives 12.57187, not 15.57187."},"concepts_required":["strong acid–strong base neutralization","molarity","pH and pOH"],"reasoning_steps":5,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_acid_base_pH","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_003427","subject":"physics","topic":"mechanics","subtopic":"projectile_range","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A projectile is launched from level ground at 77 m s⁻¹ at 60° to the horizontal. Take g=10 m s⁻². Find its horizontal range.","options":{"A":"52.34665 m","B":"385.09985 m","C":"1026.93292 m","D":"513.46646 m"},"correct_answer":"D","correct_answer_value":"513.46646 m","solution":"Range R=u²sin(2θ)/g. Substituting u=77, θ=60° and g=10 gives R=(77)²sin(120°)/10=513.46646 m.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 513.46646 m, not 52.34665 m.","B":"Incorrect. Substituting the stated values into the governing relation gives 513.46646 m, not 385.09985 m.","C":"Incorrect. Substituting the stated values into the governing relation gives 513.46646 m, not 1026.93292 m.","D":"Correct. The worked calculation gives 513.46646 m."},"concepts_required":["projectile motion","range equation","trigonometry"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"physics_projectile_range","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_001576","subject":"mathematics","topic":"calculus","subtopic":"definite_integral_polynomial","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Evaluate the definite integral ∫₀^8 (1x² +6x -5) dx.","options":{"A":"-322.66667","B":"484","C":"324.66667","D":"322.66667"},"correct_answer":"D","correct_answer_value":"322.66667","solution":"An antiderivative is (1/3)x³+(6/2)x²+(-5)x. Evaluating between 0 and 8 gives (1/3)(8)³+(6/2)(8)²+(-5)(8)=322.66667.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 322.66667, not -322.66667.","B":"Incorrect. Substituting the stated values into the governing relation gives 322.66667, not 484.","C":"Incorrect. Substituting the stated values into the governing relation gives 322.66667, not 324.66667.","D":"Correct. The worked calculation gives 322.66667."},"concepts_required":["definite integration","power rule","limits of integration"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_integral_poly","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_009738","subject":"chemistry","topic":"inorganic_chemistry","subtopic":"complex_ion_oxidation_state","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"For a hypothetical coordination ion [Fe(CN)_2]^-2, each CN ligand has charge −1. Determine the formal oxidation state of Fe.","options":{"A":"1","B":"0","C":"-3","D":"3"},"correct_answer":"B","correct_answer_value":"0","solution":"Let the formal oxidation state of Fe be x. Charge balance gives x+(2)(−1)=-2; hence x=-2+2=+0.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0, not 1.","B":"Correct. The worked calculation gives 0.","C":"Incorrect. Substituting the stated values into the governing relation gives 0, not -3.","D":"Incorrect. Substituting the stated values into the governing relation gives 0, not 3."},"concepts_required":["coordination compounds","oxidation state","charge balance"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_complex_oxidation","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_001480","subject":"mathematics","topic":"calculus","subtopic":"definite_integral_polynomial","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Evaluate the definite integral ∫₀^4 (4x² -1x -5) dx.","options":{"A":"6.73333","B":"114.66667","C":"86","D":"57.33333"},"correct_answer":"D","correct_answer_value":"57.33333","solution":"An antiderivative is (4/3)x³+(-1/2)x²+(-5)x. Evaluating between 0 and 4 gives (4/3)(4)³+(-1/2)(4)²+(-5)(4)=57.33333.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 57.33333, not 6.73333.","B":"Incorrect. Substituting the stated values into the governing relation gives 57.33333, not 114.66667.","C":"Incorrect. Substituting the stated values into the governing relation gives 57.33333, not 86.","D":"Correct. The worked calculation gives 57.33333."},"concepts_required":["definite integration","power rule","limits of integration"],"reasoning_steps":4,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"math_integral_poly","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}
+{"id":"openstem_009647","subject":"chemistry","topic":"organic_chemistry","subtopic":"degree_of_unsaturation","difficulty":"very_hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"Assuming a neutral closed-shell formula, calculate the degree of unsaturation (DBE) for C_14H_7N_2X_3. Treat X as the total number of monovalent halogen atoms.","options":{"A":"22","B":"8","C":"-11","D":"11"},"correct_answer":"D","correct_answer_value":"11","solution":"Use DBE=(2C+2+N−H−X)/2. Substitution gives (2×14+2+2−7−3)/2=11.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 11, not 22.","B":"Incorrect. Substituting the stated values into the governing relation gives 11, not 8.","C":"Incorrect. Substituting the stated values into the governing relation gives 11, not -11.","D":"Correct. The worked calculation gives 11."},"concepts_required":["molecular formula","degree of unsaturation","organic structure"],"reasoning_steps":3,"answer_type":"numeric","language":"en","origin":"synthetic_parametric","generator":"AxiloLabz deterministic STEM question generator","generator_template_id":"chem_dbe","verification_status":"template_formula_generated_structurally_validated_not_expert_reviewed","verification_note":"Answer is computed by the explicit equation in the generator template; this is not an independent expert proof.","difficulty_note":"Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1.","source_type":"original_synthetic_question","answer_validation":"correct_answer_option_matches_generated_numeric_value"}