diff --git "a/data/test.jsonl" "b/data/test.jsonl"
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+++ "b/data/test.jsonl"
@@ -0,0 +1,500 @@
+{"id":"openstem_002745","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=(9, 9, 3) and v=(2, 4, -6), calculate u·v.","options":{"A":"4.6","B":"72","C":"-36","D":"36"},"correct_answer":"D","correct_answer_value":"36","solution":"The dot product is the sum of corresponding component products: (9)(2)+(9)(4)+(3)(-6)=36.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 36, not 4.6.","B":"Incorrect. Substituting the stated values into the governing relation gives 36, not 72.","C":"Incorrect. Substituting the stated values into the governing relation gives 36, not -36.","D":"Correct. The worked calculation gives 36."},"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_007847","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 33 mmol L⁻¹, what concentration remains after 120 minutes?","options":{"A":"-0.51562 mmol L⁻¹","B":"0.51562 mmol L⁻¹","C":"3.51562 mmol L⁻¹","D":"-2.48438 mmol L⁻¹"},"correct_answer":"B","correct_answer_value":"0.51562 mmol L⁻¹","solution":"120 minutes corresponds to 6 half-lives. The remaining concentration is [A]₀(1/2)ⁿ=33/2^6=0.51562 mmol L⁻¹.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.51562 mmol L⁻¹, not -0.51562 mmol L⁻¹.","B":"Correct. The worked calculation gives 0.51562 mmol L⁻¹.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.51562 mmol L⁻¹, not 3.51562 mmol L⁻¹.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.51562 mmol L⁻¹, not -2.48438 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_006519","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 17 kg is attached to an ideal spring of force constant 899 N m⁻¹ and oscillates horizontally without damping. What is the time period divided by π?","options":{"A":"0.20627 s","B":"0.27503 s","C":"-0.72497 s","D":"0.13751 s"},"correct_answer":"B","correct_answer_value":"0.27503 s","solution":"For a mass–spring oscillator, T=2π√(m/k). Therefore T/π=2√(17/899)=0.27503 s.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.27503 s, not 0.20627 s.","B":"Correct. The worked calculation gives 0.27503 s.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.27503 s, not -0.72497 s.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.27503 s, not 0.13751 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_001639","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 ∫₀^5 (5x² -6x +7) dx.","options":{"A":"166.33333","B":"168.33333","C":"252.5","D":"84.16667"},"correct_answer":"B","correct_answer_value":"168.33333","solution":"An antiderivative is (5/3)x³+(-6/2)x²+(7)x. Evaluating between 0 and 5 gives (5/3)(5)³+(-6/2)(5)²+(7)(5)=168.33333.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 168.33333, not 166.33333.","B":"Correct. The worked calculation gives 168.33333.","C":"Incorrect. Substituting the stated values into the governing relation gives 168.33333, not 252.5.","D":"Incorrect. Substituting the stated values into the governing relation gives 168.33333, not 84.16667."},"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_006708","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 2 mol O₂ are mixed and the reaction goes to completion, how many moles of H₂O form?","options":{"A":"1.4 mol","B":"6 mol","C":"2 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 34 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 1.4 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 2 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_004736","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 11 Ω are connected in series; this combination is connected in parallel with a 5 Ω resistor across a 30 V ideal battery. Find the total current supplied by the battery.","options":{"A":"6.66667 A","B":"5.66667 A","C":"8.66667 A","D":"7.66667 A"},"correct_answer":"D","correct_answer_value":"7.66667 A","solution":"Series branch resistance=7+11=18 Ω. Parallel equivalent R_eq=(18×5)/(18+5)=3.91304 Ω. Total current I=V/R_eq=30/3.91304=7.66667 A.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 7.66667 A, not 6.66667 A.","B":"Incorrect. Substituting the stated values into the governing relation gives 7.66667 A, not 5.66667 A.","C":"Incorrect. Substituting the stated values into the governing relation gives 7.66667 A, not 8.66667 A.","D":"Correct. The worked calculation gives 7.66667 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_001080","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 +6x² +8x+7. Evaluate f′(1).","options":{"A":"4.5","B":"33","C":"52.5","D":"35"},"correct_answer":"D","correct_answer_value":"35","solution":"Differentiate: f′(x)=3(5)x²+2(6)x+(8). At x=1, f′(1)=3(5)(1)²+2(6)(1)+(8)=35.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 35, not 4.5.","B":"Incorrect. Substituting the stated values into the governing relation gives 35, not 33.","C":"Incorrect. Substituting the stated values into the governing relation gives 35, not 52.5.","D":"Correct. The worked calculation gives 35."},"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_001846","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 11 elements can be selected from a set of 68 distinct elements?","options":{"A":"1149793519368","B":"1533058025822","C":"1533058025824","D":"153305802583.39999"},"correct_answer":"C","correct_answer_value":"1533058025824","solution":"Order does not matter, so the count is C(68,11)=68!/[11!(57)!]=1533058025824.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 1533058025824, not 1149793519368.","B":"Incorrect. Substituting the stated values into the governing relation gives 1533058025824, not 1533058025822.","C":"Correct. The worked calculation gives 1533058025824.","D":"Incorrect. Substituting the stated values into the governing relation gives 1533058025824, not 153305802583.39999."},"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_003451","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 42 m s⁻¹ at 55° to the horizontal. Take g=10 m s⁻². Find its horizontal range.","options":{"A":"163.76178 m","B":"82.88089 m","C":"165.76178 m","D":"164.76178 m"},"correct_answer":"C","correct_answer_value":"165.76178 m","solution":"Range R=u²sin(2θ)/g. Substituting u=42, θ=55° and g=10 gives R=(42)²sin(110°)/10=165.76178 m.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 165.76178 m, not 163.76178 m.","B":"Incorrect. Substituting the stated values into the governing relation gives 165.76178 m, not 82.88089 m.","C":"Correct. The worked calculation gives 165.76178 m.","D":"Incorrect. Substituting the stated values into the governing relation gives 165.76178 m, not 164.76178 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_002927","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, 1) and v=(1, 3, -5), calculate u·v.","options":{"A":"-18","B":"-15","C":"-16","D":"-12"},"correct_answer":"B","correct_answer_value":"-15","solution":"The dot product is the sum of corresponding component products: (8)(1)+(-6)(3)+(1)(-5)=-15.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives -15, not -18.","B":"Correct. The worked calculation gives -15.","C":"Incorrect. Substituting the stated values into the governing relation gives -15, not -16.","D":"Incorrect. Substituting the stated values into the governing relation gives -15, 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_001310","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 +0x² +10x+7. Evaluate f′(-2).","options":{"A":"0.8","B":"-2","C":"-5","D":"-4"},"correct_answer":"B","correct_answer_value":"-2","solution":"Differentiate: f′(x)=3(-1)x²+2(0)x+(10). At x=-2, f′(-2)=3(-1)(-2)²+2(0)(-2)+(10)=-2.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives -2, not 0.8.","B":"Correct. The worked calculation gives -2.","C":"Incorrect. Substituting the stated values into the governing relation gives -2, not -5.","D":"Incorrect. Substituting the stated values into the governing relation gives -2, not -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_005482","subject":"physics","topic":"electrostatics","subtopic":"capacitor_energy","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A 68 μF capacitor is charged to a potential difference of 32 V. Calculate the energy stored, expressed in μJ.","options":{"A":"34816 μJ","B":"34814 μJ","C":"34818 μJ","D":"-34816 μJ"},"correct_answer":"A","correct_answer_value":"34816 μJ","solution":"U=½CV². Since C=68 μF, U in μJ is ½×68×(32)²=34816 μJ.","option_explanations":{"A":"Correct. The worked calculation gives 34816 μJ.","B":"Incorrect. Substituting the stated values into the governing relation gives 34816 μJ, not 34814 μJ.","C":"Incorrect. Substituting the stated values into the governing relation gives 34816 μJ, not 34818 μJ.","D":"Incorrect. Substituting the stated values into the governing relation gives 34816 μJ, not -34816 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_005210","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 +2 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":"3.6 N","B":"-1.8 N","C":"1.8 N","D":"4.8 N"},"correct_answer":"C","correct_answer_value":"1.8 N","solution":"Convert charges to coulombs and distance to metres. F=k|q₁q₂|/r²=9×10⁹(1×10⁻⁶)(2×10⁻⁶)/(10×10⁻²)²=1.8 N.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 1.8 N, not 3.6 N.","B":"Incorrect. Substituting the stated values into the governing relation gives 1.8 N, not -1.8 N.","C":"Correct. The worked calculation gives 1.8 N.","D":"Incorrect. Substituting the stated values into the governing relation gives 1.8 N, not 4.8 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_009361","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_10N_1X_3. Treat X as the total number of monovalent halogen atoms.","options":{"A":"0","B":"-1","C":"2","D":"1"},"correct_answer":"A","correct_answer_value":"0","solution":"Use DBE=(2C+2+N−H−X)/2. Substitution gives (2×5+2+1−10−3)/2=0.","option_explanations":{"A":"Correct. The worked calculation gives 0.","B":"Incorrect. Substituting the stated values into the governing relation gives 0, not -1.","C":"Incorrect. Substituting the stated values into the governing relation gives 0, not 2.","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_008245","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 13301 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":"41.76236 g","B":"87.52472 g","C":"43.76236 g","D":"45.76236 g"},"correct_answer":"C","correct_answer_value":"43.76236 g","solution":"Cu²⁺+2e⁻→Cu, so z=2. By Faraday's law m=MIt/(zF)=63.5×10×13301/(2×96500)=43.76236 g.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 43.76236 g, not 41.76236 g.","B":"Incorrect. Substituting the stated values into the governing relation gives 43.76236 g, not 87.52472 g.","C":"Correct. The worked calculation gives 43.76236 g.","D":"Incorrect. Substituting the stated values into the governing relation gives 43.76236 g, not 45.76236 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_001509","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 -8x -4) dx.","options":{"A":"-970.66667","B":"-776.53333","C":"-1456","D":"-969.66667"},"correct_answer":"A","correct_answer_value":"-970.66667","solution":"An antiderivative is (-4/3)x³+(-8/2)x²+(-4)x. Evaluating between 0 and 8 gives (-4/3)(8)³+(-8/2)(8)²+(-4)(8)=-970.66667.","option_explanations":{"A":"Correct. The worked calculation gives -970.66667.","B":"Incorrect. Substituting the stated values into the governing relation gives -970.66667, not -776.53333.","C":"Incorrect. Substituting the stated values into the governing relation gives -970.66667, not -1456.","D":"Incorrect. Substituting the stated values into the governing relation gives -970.66667, not -969.66667."},"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_003506","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 102 m s⁻¹ at 50° to the horizontal. Take g=10 m s⁻². Find its horizontal range.","options":{"A":"512.29699 m","B":"1024.59399 m","C":"1022.59399 m","D":"-1024.59399 m"},"correct_answer":"B","correct_answer_value":"1024.59399 m","solution":"Range R=u²sin(2θ)/g. Substituting u=102, θ=50° and g=10 gives R=(102)²sin(100°)/10=1024.59399 m.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 1024.59399 m, not 512.29699 m.","B":"Correct. The worked calculation gives 1024.59399 m.","C":"Incorrect. Substituting the stated values into the governing relation gives 1024.59399 m, not 1022.59399 m.","D":"Incorrect. Substituting the stated values into the governing relation gives 1024.59399 m, not -1024.59399 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_006045","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 43 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":"2.65385","B":"1.65385","C":"0.65385","D":"-2.34615"},"correct_answer":"C","correct_answer_value":"0.65385","solution":"Using distances as positive magnitudes, 1/f=1/v+1/u gives v=uf/(u−f)=(43)(17)/(43−17)=28.11538 cm. Thus |m|=v/u=28.11538/43=0.65385.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.65385, not 2.65385.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.65385, not 1.65385.","C":"Correct. The worked calculation gives 0.65385.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.65385, not -2.34615."},"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_002675","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, -1, 2) and v=(1, 4, -6), calculate u·v.","options":{"A":"-6.75","B":"-9","C":"-6","D":"-18"},"correct_answer":"B","correct_answer_value":"-9","solution":"The dot product is the sum of corresponding component products: (7)(1)+(-1)(4)+(2)(-6)=-9.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives -9, not -6.75.","B":"Correct. The worked calculation gives -9.","C":"Incorrect. Substituting the stated values into the governing relation gives -9, not -6.","D":"Incorrect. Substituting the stated values into the governing relation gives -9, not -18."},"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_002867","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, 9, 7) and v=(-2, 8, 0), calculate u·v.","options":{"A":"67.2","B":"85","C":"9.4","D":"84"},"correct_answer":"D","correct_answer_value":"84","solution":"The dot product is the sum of corresponding component products: (-6)(-2)+(9)(8)+(7)(0)=84.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 84, not 67.2.","B":"Incorrect. Substituting the stated values into the governing relation gives 84, not 85.","C":"Incorrect. Substituting the stated values into the governing relation gives 84, not 9.4.","D":"Correct. The worked calculation gives 84."},"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_006667","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 1080 N m⁻¹ and oscillates horizontally without damping. What is the time period divided by π?","options":{"A":"-2.72783 s","B":"0.27217 s","C":"-1.72783 s","D":"0.40825 s"},"correct_answer":"B","correct_answer_value":"0.27217 s","solution":"For a mass–spring oscillator, T=2π√(m/k). Therefore T/π=2√(20/1080)=0.27217 s.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.27217 s, not -2.72783 s.","B":"Correct. The worked calculation gives 0.27217 s.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.27217 s, not -1.72783 s.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.27217 s, not 0.40825 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_000828","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 -2. What is the sum of its first 10 terms?","options":{"A":"-818.4","B":"-1025","C":"-1023","D":"-1534.5"},"correct_answer":"C","correct_answer_value":"-1023","solution":"For r≠1, Sₙ=a(rⁿ−1)/(r−1). Thus Sₙ=3(-2^10−1)/(-2−1)=-1023.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives -1023, not -818.4.","B":"Incorrect. Substituting the stated values into the governing relation gives -1023, not -1025.","C":"Correct. The worked calculation gives -1023.","D":"Incorrect. Substituting the stated values into the governing relation gives -1023, not -1534.5."},"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_006543","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 2 kg is attached to an ideal spring of force constant 1099 N m⁻¹ and oscillates horizontally without damping. What is the time period divided by π?","options":{"A":"-0.08532 s","B":"0.08532 s","C":"0.12798 s","D":"1.08532 s"},"correct_answer":"B","correct_answer_value":"0.08532 s","solution":"For a mass–spring oscillator, T=2π√(m/k). Therefore T/π=2√(2/1099)=0.08532 s.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.08532 s, not -0.08532 s.","B":"Correct. The worked calculation gives 0.08532 s.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.08532 s, not 0.12798 s.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.08532 s, not 1.08532 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_006069","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 90 cm in front of a thin convex lens of focal length 18 cm. Using the Cartesian sign convention, what is the magnitude of the linear magnification?","options":{"A":"1.025","B":"-2.75","C":"-1.75","D":"0.25"},"correct_answer":"D","correct_answer_value":"0.25","solution":"Using distances as positive magnitudes, 1/f=1/v+1/u gives v=uf/(u−f)=(90)(18)/(90−18)=22.5 cm. Thus |m|=v/u=22.5/90=0.25.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.25, not 1.025.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.25, not -2.75.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.25, not -1.75.","D":"Correct. The worked calculation gives 0.25."},"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_008509","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 2 and K_b for water is 0.52 K kg mol⁻¹, find the boiling-point elevation.","options":{"A":"4.4768 K","B":"2.2152 K","C":"3.4768 K","D":"1.4768 K"},"correct_answer":"D","correct_answer_value":"1.4768 K","solution":"ΔT_b=iK_bm. Substitution gives ΔT_b=2×0.52×1.42=1.4768 K.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 1.4768 K, not 4.4768 K.","B":"Incorrect. Substituting the stated values into the governing relation gives 1.4768 K, not 2.2152 K.","C":"Incorrect. Substituting the stated values into the governing relation gives 1.4768 K, not 3.4768 K.","D":"Correct. The worked calculation gives 1.4768 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_007343","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":"90 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":"4.34523","B":"1.34523","C":"2.01785","D":"3.34523"},"correct_answer":"B","correct_answer_value":"1.34523","solution":"Excess H⁺ moles=0.10×(90−34)/1000=0.005600 mol. Total volume=0.124 L, so [H⁺]=0.045161 M and pH=−log₁₀[H⁺]=1.34523.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 1.34523, not 4.34523.","B":"Correct. The worked calculation gives 1.34523.","C":"Incorrect. Substituting the stated values into the governing relation gives 1.34523, not 2.01785.","D":"Incorrect. Substituting the stated values into the governing relation gives 1.34523, not 3.34523."},"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_004943","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 5 Ω are connected in series; this combination is connected in parallel with a 4 Ω resistor across a 6 V ideal battery. Find the total current supplied by the battery.","options":{"A":"4 A","B":"2 A","C":"3 A","D":"1 A"},"correct_answer":"B","correct_answer_value":"2 A","solution":"Series branch resistance=7+5=12 Ω. Parallel equivalent R_eq=(12×4)/(12+4)=3 Ω. Total current I=V/R_eq=6/3=2 A.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 2 A, not 4 A.","B":"Correct. The worked calculation gives 2 A.","C":"Incorrect. Substituting the stated values into the governing relation gives 2 A, not 3 A.","D":"Incorrect. Substituting the stated values into the governing relation gives 2 A, not 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_008521","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 1.36 mol kg⁻¹. If its van't Hoff factor is 2 and K_b for water is 0.52 K kg mol⁻¹, find the boiling-point elevation.","options":{"A":"1.4144 K","B":"1.0608 K","C":"3.4144 K","D":"-0.5856 K"},"correct_answer":"A","correct_answer_value":"1.4144 K","solution":"ΔT_b=iK_bm. Substitution gives ΔT_b=2×0.52×1.36=1.4144 K.","option_explanations":{"A":"Correct. The worked calculation gives 1.4144 K.","B":"Incorrect. Substituting the stated values into the governing relation gives 1.4144 K, not 1.0608 K.","C":"Incorrect. Substituting the stated values into the governing relation gives 1.4144 K, not 3.4144 K.","D":"Incorrect. Substituting the stated values into the governing relation gives 1.4144 K, not -0.5856 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_002389","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], [3, -6]].","options":{"A":"-57","B":"57","C":"-56","D":"-85.5"},"correct_answer":"A","correct_answer_value":"-57","solution":"For a 2×2 matrix [[a,b],[c,d]], det=ad−bc. Here it is (7)(-6)−(5)(3)=-57.","option_explanations":{"A":"Correct. The worked calculation gives -57.","B":"Incorrect. Substituting the stated values into the governing relation gives -57, not 57.","C":"Incorrect. Substituting the stated values into the governing relation gives -57, not -56.","D":"Incorrect. Substituting the stated values into the governing relation gives -57, not -85.5."},"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_006512","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 14 kg is attached to an ideal spring of force constant 577 N m⁻¹ and oscillates horizontally without damping. What is the time period divided by π?","options":{"A":"0.31153 s","B":"0.23365 s","C":"-1.68847 s","D":"-0.68847 s"},"correct_answer":"A","correct_answer_value":"0.31153 s","solution":"For a mass–spring oscillator, T=2π√(m/k). Therefore T/π=2√(14/577)=0.31153 s.","option_explanations":{"A":"Correct. The worked calculation gives 0.31153 s.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.31153 s, not 0.23365 s.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.31153 s, not -1.68847 s.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.31153 s, not -0.68847 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_001889","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 90 distinct elements?","options":{"A":"65923902","B":"43949268","C":"87898536","D":"35159414.4"},"correct_answer":"B","correct_answer_value":"43949268","solution":"Order does not matter, so the count is C(90,5)=90!/[5!(85)!]=43949268.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 43949268, not 65923902.","B":"Correct. The worked calculation gives 43949268.","C":"Incorrect. Substituting the stated values into the governing relation gives 43949268, not 87898536.","D":"Incorrect. Substituting the stated values into the governing relation gives 43949268, not 35159414.4."},"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_000016","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 +3=0, let its roots be α and β. What is α²+β²?","options":{"A":"1.03611","B":"0.36111","C":"2.36111","D":"-0.63889"},"correct_answer":"B","correct_answer_value":"0.36111","solution":"By Vieta's relations, α+β = -7/6 and αβ = 3/6. Then α²+β²=(α+β)²−2αβ = (-7/6)²−2(3/6) = 0.36111.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.36111, not 1.03611.","B":"Correct. The worked calculation gives 0.36111.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.36111, not 2.36111.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.36111, not -0.63889."},"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_005926","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 atm at 239 K. What is its pressure at 330 K?","options":{"A":"-2.76151 atm","B":"2.07113 atm","C":"2.76151 atm","D":"-0.23849 atm"},"correct_answer":"C","correct_answer_value":"2.76151 atm","solution":"At constant volume, P/T is constant. P₂=P₁T₂/T₁=(2)(330)/(239)=2.76151 atm.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 2.76151 atm, not -2.76151 atm.","B":"Incorrect. Substituting the stated values into the governing relation gives 2.76151 atm, not 2.07113 atm.","C":"Correct. The worked calculation gives 2.76151 atm.","D":"Incorrect. Substituting the stated values into the governing relation gives 2.76151 atm, not -0.23849 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_004358","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 29 N m about its central axis. Find its angular acceleration.","options":{"A":"-1.75833 rad s⁻²","B":"1.02417 rad s⁻²","C":"0.24167 rad s⁻²","D":"1.24167 rad s⁻²"},"correct_answer":"C","correct_answer_value":"0.24167 rad s⁻²","solution":"For a solid disc, I=½MR²=½(15)(4)²=120 kg m². Using τ=Iα, α=29/120=0.24167 rad s⁻².","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.24167 rad s⁻², not -1.75833 rad s⁻².","B":"Incorrect. Substituting the stated values into the governing relation gives 0.24167 rad s⁻², not 1.02417 rad s⁻².","C":"Correct. The worked calculation gives 0.24167 rad s⁻².","D":"Incorrect. Substituting the stated values into the governing relation gives 0.24167 rad s⁻², not 1.24167 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_006928","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 22 mol H₂ and 30 mol O₂ are mixed and the reaction goes to completion, how many moles of H₂O form?","options":{"A":"23 mol","B":"24 mol","C":"22 mol","D":"17.6 mol"},"correct_answer":"C","correct_answer_value":"22 mol","solution":"From 2 mol H₂ and 1 mol O₂, 2 mol H₂O form. H₂ can produce 22 mol H₂O; O₂ can produce 60 mol H₂O. The limiting condition is H₂; product=22 mol.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 22 mol, not 23 mol.","B":"Incorrect. Substituting the stated values into the governing relation gives 22 mol, not 24 mol.","C":"Correct. The worked calculation gives 22 mol.","D":"Incorrect. Substituting the stated values into the governing relation gives 22 mol, not 17.6 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_005458","subject":"physics","topic":"electrostatics","subtopic":"capacitor_energy","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A 9 μF capacitor is charged to a potential difference of 21 V. Calculate the energy stored, expressed in μJ.","options":{"A":"1982.5 μJ","B":"1488.375 μJ","C":"1984.5 μJ","D":"2976.75 μJ"},"correct_answer":"C","correct_answer_value":"1984.5 μJ","solution":"U=½CV². Since C=9 μF, U in μJ is ½×9×(21)²=1984.5 μJ.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 1984.5 μJ, not 1982.5 μJ.","B":"Incorrect. Substituting the stated values into the governing relation gives 1984.5 μJ, not 1488.375 μJ.","C":"Correct. The worked calculation gives 1984.5 μJ.","D":"Incorrect. Substituting the stated values into the governing relation gives 1984.5 μJ, not 2976.75 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_001600","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 (5x² +6x +2) dx.","options":{"A":"74.26667","B":"734.66667","C":"732.66667","D":"879.2"},"correct_answer":"C","correct_answer_value":"732.66667","solution":"An antiderivative is (5/3)x³+(6/2)x²+(2)x. Evaluating between 0 and 7 gives (5/3)(7)³+(6/2)(7)²+(2)(7)=732.66667.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 732.66667, not 74.26667.","B":"Incorrect. Substituting the stated values into the governing relation gives 732.66667, not 734.66667.","C":"Correct. The worked calculation gives 732.66667.","D":"Incorrect. Substituting the stated values into the governing relation gives 732.66667, not 879.2."},"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_003358","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 20 m s⁻¹ at 18° to the horizontal. Take g=10 m s⁻². Find its horizontal range.","options":{"A":"21.51141 m","B":"3.35114 m","C":"23.51141 m","D":"-23.51141 m"},"correct_answer":"C","correct_answer_value":"23.51141 m","solution":"Range R=u²sin(2θ)/g. Substituting u=20, θ=18° and g=10 gives R=(20)²sin(36°)/10=23.51141 m.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 23.51141 m, not 21.51141 m.","B":"Incorrect. Substituting the stated values into the governing relation gives 23.51141 m, not 3.35114 m.","C":"Correct. The worked calculation gives 23.51141 m.","D":"Incorrect. Substituting the stated values into the governing relation gives 23.51141 m, not -23.51141 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_006608","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 23 kg is attached to an ideal spring of force constant 105 N m⁻¹ and oscillates horizontally without damping. What is the time period divided by π?","options":{"A":"0.93605 s","B":"2.93605 s","C":"3.93605 s","D":"-0.06395 s"},"correct_answer":"A","correct_answer_value":"0.93605 s","solution":"For a mass–spring oscillator, T=2π√(m/k). Therefore T/π=2√(23/105)=0.93605 s.","option_explanations":{"A":"Correct. The worked calculation gives 0.93605 s.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.93605 s, not 2.93605 s.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.93605 s, not 3.93605 s.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.93605 s, not -0.06395 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_004763","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 6 Ω and 11 Ω are connected in series; this combination is connected in parallel with a 4 Ω resistor across a 30 V ideal battery. Find the total current supplied by the battery.","options":{"A":"-9.26471 A","B":"18.52941 A","C":"9.26471 A","D":"8.26471 A"},"correct_answer":"C","correct_answer_value":"9.26471 A","solution":"Series branch resistance=6+11=17 Ω. Parallel equivalent R_eq=(17×4)/(17+4)=3.2381 Ω. Total current I=V/R_eq=30/3.2381=9.26471 A.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 9.26471 A, not -9.26471 A.","B":"Incorrect. Substituting the stated values into the governing relation gives 9.26471 A, not 18.52941 A.","C":"Correct. The worked calculation gives 9.26471 A.","D":"Incorrect. Substituting the stated values into the governing relation gives 9.26471 A, not 8.26471 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_002915","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, 9) and v=(3, 7, -6), calculate u·v.","options":{"A":"31","B":"33","C":"24.75","D":"16.5"},"correct_answer":"B","correct_answer_value":"33","solution":"The dot product is the sum of corresponding component products: (8)(3)+(9)(7)+(9)(-6)=33.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 33, not 31.","B":"Correct. The worked calculation gives 33.","C":"Incorrect. Substituting the stated values into the governing relation gives 33, not 24.75.","D":"Incorrect. Substituting the stated values into the governing relation gives 33, not 16.5."},"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_002144","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, 10 blue and 7 green balls. Two balls are drawn uniformly at random without replacement. What is the probability that both have the same colour?","options":{"A":"0.32016","B":"2.32016","C":"3.32016","D":"-2.67984"},"correct_answer":"A","correct_answer_value":"0.32016","solution":"Favourable pairs = C(6,2)+C(10,2)+C(7,2)=81. Total pairs=C(23,2)=253. Therefore P=81/253=0.32016.","option_explanations":{"A":"Correct. The worked calculation gives 0.32016.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.32016, not 2.32016.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.32016, not 3.32016.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.32016, not -2.67984."},"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_001341","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 +0x +0) dx.","options":{"A":"-510","B":"-614.4","C":"-512","D":"-50.2"},"correct_answer":"C","correct_answer_value":"-512","solution":"An antiderivative is (-3/3)x³+(0/2)x²+(0)x. Evaluating between 0 and 8 gives (-3/3)(8)³+(0/2)(8)²+(0)(8)=-512.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives -512, not -510.","B":"Incorrect. Substituting the stated values into the governing relation gives -512, not -614.4.","C":"Correct. The worked calculation gives -512.","D":"Incorrect. Substituting the stated values into the governing relation gives -512, not -50.2."},"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_004443","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 6 kg and radius 5 m is acted on by a constant torque of 48 N m about its central axis. Find its angular acceleration.","options":{"A":"-1.36 rad s⁻²","B":"2.64 rad s⁻²","C":"-2.36 rad s⁻²","D":"0.64 rad s⁻²"},"correct_answer":"D","correct_answer_value":"0.64 rad s⁻²","solution":"For a solid disc, I=½MR²=½(6)(5)²=75 kg m². Using τ=Iα, α=48/75=0.64 rad s⁻².","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.64 rad s⁻², not -1.36 rad s⁻².","B":"Incorrect. Substituting the stated values into the governing relation gives 0.64 rad s⁻², not 2.64 rad s⁻².","C":"Incorrect. Substituting the stated values into the governing relation gives 0.64 rad s⁻², not -2.36 rad s⁻².","D":"Correct. The worked calculation gives 0.64 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_003484","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 55° to the horizontal. Take g=10 m s⁻². Find its horizontal range.","options":{"A":"236.92316 m","B":"352.38473 m","C":"235.92316 m","D":"234.92316 m"},"correct_answer":"D","correct_answer_value":"234.92316 m","solution":"Range R=u²sin(2θ)/g. Substituting u=50, θ=55° and g=10 gives R=(50)²sin(110°)/10=234.92316 m.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 234.92316 m, not 236.92316 m.","B":"Incorrect. Substituting the stated values into the governing relation gives 234.92316 m, not 352.38473 m.","C":"Incorrect. Substituting the stated values into the governing relation gives 234.92316 m, not 235.92316 m.","D":"Correct. The worked calculation gives 234.92316 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_007916","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 84 mmol L⁻¹, what concentration remains after 160 minutes?","options":{"A":"-2.67188 mmol L⁻¹","B":"0.32812 mmol L⁻¹","C":"0.16406 mmol L⁻¹","D":"0.65625 mmol L⁻¹"},"correct_answer":"B","correct_answer_value":"0.32812 mmol L⁻¹","solution":"160 minutes corresponds to 8 half-lives. The remaining concentration is [A]₀(1/2)ⁿ=84/2^8=0.32812 mmol L⁻¹.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.32812 mmol L⁻¹, not -2.67188 mmol L⁻¹.","B":"Correct. The worked calculation gives 0.32812 mmol L⁻¹.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.32812 mmol L⁻¹, not 0.16406 mmol L⁻¹.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.32812 mmol L⁻¹, not 0.65625 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_007908","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 61 mmol L⁻¹, what concentration remains after 160 minutes?","options":{"A":"-1.76172 mmol L⁻¹","B":"1.02383 mmol L⁻¹","C":"0.23828 mmol L⁻¹","D":"3.23828 mmol L⁻¹"},"correct_answer":"C","correct_answer_value":"0.23828 mmol L⁻¹","solution":"160 minutes corresponds to 8 half-lives. The remaining concentration is [A]₀(1/2)ⁿ=61/2^8=0.23828 mmol L⁻¹.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.23828 mmol L⁻¹, not -1.76172 mmol L⁻¹.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.23828 mmol L⁻¹, not 1.02383 mmol L⁻¹.","C":"Correct. The worked calculation gives 0.23828 mmol L⁻¹.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.23828 mmol L⁻¹, not 3.23828 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_004473","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 8 kg and radius 1 m is acted on by a constant torque of 57 N m about its central axis. Find its angular acceleration.","options":{"A":"14.25 rad s⁻²","B":"13.25 rad s⁻²","C":"12.25 rad s⁻²","D":"16.25 rad s⁻²"},"correct_answer":"A","correct_answer_value":"14.25 rad s⁻²","solution":"For a solid disc, I=½MR²=½(8)(1)²=4 kg m². Using τ=Iα, α=57/4=14.25 rad s⁻².","option_explanations":{"A":"Correct. The worked calculation gives 14.25 rad s⁻².","B":"Incorrect. Substituting the stated values into the governing relation gives 14.25 rad s⁻², not 13.25 rad s⁻².","C":"Incorrect. Substituting the stated values into the governing relation gives 14.25 rad s⁻², not 12.25 rad s⁻².","D":"Incorrect. Substituting the stated values into the governing relation gives 14.25 rad s⁻², not 16.25 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_006184","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 8 cm. Using the Cartesian sign convention, what is the magnitude of the linear magnification?","options":{"A":"0.51613","B":"-2.74194","C":"0.12903","D":"0.25806"},"correct_answer":"D","correct_answer_value":"0.25806","solution":"Using distances as positive magnitudes, 1/f=1/v+1/u gives v=uf/(u−f)=(39)(8)/(39−8)=10.06452 cm. Thus |m|=v/u=10.06452/39=0.25806.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.25806, not 0.51613.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.25806, not -2.74194.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.25806, not 0.12903.","D":"Correct. The worked calculation gives 0.25806."},"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_003531","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 103 m s⁻¹ at 20° to the horizontal. Take g=10 m s⁻². Find its horizontal range.","options":{"A":"818.32005 m","B":"679.93338 m","C":"1022.90006 m","D":"681.93338 m"},"correct_answer":"D","correct_answer_value":"681.93338 m","solution":"Range R=u²sin(2θ)/g. Substituting u=103, θ=20° and g=10 gives R=(103)²sin(40°)/10=681.93338 m.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 681.93338 m, not 818.32005 m.","B":"Incorrect. Substituting the stated values into the governing relation gives 681.93338 m, not 679.93338 m.","C":"Incorrect. Substituting the stated values into the governing relation gives 681.93338 m, not 1022.90006 m.","D":"Correct. The worked calculation gives 681.93338 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_000284","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 +0x -1=0, let its roots be α and β. What is α²+β²?","options":{"A":"-1","B":"1.1","C":"2","D":"1"},"correct_answer":"D","correct_answer_value":"1","solution":"By Vieta's relations, α+β = 0/2 and αβ = -1/2. Then α²+β²=(α+β)²−2αβ = (0/2)²−2(-1/2) = 1.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 1, not -1.","B":"Incorrect. Substituting the stated values into the governing relation gives 1, not 1.1.","C":"Incorrect. Substituting the stated values into the governing relation gives 1, not 2.","D":"Correct. The worked calculation gives 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_002034","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, 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":"0.31439","B":"1.31439","C":"0.1572","D":"3.31439"},"correct_answer":"A","correct_answer_value":"0.31439","solution":"Favourable pairs = C(10,2)+C(11,2)+C(12,2)=166. Total pairs=C(33,2)=528. Therefore P=166/528=0.31439.","option_explanations":{"A":"Correct. The worked calculation gives 0.31439.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.31439, not 1.31439.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.31439, not 0.1572.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.31439, not 3.31439."},"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_001654","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 +2x +2) dx.","options":{"A":"-8","B":"10","C":"8","D":"4"},"correct_answer":"C","correct_answer_value":"8","solution":"An antiderivative is (0/3)x³+(2/2)x²+(2)x. Evaluating between 0 and 2 gives (0/3)(2)³+(2/2)(2)²+(2)(2)=8.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 8, not -8.","B":"Incorrect. Substituting the stated values into the governing relation gives 8, not 10.","C":"Correct. The worked calculation gives 8.","D":"Incorrect. Substituting the stated values into the governing relation gives 8, not 4."},"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_006056","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 53 cm in front of a thin convex lens of focal length 15 cm. Using the Cartesian sign convention, what is the magnitude of the linear magnification?","options":{"A":"0.78947","B":"3.39474","C":"0.39474","D":"0.29605"},"correct_answer":"C","correct_answer_value":"0.39474","solution":"Using distances as positive magnitudes, 1/f=1/v+1/u gives v=uf/(u−f)=(53)(15)/(53−15)=20.92105 cm. Thus |m|=v/u=20.92105/53=0.39474.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.39474, not 0.78947.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.39474, not 3.39474.","C":"Correct. The worked calculation gives 0.39474.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.39474, not 0.29605."},"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_000777","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 4. What is the sum of its first 7 terms?","options":{"A":"24574.5","B":"73723.5","C":"49149","D":"49148"},"correct_answer":"C","correct_answer_value":"49149","solution":"For r≠1, Sₙ=a(rⁿ−1)/(r−1). Thus Sₙ=9(4^7−1)/(4−1)=49149.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 49149, not 24574.5.","B":"Incorrect. Substituting the stated values into the governing relation gives 49149, not 73723.5.","C":"Correct. The worked calculation gives 49149.","D":"Incorrect. Substituting the stated values into the governing relation gives 49149, not 49148."},"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_005271","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 +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":"1.72 N","B":"8.2 N","C":"7.2 N","D":"9.2 N"},"correct_answer":"C","correct_answer_value":"7.2 N","solution":"Convert charges to coulombs and distance to metres. F=k|q₁q₂|/r²=9×10⁹(8×10⁻⁶)(1×10⁻⁶)/(10×10⁻²)²=7.2 N.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 7.2 N, not 1.72 N.","B":"Incorrect. Substituting the stated values into the governing relation gives 7.2 N, not 8.2 N.","C":"Correct. The worked calculation gives 7.2 N.","D":"Incorrect. Substituting the stated values into the governing relation gives 7.2 N, not 9.2 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_002773","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, 2, -4) and v=(7, 0, -1), calculate u·v.","options":{"A":"-46.5","B":"31","C":"-31","D":"-29"},"correct_answer":"C","correct_answer_value":"-31","solution":"The dot product is the sum of corresponding component products: (-5)(7)+(2)(0)+(-4)(-1)=-31.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives -31, not -46.5.","B":"Incorrect. Substituting the stated values into the governing relation gives -31, not 31.","C":"Correct. The worked calculation gives -31.","D":"Incorrect. Substituting the stated values into the governing relation gives -31, not -29."},"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_009792","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 [Cr(OH)_2]^-2, each OH ligand has charge −1. Determine the formal oxidation state of Cr.","options":{"A":"-1","B":"-2","C":"0","D":"-3"},"correct_answer":"C","correct_answer_value":"0","solution":"Let the formal oxidation state of Cr 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":"Incorrect. Substituting the stated values into the governing relation gives 0, not -2.","C":"Correct. The worked calculation gives 0.","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_004882","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 14 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":"2.76471 A","B":"-3.76471 A","C":"5.76471 A","D":"3.76471 A"},"correct_answer":"D","correct_answer_value":"3.76471 A","solution":"Series branch resistance=3+14=17 Ω. Parallel equivalent R_eq=(17×15)/(17+15)=7.96875 Ω. Total current I=V/R_eq=30/7.96875=3.76471 A.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 3.76471 A, not 2.76471 A.","B":"Incorrect. Substituting the stated values into the governing relation gives 3.76471 A, not -3.76471 A.","C":"Incorrect. Substituting the stated values into the governing relation gives 3.76471 A, not 5.76471 A.","D":"Correct. The worked calculation gives 3.76471 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_007569","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":"60 mL of 0.10 M HCl is mixed with 44 mL of 0.10 M NaOH at 25°C. Assuming complete dissociation and additive volumes, find the final pH.","options":{"A":"0.81291","B":"3.62583","C":"1.81291","D":"-0.18709"},"correct_answer":"C","correct_answer_value":"1.81291","solution":"Excess H⁺ moles=0.10×(60−44)/1000=0.001600 mol. Total volume=0.104 L, so [H⁺]=0.015385 M and pH=−log₁₀[H⁺]=1.81291.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 1.81291, not 0.81291.","B":"Incorrect. Substituting the stated values into the governing relation gives 1.81291, not 3.62583.","C":"Correct. The worked calculation gives 1.81291.","D":"Incorrect. Substituting the stated values into the governing relation gives 1.81291, not -0.18709."},"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_007492","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":"41 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":"-12.13622","B":"12.13622","C":"11.13622","D":"18.20433"},"correct_answer":"B","correct_answer_value":"12.13622","solution":"Excess OH⁻ moles=0.10×(54−41)/1000=0.001300 mol. Total volume=0.095 L, so [OH⁻]=0.013684 M, pOH=−log₁₀[OH⁻], and pH=14−pOH=12.13622.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 12.13622, not -12.13622.","B":"Correct. The worked calculation gives 12.13622.","C":"Incorrect. Substituting the stated values into the governing relation gives 12.13622, not 11.13622.","D":"Incorrect. Substituting the stated values into the governing relation gives 12.13622, not 18.20433."},"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_002657","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, -3], [3, 2]].","options":{"A":"27","B":"40.5","C":"28","D":"54"},"correct_answer":"A","correct_answer_value":"27","solution":"For a 2×2 matrix [[a,b],[c,d]], det=ad−bc. Here it is (9)(2)−(-3)(3)=27.","option_explanations":{"A":"Correct. The worked calculation gives 27.","B":"Incorrect. Substituting the stated values into the governing relation gives 27, not 40.5.","C":"Incorrect. Substituting the stated values into the governing relation gives 27, not 28.","D":"Incorrect. Substituting the stated values into the governing relation gives 27, 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_009994","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(Cl)_2]^-2, each Cl ligand has charge −1. Determine the formal oxidation state of Mn.","options":{"A":"-2","B":"0","C":"-3","D":"1"},"correct_answer":"B","correct_answer_value":"0","solution":"Let the formal oxidation state of Mn 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 -2.","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 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_003794","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 6 m s⁻¹. A constant net force does 24 N of work over 2 m in the direction of motion. Find its final kinetic energy.","options":{"A":"191 J","B":"288 J","C":"194 J","D":"192 J"},"correct_answer":"D","correct_answer_value":"192 J","solution":"By the work–energy theorem, K_f=K_i+W=½mu²+Fs=½(8)(6)²+(24)(2)=192 J.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 192 J, not 191 J.","B":"Incorrect. Substituting the stated values into the governing relation gives 192 J, not 288 J.","C":"Incorrect. Substituting the stated values into the governing relation gives 192 J, not 194 J.","D":"Correct. The worked calculation gives 192 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_001303","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 +4x² -4x+7. Evaluate f′(-2).","options":{"A":"48","B":"63","C":"76.8","D":"64"},"correct_answer":"D","correct_answer_value":"64","solution":"Differentiate: f′(x)=3(7)x²+2(4)x+(-4). At x=-2, f′(-2)=3(7)(-2)²+2(4)(-2)+(-4)=64.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 64, not 48.","B":"Incorrect. Substituting the stated values into the governing relation gives 64, not 63.","C":"Incorrect. Substituting the stated values into the governing relation gives 64, not 76.8.","D":"Correct. The worked calculation gives 64."},"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_005614","subject":"physics","topic":"electrostatics","subtopic":"capacitor_energy","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A 51 μF capacitor is charged to a potential difference of 16 V. Calculate the energy stored, expressed in μJ.","options":{"A":"6528 μJ","B":"5222.4 μJ","C":"6526 μJ","D":"4896 μJ"},"correct_answer":"A","correct_answer_value":"6528 μJ","solution":"U=½CV². Since C=51 μF, U in μJ is ½×51×(16)²=6528 μJ.","option_explanations":{"A":"Correct. The worked calculation gives 6528 μJ.","B":"Incorrect. Substituting the stated values into the governing relation gives 6528 μJ, not 5222.4 μJ.","C":"Incorrect. Substituting the stated values into the governing relation gives 6528 μJ, not 6526 μJ.","D":"Incorrect. Substituting the stated values into the governing relation gives 6528 μJ, not 4896 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_007011","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 331 K and 1.68 atm? Use R=0.082057 L atm mol⁻¹ K⁻¹.","options":{"A":"258.67492 L","B":"388.01239 L","C":"-258.67492 L","D":"260.67492 L"},"correct_answer":"A","correct_answer_value":"258.67492 L","solution":"PV=nRT, so V=nRT/P=(16)(0.082057)(331)/(1.68)=258.67492 L.","option_explanations":{"A":"Correct. The worked calculation gives 258.67492 L.","B":"Incorrect. Substituting the stated values into the governing relation gives 258.67492 L, not 388.01239 L.","C":"Incorrect. Substituting the stated values into the governing relation gives 258.67492 L, not -258.67492 L.","D":"Incorrect. Substituting the stated values into the governing relation gives 258.67492 L, not 260.67492 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_004721","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 12 Ω and 15 Ω are connected in series; this combination is connected in parallel with a 19 Ω resistor across a 30 V ideal battery. Find the total current supplied by the battery.","options":{"A":"1.34503 A","B":"1.69006 A","C":"4.03509 A","D":"2.69006 A"},"correct_answer":"D","correct_answer_value":"2.69006 A","solution":"Series branch resistance=12+15=27 Ω. Parallel equivalent R_eq=(27×19)/(27+19)=11.15217 Ω. Total current I=V/R_eq=30/11.15217=2.69006 A.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 2.69006 A, not 1.34503 A.","B":"Incorrect. Substituting the stated values into the governing relation gives 2.69006 A, not 1.69006 A.","C":"Incorrect. Substituting the stated values into the governing relation gives 2.69006 A, not 4.03509 A.","D":"Correct. The worked calculation gives 2.69006 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_004360","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 6 m is acted on by a constant torque of 98 N m about its central axis. Find its angular acceleration.","options":{"A":"2.77778 rad s⁻²","B":"0.77778 rad s⁻²","C":"1.77778 rad s⁻²","D":"-2.22222 rad s⁻²"},"correct_answer":"B","correct_answer_value":"0.77778 rad s⁻²","solution":"For a solid disc, I=½MR²=½(7)(6)²=126 kg m². Using τ=Iα, α=98/126=0.77778 rad s⁻².","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.77778 rad s⁻², not 2.77778 rad s⁻².","B":"Correct. The worked calculation gives 0.77778 rad s⁻².","C":"Incorrect. Substituting the stated values into the governing relation gives 0.77778 rad s⁻², not 1.77778 rad s⁻².","D":"Incorrect. Substituting the stated values into the governing relation gives 0.77778 rad s⁻², not -2.22222 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_000003","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 -15x +2=0, let its roots be α and β. What is α²+β²?","options":{"A":"8.2","B":"16.4","C":"6.2","D":"12.3"},"correct_answer":"A","correct_answer_value":"8.2","solution":"By Vieta's relations, α+β = 15/5 and αβ = 2/5. Then α²+β²=(α+β)²−2αβ = (15/5)²−2(2/5) = 8.2.","option_explanations":{"A":"Correct. The worked calculation gives 8.2.","B":"Incorrect. Substituting the stated values into the governing relation gives 8.2, not 16.4.","C":"Incorrect. Substituting the stated values into the governing relation gives 8.2, not 6.2.","D":"Incorrect. Substituting the stated values into the governing relation gives 8.2, not 12.3."},"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_001886","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 52 distinct elements?","options":{"A":"133784559","B":"13378457","C":"133784560","D":"133784561"},"correct_answer":"C","correct_answer_value":"133784560","solution":"Order does not matter, so the count is C(52,7)=52!/[7!(45)!]=133784560.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 133784560, not 133784559.","B":"Incorrect. Substituting the stated values into the governing relation gives 133784560, not 13378457.","C":"Correct. The worked calculation gives 133784560.","D":"Incorrect. Substituting the stated values into the governing relation gives 133784560, not 133784561."},"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_000326","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 +0x +4=0, let its roots be α and β. What is α²+β²?","options":{"A":"-2.66667","B":"2.66667","C":"-0.66667","D":"0.33333"},"correct_answer":"A","correct_answer_value":"-2.66667","solution":"By Vieta's relations, α+β = 0/3 and αβ = 4/3. Then α²+β²=(α+β)²−2αβ = (0/3)²−2(4/3) = -2.66667.","option_explanations":{"A":"Correct. The worked calculation gives -2.66667.","B":"Incorrect. Substituting the stated values into the governing relation gives -2.66667, not 2.66667.","C":"Incorrect. Substituting the stated values into the governing relation gives -2.66667, not -0.66667.","D":"Incorrect. Substituting the stated values into the governing relation gives -2.66667, not 0.33333."},"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_001771","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 89 distinct elements?","options":{"A":"6890268571","B":"6890268573","C":"3445134286","D":"6890268572"},"correct_answer":"D","correct_answer_value":"6890268572","solution":"Order does not matter, so the count is C(89,7)=89!/[7!(82)!]=6890268572.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 6890268572, not 6890268571.","B":"Incorrect. Substituting the stated values into the governing relation gives 6890268572, not 6890268573.","C":"Incorrect. Substituting the stated values into the governing relation gives 6890268572, not 3445134286.","D":"Correct. The worked calculation gives 6890268572."},"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_002831","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=(0, 2, -1) and v=(-1, -5, 8), calculate u·v.","options":{"A":"-20","B":"-18","C":"-13.5","D":"-0.8"},"correct_answer":"B","correct_answer_value":"-18","solution":"The dot product is the sum of corresponding component products: (0)(-1)+(2)(-5)+(-1)(8)=-18.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives -18, not -20.","B":"Correct. The worked calculation gives -18.","C":"Incorrect. Substituting the stated values into the governing relation gives -18, not -13.5.","D":"Incorrect. Substituting the stated values into the governing relation gives -18, not -0.8."},"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_002004","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 6 elements can be selected from a set of 64 distinct elements?","options":{"A":"89969241.6","B":"74974369","C":"74974368","D":"74974370"},"correct_answer":"C","correct_answer_value":"74974368","solution":"Order does not matter, so the count is C(64,6)=64!/[6!(58)!]=74974368.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 74974368, not 89969241.6.","B":"Incorrect. Substituting the stated values into the governing relation gives 74974368, not 74974369.","C":"Correct. The worked calculation gives 74974368.","D":"Incorrect. Substituting the stated values into the governing relation gives 74974368, not 74974370."},"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_009572","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_4N_2X_2. Treat X as the total number of monovalent halogen atoms.","options":{"A":"5","B":"6","C":"2","D":"4"},"correct_answer":"D","correct_answer_value":"4","solution":"Use DBE=(2C+2+N−H−X)/2. Substitution gives (2×5+2+2−4−2)/2=4.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 4, not 5.","B":"Incorrect. Substituting the stated values into the governing relation gives 4, not 6.","C":"Incorrect. Substituting the stated values into the governing relation gives 4, not 2.","D":"Correct. The worked calculation gives 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_006028","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 76 cm in front of a thin convex lens of focal length 28 cm. Using the Cartesian sign convention, what is the magnitude of the linear magnification?","options":{"A":"1.16667","B":"1.58333","C":"-0.41667","D":"0.58333"},"correct_answer":"D","correct_answer_value":"0.58333","solution":"Using distances as positive magnitudes, 1/f=1/v+1/u gives v=uf/(u−f)=(76)(28)/(76−28)=44.33333 cm. Thus |m|=v/u=44.33333/76=0.58333.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.58333, not 1.16667.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.58333, not 1.58333.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.58333, not -0.41667.","D":"Correct. The worked calculation gives 0.58333."},"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_002317","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, 8 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":"3.3619","B":"0.54286","C":"0.27143","D":"0.3619"},"correct_answer":"D","correct_answer_value":"0.3619","solution":"Favourable pairs = C(10,2)+C(8,2)+C(3,2)=76. Total pairs=C(21,2)=210. Therefore P=76/210=0.3619.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.3619, not 3.3619.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.3619, not 0.54286.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.3619, not 0.27143.","D":"Correct. The worked calculation gives 0.3619."},"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_006866","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 17 mol H₂ and 21 mol O₂ are mixed and the reaction goes to completion, how many moles of H₂O form?","options":{"A":"8.5 mol","B":"17 mol","C":"19 mol","D":"2.7 mol"},"correct_answer":"B","correct_answer_value":"17 mol","solution":"From 2 mol H₂ and 1 mol O₂, 2 mol H₂O form. H₂ can produce 17 mol H₂O; O₂ can produce 42 mol H₂O. The limiting condition is H₂; product=17 mol.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 17 mol, not 8.5 mol.","B":"Correct. The worked calculation gives 17 mol.","C":"Incorrect. Substituting the stated values into the governing relation gives 17 mol, not 19 mol.","D":"Incorrect. Substituting the stated values into the governing relation gives 17 mol, not 2.7 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_003238","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=(-5+5i)/(8+8i). What is Re(z), the real part of z?","options":{"A":"2","B":"-1","C":"-3","D":"0"},"correct_answer":"D","correct_answer_value":"0","solution":"Multiply numerator and denominator by the conjugate (8-8i). Then Re(z)=(ac+bd)/(c²+d²)=((-5)(8)+(5)(8))/(8²+(8)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 -1.","C":"Incorrect. Substituting the stated values into the governing relation gives 0, not -3.","D":"Correct. The worked calculation gives 0."},"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_006534","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 12 kg is attached to an ideal spring of force constant 824 N m⁻¹ and oscillates horizontally without damping. What is the time period divided by π?","options":{"A":"-2.75864 s","B":"1.02414 s","C":"1.24136 s","D":"0.24136 s"},"correct_answer":"D","correct_answer_value":"0.24136 s","solution":"For a mass–spring oscillator, T=2π√(m/k). Therefore T/π=2√(12/824)=0.24136 s.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.24136 s, not -2.75864 s.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.24136 s, not 1.02414 s.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.24136 s, not 1.24136 s.","D":"Correct. The worked calculation gives 0.24136 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_009862","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 [Ni(OH)_6]^-1, each OH ligand has charge −1. Determine the formal oxidation state of Ni.","options":{"A":"8","B":"1.5","C":"3","D":"5"},"correct_answer":"D","correct_answer_value":"5","solution":"Let the formal oxidation state of Ni be x. Charge balance gives x+(6)(−1)=-1; hence x=-1+6=+5.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 5, not 8.","B":"Incorrect. Substituting the stated values into the governing relation gives 5, not 1.5.","C":"Incorrect. Substituting the stated values into the governing relation gives 5, not 3.","D":"Correct. The worked calculation gives 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_004991","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 14 Ω are connected in series; this combination is connected in parallel with a 3 Ω resistor across a 30 V ideal battery. Find the total current supplied by the battery.","options":{"A":"9.2 A","B":"5.6 A","C":"11.2 A","D":"12.2 A"},"correct_answer":"C","correct_answer_value":"11.2 A","solution":"Series branch resistance=11+14=25 Ω. Parallel equivalent R_eq=(25×3)/(25+3)=2.67857 Ω. Total current I=V/R_eq=30/2.67857=11.2 A.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 11.2 A, not 9.2 A.","B":"Incorrect. Substituting the stated values into the governing relation gives 11.2 A, not 5.6 A.","C":"Correct. The worked calculation gives 11.2 A.","D":"Incorrect. Substituting the stated values into the governing relation gives 11.2 A, not 12.2 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_007550","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":"58 mL of 0.10 M HCl is mixed with 10 mL of 0.10 M NaOH at 25°C. Assuming complete dissociation and additive volumes, find the final pH.","options":{"A":"1.15127","B":"1.11513","C":"4.15127","D":"-0.84873"},"correct_answer":"A","correct_answer_value":"1.15127","solution":"Excess H⁺ moles=0.10×(58−10)/1000=0.004800 mol. Total volume=0.068 L, so [H⁺]=0.070588 M and pH=−log₁₀[H⁺]=1.15127.","option_explanations":{"A":"Correct. The worked calculation gives 1.15127.","B":"Incorrect. Substituting the stated values into the governing relation gives 1.15127, not 1.11513.","C":"Incorrect. Substituting the stated values into the governing relation gives 1.15127, not 4.15127.","D":"Incorrect. Substituting the stated values into the governing relation gives 1.15127, not -0.84873."},"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_004365","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 1 m is acted on by a constant torque of 32 N m about its central axis. Find its angular acceleration.","options":{"A":"3.92308 rad s⁻²","B":"1.49231 rad s⁻²","C":"4.92308 rad s⁻²","D":"-4.92308 rad s⁻²"},"correct_answer":"C","correct_answer_value":"4.92308 rad s⁻²","solution":"For a solid disc, I=½MR²=½(13)(1)²=6.5 kg m². Using τ=Iα, α=32/6.5=4.92308 rad s⁻².","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 4.92308 rad s⁻², not 3.92308 rad s⁻².","B":"Incorrect. Substituting the stated values into the governing relation gives 4.92308 rad s⁻², not 1.49231 rad s⁻².","C":"Correct. The worked calculation gives 4.92308 rad s⁻².","D":"Incorrect. Substituting the stated values into the governing relation gives 4.92308 rad s⁻², not -4.92308 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_003435","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 78 m s⁻¹ at 75° to the horizontal. Take g=10 m s⁻². Find its horizontal range.","options":{"A":"302.2 m","B":"306.2 m","C":"305.2 m","D":"304.2 m"},"correct_answer":"D","correct_answer_value":"304.2 m","solution":"Range R=u²sin(2θ)/g. Substituting u=78, θ=75° and g=10 gives R=(78)²sin(150°)/10=304.2 m.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 304.2 m, not 302.2 m.","B":"Incorrect. Substituting the stated values into the governing relation gives 304.2 m, not 306.2 m.","C":"Incorrect. Substituting the stated values into the governing relation gives 304.2 m, not 305.2 m.","D":"Correct. The worked calculation gives 304.2 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_005320","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 10 cm in vacuum. Using k=9×10⁹ N m² C⁻², find the magnitude of the electrostatic force.","options":{"A":"10.8 N","B":"-14.4 N","C":"14.4 N","D":"11.4 N"},"correct_answer":"C","correct_answer_value":"14.4 N","solution":"Convert charges to coulombs and distance to metres. F=k|q₁q₂|/r²=9×10⁹(2×10⁻⁶)(8×10⁻⁶)/(10×10⁻²)²=14.4 N.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 14.4 N, not 10.8 N.","B":"Incorrect. Substituting the stated values into the governing relation gives 14.4 N, not -14.4 N.","C":"Correct. The worked calculation gives 14.4 N.","D":"Incorrect. Substituting the stated values into the governing relation gives 14.4 N, not 11.4 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_001847","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 69 distinct elements?","options":{"A":"-8815083648488","B":"4407541824244","C":"8815083648488","D":"17630167296976"},"correct_answer":"C","correct_answer_value":"8815083648488","solution":"Order does not matter, so the count is C(69,12)=69!/[12!(57)!]=8815083648488.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 8815083648488, not -8815083648488.","B":"Incorrect. Substituting the stated values into the governing relation gives 8815083648488, not 4407541824244.","C":"Correct. The worked calculation gives 8815083648488.","D":"Incorrect. Substituting the stated values into the governing relation gives 8815083648488, not 17630167296976."},"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_009220","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=0.50 V, transfers 1 electrons, and has reaction quotient Q=10^-2. Using E=E°−(0.0591/n)log₁₀Q, calculate Ecell.","options":{"A":"-0.3818 V","B":"0.6182 V","C":"-1.3818 V","D":"0.46365 V"},"correct_answer":"B","correct_answer_value":"0.6182 V","solution":"Apply the Nernst equation: E=0.50−(0.0591/1)(-2)=0.6182 V.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.6182 V, not -0.3818 V.","B":"Correct. The worked calculation gives 0.6182 V.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.6182 V, not -1.3818 V.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.6182 V, not 0.46365 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_002375","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 [[3, 5], [1, 6]].","options":{"A":"15","B":"14","C":"13","D":"12"},"correct_answer":"C","correct_answer_value":"13","solution":"For a 2×2 matrix [[a,b],[c,d]], det=ad−bc. Here it is (3)(6)−(5)(1)=13.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 13, not 15.","B":"Incorrect. Substituting the stated values into the governing relation gives 13, not 14.","C":"Correct. The worked calculation gives 13.","D":"Incorrect. Substituting the stated values into the governing relation gives 13, 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_007379","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":"77 mL of 0.10 M HCl is mixed with 97 mL of 0.10 M NaOH at 25°C. Assuming complete dissociation and additive volumes, find the final pH.","options":{"A":"12.06048","B":"-12.06048","C":"11.06048","D":"6.03024"},"correct_answer":"A","correct_answer_value":"12.06048","solution":"Excess OH⁻ moles=0.10×(97−77)/1000=0.002000 mol. Total volume=0.174 L, so [OH⁻]=0.011494 M, pOH=−log₁₀[OH⁻], and pH=14−pOH=12.06048.","option_explanations":{"A":"Correct. The worked calculation gives 12.06048.","B":"Incorrect. Substituting the stated values into the governing relation gives 12.06048, not -12.06048.","C":"Incorrect. Substituting the stated values into the governing relation gives 12.06048, not 11.06048.","D":"Incorrect. Substituting the stated values into the governing relation gives 12.06048, not 6.03024."},"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_004178","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 26 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":"62.4 N","B":"61.4 N","C":"93.6 N","D":"46.8 N"},"correct_answer":"A","correct_answer_value":"62.4 N","solution":"Net force=mg(sinθ−μcosθ)=26×10×(0.6−0.45×0.8)=62.4 N.","option_explanations":{"A":"Correct. The worked calculation gives 62.4 N.","B":"Incorrect. Substituting the stated values into the governing relation gives 62.4 N, not 61.4 N.","C":"Incorrect. Substituting the stated values into the governing relation gives 62.4 N, not 93.6 N.","D":"Incorrect. Substituting the stated values into the governing relation gives 62.4 N, not 46.8 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_002535","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, -8], [1, -7]].","options":{"A":"-18","B":"-20","C":"-19","D":"-40"},"correct_answer":"B","correct_answer_value":"-20","solution":"For a 2×2 matrix [[a,b],[c,d]], det=ad−bc. Here it is (4)(-7)−(-8)(1)=-20.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives -20, not -18.","B":"Correct. The worked calculation gives -20.","C":"Incorrect. Substituting the stated values into the governing relation gives -20, not -19.","D":"Incorrect. Substituting the stated values into the governing relation gives -20, not -40."},"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_009123","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=1.37 V, transfers 3 electrons, and has reaction quotient Q=10^-6. Using E=E°−(0.0591/n)log₁₀Q, calculate Ecell.","options":{"A":"1.4882 V","B":"3.4882 V","C":"0.4882 V","D":"-1.4882 V"},"correct_answer":"A","correct_answer_value":"1.4882 V","solution":"Apply the Nernst equation: E=1.37−(0.0591/3)(-6)=1.4882 V.","option_explanations":{"A":"Correct. The worked calculation gives 1.4882 V.","B":"Incorrect. Substituting the stated values into the governing relation gives 1.4882 V, not 3.4882 V.","C":"Incorrect. Substituting the stated values into the governing relation gives 1.4882 V, not 0.4882 V.","D":"Incorrect. Substituting the stated values into the governing relation gives 1.4882 V, not -1.4882 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_009888","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(Cl)_5]^-4, each Cl ligand has charge −1. Determine the formal oxidation state of Pd.","options":{"A":"-1","B":"1","C":"-2","D":"0"},"correct_answer":"B","correct_answer_value":"1","solution":"Let the formal oxidation state of Pd be x. Charge balance gives x+(5)(−1)=-4; hence x=-4+5=+1.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 1, not -1.","B":"Correct. The worked calculation gives 1.","C":"Incorrect. Substituting the stated values into the governing relation gives 1, not -2.","D":"Incorrect. Substituting the stated values into the governing relation gives 1, not 0."},"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_001580","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 (6x² +4x -2) dx.","options":{"A":"16.2","B":"152","C":"228","D":"151"},"correct_answer":"B","correct_answer_value":"152","solution":"An antiderivative is (6/3)x³+(4/2)x²+(-2)x. Evaluating between 0 and 4 gives (6/3)(4)³+(4/2)(4)²+(-2)(4)=152.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 152, not 16.2.","B":"Correct. The worked calculation gives 152.","C":"Incorrect. Substituting the stated values into the governing relation gives 152, not 228.","D":"Incorrect. Substituting the stated values into the governing relation gives 152, not 151."},"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_000837","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 4 and common ratio 4. What is the sum of its first 8 terms?","options":{"A":"-87380","B":"69904","C":"87380","D":"87382"},"correct_answer":"C","correct_answer_value":"87380","solution":"For r≠1, Sₙ=a(rⁿ−1)/(r−1). Thus Sₙ=4(4^8−1)/(4−1)=87380.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 87380, not -87380.","B":"Incorrect. Substituting the stated values into the governing relation gives 87380, not 69904.","C":"Correct. The worked calculation gives 87380.","D":"Incorrect. Substituting the stated values into the governing relation gives 87380, not 87382."},"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_002047","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, 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":"0.38961","B":"0.29221","C":"0.58442","D":"1.38961"},"correct_answer":"A","correct_answer_value":"0.38961","solution":"Favourable pairs = C(7,2)+C(12,2)+C(3,2)=90. Total pairs=C(22,2)=231. Therefore P=90/231=0.38961.","option_explanations":{"A":"Correct. The worked calculation gives 0.38961.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.38961, not 0.29221.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.38961, not 0.58442.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.38961, not 1.38961."},"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_009136","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=0.58 V, transfers 3 electrons, and has reaction quotient Q=10^4. Using E=E°−(0.0591/n)log₁₀Q, calculate Ecell.","options":{"A":"1.5012 V","B":"1.0024 V","C":"-0.5012 V","D":"0.5012 V"},"correct_answer":"D","correct_answer_value":"0.5012 V","solution":"Apply the Nernst equation: E=0.58−(0.0591/3)(4)=0.5012 V.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.5012 V, not 1.5012 V.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.5012 V, not 1.0024 V.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.5012 V, not -0.5012 V.","D":"Correct. The worked calculation gives 0.5012 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_003921","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 9 m s⁻¹. A constant net force does 15 N of work over 4 m in the direction of motion. Find its final kinetic energy.","options":{"A":"262.5 J","B":"196.875 J","C":"261.5 J","D":"525 J"},"correct_answer":"A","correct_answer_value":"262.5 J","solution":"By the work–energy theorem, K_f=K_i+W=½mu²+Fs=½(5)(9)²+(15)(4)=262.5 J.","option_explanations":{"A":"Correct. The worked calculation gives 262.5 J.","B":"Incorrect. Substituting the stated values into the governing relation gives 262.5 J, not 196.875 J.","C":"Incorrect. Substituting the stated values into the governing relation gives 262.5 J, not 261.5 J.","D":"Incorrect. Substituting the stated values into the governing relation gives 262.5 J, not 525 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_009722","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(CN)_3]^+2, each CN ligand has charge −1. Determine the formal oxidation state of Pt.","options":{"A":"10","B":"6","C":"7","D":"5"},"correct_answer":"D","correct_answer_value":"5","solution":"Let the formal oxidation state of Pt be x. Charge balance gives x+(3)(−1)=+2; hence x=2+3=+5.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 5, not 10.","B":"Incorrect. Substituting the stated values into the governing relation gives 5, not 6.","C":"Incorrect. Substituting the stated values into the governing relation gives 5, not 7.","D":"Correct. The worked calculation gives 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_009005","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=1.40 V, transfers 2 electrons, and has reaction quotient Q=10^-4. Using E=E°−(0.0591/n)log₁₀Q, calculate Ecell.","options":{"A":"1.15182 V","B":"2.2773 V","C":"3.0364 V","D":"1.5182 V"},"correct_answer":"D","correct_answer_value":"1.5182 V","solution":"Apply the Nernst equation: E=1.40−(0.0591/2)(-4)=1.5182 V.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 1.5182 V, not 1.15182 V.","B":"Incorrect. Substituting the stated values into the governing relation gives 1.5182 V, not 2.2773 V.","C":"Incorrect. Substituting the stated values into the governing relation gives 1.5182 V, not 3.0364 V.","D":"Correct. The worked calculation gives 1.5182 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_009880","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)_5]^+3, each Cl ligand has charge −1. Determine the formal oxidation state of Pt.","options":{"A":"8","B":"5","C":"1.8","D":"16"},"correct_answer":"A","correct_answer_value":"8","solution":"Let the formal oxidation state of Pt 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 5.","C":"Incorrect. Substituting the stated values into the governing relation gives 8, not 1.8.","D":"Incorrect. Substituting the stated values into the governing relation gives 8, not 16."},"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_001358","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 (-1x² -2x +5) dx.","options":{"A":"-3","B":"-4.5","C":"0","D":"-2.25"},"correct_answer":"A","correct_answer_value":"-3","solution":"An antiderivative is (-1/3)x³+(-2/2)x²+(5)x. Evaluating between 0 and 3 gives (-1/3)(3)³+(-2/2)(3)²+(5)(3)=-3.","option_explanations":{"A":"Correct. The worked calculation gives -3.","B":"Incorrect. Substituting the stated values into the governing relation gives -3, not -4.5.","C":"Incorrect. Substituting the stated values into the governing relation gives -3, not 0.","D":"Incorrect. Substituting the stated values into the governing relation gives -3, not -2.25."},"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_008683","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 1.82 mol L⁻¹ and initially B is absent. At equilibrium, 19% of A has dissociated at constant volume. Find K_c.","options":{"A":"0.48668","B":"3.32445","C":"-0.67555","D":"0.32445"},"correct_answer":"D","correct_answer_value":"0.32445","solution":"Let α=0.19. At equilibrium [A]=c₀(1−α) and [B]=2c₀α. Thus K_c=[B]²/[A]=4c₀α²/(1−α)=4(1.82)(0.19)²/(1−0.19)=0.32445.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.32445, not 0.48668.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.32445, not 3.32445.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.32445, not -0.67555.","D":"Correct. The worked calculation gives 0.32445."},"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_007372","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":"71 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":"4.469","B":"2.20351","C":"1.469","D":"2.93801"},"correct_answer":"C","correct_answer_value":"1.469","solution":"Excess H⁺ moles=0.10×(71−35)/1000=0.003600 mol. Total volume=0.106 L, so [H⁺]=0.033962 M and pH=−log₁₀[H⁺]=1.469.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 1.469, not 4.469.","B":"Incorrect. Substituting the stated values into the governing relation gives 1.469, not 2.20351.","C":"Correct. The worked calculation gives 1.469.","D":"Incorrect. Substituting the stated values into the governing relation gives 1.469, not 2.93801."},"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_004564","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 6 m is acted on by a constant torque of 98 N m about its central axis. Find its angular acceleration.","options":{"A":"0.54444 rad s⁻²","B":"3.54444 rad s⁻²","C":"0.27222 rad s⁻²","D":"-0.54444 rad s⁻²"},"correct_answer":"A","correct_answer_value":"0.54444 rad s⁻²","solution":"For a solid disc, I=½MR²=½(10)(6)²=180 kg m². Using τ=Iα, α=98/180=0.54444 rad s⁻².","option_explanations":{"A":"Correct. The worked calculation gives 0.54444 rad s⁻².","B":"Incorrect. Substituting the stated values into the governing relation gives 0.54444 rad s⁻², not 3.54444 rad s⁻².","C":"Incorrect. Substituting the stated values into the governing relation gives 0.54444 rad s⁻², not 0.27222 rad s⁻².","D":"Incorrect. Substituting the stated values into the governing relation gives 0.54444 rad s⁻², not -0.54444 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_004595","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 1 m is acted on by a constant torque of 86 N m about its central axis. Find its angular acceleration.","options":{"A":"11.23077 rad s⁻²","B":"15.23077 rad s⁻²","C":"9.92308 rad s⁻²","D":"13.23077 rad s⁻²"},"correct_answer":"D","correct_answer_value":"13.23077 rad s⁻²","solution":"For a solid disc, I=½MR²=½(13)(1)²=6.5 kg m². Using τ=Iα, α=86/6.5=13.23077 rad s⁻².","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 13.23077 rad s⁻², not 11.23077 rad s⁻².","B":"Incorrect. Substituting the stated values into the governing relation gives 13.23077 rad s⁻², not 15.23077 rad s⁻².","C":"Incorrect. Substituting the stated values into the governing relation gives 13.23077 rad s⁻², not 9.92308 rad s⁻².","D":"Correct. The worked calculation gives 13.23077 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_005938","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 2.1 atm at 263 K. What is its pressure at 585 K?","options":{"A":"3.6711 atm","B":"4.6711 atm","C":"7.00665 atm","D":"6.6711 atm"},"correct_answer":"B","correct_answer_value":"4.6711 atm","solution":"At constant volume, P/T is constant. P₂=P₁T₂/T₁=(2.1)(585)/(263)=4.6711 atm.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 4.6711 atm, not 3.6711 atm.","B":"Correct. The worked calculation gives 4.6711 atm.","C":"Incorrect. Substituting the stated values into the governing relation gives 4.6711 atm, not 7.00665 atm.","D":"Incorrect. Substituting the stated values into the governing relation gives 4.6711 atm, not 6.6711 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_000841","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 10 terms?","options":{"A":"-7324218","B":"7324218","C":"5493163.5","D":"8789061.6"},"correct_answer":"B","correct_answer_value":"7324218","solution":"For r≠1, Sₙ=a(rⁿ−1)/(r−1). Thus Sₙ=3(5^10−1)/(5−1)=7324218.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 7324218, not -7324218.","B":"Correct. The worked calculation gives 7324218.","C":"Incorrect. Substituting the stated values into the governing relation gives 7324218, not 5493163.5.","D":"Incorrect. Substituting the stated values into the governing relation gives 7324218, not 8789061.6."},"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_001011","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 -6x² -7x+7. Evaluate f′(-2).","options":{"A":"-5","B":"10","C":"2","D":"5"},"correct_answer":"D","correct_answer_value":"5","solution":"Differentiate: f′(x)=3(-1)x²+2(-6)x+(-7). At x=-2, f′(-2)=3(-1)(-2)²+2(-6)(-2)+(-7)=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 10.","C":"Incorrect. Substituting the stated values into the governing relation gives 5, not 2.","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_003223","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=(15-7i)/(5+9i). What is Re(z), the real part of z?","options":{"A":"0.0566","B":"0.11321","C":"2.11321","D":"0.22642"},"correct_answer":"B","correct_answer_value":"0.11321","solution":"Multiply numerator and denominator by the conjugate (5-9i). Then Re(z)=(ac+bd)/(c²+d²)=((15)(5)+(-7)(9))/(5²+(9)²)=0.11321.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.11321, not 0.0566.","B":"Correct. The worked calculation gives 0.11321.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.11321, not 2.11321.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.11321, not 0.22642."},"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_005114","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 +4 μC are separated by 7 cm in vacuum. Using k=9×10⁹ N m² C⁻², find the magnitude of the electrostatic force.","options":{"A":"35.26531 N","B":"44.08163 N","C":"66.12245 N","D":"42.08163 N"},"correct_answer":"B","correct_answer_value":"44.08163 N","solution":"Convert charges to coulombs and distance to metres. F=k|q₁q₂|/r²=9×10⁹(6×10⁻⁶)(4×10⁻⁶)/(7×10⁻²)²=44.08163 N.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 44.08163 N, not 35.26531 N.","B":"Correct. The worked calculation gives 44.08163 N.","C":"Incorrect. Substituting the stated values into the governing relation gives 44.08163 N, not 66.12245 N.","D":"Incorrect. Substituting the stated values into the governing relation gives 44.08163 N, not 42.08163 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_005115","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 +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":"2.625 N","B":"5.625 N","C":"1.5625 N","D":"2.8125 N"},"correct_answer":"B","correct_answer_value":"5.625 N","solution":"Convert charges to coulombs and distance to metres. F=k|q₁q₂|/r²=9×10⁹(3×10⁻⁶)(3×10⁻⁶)/(12×10⁻²)²=5.625 N.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 5.625 N, not 2.625 N.","B":"Correct. The worked calculation gives 5.625 N.","C":"Incorrect. Substituting the stated values into the governing relation gives 5.625 N, not 1.5625 N.","D":"Incorrect. Substituting the stated values into the governing relation gives 5.625 N, not 2.8125 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_006481","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 231 N m⁻¹ and oscillates horizontally without damping. What is the time period divided by π?","options":{"A":"0.37219 s","B":"-1.62781 s","C":"-0.62781 s","D":"1.37219 s"},"correct_answer":"A","correct_answer_value":"0.37219 s","solution":"For a mass–spring oscillator, T=2π√(m/k). Therefore T/π=2√(8/231)=0.37219 s.","option_explanations":{"A":"Correct. The worked calculation gives 0.37219 s.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.37219 s, not -1.62781 s.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.37219 s, not -0.62781 s.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.37219 s, not 1.37219 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_001200","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 +0x² +8x+7. Evaluate f′(-4).","options":{"A":"-184","B":"-185","C":"-220.8","D":"-182"},"correct_answer":"A","correct_answer_value":"-184","solution":"Differentiate: f′(x)=3(-4)x²+2(0)x+(8). At x=-4, f′(-4)=3(-4)(-4)²+2(0)(-4)+(8)=-184.","option_explanations":{"A":"Correct. The worked calculation gives -184.","B":"Incorrect. Substituting the stated values into the governing relation gives -184, not -185.","C":"Incorrect. Substituting the stated values into the governing relation gives -184, not -220.8.","D":"Incorrect. Substituting the stated values into the governing relation gives -184, not -182."},"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_006509","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 17 kg is attached to an ideal spring of force constant 713 N m⁻¹ and oscillates horizontally without damping. What is the time period divided by π?","options":{"A":"2.30882 s","B":"0.30882 s","C":"0.46323 s","D":"1.03088 s"},"correct_answer":"B","correct_answer_value":"0.30882 s","solution":"For a mass–spring oscillator, T=2π√(m/k). Therefore T/π=2√(17/713)=0.30882 s.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.30882 s, not 2.30882 s.","B":"Correct. The worked calculation gives 0.30882 s.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.30882 s, not 0.46323 s.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.30882 s, not 1.03088 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_000937","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 4 and common ratio 5. What is the sum of its first 11 terms?","options":{"A":"48828124","B":"-48828124","C":"48828123","D":"73242186"},"correct_answer":"A","correct_answer_value":"48828124","solution":"For r≠1, Sₙ=a(rⁿ−1)/(r−1). Thus Sₙ=4(5^11−1)/(5−1)=48828124.","option_explanations":{"A":"Correct. The worked calculation gives 48828124.","B":"Incorrect. Substituting the stated values into the governing relation gives 48828124, not -48828124.","C":"Incorrect. Substituting the stated values into the governing relation gives 48828124, not 48828123.","D":"Incorrect. Substituting the stated values into the governing relation gives 48828124, not 73242186."},"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_006126","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 26 cm. Using the Cartesian sign convention, what is the magnitude of the linear magnification?","options":{"A":"1.5","B":"-1","C":"1","D":"2"},"correct_answer":"D","correct_answer_value":"2","solution":"Using distances as positive magnitudes, 1/f=1/v+1/u gives v=uf/(u−f)=(39)(26)/(39−26)=78 cm. Thus |m|=v/u=78/39=2.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 2, not 1.5.","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 1.","D":"Correct. The worked calculation gives 2."},"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_008165","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 3049 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":"3.00951 g","B":"4.51426 g","C":"0.00951 g","D":"2.25713 g"},"correct_answer":"A","correct_answer_value":"3.00951 g","solution":"Cu²⁺+2e⁻→Cu, so z=2. By Faraday's law m=MIt/(zF)=63.5×3×3049/(2×96500)=3.00951 g.","option_explanations":{"A":"Correct. The worked calculation gives 3.00951 g.","B":"Incorrect. Substituting the stated values into the governing relation gives 3.00951 g, not 4.51426 g.","C":"Incorrect. Substituting the stated values into the governing relation gives 3.00951 g, not 0.00951 g.","D":"Incorrect. Substituting the stated values into the governing relation gives 3.00951 g, not 2.25713 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_000039","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 -3x -8=0, let its roots be α and β. What is α²+β²?","options":{"A":"25","B":"20","C":"50","D":"27"},"correct_answer":"A","correct_answer_value":"25","solution":"By Vieta's relations, α+β = 3/1 and αβ = -8/1. Then α²+β²=(α+β)²−2αβ = (3/1)²−2(-8/1) = 25.","option_explanations":{"A":"Correct. The worked calculation gives 25.","B":"Incorrect. Substituting the stated values into the governing relation gives 25, not 20.","C":"Incorrect. Substituting the stated values into the governing relation gives 25, not 50.","D":"Incorrect. Substituting the stated values into the governing relation gives 25, not 27."},"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_005061","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 +2 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":"-1.25 N","B":"-0.75 N","C":"1.25 N","D":"0.25 N"},"correct_answer":"C","correct_answer_value":"1.25 N","solution":"Convert charges to coulombs and distance to metres. F=k|q₁q₂|/r²=9×10⁹(1×10⁻⁶)(2×10⁻⁶)/(12×10⁻²)²=1.25 N.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 1.25 N, not -1.25 N.","B":"Incorrect. Substituting the stated values into the governing relation gives 1.25 N, not -0.75 N.","C":"Correct. The worked calculation gives 1.25 N.","D":"Incorrect. Substituting the stated values into the governing relation gives 1.25 N, not 0.25 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_007491","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 100 mL of 0.10 M NaOH at 25°C. Assuming complete dissociation and additive volumes, find the final pH.","options":{"A":"2.24629","B":"12.46288","C":"13.46288","D":"6.23144"},"correct_answer":"B","correct_answer_value":"12.46288","solution":"Excess OH⁻ moles=0.10×(100−55)/1000=0.004500 mol. Total volume=0.155 L, so [OH⁻]=0.029032 M, pOH=−log₁₀[OH⁻], and pH=14−pOH=12.46288.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 12.46288, not 2.24629.","B":"Correct. The worked calculation gives 12.46288.","C":"Incorrect. Substituting the stated values into the governing relation gives 12.46288, not 13.46288.","D":"Incorrect. Substituting the stated values into the governing relation gives 12.46288, not 6.23144."},"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_007036","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 382 K and 1.68 atm? Use R=0.082057 L atm mol⁻¹ K⁻¹.","options":{"A":"335.84758 L","B":"336.84758 L","C":"34.58476 L","D":"671.69516 L"},"correct_answer":"A","correct_answer_value":"335.84758 L","solution":"PV=nRT, so V=nRT/P=(18)(0.082057)(382)/(1.68)=335.84758 L.","option_explanations":{"A":"Correct. The worked calculation gives 335.84758 L.","B":"Incorrect. Substituting the stated values into the governing relation gives 335.84758 L, not 336.84758 L.","C":"Incorrect. Substituting the stated values into the governing relation gives 335.84758 L, not 34.58476 L.","D":"Incorrect. Substituting the stated values into the governing relation gives 335.84758 L, not 671.69516 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_005553","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 22 V. Calculate the energy stored, expressed in μJ.","options":{"A":"7502 μJ","B":"7501 μJ","C":"7503 μJ","D":"751.2 μJ"},"correct_answer":"A","correct_answer_value":"7502 μJ","solution":"U=½CV². Since C=31 μF, U in μJ is ½×31×(22)²=7502 μJ.","option_explanations":{"A":"Correct. The worked calculation gives 7502 μJ.","B":"Incorrect. Substituting the stated values into the governing relation gives 7502 μJ, not 7501 μJ.","C":"Incorrect. Substituting the stated values into the governing relation gives 7502 μJ, not 7503 μJ.","D":"Incorrect. Substituting the stated values into the governing relation gives 7502 μJ, not 751.2 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_002619","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], [1, 3]].","options":{"A":"3","B":"4","C":"2","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)(3)−(-1)(1)=4.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 4, not 3.","B":"Correct. The worked calculation gives 4.","C":"Incorrect. Substituting the stated values into the governing relation gives 4, not 2.","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_000018","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 +8=0, let its roots be α and β. What is α²+β²?","options":{"A":"-2.33333","B":"-1.33333","C":"1.33333","D":"-2"},"correct_answer":"B","correct_answer_value":"-1.33333","solution":"By Vieta's relations, α+β = -6/3 and αβ = 8/3. Then α²+β²=(α+β)²−2αβ = (-6/3)²−2(8/3) = -1.33333.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives -1.33333, not -2.33333.","B":"Correct. The worked calculation gives -1.33333.","C":"Incorrect. Substituting the stated values into the governing relation gives -1.33333, not 1.33333.","D":"Incorrect. Substituting the stated values into the governing relation gives -1.33333, not -2."},"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_007291","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 10 mol of gas at 243 K and 1.62 atm? Use R=0.082057 L atm mol⁻¹ K⁻¹.","options":{"A":"92.31412 L","B":"-123.0855 L","C":"123.0855 L","D":"98.4684 L"},"correct_answer":"C","correct_answer_value":"123.0855 L","solution":"PV=nRT, so V=nRT/P=(10)(0.082057)(243)/(1.62)=123.0855 L.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 123.0855 L, not 92.31412 L.","B":"Incorrect. Substituting the stated values into the governing relation gives 123.0855 L, not -123.0855 L.","C":"Correct. The worked calculation gives 123.0855 L.","D":"Incorrect. Substituting the stated values into the governing relation gives 123.0855 L, not 98.4684 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_000302","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 -11x -12=0, let its roots be α and β. What is α²+β²?","options":{"A":"23.44444","B":"22.44444","C":"20.44444","D":"21.44444"},"correct_answer":"D","correct_answer_value":"21.44444","solution":"By Vieta's relations, α+β = 11/3 and αβ = -12/3. Then α²+β²=(α+β)²−2αβ = (11/3)²−2(-12/3) = 21.44444.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 21.44444, not 23.44444.","B":"Incorrect. Substituting the stated values into the governing relation gives 21.44444, not 22.44444.","C":"Incorrect. Substituting the stated values into the governing relation gives 21.44444, not 20.44444.","D":"Correct. The worked calculation gives 21.44444."},"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_007868","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 160 minutes?","options":{"A":"1.01953 mmol L⁻¹","B":"3.19531 mmol L⁻¹","C":"0.19531 mmol L⁻¹","D":"0.09766 mmol L⁻¹"},"correct_answer":"C","correct_answer_value":"0.19531 mmol L⁻¹","solution":"160 minutes corresponds to 8 half-lives. The remaining concentration is [A]₀(1/2)ⁿ=50/2^8=0.19531 mmol L⁻¹.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.19531 mmol L⁻¹, not 1.01953 mmol L⁻¹.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.19531 mmol L⁻¹, not 3.19531 mmol L⁻¹.","C":"Correct. The worked calculation gives 0.19531 mmol L⁻¹.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.19531 mmol L⁻¹, not 0.09766 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_008380","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 0.05 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":"0.3 K","B":"3.15 K","C":"0.15 K","D":"0.075 K"},"correct_answer":"C","correct_answer_value":"0.15 K","solution":"ΔT_b=iK_bm. Substitution gives ΔT_b=3×1.0×0.05=0.15 K.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.15 K, not 0.3 K.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.15 K, not 3.15 K.","C":"Correct. The worked calculation gives 0.15 K.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.15 K, not 0.075 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_009321","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=0.86 V, transfers 2 electrons, and has reaction quotient Q=10^5. Using E=E°−(0.0591/n)log₁₀Q, calculate Ecell.","options":{"A":"0.71225 V","B":"1.06838 V","C":"-0.28775 V","D":"1.4245 V"},"correct_answer":"A","correct_answer_value":"0.71225 V","solution":"Apply the Nernst equation: E=0.86−(0.0591/2)(5)=0.71225 V.","option_explanations":{"A":"Correct. The worked calculation gives 0.71225 V.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.71225 V, not 1.06838 V.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.71225 V, not -0.28775 V.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.71225 V, not 1.4245 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_006057","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 143 cm in front of a thin convex lens of focal length 30 cm. Using the Cartesian sign convention, what is the magnitude of the linear magnification?","options":{"A":"1.26549","B":"0.26549","C":"1.02655","D":"-2.73451"},"correct_answer":"B","correct_answer_value":"0.26549","solution":"Using distances as positive magnitudes, 1/f=1/v+1/u gives v=uf/(u−f)=(143)(30)/(143−30)=37.9646 cm. Thus |m|=v/u=37.9646/143=0.26549.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.26549, not 1.26549.","B":"Correct. The worked calculation gives 0.26549.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.26549, not 1.02655.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.26549, not -2.73451."},"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_009536","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_3H_1N_1X_2. Treat X as the total number of monovalent halogen atoms.","options":{"A":"5","B":"2","C":"3","D":"1.5"},"correct_answer":"C","correct_answer_value":"3","solution":"Use DBE=(2C+2+N−H−X)/2. Substitution gives (2×3+2+1−1−2)/2=3.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 3, not 5.","B":"Incorrect. Substituting the stated values into the governing relation gives 3, not 2.","C":"Correct. The worked calculation gives 3.","D":"Incorrect. Substituting the stated values into the governing relation gives 3, not 1.5."},"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_003630","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 119 m s⁻¹ at 55° to the horizontal. Take g=10 m s⁻². Find its horizontal range.","options":{"A":"1330.69872 m","B":"1996.04808 m","C":"998.02404 m","D":"1331.69872 m"},"correct_answer":"A","correct_answer_value":"1330.69872 m","solution":"Range R=u²sin(2θ)/g. Substituting u=119, θ=55° and g=10 gives R=(119)²sin(110°)/10=1330.69872 m.","option_explanations":{"A":"Correct. The worked calculation gives 1330.69872 m.","B":"Incorrect. Substituting the stated values into the governing relation gives 1330.69872 m, not 1996.04808 m.","C":"Incorrect. Substituting the stated values into the governing relation gives 1330.69872 m, not 998.02404 m.","D":"Incorrect. Substituting the stated values into the governing relation gives 1330.69872 m, not 1331.69872 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_004956","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 6 Ω are connected in series; this combination is connected in parallel with a 12 Ω resistor across a 36 V ideal battery. Find the total current supplied by the battery.","options":{"A":"3.76923 A","B":"5.76923 A","C":"8.65385 A","D":"6.76923 A"},"correct_answer":"B","correct_answer_value":"5.76923 A","solution":"Series branch resistance=7+6=13 Ω. Parallel equivalent R_eq=(13×12)/(13+12)=6.24 Ω. Total current I=V/R_eq=36/6.24=5.76923 A.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 5.76923 A, not 3.76923 A.","B":"Correct. The worked calculation gives 5.76923 A.","C":"Incorrect. Substituting the stated values into the governing relation gives 5.76923 A, not 8.65385 A.","D":"Incorrect. Substituting the stated values into the governing relation gives 5.76923 A, not 6.76923 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_000353","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 4. Find the sum of its first 25 terms.","options":{"A":"900","B":"750","C":"1500","D":"752"},"correct_answer":"B","correct_answer_value":"750","solution":"Use Sₙ = n/2[2a+(n−1)d]. Substituting a=-18, d=4, n=25: Sₙ=25/2[2(-18)+(25−1)(4)]=750.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 750, not 900.","B":"Correct. The worked calculation gives 750.","C":"Incorrect. Substituting the stated values into the governing relation gives 750, not 1500.","D":"Incorrect. Substituting the stated values into the governing relation gives 750, not 752."},"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_002741","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, -4) and v=(2, 6, 5), calculate u·v.","options":{"A":"-52","B":"26","C":"-26","D":"-13"},"correct_answer":"C","correct_answer_value":"-26","solution":"The dot product is the sum of corresponding component products: (6)(2)+(-3)(6)+(-4)(5)=-26.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives -26, not -52.","B":"Incorrect. Substituting the stated values into the governing relation gives -26, not 26.","C":"Correct. The worked calculation gives -26.","D":"Incorrect. Substituting the stated values into the governing relation gives -26, 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_000395","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 -11 and common difference 5. Find the sum of its first 16 terms.","options":{"A":"425","B":"422","C":"424","D":"339.2"},"correct_answer":"C","correct_answer_value":"424","solution":"Use Sₙ = n/2[2a+(n−1)d]. Substituting a=-11, d=5, n=16: Sₙ=16/2[2(-11)+(16−1)(5)]=424.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 424, not 425.","B":"Incorrect. Substituting the stated values into the governing relation gives 424, not 422.","C":"Correct. The worked calculation gives 424.","D":"Incorrect. Substituting the stated values into the governing relation gives 424, not 339.2."},"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_000840","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 2. What is the sum of its first 6 terms?","options":{"A":"64","B":"630","C":"315","D":"756"},"correct_answer":"B","correct_answer_value":"630","solution":"For r≠1, Sₙ=a(rⁿ−1)/(r−1). Thus Sₙ=10(2^6−1)/(2−1)=630.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 630, not 64.","B":"Correct. The worked calculation gives 630.","C":"Incorrect. Substituting the stated values into the governing relation gives 630, not 315.","D":"Incorrect. Substituting the stated values into the governing relation gives 630, not 756."},"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_007865","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 140 minutes?","options":{"A":"0.54688 mmol L⁻¹","B":"1.54688 mmol L⁻¹","C":"1.05469 mmol L⁻¹","D":"-0.45312 mmol L⁻¹"},"correct_answer":"A","correct_answer_value":"0.54688 mmol L⁻¹","solution":"140 minutes corresponds to 7 half-lives. The remaining concentration is [A]₀(1/2)ⁿ=70/2^7=0.54688 mmol L⁻¹.","option_explanations":{"A":"Correct. The worked calculation gives 0.54688 mmol L⁻¹.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.54688 mmol L⁻¹, not 1.54688 mmol L⁻¹.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.54688 mmol L⁻¹, not 1.05469 mmol L⁻¹.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.54688 mmol L⁻¹, not -0.45312 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_000173","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 -17x +1=0, let its roots be α and β. What is α²+β²?","options":{"A":"4.61224","B":"-5.61224","C":"2.61224","D":"5.61224"},"correct_answer":"D","correct_answer_value":"5.61224","solution":"By Vieta's relations, α+β = 17/7 and αβ = 1/7. Then α²+β²=(α+β)²−2αβ = (17/7)²−2(1/7) = 5.61224.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 5.61224, not 4.61224.","B":"Incorrect. Substituting the stated values into the governing relation gives 5.61224, not -5.61224.","C":"Incorrect. Substituting the stated values into the governing relation gives 5.61224, not 2.61224.","D":"Correct. The worked calculation gives 5.61224."},"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_001757","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 87 distinct elements?","options":{"A":"211990","B":"84796","C":"105995","D":"158992.5"},"correct_answer":"C","correct_answer_value":"105995","solution":"Order does not matter, so the count is C(87,3)=87!/[3!(84)!]=105995.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 105995, not 211990.","B":"Incorrect. Substituting the stated values into the governing relation gives 105995, not 84796.","C":"Correct. The worked calculation gives 105995.","D":"Incorrect. Substituting the stated values into the governing relation gives 105995, not 158992.5."},"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_003123","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=(10+3i)/(6+5i). What is Re(z), the real part of z?","options":{"A":"0.61475","B":"1.22951","C":"4.22951","D":"0.22951"},"correct_answer":"B","correct_answer_value":"1.22951","solution":"Multiply numerator and denominator by the conjugate (6-5i). Then Re(z)=(ac+bd)/(c²+d²)=((10)(6)+(3)(5))/(6²+(5)²)=1.22951.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 1.22951, not 0.61475.","B":"Correct. The worked calculation gives 1.22951.","C":"Incorrect. Substituting the stated values into the governing relation gives 1.22951, not 4.22951.","D":"Incorrect. Substituting the stated values into the governing relation gives 1.22951, not 0.22951."},"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_004191","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 8 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":"42.96 N","B":"41.96 N","C":"40.96 N","D":"20.48 N"},"correct_answer":"C","correct_answer_value":"40.96 N","solution":"Net force=mg(sinθ−μcosθ)=8×10×(0.6−0.11×0.8)=40.96 N.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 40.96 N, not 42.96 N.","B":"Incorrect. Substituting the stated values into the governing relation gives 40.96 N, not 41.96 N.","C":"Correct. The worked calculation gives 40.96 N.","D":"Incorrect. Substituting the stated values into the governing relation gives 40.96 N, not 20.48 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_005365","subject":"physics","topic":"electrostatics","subtopic":"capacitor_energy","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A 7 μF capacitor is charged to a potential difference of 8 V. Calculate the energy stored, expressed in μJ.","options":{"A":"268.8 μJ","B":"23.4 μJ","C":"168 μJ","D":"224 μJ"},"correct_answer":"D","correct_answer_value":"224 μJ","solution":"U=½CV². Since C=7 μF, U in μJ is ½×7×(8)²=224 μJ.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 224 μJ, not 268.8 μJ.","B":"Incorrect. Substituting the stated values into the governing relation gives 224 μJ, not 23.4 μJ.","C":"Incorrect. Substituting the stated values into the governing relation gives 224 μJ, not 168 μJ.","D":"Correct. The worked calculation gives 224 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_006475","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 16 kg is attached to an ideal spring of force constant 211 N m⁻¹ and oscillates horizontally without damping. What is the time period divided by π?","options":{"A":"0.41306 s","B":"-2.44926 s","C":"0.82611 s","D":"0.55074 s"},"correct_answer":"D","correct_answer_value":"0.55074 s","solution":"For a mass–spring oscillator, T=2π√(m/k). Therefore T/π=2√(16/211)=0.55074 s.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.55074 s, not 0.41306 s.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.55074 s, not -2.44926 s.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.55074 s, not 0.82611 s.","D":"Correct. The worked calculation gives 0.55074 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_009940","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(OH)_5]^-4, each OH ligand has charge −1. Determine the formal oxidation state of Fe.","options":{"A":"1","B":"1.1","C":"2","D":"3"},"correct_answer":"A","correct_answer_value":"1","solution":"Let the formal oxidation state of Fe be x. Charge balance gives x+(5)(−1)=-4; hence x=-4+5=+1.","option_explanations":{"A":"Correct. The worked calculation gives 1.","B":"Incorrect. Substituting the stated values into the governing relation gives 1, not 1.1.","C":"Incorrect. Substituting the stated values into the governing relation gives 1, not 2.","D":"Incorrect. Substituting the stated values into the governing relation gives 1, 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_007216","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 19 mol of gas at 240 K and 1.6 atm? Use R=0.082057 L atm mol⁻¹ K⁻¹.","options":{"A":"24.38624 L","B":"231.86245 L","C":"233.86245 L","D":"175.39684 L"},"correct_answer":"C","correct_answer_value":"233.86245 L","solution":"PV=nRT, so V=nRT/P=(19)(0.082057)(240)/(1.6)=233.86245 L.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 233.86245 L, not 24.38624 L.","B":"Incorrect. Substituting the stated values into the governing relation gives 233.86245 L, not 231.86245 L.","C":"Correct. The worked calculation gives 233.86245 L.","D":"Incorrect. Substituting the stated values into the governing relation gives 233.86245 L, not 175.39684 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_008632","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 1.31 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":"2.31 K","B":"1.31 K","C":"4.31 K","D":"-1.69 K"},"correct_answer":"B","correct_answer_value":"1.31 K","solution":"ΔT_b=iK_bm. Substitution gives ΔT_b=1×1.0×1.31=1.31 K.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 1.31 K, not 2.31 K.","B":"Correct. The worked calculation gives 1.31 K.","C":"Incorrect. Substituting the stated values into the governing relation gives 1.31 K, not 4.31 K.","D":"Incorrect. Substituting the stated values into the governing relation gives 1.31 K, not -1.69 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_007467","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 48 mL of 0.10 M NaOH at 25°C. Assuming complete dissociation and additive volumes, find the final pH.","options":{"A":"-1.01201","B":"0.98799","C":"1.98799","D":"4.98799"},"correct_answer":"C","correct_answer_value":"1.98799","solution":"Excess H⁺ moles=0.10×(59−48)/1000=0.001100 mol. Total volume=0.107 L, so [H⁺]=0.010280 M and pH=−log₁₀[H⁺]=1.98799.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 1.98799, not -1.01201.","B":"Incorrect. Substituting the stated values into the governing relation gives 1.98799, not 0.98799.","C":"Correct. The worked calculation gives 1.98799.","D":"Incorrect. Substituting the stated values into the governing relation gives 1.98799, not 4.98799."},"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_002707","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, -5, -6) and v=(-3, 0, 8), calculate u·v.","options":{"A":"-51.75","B":"-69","C":"-34.5","D":"-5.9"},"correct_answer":"B","correct_answer_value":"-69","solution":"The dot product is the sum of corresponding component products: (7)(-3)+(-5)(0)+(-6)(8)=-69.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives -69, not -51.75.","B":"Correct. The worked calculation gives -69.","C":"Incorrect. Substituting the stated values into the governing relation gives -69, not -34.5.","D":"Incorrect. Substituting the stated values into the governing relation gives -69, not -5.9."},"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_001703","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 6 elements can be selected from a set of 67 distinct elements?","options":{"A":"49897848","B":"99795696","C":"99795698","D":"-99795696"},"correct_answer":"B","correct_answer_value":"99795696","solution":"Order does not matter, so the count is C(67,6)=67!/[6!(61)!]=99795696.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 99795696, not 49897848.","B":"Correct. The worked calculation gives 99795696.","C":"Incorrect. Substituting the stated values into the governing relation gives 99795696, not 99795698.","D":"Incorrect. Substituting the stated values into the governing relation gives 99795696, not -99795696."},"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_008640","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 1.62 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":"1.62 K","B":"1.162 K","C":"0.62 K","D":"4.62 K"},"correct_answer":"A","correct_answer_value":"1.62 K","solution":"ΔT_b=iK_bm. Substitution gives ΔT_b=1×1.0×1.62=1.62 K.","option_explanations":{"A":"Correct. The worked calculation gives 1.62 K.","B":"Incorrect. Substituting the stated values into the governing relation gives 1.62 K, not 1.162 K.","C":"Incorrect. Substituting the stated values into the governing relation gives 1.62 K, not 0.62 K.","D":"Incorrect. Substituting the stated values into the governing relation gives 1.62 K, not 4.62 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_008894","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 1.77 mol L⁻¹ and initially B is absent. At equilibrium, 41% of A has dissociated at constant volume. Find K_c.","options":{"A":"4.0172","B":"1.0086","C":"2.0172","D":"4.0344"},"correct_answer":"C","correct_answer_value":"2.0172","solution":"Let α=0.41. At equilibrium [A]=c₀(1−α) and [B]=2c₀α. Thus K_c=[B]²/[A]=4c₀α²/(1−α)=4(1.77)(0.41)²/(1−0.41)=2.0172.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 2.0172, not 4.0172.","B":"Incorrect. Substituting the stated values into the governing relation gives 2.0172, not 1.0086.","C":"Correct. The worked calculation gives 2.0172.","D":"Incorrect. Substituting the stated values into the governing relation gives 2.0172, not 4.0344."},"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_004755","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 9 Ω are connected in series; this combination is connected in parallel with a 20 Ω resistor across a 12 V ideal battery. Find the total current supplied by the battery.","options":{"A":"2.23158 A","B":"3.23158 A","C":"0.23158 A","D":"1.23158 A"},"correct_answer":"D","correct_answer_value":"1.23158 A","solution":"Series branch resistance=10+9=19 Ω. Parallel equivalent R_eq=(19×20)/(19+20)=9.74359 Ω. Total current I=V/R_eq=12/9.74359=1.23158 A.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 1.23158 A, not 2.23158 A.","B":"Incorrect. Substituting the stated values into the governing relation gives 1.23158 A, not 3.23158 A.","C":"Incorrect. Substituting the stated values into the governing relation gives 1.23158 A, not 0.23158 A.","D":"Correct. The worked calculation gives 1.23158 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_008987","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 0.78 mol L⁻¹ and initially B is absent. At equilibrium, 11% of A has dissociated at constant volume. Find K_c.","options":{"A":"0.04242","B":"0.08484","C":"-2.95758","D":"-0.95758"},"correct_answer":"A","correct_answer_value":"0.04242","solution":"Let α=0.11. At equilibrium [A]=c₀(1−α) and [B]=2c₀α. Thus K_c=[B]²/[A]=4c₀α²/(1−α)=4(0.78)(0.11)²/(1−0.11)=0.04242.","option_explanations":{"A":"Correct. The worked calculation gives 0.04242.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.04242, not 0.08484.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.04242, not -2.95758.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.04242, not -0.95758."},"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_003226","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+11i)/(7+9i). What is Re(z), the real part of z?","options":{"A":"-0.81538","B":"-0.18462","C":"1.08154","D":"0.81538"},"correct_answer":"D","correct_answer_value":"0.81538","solution":"Multiply numerator and denominator by the conjugate (7-9i). Then Re(z)=(ac+bd)/(c²+d²)=((1)(7)+(11)(9))/(7²+(9)²)=0.81538.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.81538, not -0.81538.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.81538, not -0.18462.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.81538, not 1.08154.","D":"Correct. The worked calculation gives 0.81538."},"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_006863","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 29 mol O₂ are mixed and the reaction goes to completion, how many moles of H₂O form?","options":{"A":"11 mol","B":"14 mol","C":"12 mol","D":"9 mol"},"correct_answer":"C","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 58 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 11 mol.","B":"Incorrect. Substituting the stated values into the governing relation gives 12 mol, not 14 mol.","C":"Correct. The worked calculation gives 12 mol.","D":"Incorrect. Substituting the stated values into the governing relation gives 12 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_001957","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 80 distinct elements?","options":{"A":"24040016","B":"2404002.6","C":"12020008","D":"28848019.2"},"correct_answer":"A","correct_answer_value":"24040016","solution":"Order does not matter, so the count is C(80,5)=80!/[5!(75)!]=24040016.","option_explanations":{"A":"Correct. The worked calculation gives 24040016.","B":"Incorrect. Substituting the stated values into the governing relation gives 24040016, not 2404002.6.","C":"Incorrect. Substituting the stated values into the governing relation gives 24040016, not 12020008.","D":"Incorrect. Substituting the stated values into the governing relation gives 24040016, not 28848019.2."},"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_008441","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 0.46 mol kg⁻¹. If its van't Hoff factor is 1 and K_b for water is 0.6 K kg mol⁻¹, find the boiling-point elevation.","options":{"A":"0.276 K","B":"2.276 K","C":"0.414 K","D":"0.138 K"},"correct_answer":"A","correct_answer_value":"0.276 K","solution":"ΔT_b=iK_bm. Substitution gives ΔT_b=1×0.6×0.46=0.276 K.","option_explanations":{"A":"Correct. The worked calculation gives 0.276 K.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.276 K, not 2.276 K.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.276 K, not 0.414 K.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.276 K, not 0.138 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_003030","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+14i)/(6+12i). What is Re(z), the real part of z?","options":{"A":"1.3","B":"2.6","C":"1.13","D":"3.3"},"correct_answer":"A","correct_answer_value":"1.3","solution":"Multiply numerator and denominator by the conjugate (6-12i). Then Re(z)=(ac+bd)/(c²+d²)=((11)(6)+(14)(12))/(6²+(12)²)=1.3.","option_explanations":{"A":"Correct. The worked calculation gives 1.3.","B":"Incorrect. Substituting the stated values into the governing relation gives 1.3, not 2.6.","C":"Incorrect. Substituting the stated values into the governing relation gives 1.3, not 1.13.","D":"Incorrect. Substituting the stated values into the governing relation gives 1.3, not 3.3."},"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_007766","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 20 mmol L⁻¹, what concentration remains after 20 minutes?","options":{"A":"9 mmol L⁻¹","B":"10 mmol L⁻¹","C":"13 mmol L⁻¹","D":"8 mmol L⁻¹"},"correct_answer":"B","correct_answer_value":"10 mmol L⁻¹","solution":"20 minutes corresponds to 1 half-lives. The remaining concentration is [A]₀(1/2)ⁿ=20/2^1=10 mmol L⁻¹.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 10 mmol L⁻¹, not 9 mmol L⁻¹.","B":"Correct. The worked calculation gives 10 mmol L⁻¹.","C":"Incorrect. Substituting the stated values into the governing relation gives 10 mmol L⁻¹, not 13 mmol L⁻¹.","D":"Incorrect. Substituting the stated values into the governing relation gives 10 mmol L⁻¹, not 8 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_008870","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 1.47 mol L⁻¹ and initially B is absent. At equilibrium, 34% of A has dissociated at constant volume. Find K_c.","options":{"A":"1.02989","B":"3.02989","C":"-1.97011","D":"0.77242"},"correct_answer":"A","correct_answer_value":"1.02989","solution":"Let α=0.34. At equilibrium [A]=c₀(1−α) and [B]=2c₀α. Thus K_c=[B]²/[A]=4c₀α²/(1−α)=4(1.47)(0.34)²/(1−0.34)=1.02989.","option_explanations":{"A":"Correct. The worked calculation gives 1.02989.","B":"Incorrect. Substituting the stated values into the governing relation gives 1.02989, not 3.02989.","C":"Incorrect. Substituting the stated values into the governing relation gives 1.02989, not -1.97011.","D":"Incorrect. Substituting the stated values into the governing relation gives 1.02989, not 0.77242."},"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_007610","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 25 mL of 0.10 M NaOH at 25°C. Assuming complete dissociation and additive volumes, find the final pH.","options":{"A":"1.42597","B":"2.42597","C":"-0.57403","D":"0.42597"},"correct_answer":"A","correct_answer_value":"1.42597","solution":"Excess H⁺ moles=0.10×(55−25)/1000=0.003000 mol. Total volume=0.080 L, so [H⁺]=0.037500 M and pH=−log₁₀[H⁺]=1.42597.","option_explanations":{"A":"Correct. The worked calculation gives 1.42597.","B":"Incorrect. Substituting the stated values into the governing relation gives 1.42597, not 2.42597.","C":"Incorrect. Substituting the stated values into the governing relation gives 1.42597, not -0.57403.","D":"Incorrect. Substituting the stated values into the governing relation gives 1.42597, not 0.42597."},"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_007646","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":"12 mL of 0.10 M HCl is mixed with 24 mL of 0.10 M NaOH at 25°C. Assuming complete dissociation and additive volumes, find the final pH.","options":{"A":"25.04576","B":"12.52288","C":"15.52288","D":"18.78432"},"correct_answer":"B","correct_answer_value":"12.52288","solution":"Excess OH⁻ moles=0.10×(24−12)/1000=0.001200 mol. Total volume=0.036 L, so [OH⁻]=0.033333 M, pOH=−log₁₀[OH⁻], and pH=14−pOH=12.52288.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 12.52288, not 25.04576.","B":"Correct. The worked calculation gives 12.52288.","C":"Incorrect. Substituting the stated values into the governing relation gives 12.52288, not 15.52288.","D":"Incorrect. Substituting the stated values into the governing relation gives 12.52288, not 18.78432."},"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_002509","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, 2], [7, 5]].","options":{"A":"11","B":"5.5","C":"8.25","D":"22"},"correct_answer":"A","correct_answer_value":"11","solution":"For a 2×2 matrix [[a,b],[c,d]], det=ad−bc. Here it is (5)(5)−(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 5.5.","C":"Incorrect. Substituting the stated values into the governing relation gives 11, not 8.25.","D":"Incorrect. Substituting the stated values into the governing relation gives 11, not 22."},"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_007236","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 373 K and 1.94 atm? Use R=0.082057 L atm mol⁻¹ K⁻¹.","options":{"A":"130.15974 L","B":"208.25559 L","C":"172.54633 L","D":"173.54633 L"},"correct_answer":"D","correct_answer_value":"173.54633 L","solution":"PV=nRT, so V=nRT/P=(11)(0.082057)(373)/(1.94)=173.54633 L.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 173.54633 L, not 130.15974 L.","B":"Incorrect. Substituting the stated values into the governing relation gives 173.54633 L, not 208.25559 L.","C":"Incorrect. Substituting the stated values into the governing relation gives 173.54633 L, not 172.54633 L.","D":"Correct. The worked calculation gives 173.54633 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_006170","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 50 cm in front of a thin convex lens of focal length 30 cm. Using the Cartesian sign convention, what is the magnitude of the linear magnification?","options":{"A":"1.5","B":"-0.5","C":"3.5","D":"4.5"},"correct_answer":"A","correct_answer_value":"1.5","solution":"Using distances as positive magnitudes, 1/f=1/v+1/u gives v=uf/(u−f)=(50)(30)/(50−30)=75 cm. Thus |m|=v/u=75/50=1.5.","option_explanations":{"A":"Correct. The worked calculation gives 1.5.","B":"Incorrect. Substituting the stated values into the governing relation gives 1.5, not -0.5.","C":"Incorrect. Substituting the stated values into the governing relation gives 1.5, not 3.5.","D":"Incorrect. Substituting the stated values into the governing relation gives 1.5, not 4.5."},"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_003052","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=(-5+2i)/(11+4i). What is Re(z), the real part of z?","options":{"A":"-0.34307","B":"-0.5146","C":"-2.34307","D":"-0.68613"},"correct_answer":"A","correct_answer_value":"-0.34307","solution":"Multiply numerator and denominator by the conjugate (11-4i). Then Re(z)=(ac+bd)/(c²+d²)=((-5)(11)+(2)(4))/(11²+(4)²)=-0.34307.","option_explanations":{"A":"Correct. The worked calculation gives -0.34307.","B":"Incorrect. Substituting the stated values into the governing relation gives -0.34307, not -0.5146.","C":"Incorrect. Substituting the stated values into the governing relation gives -0.34307, not -2.34307.","D":"Incorrect. Substituting the stated values into the governing relation gives -0.34307, not -0.68613."},"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_002102","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 6 green balls. Two balls are drawn uniformly at random without replacement. What is the probability that both have the same colour?","options":{"A":"-2.65217","B":"0.52174","C":"0.17391","D":"0.34783"},"correct_answer":"D","correct_answer_value":"0.34783","solution":"Favourable pairs = C(6,2)+C(12,2)+C(6,2)=96. Total pairs=C(24,2)=276. Therefore P=96/276=0.34783.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.34783, not -2.65217.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.34783, not 0.52174.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.34783, not 0.17391.","D":"Correct. The worked calculation gives 0.34783."},"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_007539","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":"44 mL of 0.10 M HCl is mixed with 97 mL of 0.10 M NaOH at 25°C. Assuming complete dissociation and additive volumes, find the final pH.","options":{"A":"10.57506","B":"12.57506","C":"13.57506","D":"9.57506"},"correct_answer":"B","correct_answer_value":"12.57506","solution":"Excess OH⁻ moles=0.10×(97−44)/1000=0.005300 mol. Total volume=0.141 L, so [OH⁻]=0.037589 M, pOH=−log₁₀[OH⁻], and pH=14−pOH=12.57506.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 12.57506, not 10.57506.","B":"Correct. The worked calculation gives 12.57506.","C":"Incorrect. Substituting the stated values into the governing relation gives 12.57506, not 13.57506.","D":"Incorrect. Substituting the stated values into the governing relation gives 12.57506, not 9.57506."},"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_003677","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 7 m s⁻¹. A constant net force does 29 N of work over 2 m in the direction of motion. Find its final kinetic energy.","options":{"A":"302 J","B":"304 J","C":"31.3 J","D":"303 J"},"correct_answer":"D","correct_answer_value":"303 J","solution":"By the work–energy theorem, K_f=K_i+W=½mu²+Fs=½(10)(7)²+(29)(2)=303 J.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 303 J, not 302 J.","B":"Incorrect. Substituting the stated values into the governing relation gives 303 J, not 304 J.","C":"Incorrect. Substituting the stated values into the governing relation gives 303 J, not 31.3 J.","D":"Correct. The worked calculation gives 303 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_005723","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 atm at 428 K. What is its pressure at 453 K?","options":{"A":"1.58762 atm","B":"-0.94159 atm","C":"3.05841 atm","D":"1.05841 atm"},"correct_answer":"D","correct_answer_value":"1.05841 atm","solution":"At constant volume, P/T is constant. P₂=P₁T₂/T₁=(1)(453)/(428)=1.05841 atm.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 1.05841 atm, not 1.58762 atm.","B":"Incorrect. Substituting the stated values into the governing relation gives 1.05841 atm, not -0.94159 atm.","C":"Incorrect. Substituting the stated values into the governing relation gives 1.05841 atm, not 3.05841 atm.","D":"Correct. The worked calculation gives 1.05841 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_004382","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 6 kg and radius 4 m is acted on by a constant torque of 66 N m about its central axis. Find its angular acceleration.","options":{"A":"3.375 rad s⁻²","B":"2.375 rad s⁻²","C":"-1.375 rad s⁻²","D":"1.375 rad s⁻²"},"correct_answer":"D","correct_answer_value":"1.375 rad s⁻²","solution":"For a solid disc, I=½MR²=½(6)(4)²=48 kg m². Using τ=Iα, α=66/48=1.375 rad s⁻².","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 1.375 rad s⁻², not 3.375 rad s⁻².","B":"Incorrect. Substituting the stated values into the governing relation gives 1.375 rad s⁻², not 2.375 rad s⁻².","C":"Incorrect. Substituting the stated values into the governing relation gives 1.375 rad s⁻², not -1.375 rad s⁻².","D":"Correct. The worked calculation gives 1.375 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_006155","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 95 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":"0.21284","B":"-2.71622","C":"-1.71622","D":"0.28378"},"correct_answer":"D","correct_answer_value":"0.28378","solution":"Using distances as positive magnitudes, 1/f=1/v+1/u gives v=uf/(u−f)=(95)(21)/(95−21)=26.95946 cm. Thus |m|=v/u=26.95946/95=0.28378.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.28378, not 0.21284.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.28378, not -2.71622.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.28378, not -1.71622.","D":"Correct. The worked calculation gives 0.28378."},"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_005184","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 +9 μC are separated by 4 cm in vacuum. Using k=9×10⁹ N m² C⁻², find the magnitude of the electrostatic force.","options":{"A":"202.5 N","B":"204.5 N","C":"21.25 N","D":"101.25 N"},"correct_answer":"A","correct_answer_value":"202.5 N","solution":"Convert charges to coulombs and distance to metres. F=k|q₁q₂|/r²=9×10⁹(4×10⁻⁶)(9×10⁻⁶)/(4×10⁻²)²=202.5 N.","option_explanations":{"A":"Correct. The worked calculation gives 202.5 N.","B":"Incorrect. Substituting the stated values into the governing relation gives 202.5 N, not 204.5 N.","C":"Incorrect. Substituting the stated values into the governing relation gives 202.5 N, not 21.25 N.","D":"Incorrect. Substituting the stated values into the governing relation gives 202.5 N, not 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_009507","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_18H_24N_1X_1. Treat X as the total number of monovalent halogen atoms.","options":{"A":"7","B":"9","C":"10.5","D":"1.7"},"correct_answer":"A","correct_answer_value":"7","solution":"Use DBE=(2C+2+N−H−X)/2. Substitution gives (2×18+2+1−24−1)/2=7.","option_explanations":{"A":"Correct. The worked calculation gives 7.","B":"Incorrect. Substituting the stated values into the governing relation gives 7, not 9.","C":"Incorrect. Substituting the stated values into the governing relation gives 7, not 10.5.","D":"Incorrect. Substituting the stated values into the governing relation gives 7, not 1.7."},"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_005834","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 3.9 atm at 363 K. What is its pressure at 651 K?","options":{"A":"13.98843 atm","B":"6.99421 atm","C":"5.24566 atm","D":"3.99421 atm"},"correct_answer":"B","correct_answer_value":"6.99421 atm","solution":"At constant volume, P/T is constant. P₂=P₁T₂/T₁=(3.9)(651)/(363)=6.99421 atm.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 6.99421 atm, not 13.98843 atm.","B":"Correct. The worked calculation gives 6.99421 atm.","C":"Incorrect. Substituting the stated values into the governing relation gives 6.99421 atm, not 5.24566 atm.","D":"Incorrect. Substituting the stated values into the governing relation gives 6.99421 atm, not 3.99421 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_000151","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 -9=0, let its roots be α and β. What is α²+β²?","options":{"A":"5.24","B":"4.24","C":"2.24","D":"1.424"},"correct_answer":"B","correct_answer_value":"4.24","solution":"By Vieta's relations, α+β = 4/5 and αβ = -9/5. Then α²+β²=(α+β)²−2αβ = (4/5)²−2(-9/5) = 4.24.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 4.24, not 5.24.","B":"Correct. The worked calculation gives 4.24.","C":"Incorrect. Substituting the stated values into the governing relation gives 4.24, not 2.24.","D":"Incorrect. Substituting the stated values into the governing relation gives 4.24, not 1.424."},"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_000457","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 2. Find the sum of its first 29 terms.","options":{"A":"76.4","B":"756","C":"754","D":"753"},"correct_answer":"C","correct_answer_value":"754","solution":"Use Sₙ = n/2[2a+(n−1)d]. Substituting a=-2, d=2, n=29: Sₙ=29/2[2(-2)+(29−1)(2)]=754.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 754, not 76.4.","B":"Incorrect. Substituting the stated values into the governing relation gives 754, not 756.","C":"Correct. The worked calculation gives 754.","D":"Incorrect. Substituting the stated values into the governing relation gives 754, not 753."},"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_002236","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, 4 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":"0.70588","B":"0.35294","C":"0.26471","D":"3.35294"},"correct_answer":"B","correct_answer_value":"0.35294","solution":"Favourable pairs = C(9,2)+C(4,2)+C(4,2)=48. Total pairs=C(17,2)=136. Therefore P=48/136=0.35294.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.35294, not 0.70588.","B":"Correct. The worked calculation gives 0.35294.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.35294, not 0.26471.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.35294, not 3.35294."},"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_002611","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, 0]].","options":{"A":"-42","B":"-48","C":"-80","D":"-40"},"correct_answer":"D","correct_answer_value":"-40","solution":"For a 2×2 matrix [[a,b],[c,d]], det=ad−bc. Here it is (1)(0)−(8)(5)=-40.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives -40, not -42.","B":"Incorrect. Substituting the stated values into the governing relation gives -40, not -48.","C":"Incorrect. Substituting the stated values into the governing relation gives -40, not -80.","D":"Correct. The worked calculation gives -40."},"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_000405","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 11. Find the sum of its first 8 terms.","options":{"A":"348","B":"696","C":"417.6","D":"-348"},"correct_answer":"A","correct_answer_value":"348","solution":"Use Sₙ = n/2[2a+(n−1)d]. Substituting a=5, d=11, n=8: Sₙ=8/2[2(5)+(8−1)(11)]=348.","option_explanations":{"A":"Correct. The worked calculation gives 348.","B":"Incorrect. Substituting the stated values into the governing relation gives 348, not 696.","C":"Incorrect. Substituting the stated values into the governing relation gives 348, not 417.6.","D":"Incorrect. Substituting the stated values into the governing relation gives 348, not -348."},"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_007355","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":"50 mL of 0.10 M HCl is mixed with 15 mL of 0.10 M NaOH at 25°C. Assuming complete dissociation and additive volumes, find the final pH.","options":{"A":"0.63442","B":"1.12688","C":"0.26885","D":"1.26885"},"correct_answer":"D","correct_answer_value":"1.26885","solution":"Excess H⁺ moles=0.10×(50−15)/1000=0.003500 mol. Total volume=0.065 L, so [H⁺]=0.053846 M and pH=−log₁₀[H⁺]=1.26885.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 1.26885, not 0.63442.","B":"Incorrect. Substituting the stated values into the governing relation gives 1.26885, not 1.12688.","C":"Incorrect. Substituting the stated values into the governing relation gives 1.26885, not 0.26885.","D":"Correct. The worked calculation gives 1.26885."},"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_000617","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 13. Find the sum of its first 14 terms.","options":{"A":"917","B":"915","C":"458.5","D":"1375.5"},"correct_answer":"A","correct_answer_value":"917","solution":"Use Sₙ = n/2[2a+(n−1)d]. Substituting a=-19, d=13, n=14: Sₙ=14/2[2(-19)+(14−1)(13)]=917.","option_explanations":{"A":"Correct. The worked calculation gives 917.","B":"Incorrect. Substituting the stated values into the governing relation gives 917, not 915.","C":"Incorrect. Substituting the stated values into the governing relation gives 917, not 458.5.","D":"Incorrect. Substituting the stated values into the governing relation gives 917, not 1375.5."},"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_007601","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 78 mL of 0.10 M NaOH at 25°C. Assuming complete dissociation and additive volumes, find the final pH.","options":{"A":"11.92082","B":"-11.92082","C":"8.92082","D":"9.92082"},"correct_answer":"A","correct_answer_value":"11.92082","solution":"Excess OH⁻ moles=0.10×(78−66)/1000=0.001200 mol. Total volume=0.144 L, so [OH⁻]=0.008333 M, pOH=−log₁₀[OH⁻], and pH=14−pOH=11.92082.","option_explanations":{"A":"Correct. The worked calculation gives 11.92082.","B":"Incorrect. Substituting the stated values into the governing relation gives 11.92082, not -11.92082.","C":"Incorrect. Substituting the stated values into the governing relation gives 11.92082, not 8.92082.","D":"Incorrect. Substituting the stated values into the governing relation gives 11.92082, not 9.92082."},"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_009671","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(CN)_3]^+3, each CN ligand has charge −1. Determine the formal oxidation state of Pt.","options":{"A":"3","B":"6","C":"9","D":"12"},"correct_answer":"B","correct_answer_value":"6","solution":"Let the formal oxidation state of Pt be x. Charge balance gives x+(3)(−1)=+3; hence x=3+3=+6.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 6, not 3.","B":"Correct. The worked calculation gives 6.","C":"Incorrect. Substituting the stated values into the governing relation gives 6, not 9.","D":"Incorrect. Substituting the stated values into the governing relation gives 6, not 12."},"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_002271","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, 12 blue and 7 green balls. Two balls are drawn uniformly at random without replacement. What is the probability that both have the same colour?","options":{"A":"3.36759","B":"0.36759","C":"2.36759","D":"-1.63241"},"correct_answer":"B","correct_answer_value":"0.36759","solution":"Favourable pairs = C(4,2)+C(12,2)+C(7,2)=93. Total pairs=C(23,2)=253. Therefore P=93/253=0.36759.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.36759, not 3.36759.","B":"Correct. The worked calculation gives 0.36759.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.36759, not 2.36759.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.36759, not -1.63241."},"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_007443","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":"43 mL of 0.10 M HCl is mixed with 80 mL of 0.10 M NaOH at 25°C. Assuming complete dissociation and additive volumes, find the final pH.","options":{"A":"-12.4783","B":"9.4783","C":"12.4783","D":"9.35872"},"correct_answer":"C","correct_answer_value":"12.4783","solution":"Excess OH⁻ moles=0.10×(80−43)/1000=0.003700 mol. Total volume=0.123 L, so [OH⁻]=0.030081 M, pOH=−log₁₀[OH⁻], and pH=14−pOH=12.4783.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 12.4783, not -12.4783.","B":"Incorrect. Substituting the stated values into the governing relation gives 12.4783, not 9.4783.","C":"Correct. The worked calculation gives 12.4783.","D":"Incorrect. Substituting the stated values into the governing relation gives 12.4783, not 9.35872."},"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_004626","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 8 kg and radius 5 m is acted on by a constant torque of 18 N m about its central axis. Find its angular acceleration.","options":{"A":"1.018 rad s⁻²","B":"0.18 rad s⁻²","C":"0.27 rad s⁻²","D":"0.09 rad s⁻²"},"correct_answer":"B","correct_answer_value":"0.18 rad s⁻²","solution":"For a solid disc, I=½MR²=½(8)(5)²=100 kg m². Using τ=Iα, α=18/100=0.18 rad s⁻².","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.18 rad s⁻², not 1.018 rad s⁻².","B":"Correct. The worked calculation gives 0.18 rad s⁻².","C":"Incorrect. Substituting the stated values into the governing relation gives 0.18 rad s⁻², not 0.27 rad s⁻².","D":"Incorrect. Substituting the stated values into the governing relation gives 0.18 rad s⁻², not 0.09 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_006146","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 12 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":"2","B":"-2","C":"0","D":"1"},"correct_answer":"D","correct_answer_value":"1","solution":"Using distances as positive magnitudes, 1/f=1/v+1/u gives v=uf/(u−f)=(12)(6)/(12−6)=12 cm. Thus |m|=v/u=12/12=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 -2.","C":"Incorrect. Substituting the stated values into the governing relation gives 1, not 0.","D":"Correct. The worked calculation gives 1."},"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_008609","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 1.11 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":"6.33 K","B":"3.33 K","C":"1.665 K","D":"1.333 K"},"correct_answer":"B","correct_answer_value":"3.33 K","solution":"ΔT_b=iK_bm. Substitution gives ΔT_b=3×1.0×1.11=3.33 K.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 3.33 K, not 6.33 K.","B":"Correct. The worked calculation gives 3.33 K.","C":"Incorrect. Substituting the stated values into the governing relation gives 3.33 K, not 1.665 K.","D":"Incorrect. Substituting the stated values into the governing relation gives 3.33 K, not 1.333 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_000351","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 9. Find the sum of its first 28 terms.","options":{"A":"4186","B":"4187","C":"-4186","D":"6279"},"correct_answer":"A","correct_answer_value":"4186","solution":"Use Sₙ = n/2[2a+(n−1)d]. Substituting a=28, d=9, n=28: Sₙ=28/2[2(28)+(28−1)(9)]=4186.","option_explanations":{"A":"Correct. The worked calculation gives 4186.","B":"Incorrect. Substituting the stated values into the governing relation gives 4186, not 4187.","C":"Incorrect. Substituting the stated values into the governing relation gives 4186, not -4186.","D":"Incorrect. Substituting the stated values into the governing relation gives 4186, not 6279."},"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_008443","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 0.92 mol kg⁻¹. If its van't Hoff factor is 2 and K_b for water is 0.52 K kg mol⁻¹, find the boiling-point elevation.","options":{"A":"-1.0432 K","B":"0.9568 K","C":"0.7176 K","D":"1.4352 K"},"correct_answer":"B","correct_answer_value":"0.9568 K","solution":"ΔT_b=iK_bm. Substitution gives ΔT_b=2×0.52×0.92=0.9568 K.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.9568 K, not -1.0432 K.","B":"Correct. The worked calculation gives 0.9568 K.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.9568 K, not 0.7176 K.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.9568 K, not 1.4352 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_006449","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 16 kg is attached to an ideal spring of force constant 289 N m⁻¹ and oscillates horizontally without damping. What is the time period divided by π?","options":{"A":"0.35294 s","B":"0.47059 s","C":"1.04706 s","D":"2.47059 s"},"correct_answer":"B","correct_answer_value":"0.47059 s","solution":"For a mass–spring oscillator, T=2π√(m/k). Therefore T/π=2√(16/289)=0.47059 s.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.47059 s, not 0.35294 s.","B":"Correct. The worked calculation gives 0.47059 s.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.47059 s, not 1.04706 s.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.47059 s, not 2.47059 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_002814","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, 2) and v=(-3, -6, -2), calculate u·v.","options":{"A":"17","B":"18","C":"13.6","D":"15"},"correct_answer":"A","correct_answer_value":"17","solution":"The dot product is the sum of corresponding component products: (-1)(-3)+(-3)(-6)+(2)(-2)=17.","option_explanations":{"A":"Correct. The worked calculation gives 17.","B":"Incorrect. Substituting the stated values into the governing relation gives 17, not 18.","C":"Incorrect. Substituting the stated values into the governing relation gives 17, not 13.6.","D":"Incorrect. Substituting the stated values into the governing relation gives 17, not 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_004616","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 14 kg and radius 5 m is acted on by a constant torque of 82 N m about its central axis. Find its angular acceleration.","options":{"A":"1.04686 rad s⁻²","B":"0.70286 rad s⁻²","C":"0.46857 rad s⁻²","D":"1.46857 rad s⁻²"},"correct_answer":"C","correct_answer_value":"0.46857 rad s⁻²","solution":"For a solid disc, I=½MR²=½(14)(5)²=175 kg m². Using τ=Iα, α=82/175=0.46857 rad s⁻².","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.46857 rad s⁻², not 1.04686 rad s⁻².","B":"Incorrect. Substituting the stated values into the governing relation gives 0.46857 rad s⁻², not 0.70286 rad s⁻².","C":"Correct. The worked calculation gives 0.46857 rad s⁻².","D":"Incorrect. Substituting the stated values into the governing relation gives 0.46857 rad s⁻², not 1.46857 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_009227","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=1.18 V, transfers 1 electrons, and has reaction quotient Q=10^-1. Using E=E°−(0.0591/n)log₁₀Q, calculate Ecell.","options":{"A":"2.2391 V","B":"0.61955 V","C":"2.4782 V","D":"1.2391 V"},"correct_answer":"D","correct_answer_value":"1.2391 V","solution":"Apply the Nernst equation: E=1.18−(0.0591/1)(-1)=1.2391 V.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 1.2391 V, not 2.2391 V.","B":"Incorrect. Substituting the stated values into the governing relation gives 1.2391 V, not 0.61955 V.","C":"Incorrect. Substituting the stated values into the governing relation gives 1.2391 V, not 2.4782 V.","D":"Correct. The worked calculation gives 1.2391 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_006495","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 1088 N m⁻¹ and oscillates horizontally without damping. What is the time period divided by π?","options":{"A":"-0.7156 s","B":"1.02844 s","C":"1.2844 s","D":"0.2844 s"},"correct_answer":"D","correct_answer_value":"0.2844 s","solution":"For a mass–spring oscillator, T=2π√(m/k). Therefore T/π=2√(22/1088)=0.2844 s.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.2844 s, not -0.7156 s.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.2844 s, not 1.02844 s.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.2844 s, not 1.2844 s.","D":"Correct. The worked calculation gives 0.2844 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_005279","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 +3 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":"8.66667 N","B":"10 N","C":"5.66667 N","D":"6.66667 N"},"correct_answer":"D","correct_answer_value":"6.66667 N","solution":"Convert charges to coulombs and distance to metres. F=k|q₁q₂|/r²=9×10⁹(2×10⁻⁶)(3×10⁻⁶)/(9×10⁻²)²=6.66667 N.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 6.66667 N, not 8.66667 N.","B":"Incorrect. Substituting the stated values into the governing relation gives 6.66667 N, not 10 N.","C":"Incorrect. Substituting the stated values into the governing relation gives 6.66667 N, not 5.66667 N.","D":"Correct. The worked calculation gives 6.66667 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_005105","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 +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":"8.625 N","B":"1.5625 N","C":"-5.625 N","D":"5.625 N"},"correct_answer":"D","correct_answer_value":"5.625 N","solution":"Convert charges to coulombs and distance to metres. F=k|q₁q₂|/r²=9×10⁹(9×10⁻⁶)(1×10⁻⁶)/(12×10⁻²)²=5.625 N.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 5.625 N, not 8.625 N.","B":"Incorrect. Substituting the stated values into the governing relation gives 5.625 N, not 1.5625 N.","C":"Incorrect. Substituting the stated values into the governing relation gives 5.625 N, not -5.625 N.","D":"Correct. The worked calculation gives 5.625 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_002248","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, 3 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":"3.37154","B":"2.37154","C":"0.37154","D":"0.27866"},"correct_answer":"C","correct_answer_value":"0.37154","solution":"Favourable pairs = C(9,2)+C(3,2)+C(11,2)=94. Total pairs=C(23,2)=253. Therefore P=94/253=0.37154.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.37154, not 3.37154.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.37154, not 2.37154.","C":"Correct. The worked calculation gives 0.37154.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.37154, not 0.27866."},"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_004053","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 0.16. Taking g=10 m s⁻², find the net force down the plane. Use sin37°=0.6 and cos37°=0.8.","options":{"A":"-127.44 N","B":"152.928 N","C":"127.44 N","D":"101.952 N"},"correct_answer":"C","correct_answer_value":"127.44 N","solution":"Net force=mg(sinθ−μcosθ)=27×10×(0.6−0.16×0.8)=127.44 N.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 127.44 N, not -127.44 N.","B":"Incorrect. Substituting the stated values into the governing relation gives 127.44 N, not 152.928 N.","C":"Correct. The worked calculation gives 127.44 N.","D":"Incorrect. Substituting the stated values into the governing relation gives 127.44 N, not 101.952 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_005736","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 3.4 atm at 410 K. What is its pressure at 460 K?","options":{"A":"2.86098 atm","B":"1.90732 atm","C":"3.81463 atm","D":"1.38146 atm"},"correct_answer":"C","correct_answer_value":"3.81463 atm","solution":"At constant volume, P/T is constant. P₂=P₁T₂/T₁=(3.4)(460)/(410)=3.81463 atm.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 3.81463 atm, not 2.86098 atm.","B":"Incorrect. Substituting the stated values into the governing relation gives 3.81463 atm, not 1.90732 atm.","C":"Correct. The worked calculation gives 3.81463 atm.","D":"Incorrect. Substituting the stated values into the governing relation gives 3.81463 atm, not 1.38146 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_009401","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_18H_3N_3. Treat X as the total number of monovalent halogen atoms.","options":{"A":"17","B":"19","C":"14.25","D":"20"},"correct_answer":"B","correct_answer_value":"19","solution":"Use DBE=(2C+2+N−H−X)/2. Substitution gives (2×18+2+3−3−0)/2=19.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 19, not 17.","B":"Correct. The worked calculation gives 19.","C":"Incorrect. Substituting the stated values into the governing relation gives 19, not 14.25.","D":"Incorrect. Substituting the stated values into the governing relation gives 19, not 20."},"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_004854","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 5 Ω are connected in series; this combination is connected in parallel with a 14 Ω resistor across a 6 V ideal battery. Find the total current supplied by the battery.","options":{"A":"0.72857 A","B":"0.54643 A","C":"1.72857 A","D":"-0.27143 A"},"correct_answer":"A","correct_answer_value":"0.72857 A","solution":"Series branch resistance=15+5=20 Ω. Parallel equivalent R_eq=(20×14)/(20+14)=8.23529 Ω. Total current I=V/R_eq=6/8.23529=0.72857 A.","option_explanations":{"A":"Correct. The worked calculation gives 0.72857 A.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.72857 A, not 0.54643 A.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.72857 A, not 1.72857 A.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.72857 A, not -0.27143 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_009346","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_18H_6N_1X_3. Treat X as the total number of monovalent halogen atoms.","options":{"A":"15","B":"22.5","C":"11.25","D":"30"},"correct_answer":"A","correct_answer_value":"15","solution":"Use DBE=(2C+2+N−H−X)/2. Substitution gives (2×18+2+1−6−3)/2=15.","option_explanations":{"A":"Correct. The worked calculation gives 15.","B":"Incorrect. Substituting the stated values into the governing relation gives 15, not 22.5.","C":"Incorrect. Substituting the stated values into the governing relation gives 15, not 11.25.","D":"Incorrect. Substituting the stated values into the governing relation gives 15, not 30."},"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_006851","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 10 mol O₂ are mixed and the reaction goes to completion, how many moles of H₂O form?","options":{"A":"4 mol","B":"6 mol","C":"2 mol","D":"8 mol"},"correct_answer":"A","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 20 mol H₂O. The limiting condition is H₂; product=4 mol.","option_explanations":{"A":"Correct. The worked calculation gives 4 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 2 mol.","D":"Incorrect. Substituting the stated values into the governing relation gives 4 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_001267","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 -4x² +5x+7. Evaluate f′(5).","options":{"A":"341","B":"340","C":"338","D":"342"},"correct_answer":"B","correct_answer_value":"340","solution":"Differentiate: f′(x)=3(5)x²+2(-4)x+(5). At x=5, f′(5)=3(5)(5)²+2(-4)(5)+(5)=340.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 340, not 341.","B":"Correct. The worked calculation gives 340.","C":"Incorrect. Substituting the stated values into the governing relation gives 340, not 338.","D":"Incorrect. Substituting the stated values into the governing relation gives 340, not 342."},"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_000826","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 4 and common ratio -3. What is the sum of its first 9 terms?","options":{"A":"19683","B":"19684","C":"19682","D":"39368"},"correct_answer":"B","correct_answer_value":"19684","solution":"For r≠1, Sₙ=a(rⁿ−1)/(r−1). Thus Sₙ=4(-3^9−1)/(-3−1)=19684.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 19684, not 19683.","B":"Correct. The worked calculation gives 19684.","C":"Incorrect. Substituting the stated values into the governing relation gives 19684, not 19682.","D":"Incorrect. Substituting the stated values into the governing relation gives 19684, not 39368."},"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_006521","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 400 N m⁻¹ and oscillates horizontally without damping. What is the time period divided by π?","options":{"A":"0.22361 s","B":"0.33541 s","C":"3.44721 s","D":"0.44721 s"},"correct_answer":"D","correct_answer_value":"0.44721 s","solution":"For a mass–spring oscillator, T=2π√(m/k). Therefore T/π=2√(20/400)=0.44721 s.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.44721 s, not 0.22361 s.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.44721 s, not 0.33541 s.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.44721 s, not 3.44721 s.","D":"Correct. The worked calculation gives 0.44721 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_006548","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 593 N m⁻¹ and oscillates horizontally without damping. What is the time period divided by π?","options":{"A":"0.24639 s","B":"-2.83574 s","C":"1.16426 s","D":"0.16426 s"},"correct_answer":"D","correct_answer_value":"0.16426 s","solution":"For a mass–spring oscillator, T=2π√(m/k). Therefore T/π=2√(4/593)=0.16426 s.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.16426 s, not 0.24639 s.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.16426 s, not -2.83574 s.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.16426 s, not 1.16426 s.","D":"Correct. The worked calculation gives 0.16426 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_001467","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 +2x +3) dx.","options":{"A":"-41.4","B":"-426","C":"-424","D":"-508.8"},"correct_answer":"C","correct_answer_value":"-424","solution":"An antiderivative is (-3/3)x³+(2/2)x²+(3)x. Evaluating between 0 and 8 gives (-3/3)(8)³+(2/2)(8)²+(3)(8)=-424.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives -424, not -41.4.","B":"Incorrect. Substituting the stated values into the governing relation gives -424, not -426.","C":"Correct. The worked calculation gives -424.","D":"Incorrect. Substituting the stated values into the governing relation gives -424, not -508.8."},"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_009104","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=1.03 V, transfers 1 electrons, and has reaction quotient Q=10^-2. Using E=E°−(0.0591/n)log₁₀Q, calculate Ecell.","options":{"A":"2.2964 V","B":"1.1482 V","C":"1.11482 V","D":"3.1482 V"},"correct_answer":"B","correct_answer_value":"1.1482 V","solution":"Apply the Nernst equation: E=1.03−(0.0591/1)(-2)=1.1482 V.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 1.1482 V, not 2.2964 V.","B":"Correct. The worked calculation gives 1.1482 V.","C":"Incorrect. Substituting the stated values into the governing relation gives 1.1482 V, not 1.11482 V.","D":"Incorrect. Substituting the stated values into the governing relation gives 1.1482 V, not 3.1482 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_004140","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 0.25. Taking g=10 m s⁻², find the net force down the plane. Use sin37°=0.6 and cos37°=0.8.","options":{"A":"112 N","B":"54 N","C":"56 N","D":"44.8 N"},"correct_answer":"C","correct_answer_value":"56 N","solution":"Net force=mg(sinθ−μcosθ)=14×10×(0.6−0.25×0.8)=56 N.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 56 N, not 112 N.","B":"Incorrect. Substituting the stated values into the governing relation gives 56 N, not 54 N.","C":"Correct. The worked calculation gives 56 N.","D":"Incorrect. Substituting the stated values into the governing relation gives 56 N, not 44.8 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_005535","subject":"physics","topic":"electrostatics","subtopic":"capacitor_energy","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A 60 μF capacitor is charged to a potential difference of 9 V. Calculate the energy stored, expressed in μJ.","options":{"A":"244 μJ","B":"1215 μJ","C":"2430 μJ","D":"3645 μJ"},"correct_answer":"C","correct_answer_value":"2430 μJ","solution":"U=½CV². Since C=60 μF, U in μJ is ½×60×(9)²=2430 μJ.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 2430 μJ, not 244 μJ.","B":"Incorrect. Substituting the stated values into the governing relation gives 2430 μJ, not 1215 μJ.","C":"Correct. The worked calculation gives 2430 μJ.","D":"Incorrect. Substituting the stated values into the governing relation gives 2430 μJ, not 3645 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_000977","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 4 and common ratio -3. What is the sum of its first 7 terms?","options":{"A":"2186","B":"1750.4","C":"2187","D":"2188"},"correct_answer":"D","correct_answer_value":"2188","solution":"For r≠1, Sₙ=a(rⁿ−1)/(r−1). Thus Sₙ=4(-3^7−1)/(-3−1)=2188.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 2188, not 2186.","B":"Incorrect. Substituting the stated values into the governing relation gives 2188, not 1750.4.","C":"Incorrect. Substituting the stated values into the governing relation gives 2188, not 2187.","D":"Correct. The worked calculation gives 2188."},"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_004580","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 8 kg and radius 5 m is acted on by a constant torque of 95 N m about its central axis. Find its angular acceleration.","options":{"A":"-2.05 rad s⁻²","B":"2.95 rad s⁻²","C":"0.95 rad s⁻²","D":"-0.95 rad s⁻²"},"correct_answer":"C","correct_answer_value":"0.95 rad s⁻²","solution":"For a solid disc, I=½MR²=½(8)(5)²=100 kg m². Using τ=Iα, α=95/100=0.95 rad s⁻².","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.95 rad s⁻², not -2.05 rad s⁻².","B":"Incorrect. Substituting the stated values into the governing relation gives 0.95 rad s⁻², not 2.95 rad s⁻².","C":"Correct. The worked calculation gives 0.95 rad s⁻².","D":"Incorrect. Substituting the stated values into the governing relation gives 0.95 rad s⁻², not -0.95 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_009771","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 [Co(OH)_3]^+0, each OH ligand has charge −1. Determine the formal oxidation state of Co.","options":{"A":"1.5","B":"3","C":"2.25","D":"1.3"},"correct_answer":"B","correct_answer_value":"3","solution":"Let the formal oxidation state of Co be x. Charge balance gives x+(3)(−1)=+0; hence x=0+3=+3.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 3, not 1.5.","B":"Correct. The worked calculation gives 3.","C":"Incorrect. Substituting the stated values into the governing relation gives 3, not 2.25.","D":"Incorrect. Substituting the stated values into the governing relation gives 3, not 1.3."},"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_000094","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 +4=0, let its roots be α and β. What is α²+β²?","options":{"A":"0.96","B":"-0.72","C":"-0.96","D":"-1.92"},"correct_answer":"C","correct_answer_value":"-0.96","solution":"By Vieta's relations, α+β = -4/5 and αβ = 4/5. Then α²+β²=(α+β)²−2αβ = (-4/5)²−2(4/5) = -0.96.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives -0.96, not 0.96.","B":"Incorrect. Substituting the stated values into the governing relation gives -0.96, not -0.72.","C":"Correct. The worked calculation gives -0.96.","D":"Incorrect. Substituting the stated values into the governing relation gives -0.96, not -1.92."},"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_005440","subject":"physics","topic":"electrostatics","subtopic":"capacitor_energy","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A 14 μF capacitor is charged to a potential difference of 13 V. Calculate the energy stored, expressed in μJ.","options":{"A":"1419.6 μJ","B":"946.4 μJ","C":"1183 μJ","D":"887.25 μJ"},"correct_answer":"C","correct_answer_value":"1183 μJ","solution":"U=½CV². Since C=14 μF, U in μJ is ½×14×(13)²=1183 μJ.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 1183 μJ, not 1419.6 μJ.","B":"Incorrect. Substituting the stated values into the governing relation gives 1183 μJ, not 946.4 μJ.","C":"Correct. The worked calculation gives 1183 μJ.","D":"Incorrect. Substituting the stated values into the governing relation gives 1183 μJ, not 887.25 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_009452","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_12. Treat X as the total number of monovalent halogen atoms.","options":{"A":"1","B":"4","C":"3","D":"-3"},"correct_answer":"C","correct_answer_value":"3","solution":"Use DBE=(2C+2+N−H−X)/2. Substitution gives (2×8+2+0−12−0)/2=3.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 3, not 1.","B":"Incorrect. Substituting the stated values into the governing relation gives 3, not 4.","C":"Correct. The worked calculation gives 3.","D":"Incorrect. Substituting the stated values into the governing relation gives 3, not -3."},"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_002983","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, 1, 8) and v=(6, -6, -2), calculate u·v.","options":{"A":"-20","B":"-7","C":"-10","D":"10"},"correct_answer":"C","correct_answer_value":"-10","solution":"The dot product is the sum of corresponding component products: (2)(6)+(1)(-6)+(8)(-2)=-10.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives -10, not -20.","B":"Incorrect. Substituting the stated values into the governing relation gives -10, not -7.","C":"Correct. The worked calculation gives -10.","D":"Incorrect. Substituting the stated values into the governing relation gives -10, not 10."},"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_007328","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 9 mol of gas at 611 K and 2.11 atm? Use R=0.082057 L atm mol⁻¹ K⁻¹.","options":{"A":"212.85376 L","B":"427.70753 L","C":"214.85376 L","D":"213.85376 L"},"correct_answer":"D","correct_answer_value":"213.85376 L","solution":"PV=nRT, so V=nRT/P=(9)(0.082057)(611)/(2.11)=213.85376 L.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 213.85376 L, not 212.85376 L.","B":"Incorrect. Substituting the stated values into the governing relation gives 213.85376 L, not 427.70753 L.","C":"Incorrect. Substituting the stated values into the governing relation gives 213.85376 L, not 214.85376 L.","D":"Correct. The worked calculation gives 213.85376 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_004945","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 12 Ω and 12 Ω are connected in series; this combination is connected in parallel with a 3 Ω resistor across a 9 V ideal battery. Find the total current supplied by the battery.","options":{"A":"4.375 A","B":"3.375 A","C":"6.375 A","D":"5.0625 A"},"correct_answer":"B","correct_answer_value":"3.375 A","solution":"Series branch resistance=12+12=24 Ω. Parallel equivalent R_eq=(24×3)/(24+3)=2.66667 Ω. Total current I=V/R_eq=9/2.66667=3.375 A.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 3.375 A, not 4.375 A.","B":"Correct. The worked calculation gives 3.375 A.","C":"Incorrect. Substituting the stated values into the governing relation gives 3.375 A, not 6.375 A.","D":"Incorrect. Substituting the stated values into the governing relation gives 3.375 A, not 5.0625 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_009921","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(OH)_5]^-1, each OH ligand has charge −1. Determine the formal oxidation state of Fe.","options":{"A":"2","B":"6","C":"3","D":"4"},"correct_answer":"D","correct_answer_value":"4","solution":"Let the formal oxidation state of Fe be x. Charge balance gives x+(5)(−1)=-1; hence x=-1+5=+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 6.","C":"Incorrect. Substituting the stated values into the governing relation gives 4, not 3.","D":"Correct. The worked calculation gives 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_007419","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":"49 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":"9.12731","B":"12.16975","C":"9.16975","D":"6.08488"},"correct_answer":"B","correct_answer_value":"12.16975","solution":"Excess OH⁻ moles=0.10×(66−49)/1000=0.001700 mol. Total volume=0.115 L, so [OH⁻]=0.014783 M, pOH=−log₁₀[OH⁻], and pH=14−pOH=12.16975.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 12.16975, not 9.12731.","B":"Correct. The worked calculation gives 12.16975.","C":"Incorrect. Substituting the stated values into the governing relation gives 12.16975, not 9.16975.","D":"Incorrect. Substituting the stated values into the governing relation gives 12.16975, not 6.08488."},"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_006107","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 22 cm. Using the Cartesian sign convention, what is the magnitude of the linear magnification?","options":{"A":"0.21429","B":"0.42857","C":"-0.28571","D":"0.28571"},"correct_answer":"D","correct_answer_value":"0.28571","solution":"Using distances as positive magnitudes, 1/f=1/v+1/u gives v=uf/(u−f)=(99)(22)/(99−22)=28.28571 cm. Thus |m|=v/u=28.28571/99=0.28571.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.28571, not 0.21429.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.28571, not 0.42857.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.28571, not -0.28571.","D":"Correct. The worked calculation gives 0.28571."},"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_004603","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 1 m is acted on by a constant torque of 90 N m about its central axis. Find its angular acceleration.","options":{"A":"7.5 rad s⁻²","B":"15 rad s⁻²","C":"16 rad s⁻²","D":"-15 rad s⁻²"},"correct_answer":"B","correct_answer_value":"15 rad s⁻²","solution":"For a solid disc, I=½MR²=½(12)(1)²=6 kg m². Using τ=Iα, α=90/6=15 rad s⁻².","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 15 rad s⁻², not 7.5 rad s⁻².","B":"Correct. The worked calculation gives 15 rad s⁻².","C":"Incorrect. Substituting the stated values into the governing relation gives 15 rad s⁻², not 16 rad s⁻².","D":"Incorrect. Substituting the stated values into the governing relation gives 15 rad s⁻², not -15 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_007191","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 285 K and 0.45 atm? Use R=0.082057 L atm mol⁻¹ K⁻¹.","options":{"A":"1143.32753 L","B":"569.66377 L","C":"-571.66377 L","D":"571.66377 L"},"correct_answer":"D","correct_answer_value":"571.66377 L","solution":"PV=nRT, so V=nRT/P=(11)(0.082057)(285)/(0.45)=571.66377 L.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 571.66377 L, not 1143.32753 L.","B":"Incorrect. Substituting the stated values into the governing relation gives 571.66377 L, not 569.66377 L.","C":"Incorrect. Substituting the stated values into the governing relation gives 571.66377 L, not -571.66377 L.","D":"Correct. The worked calculation gives 571.66377 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_004400","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 3 m is acted on by a constant torque of 79 N m about its central axis. Find its angular acceleration.","options":{"A":"1.46296 rad s⁻²","B":"0.46296 rad s⁻²","C":"2.92593 rad s⁻²","D":"1.09722 rad s⁻²"},"correct_answer":"A","correct_answer_value":"1.46296 rad s⁻²","solution":"For a solid disc, I=½MR²=½(12)(3)²=54 kg m². Using τ=Iα, α=79/54=1.46296 rad s⁻².","option_explanations":{"A":"Correct. The worked calculation gives 1.46296 rad s⁻².","B":"Incorrect. Substituting the stated values into the governing relation gives 1.46296 rad s⁻², not 0.46296 rad s⁻².","C":"Incorrect. Substituting the stated values into the governing relation gives 1.46296 rad s⁻², not 2.92593 rad s⁻².","D":"Incorrect. Substituting the stated values into the governing relation gives 1.46296 rad s⁻², not 1.09722 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_004905","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 5 Ω and 12 Ω are connected in series; this combination is connected in parallel with a 12 Ω resistor across a 36 V ideal battery. Find the total current supplied by the battery.","options":{"A":"6.11765 A","B":"5.11765 A","C":"3.11765 A","D":"-5.11765 A"},"correct_answer":"B","correct_answer_value":"5.11765 A","solution":"Series branch resistance=5+12=17 Ω. Parallel equivalent R_eq=(17×12)/(17+12)=7.03448 Ω. Total current I=V/R_eq=36/7.03448=5.11765 A.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 5.11765 A, not 6.11765 A.","B":"Correct. The worked calculation gives 5.11765 A.","C":"Incorrect. Substituting the stated values into the governing relation gives 5.11765 A, not 3.11765 A.","D":"Incorrect. Substituting the stated values into the governing relation gives 5.11765 A, not -5.11765 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_009090","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=1.12 V, transfers 3 electrons, and has reaction quotient Q=10^1. Using E=E°−(0.0591/n)log₁₀Q, calculate Ecell.","options":{"A":"1.11003 V","B":"1.1003 V","C":"2.1003 V","D":"1.65045 V"},"correct_answer":"B","correct_answer_value":"1.1003 V","solution":"Apply the Nernst equation: E=1.12−(0.0591/3)(1)=1.1003 V.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 1.1003 V, not 1.11003 V.","B":"Correct. The worked calculation gives 1.1003 V.","C":"Incorrect. Substituting the stated values into the governing relation gives 1.1003 V, not 2.1003 V.","D":"Incorrect. Substituting the stated values into the governing relation gives 1.1003 V, not 1.65045 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_002821","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, 7, 1) and v=(0, 6, 9), calculate u·v.","options":{"A":"51","B":"-51","C":"102","D":"53"},"correct_answer":"A","correct_answer_value":"51","solution":"The dot product is the sum of corresponding component products: (2)(0)+(7)(6)+(1)(9)=51.","option_explanations":{"A":"Correct. The worked calculation gives 51.","B":"Incorrect. Substituting the stated values into the governing relation gives 51, not -51.","C":"Incorrect. Substituting the stated values into the governing relation gives 51, not 102.","D":"Incorrect. Substituting the stated values into the governing relation gives 51, not 53."},"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_005502","subject":"physics","topic":"electrostatics","subtopic":"capacitor_energy","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A 40 μF capacitor is charged to a potential difference of 15 V. Calculate the energy stored, expressed in μJ.","options":{"A":"-4500 μJ","B":"4500 μJ","C":"5400 μJ","D":"4501 μJ"},"correct_answer":"B","correct_answer_value":"4500 μJ","solution":"U=½CV². Since C=40 μF, U in μJ is ½×40×(15)²=4500 μJ.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 4500 μJ, not -4500 μJ.","B":"Correct. The worked calculation gives 4500 μJ.","C":"Incorrect. Substituting the stated values into the governing relation gives 4500 μJ, not 5400 μJ.","D":"Incorrect. Substituting the stated values into the governing relation gives 4500 μJ, not 4501 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_000001","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 +6x +2=0, let its roots be α and β. What is α²+β²?","options":{"A":"0.96","B":"0.64","C":"0.48","D":"-0.64"},"correct_answer":"B","correct_answer_value":"0.64","solution":"By Vieta's relations, α+β = -6/5 and αβ = 2/5. Then α²+β²=(α+β)²−2αβ = (-6/5)²−2(2/5) = 0.64.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.64, not 0.96.","B":"Correct. The worked calculation gives 0.64.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.64, not 0.48.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.64, not -0.64."},"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_009673","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)_6]^+1, each OH ligand has charge −1. Determine the formal oxidation state of Cu.","options":{"A":"10.5","B":"-7","C":"7","D":"3.5"},"correct_answer":"C","correct_answer_value":"7","solution":"Let the formal oxidation state of Cu 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 10.5.","B":"Incorrect. Substituting the stated values into the governing relation gives 7, not -7.","C":"Correct. The worked calculation gives 7.","D":"Incorrect. Substituting the stated values into the governing relation gives 7, not 3.5."},"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_001222","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 -1x² -8x+7. Evaluate f′(-3).","options":{"A":"-110","B":"-109","C":"-82.5","D":"-88"},"correct_answer":"A","correct_answer_value":"-110","solution":"Differentiate: f′(x)=3(-4)x²+2(-1)x+(-8). At x=-3, f′(-3)=3(-4)(-3)²+2(-1)(-3)+(-8)=-110.","option_explanations":{"A":"Correct. The worked calculation gives -110.","B":"Incorrect. Substituting the stated values into the governing relation gives -110, not -109.","C":"Incorrect. Substituting the stated values into the governing relation gives -110, not -82.5.","D":"Incorrect. Substituting the stated values into the governing relation gives -110, not -88."},"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_001334","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 +6x+7. Evaluate f′(-4).","options":{"A":"263","B":"260","C":"131","D":"262"},"correct_answer":"D","correct_answer_value":"262","solution":"Differentiate: f′(x)=3(5)x²+2(-2)x+(6). At x=-4, f′(-4)=3(5)(-4)²+2(-2)(-4)+(6)=262.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 262, not 263.","B":"Incorrect. Substituting the stated values into the governing relation gives 262, not 260.","C":"Incorrect. Substituting the stated values into the governing relation gives 262, not 131.","D":"Correct. The worked calculation gives 262."},"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_001512","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 (6x² -4x -5) dx.","options":{"A":"856","B":"855","C":"1027.2","D":"684.8"},"correct_answer":"A","correct_answer_value":"856","solution":"An antiderivative is (6/3)x³+(-4/2)x²+(-5)x. Evaluating between 0 and 8 gives (6/3)(8)³+(-4/2)(8)²+(-5)(8)=856.","option_explanations":{"A":"Correct. The worked calculation gives 856.","B":"Incorrect. Substituting the stated values into the governing relation gives 856, not 855.","C":"Incorrect. Substituting the stated values into the governing relation gives 856, not 1027.2.","D":"Incorrect. Substituting the stated values into the governing relation gives 856, not 684.8."},"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_000914","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 7 and common ratio 5. What is the sum of its first 10 terms?","options":{"A":"17089842","B":"25634763","C":"20507810.4","D":"17089844"},"correct_answer":"A","correct_answer_value":"17089842","solution":"For r≠1, Sₙ=a(rⁿ−1)/(r−1). Thus Sₙ=7(5^10−1)/(5−1)=17089842.","option_explanations":{"A":"Correct. The worked calculation gives 17089842.","B":"Incorrect. Substituting the stated values into the governing relation gives 17089842, not 25634763.","C":"Incorrect. Substituting the stated values into the governing relation gives 17089842, not 20507810.4.","D":"Incorrect. Substituting the stated values into the governing relation gives 17089842, not 17089844."},"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_007238","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 234 K and 0.59 atm? Use R=0.082057 L atm mol⁻¹ K⁻¹.","options":{"A":"36.7991 L","B":"357.99105 L","C":"268.49329 L","D":"178.99552 L"},"correct_answer":"B","correct_answer_value":"357.99105 L","solution":"PV=nRT, so V=nRT/P=(11)(0.082057)(234)/(0.59)=357.99105 L.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 357.99105 L, not 36.7991 L.","B":"Correct. The worked calculation gives 357.99105 L.","C":"Incorrect. Substituting the stated values into the governing relation gives 357.99105 L, not 268.49329 L.","D":"Incorrect. Substituting the stated values into the governing relation gives 357.99105 L, not 178.99552 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_009418","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_9H_9X_3. Treat X as the total number of monovalent halogen atoms.","options":{"A":"-4","B":"3","C":"2","D":"4"},"correct_answer":"D","correct_answer_value":"4","solution":"Use DBE=(2C+2+N−H−X)/2. Substitution gives (2×9+2+0−9−3)/2=4.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 4, not -4.","B":"Incorrect. Substituting the stated values into the governing relation gives 4, not 3.","C":"Incorrect. Substituting the stated values into the governing relation gives 4, not 2.","D":"Correct. The worked calculation gives 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_008934","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 0.15 mol L⁻¹ and initially B is absent. At equilibrium, 78% of A has dissociated at constant volume. Find K_c.","options":{"A":"-1.65927","B":"1.65927","C":"-0.34073","D":"2.65927"},"correct_answer":"B","correct_answer_value":"1.65927","solution":"Let α=0.78. At equilibrium [A]=c₀(1−α) and [B]=2c₀α. Thus K_c=[B]²/[A]=4c₀α²/(1−α)=4(0.15)(0.78)²/(1−0.78)=1.65927.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 1.65927, not -1.65927.","B":"Correct. The worked calculation gives 1.65927.","C":"Incorrect. Substituting the stated values into the governing relation gives 1.65927, not -0.34073.","D":"Incorrect. Substituting the stated values into the governing relation gives 1.65927, not 2.65927."},"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_008174","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 7503 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":"26.68604 g","B":"24.68604 g","C":"25.68604 g","D":"18.51453 g"},"correct_answer":"B","correct_answer_value":"24.68604 g","solution":"Cu²⁺+2e⁻→Cu, so z=2. By Faraday's law m=MIt/(zF)=63.5×10×7503/(2×96500)=24.68604 g.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 24.68604 g, not 26.68604 g.","B":"Correct. The worked calculation gives 24.68604 g.","C":"Incorrect. Substituting the stated values into the governing relation gives 24.68604 g, not 25.68604 g.","D":"Incorrect. Substituting the stated values into the governing relation gives 24.68604 g, not 18.51453 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_001535","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 +2x +0) dx.","options":{"A":"1","B":"0.66667","C":"-1.33333","D":"1.33333"},"correct_answer":"D","correct_answer_value":"1.33333","solution":"An antiderivative is (-1/3)x³+(2/2)x²+(0)x. Evaluating between 0 and 2 gives (-1/3)(2)³+(2/2)(2)²+(0)(2)=1.33333.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 1.33333, not 1.","B":"Incorrect. Substituting the stated values into the governing relation gives 1.33333, not 0.66667.","C":"Incorrect. Substituting the stated values into the governing relation gives 1.33333, not -1.33333.","D":"Correct. The worked calculation gives 1.33333."},"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_001907","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 90 distinct elements?","options":{"A":"-4005","B":"4005","C":"3204","D":"4004"},"correct_answer":"B","correct_answer_value":"4005","solution":"Order does not matter, so the count is C(90,2)=90!/[2!(88)!]=4005.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 4005, not -4005.","B":"Correct. The worked calculation gives 4005.","C":"Incorrect. Substituting the stated values into the governing relation gives 4005, not 3204.","D":"Incorrect. Substituting the stated values into the governing relation gives 4005, not 4004."},"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_007665","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":"97 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":"1.32818","B":"4.32818","C":"2.65636","D":"0.99614"},"correct_answer":"A","correct_answer_value":"1.32818","solution":"Excess H⁺ moles=0.10×(97−35)/1000=0.006200 mol. Total volume=0.132 L, so [H⁺]=0.046970 M and pH=−log₁₀[H⁺]=1.32818.","option_explanations":{"A":"Correct. The worked calculation gives 1.32818.","B":"Incorrect. Substituting the stated values into the governing relation gives 1.32818, not 4.32818.","C":"Incorrect. Substituting the stated values into the governing relation gives 1.32818, not 2.65636.","D":"Incorrect. Substituting the stated values into the governing relation gives 1.32818, not 0.99614."},"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_004757","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 14 Ω and 5 Ω are connected in series; this combination is connected in parallel with a 19 Ω resistor across a 24 V ideal battery. Find the total current supplied by the battery.","options":{"A":"2.52632 A","B":"1.25263 A","C":"0.52632 A","D":"5.52632 A"},"correct_answer":"A","correct_answer_value":"2.52632 A","solution":"Series branch resistance=14+5=19 Ω. Parallel equivalent R_eq=(19×19)/(19+19)=9.5 Ω. Total current I=V/R_eq=24/9.5=2.52632 A.","option_explanations":{"A":"Correct. The worked calculation gives 2.52632 A.","B":"Incorrect. Substituting the stated values into the governing relation gives 2.52632 A, not 1.25263 A.","C":"Incorrect. Substituting the stated values into the governing relation gives 2.52632 A, not 0.52632 A.","D":"Incorrect. Substituting the stated values into the governing relation gives 2.52632 A, not 5.52632 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_004109","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 0.45. Taking g=10 m s⁻², find the net force down the plane. Use sin37°=0.6 and cos37°=0.8.","options":{"A":"50.4 N","B":"60.48 N","C":"100.8 N","D":"48.4 N"},"correct_answer":"A","correct_answer_value":"50.4 N","solution":"Net force=mg(sinθ−μcosθ)=21×10×(0.6−0.45×0.8)=50.4 N.","option_explanations":{"A":"Correct. The worked calculation gives 50.4 N.","B":"Incorrect. Substituting the stated values into the governing relation gives 50.4 N, not 60.48 N.","C":"Incorrect. Substituting the stated values into the governing relation gives 50.4 N, not 100.8 N.","D":"Incorrect. Substituting the stated values into the governing relation gives 50.4 N, not 48.4 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_009383","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_4N_2X_4. Treat X as the total number of monovalent halogen atoms.","options":{"A":"4.5","B":"2","C":"3","D":"-3"},"correct_answer":"C","correct_answer_value":"3","solution":"Use DBE=(2C+2+N−H−X)/2. Substitution gives (2×5+2+2−4−4)/2=3.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 3, not 4.5.","B":"Incorrect. Substituting the stated values into the governing relation gives 3, not 2.","C":"Correct. The worked calculation gives 3.","D":"Incorrect. Substituting the stated values into the governing relation gives 3, not -3."},"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_007476","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":"48 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":"5.96653","B":"11.93305","C":"9.93305","D":"13.93305"},"correct_answer":"B","correct_answer_value":"11.93305","solution":"Excess OH⁻ moles=0.10×(57−48)/1000=0.000900 mol. Total volume=0.105 L, so [OH⁻]=0.008571 M, pOH=−log₁₀[OH⁻], and pH=14−pOH=11.93305.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 11.93305, not 5.96653.","B":"Correct. The worked calculation gives 11.93305.","C":"Incorrect. Substituting the stated values into the governing relation gives 11.93305, not 9.93305.","D":"Incorrect. Substituting the stated values into the governing relation gives 11.93305, not 13.93305."},"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_005924","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 3.9 atm at 243 K. What is its pressure at 304 K?","options":{"A":"1.87901 atm","B":"9.75802 atm","C":"-4.87901 atm","D":"4.87901 atm"},"correct_answer":"D","correct_answer_value":"4.87901 atm","solution":"At constant volume, P/T is constant. P₂=P₁T₂/T₁=(3.9)(304)/(243)=4.87901 atm.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 4.87901 atm, not 1.87901 atm.","B":"Incorrect. Substituting the stated values into the governing relation gives 4.87901 atm, not 9.75802 atm.","C":"Incorrect. Substituting the stated values into the governing relation gives 4.87901 atm, not -4.87901 atm.","D":"Correct. The worked calculation gives 4.87901 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_000275","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 +0x +3=0, let its roots be α and β. What is α²+β²?","options":{"A":"-0.85714","B":"1.14286","C":"-0.64286","D":"-1.71429"},"correct_answer":"A","correct_answer_value":"-0.85714","solution":"By Vieta's relations, α+β = 0/7 and αβ = 3/7. Then α²+β²=(α+β)²−2αβ = (0/7)²−2(3/7) = -0.85714.","option_explanations":{"A":"Correct. The worked calculation gives -0.85714.","B":"Incorrect. Substituting the stated values into the governing relation gives -0.85714, not 1.14286.","C":"Incorrect. Substituting the stated values into the governing relation gives -0.85714, not -0.64286.","D":"Incorrect. Substituting the stated values into the governing relation gives -0.85714, not -1.71429."},"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_008717","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 0.77 mol L⁻¹ and initially B is absent. At equilibrium, 69% of A has dissociated at constant volume. Find K_c.","options":{"A":"3.73028","B":"3.54771","C":"4.73028","D":"1.47303"},"correct_answer":"C","correct_answer_value":"4.73028","solution":"Let α=0.69. At equilibrium [A]=c₀(1−α) and [B]=2c₀α. Thus K_c=[B]²/[A]=4c₀α²/(1−α)=4(0.77)(0.69)²/(1−0.69)=4.73028.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 4.73028, not 3.73028.","B":"Incorrect. Substituting the stated values into the governing relation gives 4.73028, not 3.54771.","C":"Correct. The worked calculation gives 4.73028.","D":"Incorrect. Substituting the stated values into the governing relation gives 4.73028, not 1.47303."},"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_003347","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 85 m s⁻¹ at 70° to the horizontal. Take g=10 m s⁻². Find its horizontal range.","options":{"A":"466.41405 m","B":"928.8281 m","C":"464.41405 m","D":"371.53124 m"},"correct_answer":"C","correct_answer_value":"464.41405 m","solution":"Range R=u²sin(2θ)/g. Substituting u=85, θ=70° and g=10 gives R=(85)²sin(140°)/10=464.41405 m.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 464.41405 m, not 466.41405 m.","B":"Incorrect. Substituting the stated values into the governing relation gives 464.41405 m, not 928.8281 m.","C":"Correct. The worked calculation gives 464.41405 m.","D":"Incorrect. Substituting the stated values into the governing relation gives 464.41405 m, not 371.53124 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_006620","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 511 N m⁻¹ and oscillates horizontally without damping. What is the time period divided by π?","options":{"A":"0.1595 s","B":"0.638 s","C":"1.0319 s","D":"0.319 s"},"correct_answer":"D","correct_answer_value":"0.319 s","solution":"For a mass–spring oscillator, T=2π√(m/k). Therefore T/π=2√(13/511)=0.319 s.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.319 s, not 0.1595 s.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.319 s, not 0.638 s.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.319 s, not 1.0319 s.","D":"Correct. The worked calculation gives 0.319 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_009906","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 [Ni(Cl)_3]^+0, each Cl ligand has charge −1. Determine the formal oxidation state of Ni.","options":{"A":"3","B":"1.3","C":"1","D":"4"},"correct_answer":"A","correct_answer_value":"3","solution":"Let the formal oxidation state of Ni be x. Charge balance gives x+(3)(−1)=+0; hence x=0+3=+3.","option_explanations":{"A":"Correct. The worked calculation gives 3.","B":"Incorrect. Substituting the stated values into the governing relation gives 3, not 1.3.","C":"Incorrect. Substituting the stated values into the governing relation gives 3, not 1.","D":"Incorrect. Substituting the stated values into the governing relation gives 3, 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_009397","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_8N_3X_1. Treat X as the total number of monovalent halogen atoms.","options":{"A":"4.5","B":"9","C":"8","D":"6"},"correct_answer":"D","correct_answer_value":"6","solution":"Use DBE=(2C+2+N−H−X)/2. Substitution gives (2×8+2+3−8−1)/2=6.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 6, not 4.5.","B":"Incorrect. Substituting the stated values into the governing relation gives 6, not 9.","C":"Incorrect. Substituting the stated values into the governing relation gives 6, not 8.","D":"Correct. The worked calculation gives 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_002608","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, -6], [5, 6]].","options":{"A":"63","B":"84","C":"9.4","D":"85"},"correct_answer":"B","correct_answer_value":"84","solution":"For a 2×2 matrix [[a,b],[c,d]], det=ad−bc. Here it is (9)(6)−(-6)(5)=84.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 84, not 63.","B":"Correct. The worked calculation gives 84.","C":"Incorrect. Substituting the stated values into the governing relation gives 84, not 9.4.","D":"Incorrect. Substituting the stated values into the governing relation gives 84, not 85."},"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_000396","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 25 and common difference 5. Find the sum of its first 35 terms.","options":{"A":"3848","B":"386","C":"3850","D":"3849"},"correct_answer":"C","correct_answer_value":"3850","solution":"Use Sₙ = n/2[2a+(n−1)d]. Substituting a=25, d=5, n=35: Sₙ=35/2[2(25)+(35−1)(5)]=3850.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 3850, not 3848.","B":"Incorrect. Substituting the stated values into the governing relation gives 3850, not 386.","C":"Correct. The worked calculation gives 3850.","D":"Incorrect. Substituting the stated values into the governing relation gives 3850, not 3849."},"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_008477","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 1.7 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":"2.08 K","B":"1.08 K","C":"6.12 K","D":"4.08 K"},"correct_answer":"D","correct_answer_value":"4.08 K","solution":"ΔT_b=iK_bm. Substitution gives ΔT_b=4×0.6×1.7=4.08 K.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 4.08 K, not 2.08 K.","B":"Incorrect. Substituting the stated values into the governing relation gives 4.08 K, not 1.08 K.","C":"Incorrect. Substituting the stated values into the governing relation gives 4.08 K, not 6.12 K.","D":"Correct. The worked calculation gives 4.08 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_006825","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 13 mol H₂ and 4 mol O₂ are mixed and the reaction goes to completion, how many moles of H₂O form?","options":{"A":"11 mol","B":"8 mol","C":"4 mol","D":"5 mol"},"correct_answer":"B","correct_answer_value":"8 mol","solution":"From 2 mol H₂ and 1 mol O₂, 2 mol H₂O form. H₂ can produce 13 mol H₂O; O₂ can produce 8 mol H₂O. The limiting condition is O₂; product=8 mol.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 8 mol, not 11 mol.","B":"Correct. The worked calculation gives 8 mol.","C":"Incorrect. Substituting the stated values into the governing relation gives 8 mol, not 4 mol.","D":"Incorrect. Substituting the stated values into the governing relation gives 8 mol, not 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_001376","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 +2) dx.","options":{"A":"-97","B":"-96","C":"-72","D":"96"},"correct_answer":"B","correct_answer_value":"-96","solution":"An antiderivative is (-2/3)x³+(2/2)x²+(2)x. Evaluating between 0 and 6 gives (-2/3)(6)³+(2/2)(6)²+(2)(6)=-96.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives -96, not -97.","B":"Correct. The worked calculation gives -96.","C":"Incorrect. Substituting the stated values into the governing relation gives -96, not -72.","D":"Incorrect. Substituting the stated values into the governing relation gives -96, not 96."},"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_005082","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 +5 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":"90 N","B":"225 N","C":"112.5 N","D":"113.5 N"},"correct_answer":"C","correct_answer_value":"112.5 N","solution":"Convert charges to coulombs and distance to metres. F=k|q₁q₂|/r²=9×10⁹(1×10⁻⁶)(5×10⁻⁶)/(2×10⁻²)²=112.5 N.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 112.5 N, not 90 N.","B":"Incorrect. Substituting the stated values into the governing relation gives 112.5 N, not 225 N.","C":"Correct. The worked calculation gives 112.5 N.","D":"Incorrect. Substituting the stated values into the governing relation gives 112.5 N, not 113.5 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_005193","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 +5 μC and +5 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":"90 N","B":"135 N","C":"72 N","D":"67.5 N"},"correct_answer":"A","correct_answer_value":"90 N","solution":"Convert charges to coulombs and distance to metres. F=k|q₁q₂|/r²=9×10⁹(5×10⁻⁶)(5×10⁻⁶)/(5×10⁻²)²=90 N.","option_explanations":{"A":"Correct. The worked calculation gives 90 N.","B":"Incorrect. Substituting the stated values into the governing relation gives 90 N, not 135 N.","C":"Incorrect. Substituting the stated values into the governing relation gives 90 N, not 72 N.","D":"Incorrect. Substituting the stated values into the governing relation gives 90 N, not 67.5 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_000319","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 +17x -9=0, let its roots be α and β. What is α²+β²?","options":{"A":"7.46939","B":"8.46939","C":"16.93878","D":"1.84694"},"correct_answer":"B","correct_answer_value":"8.46939","solution":"By Vieta's relations, α+β = -17/7 and αβ = -9/7. Then α²+β²=(α+β)²−2αβ = (-17/7)²−2(-9/7) = 8.46939.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 8.46939, not 7.46939.","B":"Correct. The worked calculation gives 8.46939.","C":"Incorrect. Substituting the stated values into the governing relation gives 8.46939, not 16.93878.","D":"Incorrect. Substituting the stated values into the governing relation gives 8.46939, not 1.84694."},"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_004520","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 3 m is acted on by a constant torque of 71 N m about its central axis. Find its angular acceleration.","options":{"A":"2.21368 rad s⁻²","B":"1.21368 rad s⁻²","C":"-0.78632 rad s⁻²","D":"-1.21368 rad s⁻²"},"correct_answer":"B","correct_answer_value":"1.21368 rad s⁻²","solution":"For a solid disc, I=½MR²=½(13)(3)²=58.5 kg m². Using τ=Iα, α=71/58.5=1.21368 rad s⁻².","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 1.21368 rad s⁻², not 2.21368 rad s⁻².","B":"Correct. The worked calculation gives 1.21368 rad s⁻².","C":"Incorrect. Substituting the stated values into the governing relation gives 1.21368 rad s⁻², not -0.78632 rad s⁻².","D":"Incorrect. Substituting the stated values into the governing relation gives 1.21368 rad s⁻², not -1.21368 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_003405","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 18° to the horizontal. Take g=10 m s⁻². Find its horizontal range.","options":{"A":"263.8568 m","B":"131.9284 m","C":"316.62816 m","D":"211.08544 m"},"correct_answer":"A","correct_answer_value":"263.8568 m","solution":"Range R=u²sin(2θ)/g. Substituting u=67, θ=18° and g=10 gives R=(67)²sin(36°)/10=263.8568 m.","option_explanations":{"A":"Correct. The worked calculation gives 263.8568 m.","B":"Incorrect. Substituting the stated values into the governing relation gives 263.8568 m, not 131.9284 m.","C":"Incorrect. Substituting the stated values into the governing relation gives 263.8568 m, not 316.62816 m.","D":"Incorrect. Substituting the stated values into the governing relation gives 263.8568 m, not 211.08544 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_008817","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 1.88 mol L⁻¹ and initially B is absent. At equilibrium, 53% of A has dissociated at constant volume. Find K_c.","options":{"A":"3.3708","B":"5.4944","C":"4.4944","D":"6.4944"},"correct_answer":"C","correct_answer_value":"4.4944","solution":"Let α=0.53. At equilibrium [A]=c₀(1−α) and [B]=2c₀α. Thus K_c=[B]²/[A]=4c₀α²/(1−α)=4(1.88)(0.53)²/(1−0.53)=4.4944.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 4.4944, not 3.3708.","B":"Incorrect. Substituting the stated values into the governing relation gives 4.4944, not 5.4944.","C":"Correct. The worked calculation gives 4.4944.","D":"Incorrect. Substituting the stated values into the governing relation gives 4.4944, not 6.4944."},"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_001125","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 -2x² -2x+7. Evaluate f′(-2).","options":{"A":"20","B":"18","C":"13.5","D":"-18"},"correct_answer":"B","correct_answer_value":"18","solution":"Differentiate: f′(x)=3(1)x²+2(-2)x+(-2). At x=-2, f′(-2)=3(1)(-2)²+2(-2)(-2)+(-2)=18.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 18, not 20.","B":"Correct. The worked calculation gives 18.","C":"Incorrect. Substituting the stated values into the governing relation gives 18, not 13.5.","D":"Incorrect. Substituting the stated values into the governing relation gives 18, not -18."},"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_003657","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 33 m s⁻¹ at 60° to the horizontal. Take g=10 m s⁻². Find its horizontal range.","options":{"A":"70.73262 m","B":"188.62033 m","C":"94.31017 m","D":"141.46525 m"},"correct_answer":"C","correct_answer_value":"94.31017 m","solution":"Range R=u²sin(2θ)/g. Substituting u=33, θ=60° and g=10 gives R=(33)²sin(120°)/10=94.31017 m.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 94.31017 m, not 70.73262 m.","B":"Incorrect. Substituting the stated values into the governing relation gives 94.31017 m, not 188.62033 m.","C":"Correct. The worked calculation gives 94.31017 m.","D":"Incorrect. Substituting the stated values into the governing relation gives 94.31017 m, not 141.46525 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_005773","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 atm at 263 K. What is its pressure at 545 K?","options":{"A":"3.21673 atm","B":"6.21673 atm","C":"7.21673 atm","D":"3.10837 atm"},"correct_answer":"B","correct_answer_value":"6.21673 atm","solution":"At constant volume, P/T is constant. P₂=P₁T₂/T₁=(3)(545)/(263)=6.21673 atm.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 6.21673 atm, not 3.21673 atm.","B":"Correct. The worked calculation gives 6.21673 atm.","C":"Incorrect. Substituting the stated values into the governing relation gives 6.21673 atm, not 7.21673 atm.","D":"Incorrect. Substituting the stated values into the governing relation gives 6.21673 atm, not 3.10837 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_009840","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 [Cr(Cl)_2]^+0, each Cl ligand has charge −1. Determine the formal oxidation state of Cr.","options":{"A":"4","B":"3","C":"2","D":"5"},"correct_answer":"C","correct_answer_value":"2","solution":"Let the formal oxidation state of Cr be x. Charge balance gives x+(2)(−1)=+0; hence x=0+2=+2.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 2, not 4.","B":"Incorrect. Substituting the stated values into the governing relation gives 2, not 3.","C":"Correct. The worked calculation gives 2.","D":"Incorrect. Substituting the stated values into the governing relation gives 2, 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_000874","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 4. What is the sum of its first 6 terms?","options":{"A":"10920","B":"10922","C":"10919","D":"8736"},"correct_answer":"A","correct_answer_value":"10920","solution":"For r≠1, Sₙ=a(rⁿ−1)/(r−1). Thus Sₙ=8(4^6−1)/(4−1)=10920.","option_explanations":{"A":"Correct. The worked calculation gives 10920.","B":"Incorrect. Substituting the stated values into the governing relation gives 10920, not 10922.","C":"Incorrect. Substituting the stated values into the governing relation gives 10920, not 10919.","D":"Incorrect. Substituting the stated values into the governing relation gives 10920, not 8736."},"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_004009","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 28 kg block slides down a 37° rough incline. The coefficient of kinetic friction is 0.10. Taking g=10 m s⁻², find the net force down the plane. Use sin37°=0.6 and cos37°=0.8.","options":{"A":"72.8 N","B":"291.2 N","C":"116.48 N","D":"145.6 N"},"correct_answer":"D","correct_answer_value":"145.6 N","solution":"Net force=mg(sinθ−μcosθ)=28×10×(0.6−0.10×0.8)=145.6 N.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 145.6 N, not 72.8 N.","B":"Incorrect. Substituting the stated values into the governing relation gives 145.6 N, not 291.2 N.","C":"Incorrect. Substituting the stated values into the governing relation gives 145.6 N, not 116.48 N.","D":"Correct. The worked calculation gives 145.6 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_009824","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)_3]^-3, each OH ligand has charge −1. Determine the formal oxidation state of Mn.","options":{"A":"-1","B":"-2","C":"0","D":"1"},"correct_answer":"C","correct_answer_value":"0","solution":"Let the formal oxidation state of Mn be x. Charge balance gives x+(3)(−1)=-3; hence x=-3+3=+0.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0, not -1.","B":"Incorrect. Substituting the stated values into the governing relation gives 0, not -2.","C":"Correct. The worked calculation gives 0.","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_003313","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=(-11-12i)/(11+5i). What is Re(z), the real part of z?","options":{"A":"-0.92979","B":"-1.23973","C":"-0.61986","D":"1.76027"},"correct_answer":"B","correct_answer_value":"-1.23973","solution":"Multiply numerator and denominator by the conjugate (11-5i). Then Re(z)=(ac+bd)/(c²+d²)=((-11)(11)+(-12)(5))/(11²+(5)²)=-1.23973.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives -1.23973, not -0.92979.","B":"Correct. The worked calculation gives -1.23973.","C":"Incorrect. Substituting the stated values into the governing relation gives -1.23973, not -0.61986.","D":"Incorrect. Substituting the stated values into the governing relation gives -1.23973, not 1.76027."},"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_002223","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, 8 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":"0.26383","B":"2.35178","C":"1.35178","D":"0.35178"},"correct_answer":"D","correct_answer_value":"0.35178","solution":"Favourable pairs = C(4,2)+C(8,2)+C(11,2)=89. Total pairs=C(23,2)=253. Therefore P=89/253=0.35178.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.35178, not 0.26383.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.35178, not 2.35178.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.35178, not 1.35178.","D":"Correct. The worked calculation gives 0.35178."},"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_007099","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 6 mol of gas at 409 K and 0.6 atm? Use R=0.082057 L atm mol⁻¹ K⁻¹.","options":{"A":"335.61313 L","B":"251.70985 L","C":"671.22626 L","D":"503.4197 L"},"correct_answer":"A","correct_answer_value":"335.61313 L","solution":"PV=nRT, so V=nRT/P=(6)(0.082057)(409)/(0.6)=335.61313 L.","option_explanations":{"A":"Correct. The worked calculation gives 335.61313 L.","B":"Incorrect. Substituting the stated values into the governing relation gives 335.61313 L, not 251.70985 L.","C":"Incorrect. Substituting the stated values into the governing relation gives 335.61313 L, not 671.22626 L.","D":"Incorrect. Substituting the stated values into the governing relation gives 335.61313 L, not 503.4197 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_006616","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 2 kg is attached to an ideal spring of force constant 64 N m⁻¹ and oscillates horizontally without damping. What is the time period divided by π?","options":{"A":"0.53033 s","B":"0.35355 s","C":"1.35355 s","D":"1.03536 s"},"correct_answer":"B","correct_answer_value":"0.35355 s","solution":"For a mass–spring oscillator, T=2π√(m/k). Therefore T/π=2√(2/64)=0.35355 s.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.35355 s, not 0.53033 s.","B":"Correct. The worked calculation gives 0.35355 s.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.35355 s, not 1.35355 s.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.35355 s, not 1.03536 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_004693","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 8 Ω and 6 Ω are connected in series; this combination is connected in parallel with a 2 Ω resistor across a 12 V ideal battery. Find the total current supplied by the battery.","options":{"A":"1.68571 A","B":"-6.85714 A","C":"3.42857 A","D":"6.85714 A"},"correct_answer":"D","correct_answer_value":"6.85714 A","solution":"Series branch resistance=8+6=14 Ω. Parallel equivalent R_eq=(14×2)/(14+2)=1.75 Ω. Total current I=V/R_eq=12/1.75=6.85714 A.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 6.85714 A, not 1.68571 A.","B":"Incorrect. Substituting the stated values into the governing relation gives 6.85714 A, not -6.85714 A.","C":"Incorrect. Substituting the stated values into the governing relation gives 6.85714 A, not 3.42857 A.","D":"Correct. The worked calculation gives 6.85714 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_003159","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=(9-15i)/(8+4i). What is Re(z), the real part of z?","options":{"A":"0.3","B":"-1.85","C":"0.15","D":"1.015"},"correct_answer":"C","correct_answer_value":"0.15","solution":"Multiply numerator and denominator by the conjugate (8-4i). Then Re(z)=(ac+bd)/(c²+d²)=((9)(8)+(-15)(4))/(8²+(4)²)=0.15.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.15, not 0.3.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.15, not -1.85.","C":"Correct. The worked calculation gives 0.15.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.15, not 1.015."},"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_005836","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 3.7 atm at 271 K. What is its pressure at 722 K?","options":{"A":"6.85756 atm","B":"7.39317 atm","C":"9.85756 atm","D":"4.92878 atm"},"correct_answer":"C","correct_answer_value":"9.85756 atm","solution":"At constant volume, P/T is constant. P₂=P₁T₂/T₁=(3.7)(722)/(271)=9.85756 atm.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 9.85756 atm, not 6.85756 atm.","B":"Incorrect. Substituting the stated values into the governing relation gives 9.85756 atm, not 7.39317 atm.","C":"Correct. The worked calculation gives 9.85756 atm.","D":"Incorrect. Substituting the stated values into the governing relation gives 9.85756 atm, not 4.92878 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_006479","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 372 N m⁻¹ and oscillates horizontally without damping. What is the time period divided by π?","options":{"A":"-2.51363 s","B":"1.48637 s","C":"-0.51363 s","D":"0.48637 s"},"correct_answer":"D","correct_answer_value":"0.48637 s","solution":"For a mass–spring oscillator, T=2π√(m/k). Therefore T/π=2√(22/372)=0.48637 s.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.48637 s, not -2.51363 s.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.48637 s, not 1.48637 s.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.48637 s, not -0.51363 s.","D":"Correct. The worked calculation gives 0.48637 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_003423","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 38 m s⁻¹ at 75° to the horizontal. Take g=10 m s⁻². Find its horizontal range.","options":{"A":"144.4 m","B":"54.15 m","C":"73.2 m","D":"72.2 m"},"correct_answer":"D","correct_answer_value":"72.2 m","solution":"Range R=u²sin(2θ)/g. Substituting u=38, θ=75° and g=10 gives R=(38)²sin(150°)/10=72.2 m.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 72.2 m, not 144.4 m.","B":"Incorrect. Substituting the stated values into the governing relation gives 72.2 m, not 54.15 m.","C":"Incorrect. Substituting the stated values into the governing relation gives 72.2 m, not 73.2 m.","D":"Correct. The worked calculation gives 72.2 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_003523","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 51 m s⁻¹ at 60° to the horizontal. Take g=10 m s⁻². Find its horizontal range.","options":{"A":"225.25321 m","B":"168.93991 m","C":"450.50642 m","D":"224.25321 m"},"correct_answer":"A","correct_answer_value":"225.25321 m","solution":"Range R=u²sin(2θ)/g. Substituting u=51, θ=60° and g=10 gives R=(51)²sin(120°)/10=225.25321 m.","option_explanations":{"A":"Correct. The worked calculation gives 225.25321 m.","B":"Incorrect. Substituting the stated values into the governing relation gives 225.25321 m, not 168.93991 m.","C":"Incorrect. Substituting the stated values into the governing relation gives 225.25321 m, not 450.50642 m.","D":"Incorrect. Substituting the stated values into the governing relation gives 225.25321 m, not 224.25321 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_004143","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 0.18. Taking g=10 m s⁻², find the net force down the plane. Use sin37°=0.6 and cos37°=0.8.","options":{"A":"8.296 N","B":"72.96 N","C":"-72.96 N","D":"54.72 N"},"correct_answer":"B","correct_answer_value":"72.96 N","solution":"Net force=mg(sinθ−μcosθ)=16×10×(0.6−0.18×0.8)=72.96 N.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 72.96 N, not 8.296 N.","B":"Correct. The worked calculation gives 72.96 N.","C":"Incorrect. Substituting the stated values into the governing relation gives 72.96 N, not -72.96 N.","D":"Incorrect. Substituting the stated values into the governing relation gives 72.96 N, not 54.72 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_004000","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 2 m s⁻¹. A constant net force does 10 N of work over 11 m in the direction of motion. Find its final kinetic energy.","options":{"A":"112 J","B":"138 J","C":"140 J","D":"210 J"},"correct_answer":"C","correct_answer_value":"140 J","solution":"By the work–energy theorem, K_f=K_i+W=½mu²+Fs=½(15)(2)²+(10)(11)=140 J.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 140 J, not 112 J.","B":"Incorrect. Substituting the stated values into the governing relation gives 140 J, not 138 J.","C":"Correct. The worked calculation gives 140 J.","D":"Incorrect. Substituting the stated values into the governing relation gives 140 J, not 210 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_000424","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 11 and common difference 12. Find the sum of its first 20 terms.","options":{"A":"251","B":"-2500","C":"2000","D":"2500"},"correct_answer":"D","correct_answer_value":"2500","solution":"Use Sₙ = n/2[2a+(n−1)d]. Substituting a=11, d=12, n=20: Sₙ=20/2[2(11)+(20−1)(12)]=2500.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 2500, not 251.","B":"Incorrect. Substituting the stated values into the governing relation gives 2500, not -2500.","C":"Incorrect. Substituting the stated values into the governing relation gives 2500, not 2000.","D":"Correct. The worked calculation gives 2500."},"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_008539","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 0.82 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":"6.56 K","B":"6.28 K","C":"3.28 K","D":"-3.28 K"},"correct_answer":"C","correct_answer_value":"3.28 K","solution":"ΔT_b=iK_bm. Substitution gives ΔT_b=4×1.0×0.82=3.28 K.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 3.28 K, not 6.56 K.","B":"Incorrect. Substituting the stated values into the governing relation gives 3.28 K, not 6.28 K.","C":"Correct. The worked calculation gives 3.28 K.","D":"Incorrect. Substituting the stated values into the governing relation gives 3.28 K, not -3.28 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_009113","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=1.12 V, transfers 2 electrons, and has reaction quotient Q=10^4. Using E=E°−(0.0591/n)log₁₀Q, calculate Ecell.","options":{"A":"-1.0018 V","B":"1.0018 V","C":"0.5009 V","D":"1.10018 V"},"correct_answer":"B","correct_answer_value":"1.0018 V","solution":"Apply the Nernst equation: E=1.12−(0.0591/2)(4)=1.0018 V.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 1.0018 V, not -1.0018 V.","B":"Correct. The worked calculation gives 1.0018 V.","C":"Incorrect. Substituting the stated values into the governing relation gives 1.0018 V, not 0.5009 V.","D":"Incorrect. Substituting the stated values into the governing relation gives 1.0018 V, not 1.10018 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_003481","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 97 m s⁻¹ at 65° to the horizontal. Take g=10 m s⁻². Find its horizontal range.","options":{"A":"722.77122 m","B":"1441.54243 m","C":"719.77122 m","D":"720.77122 m"},"correct_answer":"D","correct_answer_value":"720.77122 m","solution":"Range R=u²sin(2θ)/g. Substituting u=97, θ=65° and g=10 gives R=(97)²sin(130°)/10=720.77122 m.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 720.77122 m, not 722.77122 m.","B":"Incorrect. Substituting the stated values into the governing relation gives 720.77122 m, not 1441.54243 m.","C":"Incorrect. Substituting the stated values into the governing relation gives 720.77122 m, not 719.77122 m.","D":"Correct. The worked calculation gives 720.77122 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_000870","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 -2. What is the sum of its first 11 terms?","options":{"A":"2051","B":"2047","C":"2049","D":"3073.5"},"correct_answer":"C","correct_answer_value":"2049","solution":"For r≠1, Sₙ=a(rⁿ−1)/(r−1). Thus Sₙ=3(-2^11−1)/(-2−1)=2049.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 2049, not 2051.","B":"Incorrect. Substituting the stated values into the governing relation gives 2049, not 2047.","C":"Correct. The worked calculation gives 2049.","D":"Incorrect. Substituting the stated values into the governing relation gives 2049, not 3073.5."},"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_001645","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 ∫₀^5 (5x² +5x -3) dx.","options":{"A":"-255.83333","B":"127.91667","C":"255.83333","D":"256.83333"},"correct_answer":"C","correct_answer_value":"255.83333","solution":"An antiderivative is (5/3)x³+(5/2)x²+(-3)x. Evaluating between 0 and 5 gives (5/3)(5)³+(5/2)(5)²+(-3)(5)=255.83333.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 255.83333, not -255.83333.","B":"Incorrect. Substituting the stated values into the governing relation gives 255.83333, not 127.91667.","C":"Correct. The worked calculation gives 255.83333.","D":"Incorrect. Substituting the stated values into the governing relation gives 255.83333, not 256.83333."},"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_009102","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 2 electrons, and has reaction quotient Q=10^4. Using E=E°−(0.0591/n)log₁₀Q, calculate Ecell.","options":{"A":"0.4659 V","B":"0.9318 V","C":"-0.9318 V","D":"1.9318 V"},"correct_answer":"B","correct_answer_value":"0.9318 V","solution":"Apply the Nernst equation: E=1.05−(0.0591/2)(4)=0.9318 V.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.9318 V, not 0.4659 V.","B":"Correct. The worked calculation gives 0.9318 V.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.9318 V, not -0.9318 V.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.9318 V, not 1.9318 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_003750","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 10 m s⁻¹. A constant net force does 11 N of work over 13 m in the direction of motion. Find its final kinetic energy.","options":{"A":"951.6 J","B":"793 J","C":"80.3 J","D":"-793 J"},"correct_answer":"B","correct_answer_value":"793 J","solution":"By the work–energy theorem, K_f=K_i+W=½mu²+Fs=½(13)(10)²+(11)(13)=793 J.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 793 J, not 951.6 J.","B":"Correct. The worked calculation gives 793 J.","C":"Incorrect. Substituting the stated values into the governing relation gives 793 J, not 80.3 J.","D":"Incorrect. Substituting the stated values into the governing relation gives 793 J, not -793 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_000973","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 4. What is the sum of its first 12 terms?","options":{"A":"41943037.5","B":"55924050","C":"55924051","D":"55924049"},"correct_answer":"B","correct_answer_value":"55924050","solution":"For r≠1, Sₙ=a(rⁿ−1)/(r−1). Thus Sₙ=10(4^12−1)/(4−1)=55924050.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 55924050, not 41943037.5.","B":"Correct. The worked calculation gives 55924050.","C":"Incorrect. Substituting the stated values into the governing relation gives 55924050, not 55924051.","D":"Incorrect. Substituting the stated values into the governing relation gives 55924050, not 55924049."},"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_002738","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, 4) and v=(9, 4, 4), calculate u·v.","options":{"A":"-0.7","B":"-19","C":"-17","D":"17"},"correct_answer":"C","correct_answer_value":"-17","solution":"The dot product is the sum of corresponding component products: (-5)(9)+(3)(4)+(4)(4)=-17.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives -17, not -0.7.","B":"Incorrect. Substituting the stated values into the governing relation gives -17, not -19.","C":"Correct. The worked calculation gives -17.","D":"Incorrect. Substituting the stated values into the governing relation gives -17, not 17."},"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_001901","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 34 distinct elements?","options":{"A":"46376","B":"92752","C":"69564","D":"-46376"},"correct_answer":"A","correct_answer_value":"46376","solution":"Order does not matter, so the count is C(34,4)=34!/[4!(30)!]=46376.","option_explanations":{"A":"Correct. The worked calculation gives 46376.","B":"Incorrect. Substituting the stated values into the governing relation gives 46376, not 92752.","C":"Incorrect. Substituting the stated values into the governing relation gives 46376, not 69564.","D":"Incorrect. Substituting the stated values into the governing relation gives 46376, not -46376."},"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_007927","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 29 mmol L⁻¹, what concentration remains after 20 minutes?","options":{"A":"7.25 mmol L⁻¹","B":"21.75 mmol L⁻¹","C":"14.5 mmol L⁻¹","D":"16.5 mmol L⁻¹"},"correct_answer":"C","correct_answer_value":"14.5 mmol L⁻¹","solution":"20 minutes corresponds to 1 half-lives. The remaining concentration is [A]₀(1/2)ⁿ=29/2^1=14.5 mmol L⁻¹.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 14.5 mmol L⁻¹, not 7.25 mmol L⁻¹.","B":"Incorrect. Substituting the stated values into the governing relation gives 14.5 mmol L⁻¹, not 21.75 mmol L⁻¹.","C":"Correct. The worked calculation gives 14.5 mmol L⁻¹.","D":"Incorrect. Substituting the stated values into the governing relation gives 14.5 mmol L⁻¹, not 16.5 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_005595","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 12 V. Calculate the energy stored, expressed in μJ.","options":{"A":"594 μJ","B":"794 μJ","C":"792 μJ","D":"791 μJ"},"correct_answer":"C","correct_answer_value":"792 μJ","solution":"U=½CV². Since C=11 μF, U in μJ is ½×11×(12)²=792 μJ.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 792 μJ, not 594 μJ.","B":"Incorrect. Substituting the stated values into the governing relation gives 792 μJ, not 794 μJ.","C":"Correct. The worked calculation gives 792 μJ.","D":"Incorrect. Substituting the stated values into the governing relation gives 792 μJ, not 791 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_007768","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 97 mmol L⁻¹, what concentration remains after 80 minutes?","options":{"A":"4.54688 mmol L⁻¹","B":"9.0625 mmol L⁻¹","C":"-6.0625 mmol L⁻¹","D":"6.0625 mmol L⁻¹"},"correct_answer":"D","correct_answer_value":"6.0625 mmol L⁻¹","solution":"80 minutes corresponds to 4 half-lives. The remaining concentration is [A]₀(1/2)ⁿ=97/2^4=6.0625 mmol L⁻¹.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 6.0625 mmol L⁻¹, not 4.54688 mmol L⁻¹.","B":"Incorrect. Substituting the stated values into the governing relation gives 6.0625 mmol L⁻¹, not 9.0625 mmol L⁻¹.","C":"Incorrect. Substituting the stated values into the governing relation gives 6.0625 mmol L⁻¹, not -6.0625 mmol L⁻¹.","D":"Correct. The worked calculation gives 6.0625 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_009966","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(Cl)_5]^+1, each Cl ligand has charge −1. Determine the formal oxidation state of Pd.","options":{"A":"1.6","B":"6","C":"3","D":"12"},"correct_answer":"B","correct_answer_value":"6","solution":"Let the formal oxidation state of Pd be x. Charge balance gives x+(5)(−1)=+1; hence x=1+5=+6.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 6, not 1.6.","B":"Correct. The worked calculation gives 6.","C":"Incorrect. Substituting the stated values into the governing relation gives 6, not 3.","D":"Incorrect. Substituting the stated values into the governing relation gives 6, not 12."},"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_008321","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 11 A for 25139 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":"90.98234 g","B":"109.17881 g","C":"88.98234 g","D":"92.98234 g"},"correct_answer":"A","correct_answer_value":"90.98234 g","solution":"Cu²⁺+2e⁻→Cu, so z=2. By Faraday's law m=MIt/(zF)=63.5×11×25139/(2×96500)=90.98234 g.","option_explanations":{"A":"Correct. The worked calculation gives 90.98234 g.","B":"Incorrect. Substituting the stated values into the governing relation gives 90.98234 g, not 109.17881 g.","C":"Incorrect. Substituting the stated values into the governing relation gives 90.98234 g, not 88.98234 g.","D":"Incorrect. Substituting the stated values into the governing relation gives 90.98234 g, not 92.98234 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_004028","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 44 kg block slides down a 37° rough incline. The coefficient of kinetic friction is 0.30. Taking g=10 m s⁻², find the net force down the plane. Use sin37°=0.6 and cos37°=0.8.","options":{"A":"158.4 N","B":"160.4 N","C":"126.72 N","D":"237.6 N"},"correct_answer":"A","correct_answer_value":"158.4 N","solution":"Net force=mg(sinθ−μcosθ)=44×10×(0.6−0.30×0.8)=158.4 N.","option_explanations":{"A":"Correct. The worked calculation gives 158.4 N.","B":"Incorrect. Substituting the stated values into the governing relation gives 158.4 N, not 160.4 N.","C":"Incorrect. Substituting the stated values into the governing relation gives 158.4 N, not 126.72 N.","D":"Incorrect. Substituting the stated values into the governing relation gives 158.4 N, not 237.6 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_002164","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 11 green balls. Two balls are drawn uniformly at random without replacement. What is the probability that both have the same colour?","options":{"A":"0.34188","B":"-0.65812","C":"-0.34188","D":"0.17094"},"correct_answer":"A","correct_answer_value":"0.34188","solution":"Favourable pairs = C(5,2)+C(11,2)+C(11,2)=120. Total pairs=C(27,2)=351. Therefore P=120/351=0.34188.","option_explanations":{"A":"Correct. The worked calculation gives 0.34188.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.34188, not -0.65812.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.34188, not -0.34188.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.34188, not 0.17094."},"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_000376","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 22 and common difference 11. Find the sum of its first 19 terms.","options":{"A":"2297","B":"1839.2","C":"2300","D":"2299"},"correct_answer":"D","correct_answer_value":"2299","solution":"Use Sₙ = n/2[2a+(n−1)d]. Substituting a=22, d=11, n=19: Sₙ=19/2[2(22)+(19−1)(11)]=2299.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 2299, not 2297.","B":"Incorrect. Substituting the stated values into the governing relation gives 2299, not 1839.2.","C":"Incorrect. Substituting the stated values into the governing relation gives 2299, not 2300.","D":"Correct. The worked calculation gives 2299."},"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_001002","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 2 and common ratio 5. What is the sum of its first 9 terms?","options":{"A":"488281","B":"781249.6","C":"976562","D":"1953124"},"correct_answer":"C","correct_answer_value":"976562","solution":"For r≠1, Sₙ=a(rⁿ−1)/(r−1). Thus Sₙ=2(5^9−1)/(5−1)=976562.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 976562, not 488281.","B":"Incorrect. Substituting the stated values into the governing relation gives 976562, not 781249.6.","C":"Correct. The worked calculation gives 976562.","D":"Incorrect. Substituting the stated values into the governing relation gives 976562, not 1953124."},"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_009623","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_15H_20N_2X_2. Treat X as the total number of monovalent halogen atoms.","options":{"A":"-6","B":"3","C":"6","D":"9"},"correct_answer":"C","correct_answer_value":"6","solution":"Use DBE=(2C+2+N−H−X)/2. Substitution gives (2×15+2+2−20−2)/2=6.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 6, not -6.","B":"Incorrect. Substituting the stated values into the governing relation gives 6, not 3.","C":"Correct. The worked calculation gives 6.","D":"Incorrect. Substituting the stated values into the governing relation gives 6, not 9."},"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_008268","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 9571 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":"28.34107 g","B":"34.00928 g","C":"3.83411 g","D":"22.67286 g"},"correct_answer":"A","correct_answer_value":"28.34107 g","solution":"Cu²⁺+2e⁻→Cu, so z=2. By Faraday's law m=MIt/(zF)=63.5×9×9571/(2×96500)=28.34107 g.","option_explanations":{"A":"Correct. The worked calculation gives 28.34107 g.","B":"Incorrect. Substituting the stated values into the governing relation gives 28.34107 g, not 34.00928 g.","C":"Incorrect. Substituting the stated values into the governing relation gives 28.34107 g, not 3.83411 g.","D":"Incorrect. Substituting the stated values into the governing relation gives 28.34107 g, not 22.67286 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_001530","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 (5x² -6x +6) dx.","options":{"A":"711.33333","B":"709.33333","C":"-709.33333","D":"708.33333"},"correct_answer":"B","correct_answer_value":"709.33333","solution":"An antiderivative is (5/3)x³+(-6/2)x²+(6)x. Evaluating between 0 and 8 gives (5/3)(8)³+(-6/2)(8)²+(6)(8)=709.33333.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 709.33333, not 711.33333.","B":"Correct. The worked calculation gives 709.33333.","C":"Incorrect. Substituting the stated values into the governing relation gives 709.33333, not -709.33333.","D":"Incorrect. Substituting the stated values into the governing relation gives 709.33333, not 708.33333."},"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_009861","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]^+0, each CN ligand has charge −1. Determine the formal oxidation state of Cu.","options":{"A":"2.25","B":"1","C":"6","D":"3"},"correct_answer":"D","correct_answer_value":"3","solution":"Let the formal oxidation state of Cu be x. Charge balance gives x+(3)(−1)=+0; hence x=0+3=+3.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 3, not 2.25.","B":"Incorrect. Substituting the stated values into the governing relation gives 3, not 1.","C":"Incorrect. Substituting the stated values into the governing relation gives 3, not 6.","D":"Correct. The worked calculation gives 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_008687","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 1.5 mol L⁻¹ and initially B is absent. At equilibrium, 55% of A has dissociated at constant volume. Find K_c.","options":{"A":"1.03333","B":"4.03333","C":"7.03333","D":"3.03333"},"correct_answer":"B","correct_answer_value":"4.03333","solution":"Let α=0.55. At equilibrium [A]=c₀(1−α) and [B]=2c₀α. Thus K_c=[B]²/[A]=4c₀α²/(1−α)=4(1.5)(0.55)²/(1−0.55)=4.03333.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 4.03333, not 1.03333.","B":"Correct. The worked calculation gives 4.03333.","C":"Incorrect. Substituting the stated values into the governing relation gives 4.03333, not 7.03333.","D":"Incorrect. Substituting the stated values into the governing relation gives 4.03333, not 3.03333."},"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_009779","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 [Cr(CN)_3]^-1, each CN ligand has charge −1. Determine the formal oxidation state of Cr.","options":{"A":"5","B":"1","C":"3","D":"2"},"correct_answer":"D","correct_answer_value":"2","solution":"Let the formal oxidation state of Cr 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 5.","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":"Correct. The worked calculation gives 2."},"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_008911","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 0.31 mol L⁻¹ and initially B is absent. At equilibrium, 22% of A has dissociated at constant volume. Find K_c.","options":{"A":"1.07694","B":"-0.92306","C":"0.11542","D":"0.07694"},"correct_answer":"D","correct_answer_value":"0.07694","solution":"Let α=0.22. At equilibrium [A]=c₀(1−α) and [B]=2c₀α. Thus K_c=[B]²/[A]=4c₀α²/(1−α)=4(0.31)(0.22)²/(1−0.22)=0.07694.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.07694, not 1.07694.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.07694, not -0.92306.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.07694, not 0.11542.","D":"Correct. The worked calculation gives 0.07694."},"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_003232","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)/(4+8i). What is Re(z), the real part of z?","options":{"A":"-3.8","B":"-4.8","C":"-2.7","D":"-1.8"},"correct_answer":"D","correct_answer_value":"-1.8","solution":"Multiply numerator and denominator by the conjugate (4-8i). Then Re(z)=(ac+bd)/(c²+d²)=((-12)(4)+(-12)(8))/(4²+(8)²)=-1.8.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives -1.8, not -3.8.","B":"Incorrect. Substituting the stated values into the governing relation gives -1.8, not -4.8.","C":"Incorrect. Substituting the stated values into the governing relation gives -1.8, not -2.7.","D":"Correct. The worked calculation gives -1.8."},"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_005770","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 0.6 atm at 377 K. What is its pressure at 488 K?","options":{"A":"0.77666 atm","B":"-0.77666 atm","C":"-0.22334 atm","D":"-2.22334 atm"},"correct_answer":"A","correct_answer_value":"0.77666 atm","solution":"At constant volume, P/T is constant. P₂=P₁T₂/T₁=(0.6)(488)/(377)=0.77666 atm.","option_explanations":{"A":"Correct. The worked calculation gives 0.77666 atm.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.77666 atm, not -0.77666 atm.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.77666 atm, not -0.22334 atm.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.77666 atm, not -2.22334 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_005018","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 +5 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":"2.8 N","B":"18 N","C":"13.5 N","D":"9 N"},"correct_answer":"B","correct_answer_value":"18 N","solution":"Convert charges to coulombs and distance to metres. F=k|q₁q₂|/r²=9×10⁹(1×10⁻⁶)(5×10⁻⁶)/(5×10⁻²)²=18 N.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 18 N, not 2.8 N.","B":"Correct. The worked calculation gives 18 N.","C":"Incorrect. Substituting the stated values into the governing relation gives 18 N, not 13.5 N.","D":"Incorrect. Substituting the stated values into the governing relation gives 18 N, not 9 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_009788","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(Cl)_3]^-2, each Cl ligand has charge −1. Determine the formal oxidation state of Mn.","options":{"A":"-2","B":"0","C":"3","D":"1"},"correct_answer":"D","correct_answer_value":"1","solution":"Let the formal oxidation state of Mn be x. Charge balance gives x+(3)(−1)=-2; hence x=-2+3=+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 0.","C":"Incorrect. Substituting the stated values into the governing relation gives 1, not 3.","D":"Correct. The worked calculation gives 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_005878","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 242 K. What is its pressure at 415 K?","options":{"A":"9.8595 atm","B":"10.28926 atm","C":"1.68595 atm","D":"6.8595 atm"},"correct_answer":"D","correct_answer_value":"6.8595 atm","solution":"At constant volume, P/T is constant. P₂=P₁T₂/T₁=(4)(415)/(242)=6.8595 atm.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 6.8595 atm, not 9.8595 atm.","B":"Incorrect. Substituting the stated values into the governing relation gives 6.8595 atm, not 10.28926 atm.","C":"Incorrect. Substituting the stated values into the governing relation gives 6.8595 atm, not 1.68595 atm.","D":"Correct. The worked calculation gives 6.8595 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_009197","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=0.92 V, transfers 2 electrons, and has reaction quotient Q=10^2. Using E=E°−(0.0591/n)log₁₀Q, calculate Ecell.","options":{"A":"0.8609 V","B":"1.29135 V","C":"0.43045 V","D":"-0.1391 V"},"correct_answer":"A","correct_answer_value":"0.8609 V","solution":"Apply the Nernst equation: E=0.92−(0.0591/2)(2)=0.8609 V.","option_explanations":{"A":"Correct. The worked calculation gives 0.8609 V.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.8609 V, not 1.29135 V.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.8609 V, not 0.43045 V.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.8609 V, not -0.1391 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_005478","subject":"physics","topic":"electrostatics","subtopic":"capacitor_energy","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A 4 μF capacitor is charged to a potential difference of 36 V. Calculate the energy stored, expressed in μJ.","options":{"A":"260.2 μJ","B":"2592 μJ","C":"5184 μJ","D":"2590 μJ"},"correct_answer":"B","correct_answer_value":"2592 μJ","solution":"U=½CV². Since C=4 μF, U in μJ is ½×4×(36)²=2592 μJ.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 2592 μJ, not 260.2 μJ.","B":"Correct. The worked calculation gives 2592 μJ.","C":"Incorrect. Substituting the stated values into the governing relation gives 2592 μJ, not 5184 μJ.","D":"Incorrect. Substituting the stated values into the governing relation gives 2592 μJ, not 2590 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_007448","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":"16 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":"12.76264","B":"25.52528","C":"2.27626","D":"13.76264"},"correct_answer":"A","correct_answer_value":"12.76264","solution":"Excess OH⁻ moles=0.10×(60−16)/1000=0.004400 mol. Total volume=0.076 L, so [OH⁻]=0.057895 M, pOH=−log₁₀[OH⁻], and pH=14−pOH=12.76264.","option_explanations":{"A":"Correct. The worked calculation gives 12.76264.","B":"Incorrect. Substituting the stated values into the governing relation gives 12.76264, not 25.52528.","C":"Incorrect. Substituting the stated values into the governing relation gives 12.76264, not 2.27626.","D":"Incorrect. Substituting the stated values into the governing relation gives 12.76264, not 13.76264."},"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_002924","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, 0) and v=(1, 4, -3), calculate u·v.","options":{"A":"31","B":"33","C":"4.1","D":"23.25"},"correct_answer":"A","correct_answer_value":"31","solution":"The dot product is the sum of corresponding component products: (7)(1)+(6)(4)+(0)(-3)=31.","option_explanations":{"A":"Correct. The worked calculation gives 31.","B":"Incorrect. Substituting the stated values into the governing relation gives 31, not 33.","C":"Incorrect. Substituting the stated values into the governing relation gives 31, not 4.1.","D":"Incorrect. Substituting the stated values into the governing relation gives 31, not 23.25."},"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_007605","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":"40 mL of 0.10 M HCl is mixed with 85 mL of 0.10 M NaOH at 25°C. Assuming complete dissociation and additive volumes, find the final pH.","options":{"A":"6.27815","B":"11.5563","C":"10.5563","D":"12.5563"},"correct_answer":"D","correct_answer_value":"12.5563","solution":"Excess OH⁻ moles=0.10×(85−40)/1000=0.004500 mol. Total volume=0.125 L, so [OH⁻]=0.036000 M, pOH=−log₁₀[OH⁻], and pH=14−pOH=12.5563.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 12.5563, not 6.27815.","B":"Incorrect. Substituting the stated values into the governing relation gives 12.5563, not 11.5563.","C":"Incorrect. Substituting the stated values into the governing relation gives 12.5563, not 10.5563.","D":"Correct. The worked calculation gives 12.5563."},"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_008205","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 15315 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":"60.46648 g","B":"151.16619 g","C":"75.5831 g","D":"76.5831 g"},"correct_answer":"C","correct_answer_value":"75.5831 g","solution":"Cu²⁺+2e⁻→Cu, so z=2. By Faraday's law m=MIt/(zF)=63.5×15×15315/(2×96500)=75.5831 g.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 75.5831 g, not 60.46648 g.","B":"Incorrect. Substituting the stated values into the governing relation gives 75.5831 g, not 151.16619 g.","C":"Correct. The worked calculation gives 75.5831 g.","D":"Incorrect. Substituting the stated values into the governing relation gives 75.5831 g, not 76.5831 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_005412","subject":"physics","topic":"electrostatics","subtopic":"capacitor_energy","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A 73 μF capacitor is charged to a potential difference of 12 V. Calculate the energy stored, expressed in μJ.","options":{"A":"10512 μJ","B":"5256 μJ","C":"4204.8 μJ","D":"5254 μJ"},"correct_answer":"B","correct_answer_value":"5256 μJ","solution":"U=½CV². Since C=73 μF, U in μJ is ½×73×(12)²=5256 μJ.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 5256 μJ, not 10512 μJ.","B":"Correct. The worked calculation gives 5256 μJ.","C":"Incorrect. Substituting the stated values into the governing relation gives 5256 μJ, not 4204.8 μJ.","D":"Incorrect. Substituting the stated values into the governing relation gives 5256 μJ, not 5254 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_004942","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 3 Ω are connected in series; this combination is connected in parallel with a 13 Ω resistor across a 36 V ideal battery. Find the total current supplied by the battery.","options":{"A":"5.34066 A","B":"4.00549 A","C":"2.67033 A","D":"8.01099 A"},"correct_answer":"A","correct_answer_value":"5.34066 A","solution":"Series branch resistance=11+3=14 Ω. Parallel equivalent R_eq=(14×13)/(14+13)=6.74074 Ω. Total current I=V/R_eq=36/6.74074=5.34066 A.","option_explanations":{"A":"Correct. The worked calculation gives 5.34066 A.","B":"Incorrect. Substituting the stated values into the governing relation gives 5.34066 A, not 4.00549 A.","C":"Incorrect. Substituting the stated values into the governing relation gives 5.34066 A, not 2.67033 A.","D":"Incorrect. Substituting the stated values into the governing relation gives 5.34066 A, not 8.01099 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_007655","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 49 mL of 0.10 M NaOH at 25°C. Assuming complete dissociation and additive volumes, find the final pH.","options":{"A":"2.23888","B":"-0.76112","C":"1.11944","D":"1.67916"},"correct_answer":"A","correct_answer_value":"2.23888","solution":"Excess H⁺ moles=0.10×(55−49)/1000=0.000600 mol. Total volume=0.104 L, so [H⁺]=0.005769 M and pH=−log₁₀[H⁺]=2.23888.","option_explanations":{"A":"Correct. The worked calculation gives 2.23888.","B":"Incorrect. Substituting the stated values into the governing relation gives 2.23888, not -0.76112.","C":"Incorrect. Substituting the stated values into the governing relation gives 2.23888, not 1.11944.","D":"Incorrect. Substituting the stated values into the governing relation gives 2.23888, not 1.67916."},"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_002291","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, 3 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":"1.38","B":"0.19","C":"0.38","D":"0.285"},"correct_answer":"C","correct_answer_value":"0.38","solution":"Favourable pairs = C(10,2)+C(3,2)+C(12,2)=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 1.38.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.38, not 0.19.","C":"Correct. The worked calculation gives 0.38.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.38, not 0.285."},"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_005534","subject":"physics","topic":"electrostatics","subtopic":"capacitor_energy","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A 56 μF capacitor is charged to a potential difference of 11 V. Calculate the energy stored, expressed in μJ.","options":{"A":"3388 μJ","B":"2541 μJ","C":"1694 μJ","D":"339.8 μJ"},"correct_answer":"A","correct_answer_value":"3388 μJ","solution":"U=½CV². Since C=56 μF, U in μJ is ½×56×(11)²=3388 μJ.","option_explanations":{"A":"Correct. The worked calculation gives 3388 μJ.","B":"Incorrect. Substituting the stated values into the governing relation gives 3388 μJ, not 2541 μJ.","C":"Incorrect. Substituting the stated values into the governing relation gives 3388 μJ, not 1694 μJ.","D":"Incorrect. Substituting the stated values into the governing relation gives 3388 μJ, not 339.8 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_009357","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_3H_2N_1X_1. Treat X as the total number of monovalent halogen atoms.","options":{"A":"3","B":"0","C":"1.5","D":"1.3"},"correct_answer":"A","correct_answer_value":"3","solution":"Use DBE=(2C+2+N−H−X)/2. Substitution gives (2×3+2+1−2−1)/2=3.","option_explanations":{"A":"Correct. The worked calculation gives 3.","B":"Incorrect. Substituting the stated values into the governing relation gives 3, not 0.","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 1.3."},"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_006379","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 248 N m⁻¹ and oscillates horizontally without damping. What is the time period divided by π?","options":{"A":"0.35921 s","B":"-1.64079 s","C":"0.71842 s","D":"2.35921 s"},"correct_answer":"A","correct_answer_value":"0.35921 s","solution":"For a mass–spring oscillator, T=2π√(m/k). Therefore T/π=2√(8/248)=0.35921 s.","option_explanations":{"A":"Correct. The worked calculation gives 0.35921 s.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.35921 s, not -1.64079 s.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.35921 s, not 0.71842 s.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.35921 s, not 2.35921 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_009893","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(OH)_5]^-3, each OH ligand has charge −1. Determine the formal oxidation state of Pd.","options":{"A":"-1","B":"2","C":"4","D":"0"},"correct_answer":"B","correct_answer_value":"2","solution":"Let the formal oxidation state of Pd be x. Charge balance gives x+(5)(−1)=-3; hence x=-3+5=+2.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 2, not -1.","B":"Correct. The worked calculation gives 2.","C":"Incorrect. Substituting the stated values into the governing relation gives 2, not 4.","D":"Incorrect. Substituting the stated values into the governing relation gives 2, not 0."},"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_005969","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 2.7 atm at 429 K. What is its pressure at 460 K?","options":{"A":"2.17133 atm","B":"2.8951 atm","C":"3.8951 atm","D":"5.79021 atm"},"correct_answer":"B","correct_answer_value":"2.8951 atm","solution":"At constant volume, P/T is constant. P₂=P₁T₂/T₁=(2.7)(460)/(429)=2.8951 atm.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 2.8951 atm, not 2.17133 atm.","B":"Correct. The worked calculation gives 2.8951 atm.","C":"Incorrect. Substituting the stated values into the governing relation gives 2.8951 atm, not 3.8951 atm.","D":"Incorrect. Substituting the stated values into the governing relation gives 2.8951 atm, not 5.79021 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_005260","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 +5 μC and +7 μC are separated by 4 cm in vacuum. Using k=9×10⁹ N m² C⁻², find the magnitude of the electrostatic force.","options":{"A":"236.25 N","B":"98.4375 N","C":"196.875 N","D":"393.75 N"},"correct_answer":"C","correct_answer_value":"196.875 N","solution":"Convert charges to coulombs and distance to metres. F=k|q₁q₂|/r²=9×10⁹(5×10⁻⁶)(7×10⁻⁶)/(4×10⁻²)²=196.875 N.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 196.875 N, not 236.25 N.","B":"Incorrect. Substituting the stated values into the governing relation gives 196.875 N, not 98.4375 N.","C":"Correct. The worked calculation gives 196.875 N.","D":"Incorrect. Substituting the stated values into the governing relation gives 196.875 N, not 393.75 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_006503","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 1068 N m⁻¹ and oscillates horizontally without damping. What is the time period divided by π?","options":{"A":"0.27369 s","B":"0.54738 s","C":"1.27369 s","D":"0.20527 s"},"correct_answer":"A","correct_answer_value":"0.27369 s","solution":"For a mass–spring oscillator, T=2π√(m/k). Therefore T/π=2√(20/1068)=0.27369 s.","option_explanations":{"A":"Correct. The worked calculation gives 0.27369 s.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.27369 s, not 0.54738 s.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.27369 s, not 1.27369 s.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.27369 s, not 0.20527 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_009922","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)_3]^+3, each Cl ligand has charge −1. Determine the formal oxidation state of Pt.","options":{"A":"7","B":"9","C":"6","D":"-6"},"correct_answer":"C","correct_answer_value":"6","solution":"Let the formal oxidation state of Pt be x. Charge balance gives x+(3)(−1)=+3; hence x=3+3=+6.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 6, not 7.","B":"Incorrect. Substituting the stated values into the governing relation gives 6, not 9.","C":"Correct. The worked calculation gives 6.","D":"Incorrect. Substituting the stated values into the governing relation gives 6, not -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_004652","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 90 N m about its central axis. Find its angular acceleration.","options":{"A":"-1.2 rad s⁻²","B":"0.4 rad s⁻²","C":"-0.8 rad s⁻²","D":"0.8 rad s⁻²"},"correct_answer":"D","correct_answer_value":"0.8 rad s⁻²","solution":"For a solid disc, I=½MR²=½(9)(5)²=112.5 kg m². Using τ=Iα, α=90/112.5=0.8 rad s⁻².","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.8 rad s⁻², not -1.2 rad s⁻².","B":"Incorrect. Substituting the stated values into the governing relation gives 0.8 rad s⁻², not 0.4 rad s⁻².","C":"Incorrect. Substituting the stated values into the governing relation gives 0.8 rad s⁻², not -0.8 rad s⁻².","D":"Correct. The worked calculation gives 0.8 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_007785","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 54 mmol L⁻¹, what concentration remains after 140 minutes?","options":{"A":"0.42188 mmol L⁻¹","B":"2.42188 mmol L⁻¹","C":"3.42188 mmol L⁻¹","D":"1.04219 mmol L⁻¹"},"correct_answer":"A","correct_answer_value":"0.42188 mmol L⁻¹","solution":"140 minutes corresponds to 7 half-lives. The remaining concentration is [A]₀(1/2)ⁿ=54/2^7=0.42188 mmol L⁻¹.","option_explanations":{"A":"Correct. The worked calculation gives 0.42188 mmol L⁻¹.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.42188 mmol L⁻¹, not 2.42188 mmol L⁻¹.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.42188 mmol L⁻¹, not 3.42188 mmol L⁻¹.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.42188 mmol L⁻¹, not 1.04219 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_005952","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 195 K. What is its pressure at 711 K?","options":{"A":"2.20323 atm","B":"15.03231 atm","C":"12.03231 atm","D":"18.04846 atm"},"correct_answer":"C","correct_answer_value":"12.03231 atm","solution":"At constant volume, P/T is constant. P₂=P₁T₂/T₁=(3.3)(711)/(195)=12.03231 atm.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 12.03231 atm, not 2.20323 atm.","B":"Incorrect. Substituting the stated values into the governing relation gives 12.03231 atm, not 15.03231 atm.","C":"Correct. The worked calculation gives 12.03231 atm.","D":"Incorrect. Substituting the stated values into the governing relation gives 12.03231 atm, not 18.04846 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_003603","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 56 m s⁻¹ at 50° to the horizontal. Take g=10 m s⁻². Find its horizontal range.","options":{"A":"310.83571 m","B":"306.83571 m","C":"308.83571 m","D":"370.60285 m"},"correct_answer":"C","correct_answer_value":"308.83571 m","solution":"Range R=u²sin(2θ)/g. Substituting u=56, θ=50° and g=10 gives R=(56)²sin(100°)/10=308.83571 m.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 308.83571 m, not 310.83571 m.","B":"Incorrect. Substituting the stated values into the governing relation gives 308.83571 m, not 306.83571 m.","C":"Correct. The worked calculation gives 308.83571 m.","D":"Incorrect. Substituting the stated values into the governing relation gives 308.83571 m, not 370.60285 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_003330","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=(-14-13i)/(11+12i). What is Re(z), the real part of z?","options":{"A":"1.16981","B":"0.88302","C":"-1.16981","D":"-3.16981"},"correct_answer":"C","correct_answer_value":"-1.16981","solution":"Multiply numerator and denominator by the conjugate (11-12i). Then Re(z)=(ac+bd)/(c²+d²)=((-14)(11)+(-13)(12))/(11²+(12)²)=-1.16981.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives -1.16981, not 1.16981.","B":"Incorrect. Substituting the stated values into the governing relation gives -1.16981, not 0.88302.","C":"Correct. The worked calculation gives -1.16981.","D":"Incorrect. Substituting the stated values into the governing relation gives -1.16981, not -3.16981."},"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_007095","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 15 mol of gas at 402 K and 2 atm? Use R=0.082057 L atm mol⁻¹ K⁻¹.","options":{"A":"249.40186 L","B":"247.40186 L","C":"25.74019 L","D":"246.40186 L"},"correct_answer":"B","correct_answer_value":"247.40186 L","solution":"PV=nRT, so V=nRT/P=(15)(0.082057)(402)/(2)=247.40186 L.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 247.40186 L, not 249.40186 L.","B":"Correct. The worked calculation gives 247.40186 L.","C":"Incorrect. Substituting the stated values into the governing relation gives 247.40186 L, not 25.74019 L.","D":"Incorrect. Substituting the stated values into the governing relation gives 247.40186 L, not 246.40186 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_004648","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 34 N m about its central axis. Find its angular acceleration.","options":{"A":"1.0625 rad s⁻²","B":"2.125 rad s⁻²","C":"1.41667 rad s⁻²","D":"0.70833 rad s⁻²"},"correct_answer":"C","correct_answer_value":"1.41667 rad s⁻²","solution":"For a solid disc, I=½MR²=½(12)(2)²=24 kg m². Using τ=Iα, α=34/24=1.41667 rad s⁻².","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 1.41667 rad s⁻², not 1.0625 rad s⁻².","B":"Incorrect. Substituting the stated values into the governing relation gives 1.41667 rad s⁻², not 2.125 rad s⁻².","C":"Correct. The worked calculation gives 1.41667 rad s⁻².","D":"Incorrect. Substituting the stated values into the governing relation gives 1.41667 rad s⁻², not 0.70833 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_007848","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 53 mmol L⁻¹, what concentration remains after 140 minutes?","options":{"A":"0.20703 mmol L⁻¹","B":"-2.58594 mmol L⁻¹","C":"0.62109 mmol L⁻¹","D":"0.41406 mmol L⁻¹"},"correct_answer":"D","correct_answer_value":"0.41406 mmol L⁻¹","solution":"140 minutes corresponds to 7 half-lives. The remaining concentration is [A]₀(1/2)ⁿ=53/2^7=0.41406 mmol L⁻¹.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.41406 mmol L⁻¹, not 0.20703 mmol L⁻¹.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.41406 mmol L⁻¹, not -2.58594 mmol L⁻¹.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.41406 mmol L⁻¹, not 0.62109 mmol L⁻¹.","D":"Correct. The worked calculation gives 0.41406 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_006785","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 44 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":"27 mol","C":"-18 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 44 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 27 mol.","C":"Incorrect. Substituting the stated values into the governing relation gives 18 mol, not -18 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_007657","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":"69 mL of 0.10 M HCl is mixed with 22 mL of 0.10 M NaOH at 25°C. Assuming complete dissociation and additive volumes, find the final pH.","options":{"A":"-0.71306","B":"-1.71306","C":"-1.28694","D":"1.28694"},"correct_answer":"D","correct_answer_value":"1.28694","solution":"Excess H⁺ moles=0.10×(69−22)/1000=0.004700 mol. Total volume=0.091 L, so [H⁺]=0.051648 M and pH=−log₁₀[H⁺]=1.28694.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 1.28694, not -0.71306.","B":"Incorrect. Substituting the stated values into the governing relation gives 1.28694, not -1.71306.","C":"Incorrect. Substituting the stated values into the governing relation gives 1.28694, not -1.28694.","D":"Correct. The worked calculation gives 1.28694."},"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_008929","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 1.73 mol L⁻¹ and initially B is absent. At equilibrium, 32% of A has dissociated at constant volume. Find K_c.","options":{"A":"1.10421","B":"0.04207","C":"1.04207","D":"0.78155"},"correct_answer":"C","correct_answer_value":"1.04207","solution":"Let α=0.32. At equilibrium [A]=c₀(1−α) and [B]=2c₀α. Thus K_c=[B]²/[A]=4c₀α²/(1−α)=4(1.73)(0.32)²/(1−0.32)=1.04207.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 1.04207, not 1.10421.","B":"Incorrect. Substituting the stated values into the governing relation gives 1.04207, not 0.04207.","C":"Correct. The worked calculation gives 1.04207.","D":"Incorrect. Substituting the stated values into the governing relation gives 1.04207, not 0.78155."},"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_009753","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(CN)_5]^-2, each CN ligand has charge −1. Determine the formal oxidation state of Mn.","options":{"A":"-3","B":"3","C":"4.5","D":"4"},"correct_answer":"B","correct_answer_value":"3","solution":"Let the formal oxidation state of Mn be x. Charge balance gives x+(5)(−1)=-2; hence x=-2+5=+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 4.5.","D":"Incorrect. Substituting the stated values into the governing relation gives 3, 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_006133","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 57 cm in front of a thin convex lens of focal length 24 cm. Using the Cartesian sign convention, what is the magnitude of the linear magnification?","options":{"A":"0.72727","B":"2.72727","C":"3.72727","D":"1.72727"},"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)=(57)(24)/(57−24)=41.45455 cm. Thus |m|=v/u=41.45455/57=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 2.72727.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.72727, not 3.72727.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.72727, not 1.72727."},"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_004783","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 14 Ω and 2 Ω are connected in series; this combination is connected in parallel with a 7 Ω resistor across a 12 V ideal battery. Find the total current supplied by the battery.","options":{"A":"2.46429 A","B":"4.92857 A","C":"-0.53571 A","D":"4.46429 A"},"correct_answer":"A","correct_answer_value":"2.46429 A","solution":"Series branch resistance=14+2=16 Ω. Parallel equivalent R_eq=(16×7)/(16+7)=4.86957 Ω. Total current I=V/R_eq=12/4.86957=2.46429 A.","option_explanations":{"A":"Correct. The worked calculation gives 2.46429 A.","B":"Incorrect. Substituting the stated values into the governing relation gives 2.46429 A, not 4.92857 A.","C":"Incorrect. Substituting the stated values into the governing relation gives 2.46429 A, not -0.53571 A.","D":"Incorrect. Substituting the stated values into the governing relation gives 2.46429 A, not 4.46429 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_000195","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 -16x -8=0, let its roots be α and β. What is α²+β²?","options":{"A":"8.2","B":"72","C":"144","D":"57.6"},"correct_answer":"B","correct_answer_value":"72","solution":"By Vieta's relations, α+β = 16/2 and αβ = -8/2. Then α²+β²=(α+β)²−2αβ = (16/2)²−2(-8/2) = 72.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 72, not 8.2.","B":"Correct. The worked calculation gives 72.","C":"Incorrect. Substituting the stated values into the governing relation gives 72, not 144.","D":"Incorrect. Substituting the stated values into the governing relation gives 72, not 57.6."},"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_000448","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 8. Find the sum of its first 17 terms.","options":{"A":"1183.2","B":"493","C":"986","D":"1479"},"correct_answer":"C","correct_answer_value":"986","solution":"Use Sₙ = n/2[2a+(n−1)d]. Substituting a=-6, d=8, n=17: Sₙ=17/2[2(-6)+(17−1)(8)]=986.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 986, not 1183.2.","B":"Incorrect. Substituting the stated values into the governing relation gives 986, not 493.","C":"Correct. The worked calculation gives 986.","D":"Incorrect. Substituting the stated values into the governing relation gives 986, not 1479."},"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_009898","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]^-2, each CN ligand has charge −1. Determine the formal oxidation state of Pd.","options":{"A":"-4","B":"2","C":"4","D":"6"},"correct_answer":"C","correct_answer_value":"4","solution":"Let the formal oxidation state of Pd be x. Charge balance gives x+(6)(−1)=-2; hence x=-2+6=+4.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 4, not -4.","B":"Incorrect. Substituting the stated values into the governing relation gives 4, not 2.","C":"Correct. The worked calculation gives 4.","D":"Incorrect. Substituting the stated values into the governing relation gives 4, not 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_002691","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, -6, -6) and v=(-6, -2, 2), calculate u·v.","options":{"A":"-24","B":"12","C":"24","D":"23"},"correct_answer":"C","correct_answer_value":"24","solution":"The dot product is the sum of corresponding component products: (-4)(-6)+(-6)(-2)+(-6)(2)=24.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 24, not -24.","B":"Incorrect. Substituting the stated values into the governing relation gives 24, not 12.","C":"Correct. The worked calculation gives 24.","D":"Incorrect. Substituting the stated values into the governing relation gives 24, not 23."},"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_009435","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_5N_3. Treat X as the total number of monovalent halogen atoms.","options":{"A":"1.5","B":"5","C":"6","D":"10"},"correct_answer":"B","correct_answer_value":"5","solution":"Use DBE=(2C+2+N−H−X)/2. Substitution gives (2×5+2+3−5−0)/2=5.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 5, not 1.5.","B":"Correct. The worked calculation gives 5.","C":"Incorrect. Substituting the stated values into the governing relation gives 5, not 6.","D":"Incorrect. Substituting the stated values into the governing relation gives 5, not 10."},"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_000629","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 0 and common difference 10. Find the sum of its first 35 terms.","options":{"A":"4462.5","B":"4760","C":"596","D":"5950"},"correct_answer":"D","correct_answer_value":"5950","solution":"Use Sₙ = n/2[2a+(n−1)d]. Substituting a=0, d=10, n=35: Sₙ=35/2[2(0)+(35−1)(10)]=5950.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 5950, not 4462.5.","B":"Incorrect. Substituting the stated values into the governing relation gives 5950, not 4760.","C":"Incorrect. Substituting the stated values into the governing relation gives 5950, not 596.","D":"Correct. The worked calculation gives 5950."},"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_000338","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 -14 and common difference 15. Find the sum of its first 14 terms.","options":{"A":"1168","B":"1753.5","C":"2338","D":"1169"},"correct_answer":"D","correct_answer_value":"1169","solution":"Use Sₙ = n/2[2a+(n−1)d]. Substituting a=-14, d=15, n=14: Sₙ=14/2[2(-14)+(14−1)(15)]=1169.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 1169, not 1168.","B":"Incorrect. Substituting the stated values into the governing relation gives 1169, not 1753.5.","C":"Incorrect. Substituting the stated values into the governing relation gives 1169, not 2338.","D":"Correct. The worked calculation gives 1169."},"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_009602","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_0N_1X_3. Treat X as the total number of monovalent halogen atoms.","options":{"A":"20","B":"7","C":"13","D":"10"},"correct_answer":"D","correct_answer_value":"10","solution":"Use DBE=(2C+2+N−H−X)/2. Substitution gives (2×10+2+1−0−3)/2=10.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 10, not 20.","B":"Incorrect. Substituting the stated values into the governing relation gives 10, not 7.","C":"Incorrect. Substituting the stated values into the governing relation gives 10, not 13.","D":"Correct. The worked calculation gives 10."},"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_004884","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 12 Ω and 2 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":"1.88571 A","B":"1.48857 A","C":"-4.88571 A","D":"4.88571 A"},"correct_answer":"D","correct_answer_value":"4.88571 A","solution":"Series branch resistance=12+2=14 Ω. Parallel equivalent R_eq=(14×5)/(14+5)=3.68421 Ω. Total current I=V/R_eq=18/3.68421=4.88571 A.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 4.88571 A, not 1.88571 A.","B":"Incorrect. Substituting the stated values into the governing relation gives 4.88571 A, not 1.48857 A.","C":"Incorrect. Substituting the stated values into the governing relation gives 4.88571 A, not -4.88571 A.","D":"Correct. The worked calculation gives 4.88571 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_000333","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 -16x -10=0, let its roots be α and β. What is α²+β²?","options":{"A":"-35.11111","B":"35.11111","C":"70.22222","D":"37.11111"},"correct_answer":"B","correct_answer_value":"35.11111","solution":"By Vieta's relations, α+β = 16/3 and αβ = -10/3. Then α²+β²=(α+β)²−2αβ = (16/3)²−2(-10/3) = 35.11111.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 35.11111, not -35.11111.","B":"Correct. The worked calculation gives 35.11111.","C":"Incorrect. Substituting the stated values into the governing relation gives 35.11111, not 70.22222.","D":"Incorrect. Substituting the stated values into the governing relation gives 35.11111, not 37.11111."},"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_003946","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 2 m s⁻¹. A constant net force does 5 N of work over 19 m in the direction of motion. Find its final kinetic energy.","options":{"A":"157.5 J","B":"107 J","C":"105 J","D":"126 J"},"correct_answer":"C","correct_answer_value":"105 J","solution":"By the work–energy theorem, K_f=K_i+W=½mu²+Fs=½(5)(2)²+(5)(19)=105 J.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 105 J, not 157.5 J.","B":"Incorrect. Substituting the stated values into the governing relation gives 105 J, not 107 J.","C":"Correct. The worked calculation gives 105 J.","D":"Incorrect. Substituting the stated values into the governing relation gives 105 J, not 126 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_008504","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 0.46 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":"0.7176 K","B":"1.7176 K","C":"1.07176 K","D":"0.5382 K"},"correct_answer":"A","correct_answer_value":"0.7176 K","solution":"ΔT_b=iK_bm. Substitution gives ΔT_b=3×0.52×0.46=0.7176 K.","option_explanations":{"A":"Correct. The worked calculation gives 0.7176 K.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.7176 K, not 1.7176 K.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.7176 K, not 1.07176 K.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.7176 K, not 0.5382 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_004261","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 23 kg block slides down a 37° rough incline. The coefficient of kinetic friction is 0.33. Taking g=10 m s⁻², find the net force down the plane. Use sin37°=0.6 and cos37°=0.8.","options":{"A":"78.28 N","B":"57.96 N","C":"77.28 N","D":"79.28 N"},"correct_answer":"C","correct_answer_value":"77.28 N","solution":"Net force=mg(sinθ−μcosθ)=23×10×(0.6−0.33×0.8)=77.28 N.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 77.28 N, not 78.28 N.","B":"Incorrect. Substituting the stated values into the governing relation gives 77.28 N, not 57.96 N.","C":"Correct. The worked calculation gives 77.28 N.","D":"Incorrect. Substituting the stated values into the governing relation gives 77.28 N, not 79.28 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_007607","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 95 mL of 0.10 M NaOH at 25°C. Assuming complete dissociation and additive volumes, find the final pH.","options":{"A":"2.18288","B":"-11.82876","C":"11.82876","D":"8.82876"},"correct_answer":"C","correct_answer_value":"11.82876","solution":"Excess OH⁻ moles=0.10×(95−83)/1000=0.001200 mol. Total volume=0.178 L, so [OH⁻]=0.006742 M, pOH=−log₁₀[OH⁻], and pH=14−pOH=11.82876.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 11.82876, not 2.18288.","B":"Incorrect. Substituting the stated values into the governing relation gives 11.82876, not -11.82876.","C":"Correct. The worked calculation gives 11.82876.","D":"Incorrect. Substituting the stated values into the governing relation gives 11.82876, not 8.82876."},"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_004980","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 5 Ω are connected in series; this combination is connected in parallel with a 6 Ω resistor across a 24 V ideal battery. Find the total current supplied by the battery.","options":{"A":"3 A","B":"6 A","C":"12 A","D":"9 A"},"correct_answer":"B","correct_answer_value":"6 A","solution":"Series branch resistance=7+5=12 Ω. Parallel equivalent R_eq=(12×6)/(12+6)=4 Ω. Total current I=V/R_eq=24/4=6 A.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 6 A, not 3 A.","B":"Correct. The worked calculation gives 6 A.","C":"Incorrect. Substituting the stated values into the governing relation gives 6 A, not 12 A.","D":"Incorrect. Substituting the stated values into the governing relation gives 6 A, not 9 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_008220","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 22276 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":"58.6332 g","B":"56.6332 g","C":"117.2664 g","D":"46.90656 g"},"correct_answer":"A","correct_answer_value":"58.6332 g","solution":"Cu²⁺+2e⁻→Cu, so z=2. By Faraday's law m=MIt/(zF)=63.5×8×22276/(2×96500)=58.6332 g.","option_explanations":{"A":"Correct. The worked calculation gives 58.6332 g.","B":"Incorrect. Substituting the stated values into the governing relation gives 58.6332 g, not 56.6332 g.","C":"Incorrect. Substituting the stated values into the governing relation gives 58.6332 g, not 117.2664 g.","D":"Incorrect. Substituting the stated values into the governing relation gives 58.6332 g, not 46.90656 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_002019","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, 6 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":"-0.33333","B":"2.33333","C":"0.33333","D":"-1.66667"},"correct_answer":"C","correct_answer_value":"0.33333","solution":"Favourable pairs = C(10,2)+C(6,2)+C(5,2)=70. Total pairs=C(21,2)=210. Therefore P=70/210=0.33333.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.33333, not -0.33333.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.33333, not 2.33333.","C":"Correct. The worked calculation gives 0.33333.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.33333, not -1.66667."},"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_002253","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, 3 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":"0.36364","B":"0.27273","C":"0.72727","D":"1.03636"},"correct_answer":"A","correct_answer_value":"0.36364","solution":"Favourable pairs = C(9,2)+C(3,2)+C(10,2)=84. Total pairs=C(22,2)=231. Therefore P=84/231=0.36364.","option_explanations":{"A":"Correct. The worked calculation gives 0.36364.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.36364, not 0.27273.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.36364, not 0.72727.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.36364, not 1.03636."},"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_006873","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 39 mol H₂ and 32 mol O₂ are mixed and the reaction goes to completion, how many moles of H₂O form?","options":{"A":"40 mol","B":"39 mol","C":"46.8 mol","D":"31.2 mol"},"correct_answer":"B","correct_answer_value":"39 mol","solution":"From 2 mol H₂ and 1 mol O₂, 2 mol H₂O form. H₂ can produce 39 mol H₂O; O₂ can produce 64 mol H₂O. The limiting condition is H₂; product=39 mol.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 39 mol, not 40 mol.","B":"Correct. The worked calculation gives 39 mol.","C":"Incorrect. Substituting the stated values into the governing relation gives 39 mol, not 46.8 mol.","D":"Incorrect. Substituting the stated values into the governing relation gives 39 mol, not 31.2 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_005264","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 6 cm in vacuum. Using k=9×10⁹ N m² C⁻², find the magnitude of the electrostatic force.","options":{"A":"67.5 N","B":"89 N","C":"90 N","D":"108 N"},"correct_answer":"C","correct_answer_value":"90 N","solution":"Convert charges to coulombs and distance to metres. F=k|q₁q₂|/r²=9×10⁹(6×10⁻⁶)(6×10⁻⁶)/(6×10⁻²)²=90 N.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 90 N, not 67.5 N.","B":"Incorrect. Substituting the stated values into the governing relation gives 90 N, not 89 N.","C":"Correct. The worked calculation gives 90 N.","D":"Incorrect. Substituting the stated values into the governing relation gives 90 N, not 108 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_006211","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 22 cm. Using the Cartesian sign convention, what is the magnitude of the linear magnification?","options":{"A":"3.36667","B":"1.03667","C":"0.73333","D":"0.36667"},"correct_answer":"D","correct_answer_value":"0.36667","solution":"Using distances as positive magnitudes, 1/f=1/v+1/u gives v=uf/(u−f)=(82)(22)/(82−22)=30.06667 cm. Thus |m|=v/u=30.06667/82=0.36667.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.36667, not 3.36667.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.36667, not 1.03667.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.36667, not 0.73333.","D":"Correct. The worked calculation gives 0.36667."},"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_002564","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], [8, 0]].","options":{"A":"62","B":"64","C":"76.8","D":"63"},"correct_answer":"B","correct_answer_value":"64","solution":"For a 2×2 matrix [[a,b],[c,d]], det=ad−bc. Here it is (1)(0)−(-8)(8)=64.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 64, not 62.","B":"Correct. The worked calculation gives 64.","C":"Incorrect. Substituting the stated values into the governing relation gives 64, not 76.8.","D":"Incorrect. Substituting the stated values into the governing relation gives 64, not 63."},"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_000097","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 +17x +1=0, let its roots be α and β. What is α²+β²?","options":{"A":"287","B":"574","C":"288","D":"430.5"},"correct_answer":"A","correct_answer_value":"287","solution":"By Vieta's relations, α+β = -17/1 and αβ = 1/1. Then α²+β²=(α+β)²−2αβ = (-17/1)²−2(1/1) = 287.","option_explanations":{"A":"Correct. The worked calculation gives 287.","B":"Incorrect. Substituting the stated values into the governing relation gives 287, not 574.","C":"Incorrect. Substituting the stated values into the governing relation gives 287, not 288.","D":"Incorrect. Substituting the stated values into the governing relation gives 287, not 430.5."},"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_003607","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 52 m s⁻¹ at 18° to the horizontal. Take g=10 m s⁻². Find its horizontal range.","options":{"A":"157.93713 m","B":"156.93713 m","C":"238.4057 m","D":"158.93713 m"},"correct_answer":"D","correct_answer_value":"158.93713 m","solution":"Range R=u²sin(2θ)/g. Substituting u=52, θ=18° and g=10 gives R=(52)²sin(36°)/10=158.93713 m.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 158.93713 m, not 157.93713 m.","B":"Incorrect. Substituting the stated values into the governing relation gives 158.93713 m, not 156.93713 m.","C":"Incorrect. Substituting the stated values into the governing relation gives 158.93713 m, not 238.4057 m.","D":"Correct. The worked calculation gives 158.93713 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_001615","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 (-4x² +7x -1) dx.","options":{"A":"1.33333","B":"-1.66667","C":"0.33333","D":"-0.66667"},"correct_answer":"A","correct_answer_value":"1.33333","solution":"An antiderivative is (-4/3)x³+(7/2)x²+(-1)x. Evaluating between 0 and 2 gives (-4/3)(2)³+(7/2)(2)²+(-1)(2)=1.33333.","option_explanations":{"A":"Correct. The worked calculation gives 1.33333.","B":"Incorrect. Substituting the stated values into the governing relation gives 1.33333, not -1.66667.","C":"Incorrect. Substituting the stated values into the governing relation gives 1.33333, not 0.33333.","D":"Incorrect. Substituting the stated values into the governing relation gives 1.33333, not -0.66667."},"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_001115","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 +8x² +2x+7. Evaluate f′(1).","options":{"A":"22.5","B":"15","C":"-30","D":"30"},"correct_answer":"D","correct_answer_value":"30","solution":"Differentiate: f′(x)=3(4)x²+2(8)x+(2). At x=1, f′(1)=3(4)(1)²+2(8)(1)+(2)=30.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 30, not 22.5.","B":"Incorrect. Substituting the stated values into the governing relation gives 30, not 15.","C":"Incorrect. Substituting the stated values into the governing relation gives 30, not -30.","D":"Correct. The worked calculation gives 30."},"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_005211","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 +1 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":"2.10938 N","B":"-1.59375 N","C":"1.40625 N","D":"4.40625 N"},"correct_answer":"C","correct_answer_value":"1.40625 N","solution":"Convert charges to coulombs and distance to metres. F=k|q₁q₂|/r²=9×10⁹(1×10⁻⁶)(1×10⁻⁶)/(8×10⁻²)²=1.40625 N.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 1.40625 N, not 2.10938 N.","B":"Incorrect. Substituting the stated values into the governing relation gives 1.40625 N, not -1.59375 N.","C":"Correct. The worked calculation gives 1.40625 N.","D":"Incorrect. Substituting the stated values into the governing relation gives 1.40625 N, not 4.40625 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_003725","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 11 m s⁻¹. A constant net force does 11 N of work over 14 m in the direction of motion. Find its final kinetic energy.","options":{"A":"550 J","B":"330 J","C":"-275 J","D":"275 J"},"correct_answer":"D","correct_answer_value":"275 J","solution":"By the work–energy theorem, K_f=K_i+W=½mu²+Fs=½(2)(11)²+(11)(14)=275 J.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 275 J, not 550 J.","B":"Incorrect. Substituting the stated values into the governing relation gives 275 J, not 330 J.","C":"Incorrect. Substituting the stated values into the governing relation gives 275 J, not -275 J.","D":"Correct. The worked calculation gives 275 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_001104","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)=0x³ +1x² -8x+7. Evaluate f′(2).","options":{"A":"-5","B":"-4","C":"-8","D":"-6"},"correct_answer":"B","correct_answer_value":"-4","solution":"Differentiate: f′(x)=3(0)x²+2(1)x+(-8). At x=2, f′(2)=3(0)(2)²+2(1)(2)+(-8)=-4.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives -4, not -5.","B":"Correct. The worked calculation gives -4.","C":"Incorrect. Substituting the stated values into the governing relation gives -4, not -8.","D":"Incorrect. Substituting the stated values into the governing relation gives -4, not -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_001332","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)=0x³ +8x² +10x+7. Evaluate f′(-4).","options":{"A":"-52","B":"-4.4","C":"-54","D":"-43.2"},"correct_answer":"C","correct_answer_value":"-54","solution":"Differentiate: f′(x)=3(0)x²+2(8)x+(10). At x=-4, f′(-4)=3(0)(-4)²+2(8)(-4)+(10)=-54.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives -54, not -52.","B":"Incorrect. Substituting the stated values into the governing relation gives -54, not -4.4.","C":"Correct. The worked calculation gives -54.","D":"Incorrect. Substituting the stated values into the governing relation gives -54, not -43.2."},"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_007567","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":"29 mL of 0.10 M HCl is mixed with 79 mL of 0.10 M NaOH at 25°C. Assuming complete dissociation and additive volumes, find the final pH.","options":{"A":"15.66555","B":"13.66555","C":"12.66555","D":"-12.66555"},"correct_answer":"C","correct_answer_value":"12.66555","solution":"Excess OH⁻ moles=0.10×(79−29)/1000=0.005000 mol. Total volume=0.108 L, so [OH⁻]=0.046296 M, pOH=−log₁₀[OH⁻], and pH=14−pOH=12.66555.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 12.66555, not 15.66555.","B":"Incorrect. Substituting the stated values into the governing relation gives 12.66555, not 13.66555.","C":"Correct. The worked calculation gives 12.66555.","D":"Incorrect. Substituting the stated values into the governing relation gives 12.66555, not -12.66555."},"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_005561","subject":"physics","topic":"electrostatics","subtopic":"capacitor_energy","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A 22 μF capacitor is charged to a potential difference of 12 V. Calculate the energy stored, expressed in μJ.","options":{"A":"1583 μJ","B":"1584 μJ","C":"1582 μJ","D":"1267.2 μJ"},"correct_answer":"B","correct_answer_value":"1584 μJ","solution":"U=½CV². Since C=22 μF, U in μJ is ½×22×(12)²=1584 μJ.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 1584 μJ, not 1583 μJ.","B":"Correct. The worked calculation gives 1584 μJ.","C":"Incorrect. Substituting the stated values into the governing relation gives 1584 μJ, not 1582 μJ.","D":"Incorrect. Substituting the stated values into the governing relation gives 1584 μJ, not 1267.2 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_009988","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 [Co(Cl)_4]^+1, each Cl ligand has charge −1. Determine the formal oxidation state of Co.","options":{"A":"7","B":"1.5","C":"6","D":"5"},"correct_answer":"D","correct_answer_value":"5","solution":"Let the formal oxidation state of Co be x. Charge balance gives x+(4)(−1)=+1; hence x=1+4=+5.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 5, not 7.","B":"Incorrect. Substituting the stated values into the governing relation gives 5, not 1.5.","C":"Incorrect. Substituting the stated values into the governing relation gives 5, not 6.","D":"Correct. The worked calculation gives 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_000054","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 -12x +1=0, let its roots be α and β. What is α²+β²?","options":{"A":"142","B":"106.5","C":"144","D":"140"},"correct_answer":"A","correct_answer_value":"142","solution":"By Vieta's relations, α+β = 12/1 and αβ = 1/1. Then α²+β²=(α+β)²−2αβ = (12/1)²−2(1/1) = 142.","option_explanations":{"A":"Correct. The worked calculation gives 142.","B":"Incorrect. Substituting the stated values into the governing relation gives 142, not 106.5.","C":"Incorrect. Substituting the stated values into the governing relation gives 142, not 144.","D":"Incorrect. Substituting the stated values into the governing relation gives 142, not 140."},"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_004593","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 6 m is acted on by a constant torque of 16 N m about its central axis. Find its angular acceleration.","options":{"A":"0.11111 rad s⁻²","B":"-0.77778 rad s⁻²","C":"-0.22222 rad s⁻²","D":"0.22222 rad s⁻²"},"correct_answer":"D","correct_answer_value":"0.22222 rad s⁻²","solution":"For a solid disc, I=½MR²=½(4)(6)²=72 kg m². Using τ=Iα, α=16/72=0.22222 rad s⁻².","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.22222 rad s⁻², not 0.11111 rad s⁻².","B":"Incorrect. Substituting the stated values into the governing relation gives 0.22222 rad s⁻², not -0.77778 rad s⁻².","C":"Incorrect. Substituting the stated values into the governing relation gives 0.22222 rad s⁻², not -0.22222 rad s⁻².","D":"Correct. The worked calculation gives 0.22222 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_003381","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 41 m s⁻¹ at 75° to the horizontal. Take g=10 m s⁻². Find its horizontal range.","options":{"A":"83.05 m","B":"84.05 m","C":"42.025 m","D":"9.405 m"},"correct_answer":"B","correct_answer_value":"84.05 m","solution":"Range R=u²sin(2θ)/g. Substituting u=41, θ=75° and g=10 gives R=(41)²sin(150°)/10=84.05 m.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 84.05 m, not 83.05 m.","B":"Correct. The worked calculation gives 84.05 m.","C":"Incorrect. Substituting the stated values into the governing relation gives 84.05 m, not 42.025 m.","D":"Incorrect. Substituting the stated values into the governing relation gives 84.05 m, not 9.405 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_005508","subject":"physics","topic":"electrostatics","subtopic":"capacitor_energy","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A 65 μF capacitor is charged to a potential difference of 7 V. Calculate the energy stored, expressed in μJ.","options":{"A":"1590.5 μJ","B":"1592.5 μJ","C":"160.25 μJ","D":"3185 μJ"},"correct_answer":"B","correct_answer_value":"1592.5 μJ","solution":"U=½CV². Since C=65 μF, U in μJ is ½×65×(7)²=1592.5 μJ.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 1592.5 μJ, not 1590.5 μJ.","B":"Correct. The worked calculation gives 1592.5 μJ.","C":"Incorrect. Substituting the stated values into the governing relation gives 1592.5 μJ, not 160.25 μJ.","D":"Incorrect. Substituting the stated values into the governing relation gives 1592.5 μJ, not 3185 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_004529","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 5 m is acted on by a constant torque of 45 N m about its central axis. Find its angular acceleration.","options":{"A":"1.08 rad s⁻²","B":"0.54 rad s⁻²","C":"-2.28 rad s⁻²","D":"0.72 rad s⁻²"},"correct_answer":"D","correct_answer_value":"0.72 rad s⁻²","solution":"For a solid disc, I=½MR²=½(5)(5)²=62.5 kg m². Using τ=Iα, α=45/62.5=0.72 rad s⁻².","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.72 rad s⁻², not 1.08 rad s⁻².","B":"Incorrect. Substituting the stated values into the governing relation gives 0.72 rad s⁻², not 0.54 rad s⁻².","C":"Incorrect. Substituting the stated values into the governing relation gives 0.72 rad s⁻², not -2.28 rad s⁻².","D":"Correct. The worked calculation gives 0.72 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_006122","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 47 cm in front of a thin convex lens of focal length 28 cm. Using the Cartesian sign convention, what is the magnitude of the linear magnification?","options":{"A":"1.47368","B":"2.47368","C":"4.47368","D":"2.94737"},"correct_answer":"A","correct_answer_value":"1.47368","solution":"Using distances as positive magnitudes, 1/f=1/v+1/u gives v=uf/(u−f)=(47)(28)/(47−28)=69.26316 cm. Thus |m|=v/u=69.26316/47=1.47368.","option_explanations":{"A":"Correct. The worked calculation gives 1.47368.","B":"Incorrect. Substituting the stated values into the governing relation gives 1.47368, not 2.47368.","C":"Incorrect. Substituting the stated values into the governing relation gives 1.47368, not 4.47368.","D":"Incorrect. Substituting the stated values into the governing relation gives 1.47368, not 2.94737."},"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_002964","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, 2, 6) and v=(-5, 7, -1), calculate u·v.","options":{"A":"-1","B":"-2","C":"2","D":"0.8"},"correct_answer":"B","correct_answer_value":"-2","solution":"The dot product is the sum of corresponding component products: (2)(-5)+(2)(7)+(6)(-1)=-2.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives -2, not -1.","B":"Correct. The worked calculation gives -2.","C":"Incorrect. Substituting the stated values into the governing relation gives -2, not 2.","D":"Incorrect. Substituting the stated values into the governing relation gives -2, not 0.8."},"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_004142","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 0.34. Taking g=10 m s⁻², find the net force down the plane. Use sin37°=0.6 and cos37°=0.8.","options":{"A":"66.88 N","B":"137.76 N","C":"7.888 N","D":"68.88 N"},"correct_answer":"D","correct_answer_value":"68.88 N","solution":"Net force=mg(sinθ−μcosθ)=21×10×(0.6−0.34×0.8)=68.88 N.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 68.88 N, not 66.88 N.","B":"Incorrect. Substituting the stated values into the governing relation gives 68.88 N, not 137.76 N.","C":"Incorrect. Substituting the stated values into the governing relation gives 68.88 N, not 7.888 N.","D":"Correct. The worked calculation gives 68.88 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_003291","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=(10+9i)/(2+2i). What is Re(z), the real part of z?","options":{"A":"5.75","B":"3.75","C":"4.75","D":"-4.75"},"correct_answer":"C","correct_answer_value":"4.75","solution":"Multiply numerator and denominator by the conjugate (2-2i). Then Re(z)=(ac+bd)/(c²+d²)=((10)(2)+(9)(2))/(2²+(2)²)=4.75.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 4.75, not 5.75.","B":"Incorrect. Substituting the stated values into the governing relation gives 4.75, not 3.75.","C":"Correct. The worked calculation gives 4.75.","D":"Incorrect. Substituting the stated values into the governing relation gives 4.75, not -4.75."},"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_005280","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 +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":"3.3 N","B":"9.45 N","C":"6.3 N","D":"7.3 N"},"correct_answer":"C","correct_answer_value":"6.3 N","solution":"Convert charges to coulombs and distance to metres. F=k|q₁q₂|/r²=9×10⁹(1×10⁻⁶)(7×10⁻⁶)/(10×10⁻²)²=6.3 N.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 6.3 N, not 3.3 N.","B":"Incorrect. Substituting the stated values into the governing relation gives 6.3 N, not 9.45 N.","C":"Correct. The worked calculation gives 6.3 N.","D":"Incorrect. Substituting the stated values into the governing relation gives 6.3 N, not 7.3 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_006910","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 21 mol H₂ and 19 mol O₂ are mixed and the reaction goes to completion, how many moles of H₂O form?","options":{"A":"-21 mol","B":"31.5 mol","C":"21 mol","D":"19 mol"},"correct_answer":"C","correct_answer_value":"21 mol","solution":"From 2 mol H₂ and 1 mol O₂, 2 mol H₂O form. H₂ can produce 21 mol H₂O; O₂ can produce 38 mol H₂O. The limiting condition is H₂; product=21 mol.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 21 mol, not -21 mol.","B":"Incorrect. Substituting the stated values into the governing relation gives 21 mol, not 31.5 mol.","C":"Correct. The worked calculation gives 21 mol.","D":"Incorrect. Substituting the stated values into the governing relation gives 21 mol, not 19 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_002694","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, 1, -2) and v=(9, 8, -2), calculate u·v.","options":{"A":"168","B":"84","C":"42","D":"-84"},"correct_answer":"B","correct_answer_value":"84","solution":"The dot product is the sum of corresponding component products: (8)(9)+(1)(8)+(-2)(-2)=84.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 84, not 168.","B":"Correct. The worked calculation gives 84.","C":"Incorrect. Substituting the stated values into the governing relation gives 84, not 42.","D":"Incorrect. Substituting the stated values into the governing relation gives 84, not -84."},"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_008506","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 1.98 mol kg⁻¹. If its van't Hoff factor is 2 and K_b for water is 0.6 K kg mol⁻¹, find the boiling-point elevation.","options":{"A":"2.376 K","B":"1.188 K","C":"-0.624 K","D":"0.376 K"},"correct_answer":"A","correct_answer_value":"2.376 K","solution":"ΔT_b=iK_bm. Substitution gives ΔT_b=2×0.6×1.98=2.376 K.","option_explanations":{"A":"Correct. The worked calculation gives 2.376 K.","B":"Incorrect. Substituting the stated values into the governing relation gives 2.376 K, not 1.188 K.","C":"Incorrect. Substituting the stated values into the governing relation gives 2.376 K, not -0.624 K.","D":"Incorrect. Substituting the stated values into the governing relation gives 2.376 K, not 0.376 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_001606","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 (3x² +6x +0) dx.","options":{"A":"322","B":"324","C":"323","D":"162"},"correct_answer":"B","correct_answer_value":"324","solution":"An antiderivative is (3/3)x³+(6/2)x²+(0)x. Evaluating between 0 and 6 gives (3/3)(6)³+(6/2)(6)²+(0)(6)=324.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 324, not 322.","B":"Correct. The worked calculation gives 324.","C":"Incorrect. Substituting the stated values into the governing relation gives 324, not 323.","D":"Incorrect. Substituting the stated values into the governing relation gives 324, not 162."},"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_005988","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 2.9 atm at 291 K. What is its pressure at 753 K?","options":{"A":"8.50412 atm","B":"10.50412 atm","C":"4.50412 atm","D":"7.50412 atm"},"correct_answer":"D","correct_answer_value":"7.50412 atm","solution":"At constant volume, P/T is constant. P₂=P₁T₂/T₁=(2.9)(753)/(291)=7.50412 atm.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 7.50412 atm, not 8.50412 atm.","B":"Incorrect. Substituting the stated values into the governing relation gives 7.50412 atm, not 10.50412 atm.","C":"Incorrect. Substituting the stated values into the governing relation gives 7.50412 atm, not 4.50412 atm.","D":"Correct. The worked calculation gives 7.50412 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_004132","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 35 kg block slides down a 37° rough incline. The coefficient of kinetic friction is 0.21. Taking g=10 m s⁻², find the net force down the plane. Use sin37°=0.6 and cos37°=0.8.","options":{"A":"150.2 N","B":"120.96 N","C":"151.2 N","D":"226.8 N"},"correct_answer":"C","correct_answer_value":"151.2 N","solution":"Net force=mg(sinθ−μcosθ)=35×10×(0.6−0.21×0.8)=151.2 N.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 151.2 N, not 150.2 N.","B":"Incorrect. Substituting the stated values into the governing relation gives 151.2 N, not 120.96 N.","C":"Correct. The worked calculation gives 151.2 N.","D":"Incorrect. Substituting the stated values into the governing relation gives 151.2 N, not 226.8 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_000913","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 4. What is the sum of its first 10 terms?","options":{"A":"5592400","B":"3355440","C":"2796198","D":"2796200"},"correct_answer":"D","correct_answer_value":"2796200","solution":"For r≠1, Sₙ=a(rⁿ−1)/(r−1). Thus Sₙ=8(4^10−1)/(4−1)=2796200.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 2796200, not 5592400.","B":"Incorrect. Substituting the stated values into the governing relation gives 2796200, not 3355440.","C":"Incorrect. Substituting the stated values into the governing relation gives 2796200, not 2796198.","D":"Correct. The worked calculation gives 2796200."},"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_000482","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 -8 and common difference 9. Find the sum of its first 18 terms.","options":{"A":"-1233","B":"1233","C":"2466","D":"124.3"},"correct_answer":"B","correct_answer_value":"1233","solution":"Use Sₙ = n/2[2a+(n−1)d]. Substituting a=-8, d=9, n=18: Sₙ=18/2[2(-8)+(18−1)(9)]=1233.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 1233, not -1233.","B":"Correct. The worked calculation gives 1233.","C":"Incorrect. Substituting the stated values into the governing relation gives 1233, not 2466.","D":"Incorrect. Substituting the stated values into the governing relation gives 1233, not 124.3."},"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_000524","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 6. Find the sum of its first 35 terms.","options":{"A":"4025","B":"4830","C":"-4025","D":"3220"},"correct_answer":"A","correct_answer_value":"4025","solution":"Use Sₙ = n/2[2a+(n−1)d]. Substituting a=13, d=6, n=35: Sₙ=35/2[2(13)+(35−1)(6)]=4025.","option_explanations":{"A":"Correct. The worked calculation gives 4025.","B":"Incorrect. Substituting the stated values into the governing relation gives 4025, not 4830.","C":"Incorrect. Substituting the stated values into the governing relation gives 4025, not -4025.","D":"Incorrect. Substituting the stated values into the governing relation gives 4025, not 3220."},"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_002491","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, -8], [3, 7]].","options":{"A":"87","B":"85","C":"43.5","D":"65.25"},"correct_answer":"A","correct_answer_value":"87","solution":"For a 2×2 matrix [[a,b],[c,d]], det=ad−bc. Here it is (9)(7)−(-8)(3)=87.","option_explanations":{"A":"Correct. The worked calculation gives 87.","B":"Incorrect. Substituting the stated values into the governing relation gives 87, not 85.","C":"Incorrect. Substituting the stated values into the governing relation gives 87, not 43.5.","D":"Incorrect. Substituting the stated values into the governing relation gives 87, not 65.25."},"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_007637","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":"33 mL of 0.10 M HCl is mixed with 36 mL of 0.10 M NaOH at 25°C. Assuming complete dissociation and additive volumes, find the final pH.","options":{"A":"10.63827","B":"11.63827","C":"2.16383","D":"5.81914"},"correct_answer":"B","correct_answer_value":"11.63827","solution":"Excess OH⁻ moles=0.10×(36−33)/1000=0.000300 mol. Total volume=0.069 L, so [OH⁻]=0.004348 M, pOH=−log₁₀[OH⁻], and pH=14−pOH=11.63827.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 11.63827, not 10.63827.","B":"Correct. The worked calculation gives 11.63827.","C":"Incorrect. Substituting the stated values into the governing relation gives 11.63827, not 2.16383.","D":"Incorrect. Substituting the stated values into the governing relation gives 11.63827, not 5.81914."},"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_001665","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 +7x +5) dx.","options":{"A":"18.66667","B":"22.4","C":"14.93333","D":"16.66667"},"correct_answer":"A","correct_answer_value":"18.66667","solution":"An antiderivative is (-2/3)x³+(7/2)x²+(5)x. Evaluating between 0 and 2 gives (-2/3)(2)³+(7/2)(2)²+(5)(2)=18.66667.","option_explanations":{"A":"Correct. The worked calculation gives 18.66667.","B":"Incorrect. Substituting the stated values into the governing relation gives 18.66667, not 22.4.","C":"Incorrect. Substituting the stated values into the governing relation gives 18.66667, not 14.93333.","D":"Incorrect. Substituting the stated values into the governing relation gives 18.66667, not 16.66667."},"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_004550","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 44 N m about its central axis. Find its angular acceleration.","options":{"A":"1.12222 rad s⁻²","B":"2.22222 rad s⁻²","C":"1.22222 rad s⁻²","D":"0.61111 rad s⁻²"},"correct_answer":"C","correct_answer_value":"1.22222 rad s⁻²","solution":"For a solid disc, I=½MR²=½(2)(6)²=36 kg m². Using τ=Iα, α=44/36=1.22222 rad s⁻².","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 1.22222 rad s⁻², not 1.12222 rad s⁻².","B":"Incorrect. Substituting the stated values into the governing relation gives 1.22222 rad s⁻², not 2.22222 rad s⁻².","C":"Correct. The worked calculation gives 1.22222 rad s⁻².","D":"Incorrect. Substituting the stated values into the governing relation gives 1.22222 rad s⁻², not 0.61111 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_007512","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":"45 mL of 0.10 M HCl is mixed with 61 mL of 0.10 M NaOH at 25°C. Assuming complete dissociation and additive volumes, find the final pH.","options":{"A":"15.17881","B":"12.17881","C":"11.17881","D":"13.17881"},"correct_answer":"B","correct_answer_value":"12.17881","solution":"Excess OH⁻ moles=0.10×(61−45)/1000=0.001600 mol. Total volume=0.106 L, so [OH⁻]=0.015094 M, pOH=−log₁₀[OH⁻], and pH=14−pOH=12.17881.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 12.17881, not 15.17881.","B":"Correct. The worked calculation gives 12.17881.","C":"Incorrect. Substituting the stated values into the governing relation gives 12.17881, not 11.17881.","D":"Incorrect. Substituting the stated values into the governing relation gives 12.17881, not 13.17881."},"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_001041","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)=0x³ +10x² +8x+7. Evaluate f′(-2).","options":{"A":"-32","B":"-48","C":"-31","D":"32"},"correct_answer":"A","correct_answer_value":"-32","solution":"Differentiate: f′(x)=3(0)x²+2(10)x+(8). At x=-2, f′(-2)=3(0)(-2)²+2(10)(-2)+(8)=-32.","option_explanations":{"A":"Correct. The worked calculation gives -32.","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 -31.","D":"Incorrect. Substituting the stated values into the governing relation gives -32, not 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_003174","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=(10-5i)/(11+9i). What is Re(z), the real part of z?","options":{"A":"0.48267","B":"0.24134","C":"2.32178","D":"0.32178"},"correct_answer":"D","correct_answer_value":"0.32178","solution":"Multiply numerator and denominator by the conjugate (11-9i). Then Re(z)=(ac+bd)/(c²+d²)=((10)(11)+(-5)(9))/(11²+(9)²)=0.32178.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.32178, not 0.48267.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.32178, not 0.24134.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.32178, not 2.32178.","D":"Correct. The worked calculation gives 0.32178."},"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_009371","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_12N_1X_3. Treat X as the total number of monovalent halogen atoms.","options":{"A":"0","B":"3","C":"2","D":"4"},"correct_answer":"C","correct_answer_value":"2","solution":"Use DBE=(2C+2+N−H−X)/2. Substitution gives (2×8+2+1−12−3)/2=2.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 2, not 0.","B":"Incorrect. Substituting the stated values into the governing relation gives 2, not 3.","C":"Correct. The worked calculation gives 2.","D":"Incorrect. Substituting the stated values into the governing relation gives 2, 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_009026","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=0.56 V, transfers 1 electrons, and has reaction quotient Q=10^1. Using E=E°−(0.0591/n)log₁₀Q, calculate Ecell.","options":{"A":"0.5009 V","B":"0.75135 V","C":"-1.4991 V","D":"3.5009 V"},"correct_answer":"A","correct_answer_value":"0.5009 V","solution":"Apply the Nernst equation: E=0.56−(0.0591/1)(1)=0.5009 V.","option_explanations":{"A":"Correct. The worked calculation gives 0.5009 V.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.5009 V, not 0.75135 V.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.5009 V, not -1.4991 V.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.5009 V, not 3.5009 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_009869","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 [Ni(CN)_6]^+3, each CN ligand has charge −1. Determine the formal oxidation state of Ni.","options":{"A":"1.9","B":"12","C":"10","D":"9"},"correct_answer":"D","correct_answer_value":"9","solution":"Let the formal oxidation state of Ni be x. Charge balance gives x+(6)(−1)=+3; hence x=3+6=+9.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 9, not 1.9.","B":"Incorrect. Substituting the stated values into the governing relation gives 9, not 12.","C":"Incorrect. Substituting the stated values into the governing relation gives 9, not 10.","D":"Correct. The worked calculation gives 9."},"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_007748","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 59 mmol L⁻¹, what concentration remains after 40 minutes?","options":{"A":"22.125 mmol L⁻¹","B":"29.5 mmol L⁻¹","C":"15.75 mmol L⁻¹","D":"14.75 mmol L⁻¹"},"correct_answer":"D","correct_answer_value":"14.75 mmol L⁻¹","solution":"40 minutes corresponds to 2 half-lives. The remaining concentration is [A]₀(1/2)ⁿ=59/2^2=14.75 mmol L⁻¹.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 14.75 mmol L⁻¹, not 22.125 mmol L⁻¹.","B":"Incorrect. Substituting the stated values into the governing relation gives 14.75 mmol L⁻¹, not 29.5 mmol L⁻¹.","C":"Incorrect. Substituting the stated values into the governing relation gives 14.75 mmol L⁻¹, not 15.75 mmol L⁻¹.","D":"Correct. The worked calculation gives 14.75 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_007221","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 285 K and 0.17 atm? Use R=0.082057 L atm mol⁻¹ K⁻¹.","options":{"A":"165.07938 L","B":"137.56615 L","C":"68.78307 L","D":"275.13229 L"},"correct_answer":"B","correct_answer_value":"137.56615 L","solution":"PV=nRT, so V=nRT/P=(1)(0.082057)(285)/(0.17)=137.56615 L.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 137.56615 L, not 165.07938 L.","B":"Correct. The worked calculation gives 137.56615 L.","C":"Incorrect. Substituting the stated values into the governing relation gives 137.56615 L, not 68.78307 L.","D":"Incorrect. Substituting the stated values into the governing relation gives 137.56615 L, not 275.13229 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_002705","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, 2) and v=(-7, -5, -5), calculate u·v.","options":{"A":"-96","B":"-144","C":"-94","D":"-72"},"correct_answer":"A","correct_answer_value":"-96","solution":"The dot product is the sum of corresponding component products: (8)(-7)+(6)(-5)+(2)(-5)=-96.","option_explanations":{"A":"Correct. The worked calculation gives -96.","B":"Incorrect. Substituting the stated values into the governing relation gives -96, not -144.","C":"Incorrect. Substituting the stated values into the governing relation gives -96, not -94.","D":"Incorrect. Substituting the stated values into the governing relation gives -96, not -72."},"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_008809","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, 78% of A has dissociated at constant volume. Find K_c.","options":{"A":"5.76545","B":"4.76545","C":"4.14818","D":"2.76545"},"correct_answer":"D","correct_answer_value":"2.76545","solution":"Let α=0.78. At equilibrium [A]=c₀(1−α) and [B]=2c₀α. Thus K_c=[B]²/[A]=4c₀α²/(1−α)=4(0.25)(0.78)²/(1−0.78)=2.76545.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 2.76545, not 5.76545.","B":"Incorrect. Substituting the stated values into the governing relation gives 2.76545, not 4.76545.","C":"Incorrect. Substituting the stated values into the governing relation gives 2.76545, not 4.14818.","D":"Correct. The worked calculation gives 2.76545."},"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_002940","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, 2, -2) and v=(-4, 0, 6), calculate u·v.","options":{"A":"3","B":"0","C":"1","D":"-3"},"correct_answer":"B","correct_answer_value":"0","solution":"The dot product is the sum of corresponding component products: (-3)(-4)+(2)(0)+(-2)(6)=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 -3."},"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_009041","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=0.84 V, transfers 2 electrons, and has reaction quotient Q=10^1. Using E=E°−(0.0591/n)log₁₀Q, calculate Ecell.","options":{"A":"1.81045 V","B":"1.21568 V","C":"0.81045 V","D":"1.6209 V"},"correct_answer":"C","correct_answer_value":"0.81045 V","solution":"Apply the Nernst equation: E=0.84−(0.0591/2)(1)=0.81045 V.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.81045 V, not 1.81045 V.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.81045 V, not 1.21568 V.","C":"Correct. The worked calculation gives 0.81045 V.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.81045 V, not 1.6209 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_002829","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, -7) and v=(9, -1, -3), calculate u·v.","options":{"A":"-54","B":"-2.6","C":"-36","D":"-18"},"correct_answer":"C","correct_answer_value":"-36","solution":"The dot product is the sum of corresponding component products: (-7)(9)+(-6)(-1)+(-7)(-3)=-36.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives -36, not -54.","B":"Incorrect. Substituting the stated values into the governing relation gives -36, not -2.6.","C":"Correct. The worked calculation gives -36.","D":"Incorrect. Substituting the stated values into the governing relation gives -36, not -18."},"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_000120","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 +2x -9=0, let its roots be α and β. What is α²+β²?","options":{"A":"5.30612","B":"2.65306","C":"5.65306","D":"3.97959"},"correct_answer":"B","correct_answer_value":"2.65306","solution":"By Vieta's relations, α+β = -2/7 and αβ = -9/7. Then α²+β²=(α+β)²−2αβ = (-2/7)²−2(-9/7) = 2.65306.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 2.65306, not 5.30612.","B":"Correct. The worked calculation gives 2.65306.","C":"Incorrect. Substituting the stated values into the governing relation gives 2.65306, not 5.65306.","D":"Incorrect. Substituting the stated values into the governing relation gives 2.65306, not 3.97959."},"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_006340","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 10 kg is attached to an ideal spring of force constant 678 N m⁻¹ and oscillates horizontally without damping. What is the time period divided by π?","options":{"A":"-1.75711 s","B":"1.02429 s","C":"-2.75711 s","D":"0.24289 s"},"correct_answer":"D","correct_answer_value":"0.24289 s","solution":"For a mass–spring oscillator, T=2π√(m/k). Therefore T/π=2√(10/678)=0.24289 s.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.24289 s, not -1.75711 s.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.24289 s, not 1.02429 s.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.24289 s, not -2.75711 s.","D":"Correct. The worked calculation gives 0.24289 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_007571","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 20 mL of 0.10 M NaOH at 25°C. Assuming complete dissociation and additive volumes, find the final pH.","options":{"A":"-0.04576","B":"1.95424","C":"-1.04576","D":"1.46568"},"correct_answer":"B","correct_answer_value":"1.95424","solution":"Excess H⁺ moles=0.10×(25−20)/1000=0.000500 mol. Total volume=0.045 L, so [H⁺]=0.011111 M and pH=−log₁₀[H⁺]=1.95424.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 1.95424, not -0.04576.","B":"Correct. The worked calculation gives 1.95424.","C":"Incorrect. Substituting the stated values into the governing relation gives 1.95424, not -1.04576.","D":"Incorrect. Substituting the stated values into the governing relation gives 1.95424, not 1.46568."},"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_005935","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 340 K. What is its pressure at 504 K?","options":{"A":"1.20753 atm","B":"2.07529 atm","C":"4.15059 atm","D":"4.07529 atm"},"correct_answer":"B","correct_answer_value":"2.07529 atm","solution":"At constant volume, P/T is constant. P₂=P₁T₂/T₁=(1.4)(504)/(340)=2.07529 atm.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 2.07529 atm, not 1.20753 atm.","B":"Correct. The worked calculation gives 2.07529 atm.","C":"Incorrect. Substituting the stated values into the governing relation gives 2.07529 atm, not 4.15059 atm.","D":"Incorrect. Substituting the stated values into the governing relation gives 2.07529 atm, not 4.07529 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_000753","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 3. What is the sum of its first 7 terms?","options":{"A":"9017.25","B":"12023","C":"24046","D":"18034.5"},"correct_answer":"B","correct_answer_value":"12023","solution":"For r≠1, Sₙ=a(rⁿ−1)/(r−1). Thus Sₙ=11(3^7−1)/(3−1)=12023.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 12023, not 9017.25.","B":"Correct. The worked calculation gives 12023.","C":"Incorrect. Substituting the stated values into the governing relation gives 12023, not 24046.","D":"Incorrect. Substituting the stated values into the governing relation gives 12023, not 18034.5."},"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_009236","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=0.78 V, transfers 2 electrons, and has reaction quotient Q=10^2. Using E=E°−(0.0591/n)log₁₀Q, calculate Ecell.","options":{"A":"1.07209 V","B":"3.7209 V","C":"1.08135 V","D":"0.7209 V"},"correct_answer":"D","correct_answer_value":"0.7209 V","solution":"Apply the Nernst equation: E=0.78−(0.0591/2)(2)=0.7209 V.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.7209 V, not 1.07209 V.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.7209 V, not 3.7209 V.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.7209 V, not 1.08135 V.","D":"Correct. The worked calculation gives 0.7209 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_002978","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, -4, -2) and v=(-5, -5, -3), calculate u·v.","options":{"A":"16","B":"32","C":"15","D":"14"},"correct_answer":"A","correct_answer_value":"16","solution":"The dot product is the sum of corresponding component products: (2)(-5)+(-4)(-5)+(-2)(-3)=16.","option_explanations":{"A":"Correct. The worked calculation gives 16.","B":"Incorrect. Substituting the stated values into the governing relation gives 16, not 32.","C":"Incorrect. Substituting the stated values into the governing relation gives 16, not 15.","D":"Incorrect. Substituting the stated values into the governing relation gives 16, not 14."},"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_004403","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 1 m is acted on by a constant torque of 22 N m about its central axis. Find its angular acceleration.","options":{"A":"5.93333 rad s⁻²","B":"1.93333 rad s⁻²","C":"2.93333 rad s⁻²","D":"1.29333 rad s⁻²"},"correct_answer":"C","correct_answer_value":"2.93333 rad s⁻²","solution":"For a solid disc, I=½MR²=½(15)(1)²=7.5 kg m². Using τ=Iα, α=22/7.5=2.93333 rad s⁻².","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 2.93333 rad s⁻², not 5.93333 rad s⁻².","B":"Incorrect. Substituting the stated values into the governing relation gives 2.93333 rad s⁻², not 1.93333 rad s⁻².","C":"Correct. The worked calculation gives 2.93333 rad s⁻².","D":"Incorrect. Substituting the stated values into the governing relation gives 2.93333 rad s⁻², not 1.29333 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_007025","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 286 K and 1.57 atm? Use R=0.082057 L atm mol⁻¹ K⁻¹.","options":{"A":"388.64704 L","B":"193.32352 L","C":"192.32352 L","D":"194.32352 L"},"correct_answer":"D","correct_answer_value":"194.32352 L","solution":"PV=nRT, so V=nRT/P=(13)(0.082057)(286)/(1.57)=194.32352 L.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 194.32352 L, not 388.64704 L.","B":"Incorrect. Substituting the stated values into the governing relation gives 194.32352 L, not 193.32352 L.","C":"Incorrect. Substituting the stated values into the governing relation gives 194.32352 L, not 192.32352 L.","D":"Correct. The worked calculation gives 194.32352 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_007608","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":"89 mL of 0.10 M HCl is mixed with 40 mL of 0.10 M NaOH at 25°C. Assuming complete dissociation and additive volumes, find the final pH.","options":{"A":"0.42039","B":"2.42039","C":"0.7102","D":"1.42039"},"correct_answer":"D","correct_answer_value":"1.42039","solution":"Excess H⁺ moles=0.10×(89−40)/1000=0.004900 mol. Total volume=0.129 L, so [H⁺]=0.037984 M and pH=−log₁₀[H⁺]=1.42039.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 1.42039, not 0.42039.","B":"Incorrect. Substituting the stated values into the governing relation gives 1.42039, not 2.42039.","C":"Incorrect. Substituting the stated values into the governing relation gives 1.42039, not 0.7102.","D":"Correct. The worked calculation gives 1.42039."},"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_005005","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 4 Ω are connected in series; this combination is connected in parallel with a 15 Ω resistor across a 24 V ideal battery. Find the total current supplied by the battery.","options":{"A":"8.02857 A","B":"2.02857 A","C":"6.02857 A","D":"5.02857 A"},"correct_answer":"D","correct_answer_value":"5.02857 A","solution":"Series branch resistance=3+4=7 Ω. Parallel equivalent R_eq=(7×15)/(7+15)=4.77273 Ω. Total current I=V/R_eq=24/4.77273=5.02857 A.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 5.02857 A, not 8.02857 A.","B":"Incorrect. Substituting the stated values into the governing relation gives 5.02857 A, not 2.02857 A.","C":"Incorrect. Substituting the stated values into the governing relation gives 5.02857 A, not 6.02857 A.","D":"Correct. The worked calculation gives 5.02857 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_007706","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 43 mmol L⁻¹, what concentration remains after 40 minutes?","options":{"A":"-10.75 mmol L⁻¹","B":"5.375 mmol L⁻¹","C":"21.5 mmol L⁻¹","D":"10.75 mmol L⁻¹"},"correct_answer":"D","correct_answer_value":"10.75 mmol L⁻¹","solution":"40 minutes corresponds to 2 half-lives. The remaining concentration is [A]₀(1/2)ⁿ=43/2^2=10.75 mmol L⁻¹.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 10.75 mmol L⁻¹, not -10.75 mmol L⁻¹.","B":"Incorrect. Substituting the stated values into the governing relation gives 10.75 mmol L⁻¹, not 5.375 mmol L⁻¹.","C":"Incorrect. Substituting the stated values into the governing relation gives 10.75 mmol L⁻¹, not 21.5 mmol L⁻¹.","D":"Correct. The worked calculation gives 10.75 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_001475","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 -8x -2) dx.","options":{"A":"-420","B":"-105","C":"-157.5","D":"-210"},"correct_answer":"D","correct_answer_value":"-210","solution":"An antiderivative is (0/3)x³+(-8/2)x²+(-2)x. Evaluating between 0 and 7 gives (0/3)(7)³+(-8/2)(7)²+(-2)(7)=-210.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives -210, not -420.","B":"Incorrect. Substituting the stated values into the governing relation gives -210, not -105.","C":"Incorrect. Substituting the stated values into the governing relation gives -210, not -157.5.","D":"Correct. The worked calculation gives -210."},"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_002646","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, -7], [8, 7]].","options":{"A":"84","B":"86","C":"42","D":"67.2"},"correct_answer":"A","correct_answer_value":"84","solution":"For a 2×2 matrix [[a,b],[c,d]], det=ad−bc. Here it is (4)(7)−(-7)(8)=84.","option_explanations":{"A":"Correct. The worked calculation gives 84.","B":"Incorrect. Substituting the stated values into the governing relation gives 84, not 86.","C":"Incorrect. Substituting the stated values into the governing relation gives 84, not 42.","D":"Incorrect. Substituting the stated values into the governing relation gives 84, not 67.2."},"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_000967","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 1 and common ratio 2. What is the sum of its first 7 terms?","options":{"A":"125","B":"152.4","C":"127","D":"63.5"},"correct_answer":"C","correct_answer_value":"127","solution":"For r≠1, Sₙ=a(rⁿ−1)/(r−1). Thus Sₙ=1(2^7−1)/(2−1)=127.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 127, not 125.","B":"Incorrect. Substituting the stated values into the governing relation gives 127, not 152.4.","C":"Correct. The worked calculation gives 127.","D":"Incorrect. Substituting the stated values into the governing relation gives 127, not 63.5."},"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_004462","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 6 m is acted on by a constant torque of 13 N m about its central axis. Find its angular acceleration.","options":{"A":"-0.81944 rad s⁻²","B":"0.13542 rad s⁻²","C":"0.18056 rad s⁻²","D":"3.18056 rad s⁻²"},"correct_answer":"C","correct_answer_value":"0.18056 rad s⁻²","solution":"For a solid disc, I=½MR²=½(4)(6)²=72 kg m². Using τ=Iα, α=13/72=0.18056 rad s⁻².","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.18056 rad s⁻², not -0.81944 rad s⁻².","B":"Incorrect. Substituting the stated values into the governing relation gives 0.18056 rad s⁻², not 0.13542 rad s⁻².","C":"Correct. The worked calculation gives 0.18056 rad s⁻².","D":"Incorrect. Substituting the stated values into the governing relation gives 0.18056 rad s⁻², not 3.18056 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_005373","subject":"physics","topic":"electrostatics","subtopic":"capacitor_energy","difficulty":"hard","difficulty_labeling_method":"provisional_template_level_label","question_type":"single_correct_mcq","question":"A 1 μF capacitor is charged to a potential difference of 33 V. Calculate the energy stored, expressed in μJ.","options":{"A":"816.75 μJ","B":"543.5 μJ","C":"272.25 μJ","D":"544.5 μJ"},"correct_answer":"D","correct_answer_value":"544.5 μJ","solution":"U=½CV². Since C=1 μF, U in μJ is ½×1×(33)²=544.5 μJ.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 544.5 μJ, not 816.75 μJ.","B":"Incorrect. Substituting the stated values into the governing relation gives 544.5 μJ, not 543.5 μJ.","C":"Incorrect. Substituting the stated values into the governing relation gives 544.5 μJ, not 272.25 μJ.","D":"Correct. The worked calculation gives 544.5 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_001862","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 37 distinct elements?","options":{"A":"11655","B":"7772","C":"5827.5","D":"7770"},"correct_answer":"D","correct_answer_value":"7770","solution":"Order does not matter, so the count is C(37,3)=37!/[3!(34)!]=7770.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 7770, not 11655.","B":"Incorrect. Substituting the stated values into the governing relation gives 7770, not 7772.","C":"Incorrect. Substituting the stated values into the governing relation gives 7770, not 5827.5.","D":"Correct. The worked calculation gives 7770."},"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_001218","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 +4x² -8x+7. Evaluate f′(0).","options":{"A":"-8","B":"8","C":"-7","D":"-16"},"correct_answer":"A","correct_answer_value":"-8","solution":"Differentiate: f′(x)=3(2)x²+2(4)x+(-8). At x=0, f′(0)=3(2)(0)²+2(4)(0)+(-8)=-8.","option_explanations":{"A":"Correct. The worked calculation gives -8.","B":"Incorrect. Substituting the stated values into the governing relation gives -8, not 8.","C":"Incorrect. Substituting the stated values into the governing relation gives -8, not -7.","D":"Incorrect. Substituting the stated values into the governing relation gives -8, not -16."},"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_005178","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 12 cm in vacuum. Using k=9×10⁹ N m² C⁻², find the magnitude of the electrostatic force.","options":{"A":"7.5 N","B":"5.5 N","C":"15 N","D":"8.5 N"},"correct_answer":"A","correct_answer_value":"7.5 N","solution":"Convert charges to coulombs and distance to metres. F=k|q₁q₂|/r²=9×10⁹(2×10⁻⁶)(6×10⁻⁶)/(12×10⁻²)²=7.5 N.","option_explanations":{"A":"Correct. The worked calculation gives 7.5 N.","B":"Incorrect. Substituting the stated values into the governing relation gives 7.5 N, not 5.5 N.","C":"Incorrect. Substituting the stated values into the governing relation gives 7.5 N, not 15 N.","D":"Incorrect. Substituting the stated values into the governing relation gives 7.5 N, not 8.5 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_004158","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 0.17. Taking g=10 m s⁻², find the net force down the plane. Use sin37°=0.6 and cos37°=0.8.","options":{"A":"380.48 N","B":"188.24 N","C":"20.024 N","D":"190.24 N"},"correct_answer":"D","correct_answer_value":"190.24 N","solution":"Net force=mg(sinθ−μcosθ)=41×10×(0.6−0.17×0.8)=190.24 N.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 190.24 N, not 380.48 N.","B":"Incorrect. Substituting the stated values into the governing relation gives 190.24 N, not 188.24 N.","C":"Incorrect. Substituting the stated values into the governing relation gives 190.24 N, not 20.024 N.","D":"Correct. The worked calculation gives 190.24 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_001447","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 -4x +3) dx.","options":{"A":"-266","B":"199.5","C":"319.2","D":"266"},"correct_answer":"D","correct_answer_value":"266","solution":"An antiderivative is (3/3)x³+(-4/2)x²+(3)x. Evaluating between 0 and 7 gives (3/3)(7)³+(-4/2)(7)²+(3)(7)=266.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 266, not -266.","B":"Incorrect. Substituting the stated values into the governing relation gives 266, not 199.5.","C":"Incorrect. Substituting the stated values into the governing relation gives 266, not 319.2.","D":"Correct. The worked calculation gives 266."},"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_001879","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 31 distinct elements?","options":{"A":"141120525","B":"112896420","C":"141120527","D":"70560262.5"},"correct_answer":"A","correct_answer_value":"141120525","solution":"Order does not matter, so the count is C(31,12)=31!/[12!(19)!]=141120525.","option_explanations":{"A":"Correct. The worked calculation gives 141120525.","B":"Incorrect. Substituting the stated values into the governing relation gives 141120525, not 112896420.","C":"Incorrect. Substituting the stated values into the governing relation gives 141120525, not 141120527.","D":"Incorrect. Substituting the stated values into the governing relation gives 141120525, not 70560262.5."},"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_002380","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], [1, -7]].","options":{"A":"-15","B":"15","C":"-18","D":"-0.5"},"correct_answer":"A","correct_answer_value":"-15","solution":"For a 2×2 matrix [[a,b],[c,d]], det=ad−bc. Here it is (1)(-7)−(8)(1)=-15.","option_explanations":{"A":"Correct. The worked calculation gives -15.","B":"Incorrect. Substituting the stated values into the governing relation gives -15, not 15.","C":"Incorrect. Substituting the stated values into the governing relation gives -15, not -18.","D":"Incorrect. Substituting the stated values into the governing relation gives -15, not -0.5."},"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_001740","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 74 distinct elements?","options":{"A":"1799579064","B":"1799579065","C":"1799579062","D":"1349684298"},"correct_answer":"A","correct_answer_value":"1799579064","solution":"Order does not matter, so the count is C(74,7)=74!/[7!(67)!]=1799579064.","option_explanations":{"A":"Correct. The worked calculation gives 1799579064.","B":"Incorrect. Substituting the stated values into the governing relation gives 1799579064, not 1799579065.","C":"Incorrect. Substituting the stated values into the governing relation gives 1799579064, not 1799579062.","D":"Incorrect. Substituting the stated values into the governing relation gives 1799579064, not 1349684298."},"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_002561","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, -6], [3, 1]].","options":{"A":"3.3","B":"18.4","C":"23","D":"24"},"correct_answer":"C","correct_answer_value":"23","solution":"For a 2×2 matrix [[a,b],[c,d]], det=ad−bc. Here it is (5)(1)−(-6)(3)=23.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 23, not 3.3.","B":"Incorrect. Substituting the stated values into the governing relation gives 23, not 18.4.","C":"Correct. The worked calculation gives 23.","D":"Incorrect. Substituting the stated values into the governing relation gives 23, not 24."},"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_007235","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 383 K and 2.78 atm? Use R=0.082057 L atm mol⁻¹ K⁻¹.","options":{"A":"-146.96468 L","B":"148.96468 L","C":"146.96468 L","D":"144.96468 L"},"correct_answer":"C","correct_answer_value":"146.96468 L","solution":"PV=nRT, so V=nRT/P=(13)(0.082057)(383)/(2.78)=146.96468 L.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 146.96468 L, not -146.96468 L.","B":"Incorrect. Substituting the stated values into the governing relation gives 146.96468 L, not 148.96468 L.","C":"Correct. The worked calculation gives 146.96468 L.","D":"Incorrect. Substituting the stated values into the governing relation gives 146.96468 L, not 144.96468 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_000946","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 3. What is the sum of its first 8 terms?","options":{"A":"19680","B":"9840","C":"15744","D":"19678"},"correct_answer":"A","correct_answer_value":"19680","solution":"For r≠1, Sₙ=a(rⁿ−1)/(r−1). Thus Sₙ=6(3^8−1)/(3−1)=19680.","option_explanations":{"A":"Correct. The worked calculation gives 19680.","B":"Incorrect. Substituting the stated values into the governing relation gives 19680, not 9840.","C":"Incorrect. Substituting the stated values into the governing relation gives 19680, not 15744.","D":"Incorrect. Substituting the stated values into the governing relation gives 19680, not 19678."},"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_004115","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 39 kg block slides down a 37° rough incline. The coefficient of kinetic friction is 0.42. Taking g=10 m s⁻², find the net force down the plane. Use sin37°=0.6 and cos37°=0.8.","options":{"A":"101.96 N","B":"205.92 N","C":"102.96 N","D":"77.22 N"},"correct_answer":"C","correct_answer_value":"102.96 N","solution":"Net force=mg(sinθ−μcosθ)=39×10×(0.6−0.42×0.8)=102.96 N.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 102.96 N, not 101.96 N.","B":"Incorrect. Substituting the stated values into the governing relation gives 102.96 N, not 205.92 N.","C":"Correct. The worked calculation gives 102.96 N.","D":"Incorrect. Substituting the stated values into the governing relation gives 102.96 N, not 77.22 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_001914","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 47 distinct elements?","options":{"A":"178365","B":"267547.5","C":"356730","D":"142692"},"correct_answer":"A","correct_answer_value":"178365","solution":"Order does not matter, so the count is C(47,4)=47!/[4!(43)!]=178365.","option_explanations":{"A":"Correct. The worked calculation gives 178365.","B":"Incorrect. Substituting the stated values into the governing relation gives 178365, not 267547.5.","C":"Incorrect. Substituting the stated values into the governing relation gives 178365, not 356730.","D":"Incorrect. Substituting the stated values into the governing relation gives 178365, not 142692."},"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_007652","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":"45 mL of 0.10 M HCl is mixed with 38 mL of 0.10 M NaOH at 25°C. Assuming complete dissociation and additive volumes, find the final pH.","options":{"A":"1.03699","B":"2.07398","C":"4.14796","D":"4.07398"},"correct_answer":"B","correct_answer_value":"2.07398","solution":"Excess H⁺ moles=0.10×(45−38)/1000=0.000700 mol. Total volume=0.083 L, so [H⁺]=0.008434 M and pH=−log₁₀[H⁺]=2.07398.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 2.07398, not 1.03699.","B":"Correct. The worked calculation gives 2.07398.","C":"Incorrect. Substituting the stated values into the governing relation gives 2.07398, not 4.14796.","D":"Incorrect. Substituting the stated values into the governing relation gives 2.07398, not 4.07398."},"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_008371","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 1.32 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":"1.20275 K","B":"-2.02752 K","C":"1.52064 K","D":"2.02752 K"},"correct_answer":"D","correct_answer_value":"2.02752 K","solution":"ΔT_b=iK_bm. Substitution gives ΔT_b=3×0.512×1.32=2.02752 K.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 2.02752 K, not 1.20275 K.","B":"Incorrect. Substituting the stated values into the governing relation gives 2.02752 K, not -2.02752 K.","C":"Incorrect. Substituting the stated values into the governing relation gives 2.02752 K, not 1.52064 K.","D":"Correct. The worked calculation gives 2.02752 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_001905","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 6 elements can be selected from a set of 71 distinct elements?","options":{"A":"107414249.25","B":"71609499.5","C":"214828498.5","D":"143218999"},"correct_answer":"D","correct_answer_value":"143218999","solution":"Order does not matter, so the count is C(71,6)=71!/[6!(65)!]=143218999.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 143218999, not 107414249.25.","B":"Incorrect. Substituting the stated values into the governing relation gives 143218999, not 71609499.5.","C":"Incorrect. Substituting the stated values into the governing relation gives 143218999, not 214828498.5.","D":"Correct. The worked calculation gives 143218999."},"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_007239","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 548 K and 0.31 atm? Use R=0.082057 L atm mol⁻¹ K⁻¹.","options":{"A":"2320.8896 L","B":"870.3336 L","C":"1158.4448 L","D":"1160.4448 L"},"correct_answer":"D","correct_answer_value":"1160.4448 L","solution":"PV=nRT, so V=nRT/P=(8)(0.082057)(548)/(0.31)=1160.4448 L.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 1160.4448 L, not 2320.8896 L.","B":"Incorrect. Substituting the stated values into the governing relation gives 1160.4448 L, not 870.3336 L.","C":"Incorrect. Substituting the stated values into the governing relation gives 1160.4448 L, not 1158.4448 L.","D":"Correct. The worked calculation gives 1160.4448 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_008383","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 0.47 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":"-2.53 K","B":"0.47 K","C":"2.47 K","D":"0.705 K"},"correct_answer":"B","correct_answer_value":"0.47 K","solution":"ΔT_b=iK_bm. Substitution gives ΔT_b=1×1.0×0.47=0.47 K.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.47 K, not -2.53 K.","B":"Correct. The worked calculation gives 0.47 K.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.47 K, not 2.47 K.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.47 K, not 0.705 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_009215","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=1.04 V, transfers 3 electrons, and has reaction quotient Q=10^1. Using E=E°−(0.0591/n)log₁₀Q, calculate Ecell.","options":{"A":"4.0203 V","B":"1.0203 V","C":"2.0406 V","D":"0.0203 V"},"correct_answer":"B","correct_answer_value":"1.0203 V","solution":"Apply the Nernst equation: E=1.04−(0.0591/3)(1)=1.0203 V.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 1.0203 V, not 4.0203 V.","B":"Correct. The worked calculation gives 1.0203 V.","C":"Incorrect. Substituting the stated values into the governing relation gives 1.0203 V, not 2.0406 V.","D":"Incorrect. Substituting the stated values into the governing relation gives 1.0203 V, not 0.0203 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_000209","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 -15x -4=0, let its roots be α and β. What is α²+β²?","options":{"A":"3.79167","B":"1.75833","C":"-7.58333","D":"7.58333"},"correct_answer":"D","correct_answer_value":"7.58333","solution":"By Vieta's relations, α+β = 15/6 and αβ = -4/6. Then α²+β²=(α+β)²−2αβ = (15/6)²−2(-4/6) = 7.58333.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 7.58333, not 3.79167.","B":"Incorrect. Substituting the stated values into the governing relation gives 7.58333, not 1.75833.","C":"Incorrect. Substituting the stated values into the governing relation gives 7.58333, not -7.58333.","D":"Correct. The worked calculation gives 7.58333."},"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_000077","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 -7x -12=0, let its roots be α and β. What is α²+β²?","options":{"A":"6.76","B":"13.52","C":"3.76","D":"10.14"},"correct_answer":"A","correct_answer_value":"6.76","solution":"By Vieta's relations, α+β = 7/5 and αβ = -12/5. Then α²+β²=(α+β)²−2αβ = (7/5)²−2(-12/5) = 6.76.","option_explanations":{"A":"Correct. The worked calculation gives 6.76.","B":"Incorrect. Substituting the stated values into the governing relation gives 6.76, not 13.52.","C":"Incorrect. Substituting the stated values into the governing relation gives 6.76, not 3.76.","D":"Incorrect. Substituting the stated values into the governing relation gives 6.76, not 10.14."},"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_009506","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_18H_1N_1X_4. Treat X as the total number of monovalent halogen atoms.","options":{"A":"17","B":"18","C":"15","D":"8.5"},"correct_answer":"A","correct_answer_value":"17","solution":"Use DBE=(2C+2+N−H−X)/2. Substitution gives (2×18+2+1−1−4)/2=17.","option_explanations":{"A":"Correct. The worked calculation gives 17.","B":"Incorrect. Substituting the stated values into the governing relation gives 17, not 18.","C":"Incorrect. Substituting the stated values into the governing relation gives 17, not 15.","D":"Incorrect. Substituting the stated values into the governing relation gives 17, not 8.5."},"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_007888","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 61 mmol L⁻¹, what concentration remains after 120 minutes?","options":{"A":"0.47656 mmol L⁻¹","B":"-2.04688 mmol L⁻¹","C":"-0.95312 mmol L⁻¹","D":"0.95312 mmol L⁻¹"},"correct_answer":"D","correct_answer_value":"0.95312 mmol L⁻¹","solution":"120 minutes corresponds to 6 half-lives. The remaining concentration is [A]₀(1/2)ⁿ=61/2^6=0.95312 mmol L⁻¹.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.95312 mmol L⁻¹, not 0.47656 mmol L⁻¹.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.95312 mmol L⁻¹, not -2.04688 mmol L⁻¹.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.95312 mmol L⁻¹, not -0.95312 mmol L⁻¹.","D":"Correct. The worked calculation gives 0.95312 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_003568","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 99 m s⁻¹ at 30° to the horizontal. Take g=10 m s⁻². Find its horizontal range.","options":{"A":"636.59362 m","B":"848.7915 m","C":"1018.5498 m","D":"850.7915 m"},"correct_answer":"B","correct_answer_value":"848.7915 m","solution":"Range R=u²sin(2θ)/g. Substituting u=99, θ=30° and g=10 gives R=(99)²sin(60°)/10=848.7915 m.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 848.7915 m, not 636.59362 m.","B":"Correct. The worked calculation gives 848.7915 m.","C":"Incorrect. Substituting the stated values into the governing relation gives 848.7915 m, not 1018.5498 m.","D":"Incorrect. Substituting the stated values into the governing relation gives 848.7915 m, not 850.7915 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_002110","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, 11 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":"-0.5817","B":"0.31373","C":"0.4183","D":"-1.5817"},"correct_answer":"C","correct_answer_value":"0.4183","solution":"Favourable pairs = C(4,2)+C(11,2)+C(3,2)=64. Total pairs=C(18,2)=153. Therefore P=64/153=0.4183.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.4183, not -0.5817.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.4183, not 0.31373.","C":"Correct. The worked calculation gives 0.4183.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.4183, not -1.5817."},"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_009061","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=1.15 V, transfers 1 electrons, and has reaction quotient Q=10^-1. Using E=E°−(0.0591/n)log₁₀Q, calculate Ecell.","options":{"A":"3.2091 V","B":"1.12091 V","C":"1.2091 V","D":"-1.7909 V"},"correct_answer":"C","correct_answer_value":"1.2091 V","solution":"Apply the Nernst equation: E=1.15−(0.0591/1)(-1)=1.2091 V.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 1.2091 V, not 3.2091 V.","B":"Incorrect. Substituting the stated values into the governing relation gives 1.2091 V, not 1.12091 V.","C":"Correct. The worked calculation gives 1.2091 V.","D":"Incorrect. Substituting the stated values into the governing relation gives 1.2091 V, not -1.7909 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_008676","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 1.42 mol L⁻¹ and initially B is absent. At equilibrium, 21% of A has dissociated at constant volume. Find K_c.","options":{"A":"-0.31707","B":"0.31707","C":"1.31707","D":"0.63415"},"correct_answer":"B","correct_answer_value":"0.31707","solution":"Let α=0.21. At equilibrium [A]=c₀(1−α) and [B]=2c₀α. Thus K_c=[B]²/[A]=4c₀α²/(1−α)=4(1.42)(0.21)²/(1−0.21)=0.31707.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.31707, not -0.31707.","B":"Correct. The worked calculation gives 0.31707.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.31707, not 1.31707.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.31707, not 0.63415."},"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_007856","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 47 mmol L⁻¹, what concentration remains after 40 minutes?","options":{"A":"11.75 mmol L⁻¹","B":"-11.75 mmol L⁻¹","C":"2.175 mmol L⁻¹","D":"10.75 mmol L⁻¹"},"correct_answer":"A","correct_answer_value":"11.75 mmol L⁻¹","solution":"40 minutes corresponds to 2 half-lives. The remaining concentration is [A]₀(1/2)ⁿ=47/2^2=11.75 mmol L⁻¹.","option_explanations":{"A":"Correct. The worked calculation gives 11.75 mmol L⁻¹.","B":"Incorrect. Substituting the stated values into the governing relation gives 11.75 mmol L⁻¹, not -11.75 mmol L⁻¹.","C":"Incorrect. Substituting the stated values into the governing relation gives 11.75 mmol L⁻¹, not 2.175 mmol L⁻¹.","D":"Incorrect. Substituting the stated values into the governing relation gives 11.75 mmol L⁻¹, not 10.75 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_008120","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 2572 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":"13.84719 g","B":"10.84719 g","C":"11.84719 g","D":"8.84719 g"},"correct_answer":"C","correct_answer_value":"11.84719 g","solution":"Cu²⁺+2e⁻→Cu, so z=2. By Faraday's law m=MIt/(zF)=63.5×14×2572/(2×96500)=11.84719 g.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 11.84719 g, not 13.84719 g.","B":"Incorrect. Substituting the stated values into the governing relation gives 11.84719 g, not 10.84719 g.","C":"Correct. The worked calculation gives 11.84719 g.","D":"Incorrect. Substituting the stated values into the governing relation gives 11.84719 g, not 8.84719 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_004182","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 17 kg block slides down a 37° rough incline. The coefficient of kinetic friction is 0.17. Taking g=10 m s⁻², find the net force down the plane. Use sin37°=0.6 and cos37°=0.8.","options":{"A":"77.88 N","B":"78.88 N","C":"94.656 N","D":"157.76 N"},"correct_answer":"B","correct_answer_value":"78.88 N","solution":"Net force=mg(sinθ−μcosθ)=17×10×(0.6−0.17×0.8)=78.88 N.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 78.88 N, not 77.88 N.","B":"Correct. The worked calculation gives 78.88 N.","C":"Incorrect. Substituting the stated values into the governing relation gives 78.88 N, not 94.656 N.","D":"Incorrect. Substituting the stated values into the governing relation gives 78.88 N, not 157.76 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_003388","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 13 m s⁻¹ at 75° to the horizontal. Take g=10 m s⁻². Find its horizontal range.","options":{"A":"8.45 m","B":"5.45 m","C":"7.45 m","D":"1.845 m"},"correct_answer":"A","correct_answer_value":"8.45 m","solution":"Range R=u²sin(2θ)/g. Substituting u=13, θ=75° and g=10 gives R=(13)²sin(150°)/10=8.45 m.","option_explanations":{"A":"Correct. The worked calculation gives 8.45 m.","B":"Incorrect. Substituting the stated values into the governing relation gives 8.45 m, not 5.45 m.","C":"Incorrect. Substituting the stated values into the governing relation gives 8.45 m, not 7.45 m.","D":"Incorrect. Substituting the stated values into the governing relation gives 8.45 m, not 1.845 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_005076","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 7 cm in vacuum. Using k=9×10⁹ N m² C⁻², find the magnitude of the electrostatic force.","options":{"A":"105.79592 N","B":"158.69388 N","C":"134.2449 N","D":"132.2449 N"},"correct_answer":"D","correct_answer_value":"132.2449 N","solution":"Convert charges to coulombs and distance to metres. F=k|q₁q₂|/r²=9×10⁹(9×10⁻⁶)(8×10⁻⁶)/(7×10⁻²)²=132.2449 N.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 132.2449 N, not 105.79592 N.","B":"Incorrect. Substituting the stated values into the governing relation gives 132.2449 N, not 158.69388 N.","C":"Incorrect. Substituting the stated values into the governing relation gives 132.2449 N, not 134.2449 N.","D":"Correct. The worked calculation gives 132.2449 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_006527","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 11 kg is attached to an ideal spring of force constant 897 N m⁻¹ and oscillates horizontally without damping. What is the time period divided by π?","options":{"A":"2.22148 s","B":"3.22148 s","C":"0.22148 s","D":"1.02215 s"},"correct_answer":"C","correct_answer_value":"0.22148 s","solution":"For a mass–spring oscillator, T=2π√(m/k). Therefore T/π=2√(11/897)=0.22148 s.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.22148 s, not 2.22148 s.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.22148 s, not 3.22148 s.","C":"Correct. The worked calculation gives 0.22148 s.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.22148 s, not 1.02215 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_000061","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 -8x -10=0, let its roots be α and β. What is α²+β²?","options":{"A":"52","B":"-26","C":"26","D":"39"},"correct_answer":"C","correct_answer_value":"26","solution":"By Vieta's relations, α+β = 8/2 and αβ = -10/2. Then α²+β²=(α+β)²−2αβ = (8/2)²−2(-10/2) = 26.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 26, not 52.","B":"Incorrect. Substituting the stated values into the governing relation gives 26, not -26.","C":"Correct. The worked calculation gives 26.","D":"Incorrect. Substituting the stated values into the governing relation gives 26, not 39."},"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_009098","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=10^6. Using E=E°−(0.0591/n)log₁₀Q, calculate Ecell.","options":{"A":"1.05618 V","B":"2.5618 V","C":"-0.4382 V","D":"0.5618 V"},"correct_answer":"D","correct_answer_value":"0.5618 V","solution":"Apply the Nernst equation: E=0.68−(0.0591/3)(6)=0.5618 V.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.5618 V, not 1.05618 V.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.5618 V, not 2.5618 V.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.5618 V, not -0.4382 V.","D":"Correct. The worked calculation gives 0.5618 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_007121","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 317 K and 0.48 atm? Use R=0.082057 L atm mol⁻¹ K⁻¹.","options":{"A":"758.68535 L","B":"-758.68535 L","C":"760.68535 L","D":"569.01401 L"},"correct_answer":"A","correct_answer_value":"758.68535 L","solution":"PV=nRT, so V=nRT/P=(14)(0.082057)(317)/(0.48)=758.68535 L.","option_explanations":{"A":"Correct. The worked calculation gives 758.68535 L.","B":"Incorrect. Substituting the stated values into the governing relation gives 758.68535 L, not -758.68535 L.","C":"Incorrect. Substituting the stated values into the governing relation gives 758.68535 L, not 760.68535 L.","D":"Incorrect. Substituting the stated values into the governing relation gives 758.68535 L, not 569.01401 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_007075","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 317 K and 2.09 atm? Use R=0.082057 L atm mol⁻¹ K⁻¹.","options":{"A":"49.78386 L","B":"62.22983 L","C":"64.22983 L","D":"74.6758 L"},"correct_answer":"B","correct_answer_value":"62.22983 L","solution":"PV=nRT, so V=nRT/P=(5)(0.082057)(317)/(2.09)=62.22983 L.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 62.22983 L, not 49.78386 L.","B":"Correct. The worked calculation gives 62.22983 L.","C":"Incorrect. Substituting the stated values into the governing relation gives 62.22983 L, not 64.22983 L.","D":"Incorrect. Substituting the stated values into the governing relation gives 62.22983 L, not 74.6758 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_003853","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 1 m s⁻¹. A constant net force does 9 N of work over 5 m in the direction of motion. Find its final kinetic energy.","options":{"A":"51.5 J","B":"38.625 J","C":"77.25 J","D":"61.8 J"},"correct_answer":"A","correct_answer_value":"51.5 J","solution":"By the work–energy theorem, K_f=K_i+W=½mu²+Fs=½(13)(1)²+(9)(5)=51.5 J.","option_explanations":{"A":"Correct. The worked calculation gives 51.5 J.","B":"Incorrect. Substituting the stated values into the governing relation gives 51.5 J, not 38.625 J.","C":"Incorrect. Substituting the stated values into the governing relation gives 51.5 J, not 77.25 J.","D":"Incorrect. Substituting the stated values into the governing relation gives 51.5 J, not 61.8 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_008853","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 1.13 mol L⁻¹ and initially B is absent. At equilibrium, 38% of A has dissociated at constant volume. Find K_c.","options":{"A":"2.10545","B":"-0.94728","C":"1.05272","D":"1.57908"},"correct_answer":"C","correct_answer_value":"1.05272","solution":"Let α=0.38. At equilibrium [A]=c₀(1−α) and [B]=2c₀α. Thus K_c=[B]²/[A]=4c₀α²/(1−α)=4(1.13)(0.38)²/(1−0.38)=1.05272.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 1.05272, not 2.10545.","B":"Incorrect. Substituting the stated values into the governing relation gives 1.05272, not -0.94728.","C":"Correct. The worked calculation gives 1.05272.","D":"Incorrect. Substituting the stated values into the governing relation gives 1.05272, not 1.57908."},"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_009707","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 [Cr(CN)_3]^-4, each CN ligand has charge −1. Determine the formal oxidation state of Cr.","options":{"A":"-4","B":"1","C":"-1","D":"0.9"},"correct_answer":"C","correct_answer_value":"-1","solution":"Let the formal oxidation state of Cr be x. Charge balance gives x+(3)(−1)=-4; hence x=-4+3=-1.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives -1, not -4.","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 0.9."},"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_004193","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 38 kg block slides down a 37° rough incline. The coefficient of kinetic friction is 0.35. Taking g=10 m s⁻², find the net force down the plane. Use sin37°=0.6 and cos37°=0.8.","options":{"A":"121.6 N","B":"122.6 N","C":"182.4 N","D":"97.28 N"},"correct_answer":"A","correct_answer_value":"121.6 N","solution":"Net force=mg(sinθ−μcosθ)=38×10×(0.6−0.35×0.8)=121.6 N.","option_explanations":{"A":"Correct. The worked calculation gives 121.6 N.","B":"Incorrect. Substituting the stated values into the governing relation gives 121.6 N, not 122.6 N.","C":"Incorrect. Substituting the stated values into the governing relation gives 121.6 N, not 182.4 N.","D":"Incorrect. Substituting the stated values into the governing relation gives 121.6 N, not 97.28 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_009202","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=1.09 V, transfers 2 electrons, and has reaction quotient Q=10^-2. Using E=E°−(0.0591/n)log₁₀Q, calculate Ecell.","options":{"A":"1.1491 V","B":"2.2982 V","C":"0.1491 V","D":"2.1491 V"},"correct_answer":"A","correct_answer_value":"1.1491 V","solution":"Apply the Nernst equation: E=1.09−(0.0591/2)(-2)=1.1491 V.","option_explanations":{"A":"Correct. The worked calculation gives 1.1491 V.","B":"Incorrect. Substituting the stated values into the governing relation gives 1.1491 V, not 2.2982 V.","C":"Incorrect. Substituting the stated values into the governing relation gives 1.1491 V, not 0.1491 V.","D":"Incorrect. Substituting the stated values into the governing relation gives 1.1491 V, not 2.1491 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_008762","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 1.23 mol L⁻¹ and initially B is absent. At equilibrium, 22% of A has dissociated at constant volume. Find K_c.","options":{"A":"-1.69471","B":"0.15265","C":"0.30529","D":"0.61058"},"correct_answer":"C","correct_answer_value":"0.30529","solution":"Let α=0.22. At equilibrium [A]=c₀(1−α) and [B]=2c₀α. Thus K_c=[B]²/[A]=4c₀α²/(1−α)=4(1.23)(0.22)²/(1−0.22)=0.30529.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.30529, not -1.69471.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.30529, not 0.15265.","C":"Correct. The worked calculation gives 0.30529.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.30529, not 0.61058."},"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_003220","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+9i)/(9+11i). What is Re(z), the real part of z?","options":{"A":"3.62376","B":"0.62376","C":"0.93564","D":"1.24752"},"correct_answer":"B","correct_answer_value":"0.62376","solution":"Multiply numerator and denominator by the conjugate (9-11i). Then Re(z)=(ac+bd)/(c²+d²)=((3)(9)+(9)(11))/(9²+(11)²)=0.62376.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.62376, not 3.62376.","B":"Correct. The worked calculation gives 0.62376.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.62376, not 0.93564.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.62376, not 1.24752."},"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_004972","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 6 Ω and 2 Ω are connected in series; this combination is connected in parallel with a 16 Ω resistor across a 24 V ideal battery. Find the total current supplied by the battery.","options":{"A":"6.5 A","B":"5.5 A","C":"2.25 A","D":"4.5 A"},"correct_answer":"D","correct_answer_value":"4.5 A","solution":"Series branch resistance=6+2=8 Ω. Parallel equivalent R_eq=(8×16)/(8+16)=5.33333 Ω. Total current I=V/R_eq=24/5.33333=4.5 A.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 4.5 A, not 6.5 A.","B":"Incorrect. Substituting the stated values into the governing relation gives 4.5 A, not 5.5 A.","C":"Incorrect. Substituting the stated values into the governing relation gives 4.5 A, not 2.25 A.","D":"Correct. The worked calculation gives 4.5 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_001583","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 (5x² +4x +4) dx.","options":{"A":"37.5","B":"60","C":"-75","D":"75"},"correct_answer":"D","correct_answer_value":"75","solution":"An antiderivative is (5/3)x³+(4/2)x²+(4)x. Evaluating between 0 and 3 gives (5/3)(3)³+(4/2)(3)²+(4)(3)=75.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 75, not 37.5.","B":"Incorrect. Substituting the stated values into the governing relation gives 75, not 60.","C":"Incorrect. Substituting the stated values into the governing relation gives 75, not -75.","D":"Correct. The worked calculation gives 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_001338","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 (-1x² -7x +0) dx.","options":{"A":"-38.66667","B":"-77.33333","C":"-79.33333","D":"-61.86667"},"correct_answer":"B","correct_answer_value":"-77.33333","solution":"An antiderivative is (-1/3)x³+(-7/2)x²+(0)x. Evaluating between 0 and 4 gives (-1/3)(4)³+(-7/2)(4)²+(0)(4)=-77.33333.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives -77.33333, not -38.66667.","B":"Correct. The worked calculation gives -77.33333.","C":"Incorrect. Substituting the stated values into the governing relation gives -77.33333, not -79.33333.","D":"Incorrect. Substituting the stated values into the governing relation gives -77.33333, not -61.86667."},"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_001416","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 -3x +6) dx.","options":{"A":"-145.83333","B":"-143.83333","C":"-116.66667","D":"-175"},"correct_answer":"A","correct_answer_value":"-145.83333","solution":"An antiderivative is (-1/3)x³+(-3/2)x²+(6)x. Evaluating between 0 and 7 gives (-1/3)(7)³+(-3/2)(7)²+(6)(7)=-145.83333.","option_explanations":{"A":"Correct. The worked calculation gives -145.83333.","B":"Incorrect. Substituting the stated values into the governing relation gives -145.83333, not -143.83333.","C":"Incorrect. Substituting the stated values into the governing relation gives -145.83333, not -116.66667.","D":"Incorrect. Substituting the stated values into the governing relation gives -145.83333, not -175."},"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_006402","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 2 kg is attached to an ideal spring of force constant 123 N m⁻¹ and oscillates horizontally without damping. What is the time period divided by π?","options":{"A":"0.51006 s","B":"1.0255 s","C":"0.25503 s","D":"0.12752 s"},"correct_answer":"C","correct_answer_value":"0.25503 s","solution":"For a mass–spring oscillator, T=2π√(m/k). Therefore T/π=2√(2/123)=0.25503 s.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.25503 s, not 0.51006 s.","B":"Incorrect. Substituting the stated values into the governing relation gives 0.25503 s, not 1.0255 s.","C":"Correct. The worked calculation gives 0.25503 s.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.25503 s, not 0.12752 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_009787","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(Cl)_6]^-2, each Cl ligand has charge −1. Determine the formal oxidation state of Cu.","options":{"A":"3","B":"4","C":"2","D":"1.4"},"correct_answer":"B","correct_answer_value":"4","solution":"Let the formal oxidation state of Cu be x. Charge balance gives x+(6)(−1)=-2; hence x=-2+6=+4.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 4, not 3.","B":"Correct. The worked calculation gives 4.","C":"Incorrect. Substituting the stated values into the governing relation gives 4, not 2.","D":"Incorrect. Substituting the stated values into the governing relation gives 4, not 1.4."},"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_001564","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 ∫₀^5 (-4x² +7x +3) dx.","options":{"A":"-65.16667","B":"-64.16667","C":"64.16667","D":"-77"},"correct_answer":"B","correct_answer_value":"-64.16667","solution":"An antiderivative is (-4/3)x³+(7/2)x²+(3)x. Evaluating between 0 and 5 gives (-4/3)(5)³+(7/2)(5)²+(3)(5)=-64.16667.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives -64.16667, not -65.16667.","B":"Correct. The worked calculation gives -64.16667.","C":"Incorrect. Substituting the stated values into the governing relation gives -64.16667, not 64.16667.","D":"Incorrect. Substituting the stated values into the governing relation gives -64.16667, not -77."},"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_006109","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 80 cm in front of a thin convex lens of focal length 18 cm. Using the Cartesian sign convention, what is the magnitude of the linear magnification?","options":{"A":"0.58065","B":"0.29032","C":"0.14516","D":"-2.70968"},"correct_answer":"B","correct_answer_value":"0.29032","solution":"Using distances as positive magnitudes, 1/f=1/v+1/u gives v=uf/(u−f)=(80)(18)/(80−18)=23.22581 cm. Thus |m|=v/u=23.22581/80=0.29032.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 0.29032, not 0.58065.","B":"Correct. The worked calculation gives 0.29032.","C":"Incorrect. Substituting the stated values into the governing relation gives 0.29032, not 0.14516.","D":"Incorrect. Substituting the stated values into the governing relation gives 0.29032, not -2.70968."},"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_001742","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 18 distinct elements?","options":{"A":"6854.4","B":"-8568","C":"8568","D":"8567"},"correct_answer":"C","correct_answer_value":"8568","solution":"Order does not matter, so the count is C(18,5)=18!/[5!(13)!]=8568.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 8568, not 6854.4.","B":"Incorrect. Substituting the stated values into the governing relation gives 8568, not -8568.","C":"Correct. The worked calculation gives 8568.","D":"Incorrect. Substituting the stated values into the governing relation gives 8568, not 8567."},"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_007308","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 322 K and 2.52 atm? Use R=0.082057 L atm mol⁻¹ K⁻¹.","options":{"A":"26.21265 L","B":"6.24253 L","C":"51.42531 L","D":"52.42531 L"},"correct_answer":"D","correct_answer_value":"52.42531 L","solution":"PV=nRT, so V=nRT/P=(5)(0.082057)(322)/(2.52)=52.42531 L.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 52.42531 L, not 26.21265 L.","B":"Incorrect. Substituting the stated values into the governing relation gives 52.42531 L, not 6.24253 L.","C":"Incorrect. Substituting the stated values into the governing relation gives 52.42531 L, not 51.42531 L.","D":"Correct. The worked calculation gives 52.42531 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_007114","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 2 mol of gas at 498 K and 1.01 atm? Use R=0.082057 L atm mol⁻¹ K⁻¹.","options":{"A":"80.91958 L","B":"40.45979 L","C":"78.91958 L","D":"161.83915 L"},"correct_answer":"A","correct_answer_value":"80.91958 L","solution":"PV=nRT, so V=nRT/P=(2)(0.082057)(498)/(1.01)=80.91958 L.","option_explanations":{"A":"Correct. The worked calculation gives 80.91958 L.","B":"Incorrect. Substituting the stated values into the governing relation gives 80.91958 L, not 40.45979 L.","C":"Incorrect. Substituting the stated values into the governing relation gives 80.91958 L, not 78.91958 L.","D":"Incorrect. Substituting the stated values into the governing relation gives 80.91958 L, not 161.83915 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_008286","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 9966 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":"39.34763 g","B":"19.67381 g","C":"9.83691 g","D":"15.73905 g"},"correct_answer":"B","correct_answer_value":"19.67381 g","solution":"Cu²⁺+2e⁻→Cu, so z=2. By Faraday's law m=MIt/(zF)=63.5×6×9966/(2×96500)=19.67381 g.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 19.67381 g, not 39.34763 g.","B":"Correct. The worked calculation gives 19.67381 g.","C":"Incorrect. Substituting the stated values into the governing relation gives 19.67381 g, not 9.83691 g.","D":"Incorrect. Substituting the stated values into the governing relation gives 19.67381 g, not 15.73905 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_004123","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 0.35. Taking g=10 m s⁻², find the net force down the plane. Use sin37°=0.6 and cos37°=0.8.","options":{"A":"128 N","B":"-128 N","C":"13.8 N","D":"126 N"},"correct_answer":"A","correct_answer_value":"128 N","solution":"Net force=mg(sinθ−μcosθ)=40×10×(0.6−0.35×0.8)=128 N.","option_explanations":{"A":"Correct. The worked calculation gives 128 N.","B":"Incorrect. Substituting the stated values into the governing relation gives 128 N, not -128 N.","C":"Incorrect. Substituting the stated values into the governing relation gives 128 N, not 13.8 N.","D":"Incorrect. Substituting the stated values into the governing relation gives 128 N, not 126 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_001028","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 -6x+7. Evaluate f′(-2).","options":{"A":"2","B":"3","C":"-1","D":"1"},"correct_answer":"A","correct_answer_value":"2","solution":"Differentiate: f′(x)=3(1)x²+2(1)x+(-6). At x=-2, f′(-2)=3(1)(-2)²+2(1)(-2)+(-6)=2.","option_explanations":{"A":"Correct. The worked calculation gives 2.","B":"Incorrect. Substituting the stated values into the governing relation gives 2, not 3.","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 1."},"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_004747","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 8 Ω and 12 Ω are connected in series; this combination is connected in parallel with a 9 Ω resistor across a 36 V ideal battery. Find the total current supplied by the battery.","options":{"A":"3.8 A","B":"5.8 A","C":"-5.8 A","D":"8.7 A"},"correct_answer":"B","correct_answer_value":"5.8 A","solution":"Series branch resistance=8+12=20 Ω. Parallel equivalent R_eq=(20×9)/(20+9)=6.2069 Ω. Total current I=V/R_eq=36/6.2069=5.8 A.","option_explanations":{"A":"Incorrect. Substituting the stated values into the governing relation gives 5.8 A, not 3.8 A.","B":"Correct. The worked calculation gives 5.8 A.","C":"Incorrect. Substituting the stated values into the governing relation gives 5.8 A, not -5.8 A.","D":"Incorrect. Substituting the stated values into the governing relation gives 5.8 A, not 8.7 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_007138","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 4 mol of gas at 300 K and 2.08 atm? Use R=0.082057 L atm mol⁻¹ K⁻¹.","options":{"A":"47.34058 L","B":"48.34058 L","C":"49.34058 L","D":"23.67029 L"},"correct_answer":"A","correct_answer_value":"47.34058 L","solution":"PV=nRT, so V=nRT/P=(4)(0.082057)(300)/(2.08)=47.34058 L.","option_explanations":{"A":"Correct. The worked calculation gives 47.34058 L.","B":"Incorrect. Substituting the stated values into the governing relation gives 47.34058 L, not 48.34058 L.","C":"Incorrect. Substituting the stated values into the governing relation gives 47.34058 L, not 49.34058 L.","D":"Incorrect. Substituting the stated values into the governing relation gives 47.34058 L, not 23.67029 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"}