id string | subject string | topic string | subtopic string | difficulty string | difficulty_labeling_method string | question_type string | question string | options dict | correct_answer string | correct_answer_value string | solution string | option_explanations dict | concepts_required list | reasoning_steps int64 | answer_type string | language string | origin string | generator string | generator_template_id string | verification_status string | verification_note string | difficulty_note string | source_type string | answer_validation string |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
openstem_001458 | mathematics | calculus | definite_integral_polynomial | very_hard | provisional_template_level_label | single_correct_mcq | Evaluate the definite integral ∫₀^8 (-1x² +0x +0) dx. | {
"A": "-170.66667",
"B": "-172.66667",
"C": "-168.66667",
"D": "170.66667"
} | A | -170.66667 | An antiderivative is (-1/3)x³+(0/2)x²+(0)x. Evaluating between 0 and 8 gives (-1/3)(8)³+(0/2)(8)²+(0)(8)=-170.66667. | {
"A": "Correct. The worked calculation gives -170.66667.",
"B": "Incorrect. Substituting the stated values into the governing relation gives -170.66667, not -172.66667.",
"C": "Incorrect. Substituting the stated values into the governing relation gives -170.66667, not -168.66667.",
"D": "Incorrect. Substitutin... | [
"definite integration",
"power rule",
"limits of integration"
] | 4 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | math_integral_poly | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_001908 | mathematics | combinatorics | binomial_coefficient | hard | provisional_template_level_label | single_correct_mcq | How many distinct subsets containing exactly 9 elements can be selected from a set of 58 distinct elements? | {
"A": "1064887396",
"B": "10648873950",
"C": "10648873949",
"D": "15973310925"
} | B | 10648873950 | Order does not matter, so the count is C(58,9)=58!/[9!(49)!]=10648873950. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 10648873950, not 1064887396.",
"B": "Correct. The worked calculation gives 10648873950.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 10648873950, not 10648873949.",
"D": "Incorrect. Substit... | [
"combinations",
"factorials"
] | 2 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | math_combinations | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_002634 | mathematics | linear_algebra | 2x2_determinant | hard | provisional_template_level_label | single_correct_mcq | Find the determinant of the matrix [[1, -1], [6, -8]]. | {
"A": "-3",
"B": "-4",
"C": "-2",
"D": "-1"
} | C | -2 | For a 2×2 matrix [[a,b],[c,d]], det=ad−bc. Here it is (1)(-8)−(-1)(6)=-2. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives -2, not -3.",
"B": "Incorrect. Substituting the stated values into the governing relation gives -2, not -4.",
"C": "Correct. The worked calculation gives -2.",
"D": "Incorrect. Substituting the stated values into the governing r... | [
"determinants",
"matrix operations"
] | 2 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | math_determinant_2x2 | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_000487 | mathematics | sequences_and_series | arithmetic_progression | hard | provisional_template_level_label | single_correct_mcq | An arithmetic progression has first term 10 and common difference 9. Find the sum of its first 9 terms. | {
"A": "415",
"B": "414",
"C": "331.2",
"D": "413"
} | B | 414 | Use Sₙ = n/2[2a+(n−1)d]. Substituting a=10, d=9, n=9: Sₙ=9/2[2(10)+(9−1)(9)]=414. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 414, not 415.",
"B": "Correct. The worked calculation gives 414.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 414, not 331.2.",
"D": "Incorrect. Substituting the stated values into the gove... | [
"arithmetic progression",
"finite series"
] | 3 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | math_ap_sum | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_004828 | physics | electricity | series_parallel_circuit | very_hard | provisional_template_level_label | single_correct_mcq | Resistors 12 Ω and 9 Ω are connected in series; this combination is connected in parallel with a 16 Ω resistor across a 36 V ideal battery. Find the total current supplied by the battery. | {
"A": "1.39643 A",
"B": "3.96429 A",
"C": "1.98214 A",
"D": "1.96429 A"
} | B | 3.96429 A | Series branch resistance=12+9=21 Ω. Parallel equivalent R_eq=(21×16)/(21+16)=9.08108 Ω. Total current I=V/R_eq=36/9.08108=3.96429 A. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 3.96429 A, not 1.39643 A.",
"B": "Correct. The worked calculation gives 3.96429 A.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 3.96429 A, not 1.98214 A.",
"D": "Incorrect. Substituting the... | [
"series resistors",
"parallel resistors",
"Ohm's law"
] | 4 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | physics_series_parallel | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_008607 | chemistry | physical_chemistry | boiling_point_elevation | hard | provisional_template_level_label | single_correct_mcq | A non-volatile solute is dissolved in water to produce a solution of molality 1.87 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. | {
"A": "1.122 K",
"B": "1.1122 K",
"C": "2.244 K",
"D": "-1.878 K"
} | A | 1.122 K | ΔT_b=iK_bm. Substitution gives ΔT_b=1×0.6×1.87=1.122 K. | {
"A": "Correct. The worked calculation gives 1.122 K.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 1.122 K, not 1.1122 K.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 1.122 K, not 2.244 K.",
"D": "Incorrect. Substituting the stated v... | [
"colligative properties",
"van't Hoff factor",
"molality"
] | 3 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | chem_boiling_elevation | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_000598 | mathematics | sequences_and_series | arithmetic_progression | hard | provisional_template_level_label | single_correct_mcq | An arithmetic progression has first term 15 and common difference 12. Find the sum of its first 31 terms. | {
"A": "7254",
"B": "4836",
"C": "6045",
"D": "6046"
} | C | 6045 | Use Sₙ = n/2[2a+(n−1)d]. Substituting a=15, d=12, n=31: Sₙ=31/2[2(15)+(31−1)(12)]=6045. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 6045, not 7254.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 6045, not 4836.",
"C": "Correct. The worked calculation gives 6045.",
"D": "Incorrect. Substituting the stated values into the g... | [
"arithmetic progression",
"finite series"
] | 3 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | math_ap_sum | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_003952 | physics | mechanics | work_energy_theorem | very_hard | provisional_template_level_label | single_correct_mcq | A 2 kg block initially moves at 1 m s⁻¹. A constant net force does 29 N of work over 5 m in the direction of motion. Find its final kinetic energy. | {
"A": "146 J",
"B": "109.5 J",
"C": "144 J",
"D": "147 J"
} | A | 146 J | By the work–energy theorem, K_f=K_i+W=½mu²+Fs=½(2)(1)²+(29)(5)=146 J. | {
"A": "Correct. The worked calculation gives 146 J.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 146 J, not 109.5 J.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 146 J, not 144 J.",
"D": "Incorrect. Substituting the stated values int... | [
"work–energy theorem",
"kinetic energy",
"work by constant force"
] | 4 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | physics_work_energy | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_003669 | physics | mechanics | projectile_range | hard | provisional_template_level_label | single_correct_mcq | A projectile is launched from level ground at 15 m s⁻¹ at 65° to the horizontal. Take g=10 m s⁻². Find its horizontal range. | {
"A": "12.927 m",
"B": "8.618 m",
"C": "17.236 m",
"D": "34.472 m"
} | C | 17.236 m | Range R=u²sin(2θ)/g. Substituting u=15, θ=65° and g=10 gives R=(15)²sin(130°)/10=17.236 m. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 17.236 m, not 12.927 m.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 17.236 m, not 8.618 m.",
"C": "Correct. The worked calculation gives 17.236 m.",
"D": "Incorrect. Substituting the state... | [
"projectile motion",
"range equation",
"trigonometry"
] | 4 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | physics_projectile_range | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_001082 | mathematics | calculus | polynomial_differentiation | hard | provisional_template_level_label | single_correct_mcq | Let f(x)=6x³ -6x² -10x+7. Evaluate f′(4). | {
"A": "230",
"B": "115",
"C": "231",
"D": "229"
} | A | 230 | Differentiate: f′(x)=3(6)x²+2(-6)x+(-10). At x=4, f′(4)=3(6)(4)²+2(-6)(4)+(-10)=230. | {
"A": "Correct. The worked calculation gives 230.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 230, not 115.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 230, not 231.",
"D": "Incorrect. Substituting the stated values into the govern... | [
"power rule",
"polynomial derivative"
] | 3 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | math_derivative_cubic | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_005402 | physics | electrostatics | capacitor_energy | hard | provisional_template_level_label | single_correct_mcq | A 49 μF capacitor is charged to a potential difference of 9 V. Calculate the energy stored, expressed in μJ. | {
"A": "2381.4 μJ",
"B": "199.45 μJ",
"C": "1984.5 μJ",
"D": "1983.5 μJ"
} | C | 1984.5 μJ | U=½CV². Since C=49 μF, U in μJ is ½×49×(9)²=1984.5 μJ. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 1984.5 μJ, not 2381.4 μJ.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 1984.5 μJ, not 199.45 μJ.",
"C": "Correct. The worked calculation gives 1984.5 μJ.",
"D": "Incorrect. Substituting the... | [
"capacitance",
"energy stored in a capacitor"
] | 3 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | physics_capacitor_energy | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_006638 | physics | oscillations | mass_spring_period_over_pi | hard | provisional_template_level_label | single_correct_mcq | A mass of 10 kg is attached to an ideal spring of force constant 238 N m⁻¹ and oscillates horizontally without damping. What is the time period divided by π? | {
"A": "-0.59004 s",
"B": "3.40996 s",
"C": "-1.59004 s",
"D": "0.40996 s"
} | D | 0.40996 s | For a mass–spring oscillator, T=2π√(m/k). Therefore T/π=2√(10/238)=0.40996 s. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 0.40996 s, not -0.59004 s.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 0.40996 s, not 3.40996 s.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 0.40996 s... | [
"simple harmonic motion",
"mass–spring oscillator",
"time period"
] | 3 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | physics_shm_period | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_006880 | chemistry | stoichiometry | limiting_reagent_water | very_hard | provisional_template_level_label | single_correct_mcq | Hydrogen reacts with oxygen according to 2H₂ + O₂ → 2H₂O. If 31 mol H₂ and 9 mol O₂ are mixed and the reaction goes to completion, how many moles of H₂O form? | {
"A": "21.6 mol",
"B": "13.5 mol",
"C": "18 mol",
"D": "2.8 mol"
} | C | 18 mol | From 2 mol H₂ and 1 mol O₂, 2 mol H₂O form. H₂ can produce 31 mol H₂O; O₂ can produce 18 mol H₂O. The limiting condition is O₂; product=18 mol. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 18 mol, not 21.6 mol.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 18 mol, not 13.5 mol.",
"C": "Correct. The worked calculation gives 18 mol.",
"D": "Incorrect. Substituting the stated val... | [
"balanced chemical equations",
"limiting reagent",
"mole ratios"
] | 4 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | chem_limiting_reagent | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_008546 | chemistry | physical_chemistry | boiling_point_elevation | hard | provisional_template_level_label | single_correct_mcq | A non-volatile solute is dissolved in water to produce a solution of molality 0.69 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. | {
"A": "1.76 K",
"B": "2.07 K",
"C": "2.76 K",
"D": "5.76 K"
} | C | 2.76 K | ΔT_b=iK_bm. Substitution gives ΔT_b=4×1.0×0.69=2.76 K. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 2.76 K, not 1.76 K.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 2.76 K, not 2.07 K.",
"C": "Correct. The worked calculation gives 2.76 K.",
"D": "Incorrect. Substituting the stated values ... | [
"colligative properties",
"van't Hoff factor",
"molality"
] | 3 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | chem_boiling_elevation | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_009686 | chemistry | inorganic_chemistry | complex_ion_oxidation_state | hard | provisional_template_level_label | single_correct_mcq | For a hypothetical coordination ion [Cu(OH)_6]^-2, each OH ligand has charge −1. Determine the formal oxidation state of Cu. | {
"A": "8",
"B": "2",
"C": "3",
"D": "4"
} | D | 4 | Let the formal oxidation state of Cu be x. Charge balance gives x+(6)(−1)=-2; hence x=-2+6=+4. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 4, not 8.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 4, not 2.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 4, not 3.",
"D": "Correct. The worked ca... | [
"coordination compounds",
"oxidation state",
"charge balance"
] | 3 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | chem_complex_oxidation | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_008478 | chemistry | physical_chemistry | boiling_point_elevation | hard | provisional_template_level_label | single_correct_mcq | A non-volatile solute is dissolved in water to produce a solution of molality 1.86 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. | {
"A": "5.85696 K",
"B": "2.85696 K",
"C": "4.28544 K",
"D": "1.2857 K"
} | B | 2.85696 K | ΔT_b=iK_bm. Substitution gives ΔT_b=3×0.512×1.86=2.85696 K. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 2.85696 K, not 5.85696 K.",
"B": "Correct. The worked calculation gives 2.85696 K.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 2.85696 K, not 4.28544 K.",
"D": "Incorrect. Substituting the... | [
"colligative properties",
"van't Hoff factor",
"molality"
] | 3 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | chem_boiling_elevation | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_000069 | mathematics | algebra | quadratic_roots | hard | provisional_template_level_label | single_correct_mcq | For the quadratic equation 5x^2 +2x +7=0, let its roots be α and β. What is α²+β²? | {
"A": "-1.32",
"B": "-5.64",
"C": "-3.96",
"D": "-2.64"
} | D | -2.64 | By Vieta's relations, α+β = -2/5 and αβ = 7/5. Then α²+β²=(α+β)²−2αβ = (-2/5)²−2(7/5) = -2.64. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives -2.64, not -1.32.",
"B": "Incorrect. Substituting the stated values into the governing relation gives -2.64, not -5.64.",
"C": "Incorrect. Substituting the stated values into the governing relation gives -2.64, not -3.96.",
"D":... | [
"Vieta's formulas",
"symmetric polynomial expressions"
] | 3 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | math_quadratic_vieta | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_001204 | mathematics | calculus | polynomial_differentiation | hard | provisional_template_level_label | single_correct_mcq | Let f(x)=2x³ -2x² -9x+7. Evaluate f′(-1). | {
"A": "-2",
"B": "0",
"C": "-1",
"D": "1"
} | D | 1 | Differentiate: f′(x)=3(2)x²+2(-2)x+(-9). At x=-1, f′(-1)=3(2)(-1)²+2(-2)(-1)+(-9)=1. | {
"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 -1.",
"D": "Correct. The worked ... | [
"power rule",
"polynomial derivative"
] | 3 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | math_derivative_cubic | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_003574 | physics | mechanics | projectile_range | hard | provisional_template_level_label | single_correct_mcq | A projectile is launched from level ground at 12 m s⁻¹ at 20° to the horizontal. Take g=10 m s⁻². Find its horizontal range. | {
"A": "18.51228 m",
"B": "6.25614 m",
"C": "9.25614 m",
"D": "11.25614 m"
} | C | 9.25614 m | Range R=u²sin(2θ)/g. Substituting u=12, θ=20° and g=10 gives R=(12)²sin(40°)/10=9.25614 m. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 9.25614 m, not 18.51228 m.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 9.25614 m, not 6.25614 m.",
"C": "Correct. The worked calculation gives 9.25614 m.",
"D": "Incorrect. Substituting th... | [
"projectile motion",
"range equation",
"trigonometry"
] | 4 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | physics_projectile_range | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_005999 | physics | thermodynamics | ideal_gas_isochoric_pressure | hard | provisional_template_level_label | single_correct_mcq | A fixed amount of ideal gas is kept in a rigid container. Its pressure is 1.8 atm at 247 K. What is its pressure at 747 K? | {
"A": "8.44372 atm",
"B": "7.44372 atm",
"C": "5.44372 atm",
"D": "3.44372 atm"
} | C | 5.44372 atm | At constant volume, P/T is constant. P₂=P₁T₂/T₁=(1.8)(747)/(247)=5.44372 atm. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 5.44372 atm, not 8.44372 atm.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 5.44372 atm, not 7.44372 atm.",
"C": "Correct. The worked calculation gives 5.44372 atm.",
"D": "Incorrect. Substi... | [
"ideal gas law",
"isochoric process",
"absolute temperature"
] | 3 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | physics_isochoric_gas | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_003119 | mathematics | complex_numbers | real_part_of_complex_quotient | very_hard | provisional_template_level_label | single_correct_mcq | Let z=(-8+15i)/(11+9i). What is Re(z), the real part of z? | {
"A": "2.23267",
"B": "0.23267",
"C": "1.23267",
"D": "0.46535"
} | B | 0.23267 | Multiply numerator and denominator by the conjugate (11-9i). Then Re(z)=(ac+bd)/(c²+d²)=((-8)(11)+(15)(9))/(11²+(9)²)=0.23267. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 0.23267, not 2.23267.",
"B": "Correct. The worked calculation gives 0.23267.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 0.23267, not 1.23267.",
"D": "Incorrect. Substituting the stated va... | [
"complex conjugate",
"complex division",
"real and imaginary parts"
] | 4 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | math_complex_quotient | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_007155 | chemistry | physical_chemistry | ideal_gas_volume | hard | provisional_template_level_label | single_correct_mcq | Assuming ideal behaviour, what volume is occupied by 5 mol of gas at 534 K and 2.74 atm? Use R=0.082057 L atm mol⁻¹ K⁻¹. | {
"A": "8.99607 L",
"B": "39.98033 L",
"C": "79.96065 L",
"D": "-79.96065 L"
} | C | 79.96065 L | PV=nRT, so V=nRT/P=(5)(0.082057)(534)/(2.74)=79.96065 L. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 79.96065 L, not 8.99607 L.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 79.96065 L, not 39.98033 L.",
"C": "Correct. The worked calculation gives 79.96065 L.",
"D": "Incorrect. Substituting... | [
"ideal gas equation",
"unit consistency"
] | 3 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | chem_ideal_gas | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_006959 | chemistry | stoichiometry | limiting_reagent_water | very_hard | provisional_template_level_label | single_correct_mcq | Hydrogen reacts with oxygen according to 2H₂ + O₂ → 2H₂O. If 15 mol H₂ and 29 mol O₂ are mixed and the reaction goes to completion, how many moles of H₂O form? | {
"A": "11.25 mol",
"B": "22.5 mol",
"C": "15 mol",
"D": "18 mol"
} | C | 15 mol | From 2 mol H₂ and 1 mol O₂, 2 mol H₂O form. H₂ can produce 15 mol H₂O; O₂ can produce 58 mol H₂O. The limiting condition is H₂; product=15 mol. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 15 mol, not 11.25 mol.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 15 mol, not 22.5 mol.",
"C": "Correct. The worked calculation gives 15 mol.",
"D": "Incorrect. Substituting the stated va... | [
"balanced chemical equations",
"limiting reagent",
"mole ratios"
] | 4 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | chem_limiting_reagent | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_001975 | mathematics | combinatorics | binomial_coefficient | hard | provisional_template_level_label | single_correct_mcq | How many distinct subsets containing exactly 5 elements can be selected from a set of 70 distinct elements? | {
"A": "12103012",
"B": "12103013",
"C": "12103014",
"D": "12103016"
} | C | 12103014 | Order does not matter, so the count is C(70,5)=70!/[5!(65)!]=12103014. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 12103014, not 12103012.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 12103014, not 12103013.",
"C": "Correct. The worked calculation gives 12103014.",
"D": "Incorrect. Substituting the stat... | [
"combinations",
"factorials"
] | 2 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | math_combinations | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_000122 | mathematics | algebra | quadratic_roots | hard | provisional_template_level_label | single_correct_mcq | For the quadratic equation 1x^2 +14x +3=0, let its roots be α and β. What is α²+β²? | {
"A": "-190",
"B": "95",
"C": "190",
"D": "152"
} | C | 190 | By Vieta's relations, α+β = -14/1 and αβ = 3/1. Then α²+β²=(α+β)²−2αβ = (-14/1)²−2(3/1) = 190. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 190, not -190.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 190, not 95.",
"C": "Correct. The worked calculation gives 190.",
"D": "Incorrect. Substituting the stated values into the govern... | [
"Vieta's formulas",
"symmetric polynomial expressions"
] | 3 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | math_quadratic_vieta | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_002489 | mathematics | linear_algebra | 2x2_determinant | hard | provisional_template_level_label | single_correct_mcq | Find the determinant of the matrix [[7, 4], [5, 0]]. | {
"A": "-21",
"B": "-1",
"C": "-20",
"D": "-18"
} | C | -20 | For a 2×2 matrix [[a,b],[c,d]], det=ad−bc. Here it is (7)(0)−(4)(5)=-20. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives -20, not -21.",
"B": "Incorrect. Substituting the stated values into the governing relation gives -20, not -1.",
"C": "Correct. The worked calculation gives -20.",
"D": "Incorrect. Substituting the stated values into the governi... | [
"determinants",
"matrix operations"
] | 2 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | math_determinant_2x2 | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_002173 | mathematics | probability | same_colour_without_replacement | very_hard | provisional_template_level_label | single_correct_mcq | A bag contains 11 red, 8 blue and 4 green balls. Two balls are drawn uniformly at random without replacement. What is the probability that both have the same colour? | {
"A": "0.70356",
"B": "-1.64822",
"C": "0.35178",
"D": "1.03518"
} | C | 0.35178 | Favourable pairs = C(11,2)+C(8,2)+C(4,2)=89. Total pairs=C(23,2)=253. Therefore P=89/253=0.35178. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 0.35178, not 0.70356.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 0.35178, not -1.64822.",
"C": "Correct. The worked calculation gives 0.35178.",
"D": "Incorrect. Substituting the stated v... | [
"classical probability",
"combinations",
"without replacement"
] | 4 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | math_probability_colours | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_003312 | mathematics | complex_numbers | real_part_of_complex_quotient | very_hard | provisional_template_level_label | single_correct_mcq | Let z=(6-9i)/(9+7i). What is Re(z), the real part of z? | {
"A": "-0.05192",
"B": "-3.06923",
"C": "-0.06923",
"D": "-2.06923"
} | C | -0.06923 | Multiply numerator and denominator by the conjugate (9-7i). Then Re(z)=(ac+bd)/(c²+d²)=((6)(9)+(-9)(7))/(9²+(7)²)=-0.06923. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives -0.06923, not -0.05192.",
"B": "Incorrect. Substituting the stated values into the governing relation gives -0.06923, not -3.06923.",
"C": "Correct. The worked calculation gives -0.06923.",
"D": "Incorrect. Substituting the stat... | [
"complex conjugate",
"complex division",
"real and imaginary parts"
] | 4 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | math_complex_quotient | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_002050 | mathematics | probability | same_colour_without_replacement | very_hard | provisional_template_level_label | single_correct_mcq | A bag contains 3 red, 9 blue and 7 green balls. Two balls are drawn uniformly at random without replacement. What is the probability that both have the same colour? | {
"A": "-2.64912",
"B": "0.17544",
"C": "0.35088",
"D": "1.35088"
} | C | 0.35088 | Favourable pairs = C(3,2)+C(9,2)+C(7,2)=60. Total pairs=C(19,2)=171. Therefore P=60/171=0.35088. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 0.35088, not -2.64912.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 0.35088, not 0.17544.",
"C": "Correct. The worked calculation gives 0.35088.",
"D": "Incorrect. Substituting the stated v... | [
"classical probability",
"combinations",
"without replacement"
] | 4 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | math_probability_colours | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_002324 | mathematics | probability | same_colour_without_replacement | very_hard | provisional_template_level_label | single_correct_mcq | A bag contains 5 red, 8 blue and 5 green balls. Two balls are drawn uniformly at random without replacement. What is the probability that both have the same colour? | {
"A": "0.23529",
"B": "0.62745",
"C": "0.31373",
"D": "0.47059"
} | C | 0.31373 | Favourable pairs = C(5,2)+C(8,2)+C(5,2)=48. Total pairs=C(18,2)=153. Therefore P=48/153=0.31373. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 0.31373, not 0.23529.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 0.31373, not 0.62745.",
"C": "Correct. The worked calculation gives 0.31373.",
"D": "Incorrect. Substituting the stated va... | [
"classical probability",
"combinations",
"without replacement"
] | 4 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | math_probability_colours | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_000184 | mathematics | algebra | quadratic_roots | hard | provisional_template_level_label | single_correct_mcq | For the quadratic equation 1x^2 +5x +11=0, let its roots be α and β. What is α²+β²? | {
"A": "3",
"B": "1",
"C": "6",
"D": "2.25"
} | A | 3 | By Vieta's relations, α+β = -5/1 and αβ = 11/1. Then α²+β²=(α+β)²−2αβ = (-5/1)²−2(11/1) = 3. | {
"A": "Correct. The worked calculation gives 3.",
"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": "Incorrect. Substituting the stated values into the governing relati... | [
"Vieta's formulas",
"symmetric polynomial expressions"
] | 3 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | math_quadratic_vieta | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_002544 | mathematics | linear_algebra | 2x2_determinant | hard | provisional_template_level_label | single_correct_mcq | Find the determinant of the matrix [[9, -5], [1, -4]]. | {
"A": "-15.5",
"B": "-46.5",
"C": "-31",
"D": "-33"
} | C | -31 | For a 2×2 matrix [[a,b],[c,d]], det=ad−bc. Here it is (9)(-4)−(-5)(1)=-31. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives -31, not -15.5.",
"B": "Incorrect. Substituting the stated values into the governing relation gives -31, not -46.5.",
"C": "Correct. The worked calculation gives -31.",
"D": "Incorrect. Substituting the stated values into the go... | [
"determinants",
"matrix operations"
] | 2 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | math_determinant_2x2 | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_009394 | chemistry | organic_chemistry | degree_of_unsaturation | very_hard | provisional_template_level_label | single_correct_mcq | Assuming a neutral closed-shell formula, calculate the degree of unsaturation (DBE) for C_16H_32N_1X_3. Treat X as the total number of monovalent halogen atoms. | {
"A": "2",
"B": "1",
"C": "-3",
"D": "0"
} | D | 0 | Use DBE=(2C+2+N−H−X)/2. Substitution gives (2×16+2+1−32−3)/2=0. | {
"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 c... | [
"molecular formula",
"degree of unsaturation",
"organic structure"
] | 3 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | chem_dbe | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_007603 | chemistry | physical_chemistry | strong_acid_base_mixture | very_hard | provisional_template_level_label | single_correct_mcq | 28 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. | {
"A": "10.56067",
"B": "11.56067",
"C": "13.56067",
"D": "12.56067"
} | D | 12.56067 | Excess OH⁻ moles=0.10×(60−28)/1000=0.003200 mol. Total volume=0.088 L, so [OH⁻]=0.036364 M, pOH=−log₁₀[OH⁻], and pH=14−pOH=12.56067. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 12.56067, not 10.56067.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 12.56067, not 11.56067.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 12.56067, not ... | [
"strong acid–strong base neutralization",
"molarity",
"pH and pOH"
] | 5 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | chem_acid_base_pH | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_008519 | chemistry | physical_chemistry | boiling_point_elevation | hard | provisional_template_level_label | single_correct_mcq | A non-volatile solute is dissolved in water to produce a solution of molality 1.54 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. | {
"A": "1.0924 K",
"B": "1.848 K",
"C": "0.924 K",
"D": "-0.924 K"
} | C | 0.924 K | ΔT_b=iK_bm. Substitution gives ΔT_b=1×0.6×1.54=0.924 K. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 0.924 K, not 1.0924 K.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 0.924 K, not 1.848 K.",
"C": "Correct. The worked calculation gives 0.924 K.",
"D": "Incorrect. Substituting the stated v... | [
"colligative properties",
"van't Hoff factor",
"molality"
] | 3 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | chem_boiling_elevation | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_007349 | chemistry | physical_chemistry | strong_acid_base_mixture | very_hard | provisional_template_level_label | single_correct_mcq | 66 mL of 0.10 M HCl is mixed with 53 mL of 0.10 M NaOH at 25°C. Assuming complete dissociation and additive volumes, find the final pH. | {
"A": "1.19616",
"B": "1.9616",
"C": "-1.0384",
"D": "2.9616"
} | B | 1.9616 | Excess H⁺ moles=0.10×(66−53)/1000=0.001300 mol. Total volume=0.119 L, so [H⁺]=0.010924 M and pH=−log₁₀[H⁺]=1.9616. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 1.9616, not 1.19616.",
"B": "Correct. The worked calculation gives 1.9616.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 1.9616, not -1.0384.",
"D": "Incorrect. Substituting the stated value... | [
"strong acid–strong base neutralization",
"molarity",
"pH and pOH"
] | 5 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | chem_acid_base_pH | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_005607 | physics | electrostatics | capacitor_energy | hard | provisional_template_level_label | single_correct_mcq | A 15 μF capacitor is charged to a potential difference of 13 V. Calculate the energy stored, expressed in μJ. | {
"A": "1267.5 μJ",
"B": "1269.5 μJ",
"C": "1266.5 μJ",
"D": "1268.5 μJ"
} | A | 1267.5 μJ | U=½CV². Since C=15 μF, U in μJ is ½×15×(13)²=1267.5 μJ. | {
"A": "Correct. The worked calculation gives 1267.5 μJ.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 1267.5 μJ, not 1269.5 μJ.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 1267.5 μJ, not 1266.5 μJ.",
"D": "Incorrect. Substituting the... | [
"capacitance",
"energy stored in a capacitor"
] | 3 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | physics_capacitor_energy | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_002837 | mathematics | vectors | dot_product | hard | provisional_template_level_label | single_correct_mcq | Given vectors u=(-6, -2, 1) and v=(1, -5, -6), calculate u·v. | {
"A": "-1.5",
"B": "-2",
"C": "-1",
"D": "0.8"
} | B | -2 | The dot product is the sum of corresponding component products: (-6)(1)+(-2)(-5)+(1)(-6)=-2. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives -2, not -1.5.",
"B": "Correct. The worked calculation gives -2.",
"C": "Incorrect. Substituting the stated values into the governing relation gives -2, not -1.",
"D": "Incorrect. Substituting the stated values into the governing... | [
"three-dimensional vectors",
"scalar product"
] | 3 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | math_vector_dot | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_008494 | chemistry | physical_chemistry | boiling_point_elevation | hard | provisional_template_level_label | single_correct_mcq | A non-volatile solute is dissolved in water to produce a solution of molality 1.83 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. | {
"A": "2.8548 K",
"B": "1.4274 K",
"C": "1.28548 K",
"D": "-0.1452 K"
} | A | 2.8548 K | ΔT_b=iK_bm. Substitution gives ΔT_b=3×0.52×1.83=2.8548 K. | {
"A": "Correct. The worked calculation gives 2.8548 K.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 2.8548 K, not 1.4274 K.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 2.8548 K, not 1.28548 K.",
"D": "Incorrect. Substituting the sta... | [
"colligative properties",
"van't Hoff factor",
"molality"
] | 3 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | chem_boiling_elevation | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_003434 | physics | mechanics | projectile_range | hard | provisional_template_level_label | single_correct_mcq | A projectile is launched from level ground at 69 m s⁻¹ at 55° to the horizontal. Take g=10 m s⁻². Find its horizontal range. | {
"A": "446.38766 m",
"B": "335.54074 m",
"C": "45.73877 m",
"D": "447.38766 m"
} | D | 447.38766 m | Range R=u²sin(2θ)/g. Substituting u=69, θ=55° and g=10 gives R=(69)²sin(110°)/10=447.38766 m. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 447.38766 m, not 446.38766 m.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 447.38766 m, not 335.54074 m.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 44... | [
"projectile motion",
"range equation",
"trigonometry"
] | 4 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | physics_projectile_range | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_000360 | mathematics | sequences_and_series | arithmetic_progression | hard | provisional_template_level_label | single_correct_mcq | An arithmetic progression has first term -2 and common difference 4. Find the sum of its first 20 terms. | {
"A": "864",
"B": "1080",
"C": "-720",
"D": "720"
} | D | 720 | Use Sₙ = n/2[2a+(n−1)d]. Substituting a=-2, d=4, n=20: Sₙ=20/2[2(-2)+(20−1)(4)]=720. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 720, not 864.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 720, not 1080.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 720, not -720.",
"D": "Correct.... | [
"arithmetic progression",
"finite series"
] | 3 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | math_ap_sum | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_004808 | physics | electricity | series_parallel_circuit | very_hard | provisional_template_level_label | single_correct_mcq | Resistors 9 Ω and 6 Ω are connected in series; this combination is connected in parallel with a 2 Ω resistor across a 30 V ideal battery. Find the total current supplied by the battery. | {
"A": "8.5 A",
"B": "2.7 A",
"C": "18 A",
"D": "17 A"
} | D | 17 A | Series branch resistance=9+6=15 Ω. Parallel equivalent R_eq=(15×2)/(15+2)=1.76471 Ω. Total current I=V/R_eq=30/1.76471=17 A. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 17 A, not 8.5 A.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 17 A, not 2.7 A.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 17 A, not 18 A.",
"D": "Co... | [
"series resistors",
"parallel resistors",
"Ohm's law"
] | 4 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | physics_series_parallel | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_006634 | physics | oscillations | mass_spring_period_over_pi | hard | provisional_template_level_label | single_correct_mcq | A mass of 15 kg is attached to an ideal spring of force constant 63 N m⁻¹ and oscillates horizontally without damping. What is the time period divided by π? | {
"A": "1.9759 s",
"B": "-0.9759 s",
"C": "0.9759 s",
"D": "0.48795 s"
} | C | 0.9759 s | For a mass–spring oscillator, T=2π√(m/k). Therefore T/π=2√(15/63)=0.9759 s. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 0.9759 s, not 1.9759 s.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 0.9759 s, not -0.9759 s.",
"C": "Correct. The worked calculation gives 0.9759 s.",
"D": "Incorrect. Substituting the sta... | [
"simple harmonic motion",
"mass–spring oscillator",
"time period"
] | 3 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | physics_shm_period | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_003304 | mathematics | complex_numbers | real_part_of_complex_quotient | very_hard | provisional_template_level_label | single_correct_mcq | Let z=(-11+9i)/(2+3i). What is Re(z), the real part of z? | {
"A": "0.76923",
"B": "-1.61538",
"C": "3.38462",
"D": "0.38462"
} | D | 0.38462 | Multiply numerator and denominator by the conjugate (2-3i). Then Re(z)=(ac+bd)/(c²+d²)=((-11)(2)+(9)(3))/(2²+(3)²)=0.38462. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 0.38462, not 0.76923.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 0.38462, not -1.61538.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 0.38462, not 3.38... | [
"complex conjugate",
"complex division",
"real and imaginary parts"
] | 4 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | math_complex_quotient | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_007714 | chemistry | physical_chemistry | first_order_half_life | hard | provisional_template_level_label | single_correct_mcq | A first-order reaction has a half-life of 20 minutes. If the initial reactant concentration is 10 mmol L⁻¹, what concentration remains after 80 minutes? | {
"A": "-0.625 mmol L⁻¹",
"B": "0.46875 mmol L⁻¹",
"C": "-1.375 mmol L⁻¹",
"D": "0.625 mmol L⁻¹"
} | D | 0.625 mmol L⁻¹ | 80 minutes corresponds to 4 half-lives. The remaining concentration is [A]₀(1/2)ⁿ=10/2^4=0.625 mmol L⁻¹. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 0.625 mmol L⁻¹, not -0.625 mmol L⁻¹.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 0.625 mmol L⁻¹, not 0.46875 mmol L⁻¹.",
"C": "Incorrect. Substituting the stated values into the governing re... | [
"first-order kinetics",
"half-life",
"exponential decay"
] | 3 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | chem_first_order_halflife | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_001811 | mathematics | combinatorics | binomial_coefficient | hard | provisional_template_level_label | single_correct_mcq | How many distinct subsets containing exactly 2 elements can be selected from a set of 27 distinct elements? | {
"A": "352",
"B": "353",
"C": "349",
"D": "351"
} | D | 351 | Order does not matter, so the count is C(27,2)=27!/[2!(25)!]=351. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 351, not 352.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 351, not 353.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 351, not 349.",
"D": "Correct. T... | [
"combinations",
"factorials"
] | 2 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | math_combinations | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_009359 | chemistry | organic_chemistry | degree_of_unsaturation | very_hard | provisional_template_level_label | single_correct_mcq | Assuming a neutral closed-shell formula, calculate the degree of unsaturation (DBE) for C_13H_20N_2X_4. Treat X as the total number of monovalent halogen atoms. | {
"A": "6",
"B": "-3",
"C": "4",
"D": "3"
} | D | 3 | Use DBE=(2C+2+N−H−X)/2. Substitution gives (2×13+2+2−20−4)/2=3. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 3, not 6.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 3, not -3.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 3, not 4.",
"D": "Correct. The worked c... | [
"molecular formula",
"degree of unsaturation",
"organic structure"
] | 3 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | chem_dbe | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_005983 | physics | thermodynamics | ideal_gas_isochoric_pressure | hard | provisional_template_level_label | single_correct_mcq | A fixed amount of ideal gas is kept in a rigid container. Its pressure is 4.6 atm at 221 K. What is its pressure at 363 K? | {
"A": "3.77783 atm",
"B": "8.55566 atm",
"C": "7.55566 atm",
"D": "11.33348 atm"
} | C | 7.55566 atm | At constant volume, P/T is constant. P₂=P₁T₂/T₁=(4.6)(363)/(221)=7.55566 atm. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 7.55566 atm, not 3.77783 atm.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 7.55566 atm, not 8.55566 atm.",
"C": "Correct. The worked calculation gives 7.55566 atm.",
"D": "Incorrect. Substi... | [
"ideal gas law",
"isochoric process",
"absolute temperature"
] | 3 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | physics_isochoric_gas | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_005399 | physics | electrostatics | capacitor_energy | hard | provisional_template_level_label | single_correct_mcq | A 63 μF capacitor is charged to a potential difference of 11 V. Calculate the energy stored, expressed in μJ. | {
"A": "3811.5 μJ",
"B": "3049.2 μJ",
"C": "4573.8 μJ",
"D": "7623 μJ"
} | A | 3811.5 μJ | U=½CV². Since C=63 μF, U in μJ is ½×63×(11)²=3811.5 μJ. | {
"A": "Correct. The worked calculation gives 3811.5 μJ.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 3811.5 μJ, not 3049.2 μJ.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 3811.5 μJ, not 4573.8 μJ.",
"D": "Incorrect. Substituting the... | [
"capacitance",
"energy stored in a capacitor"
] | 3 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | physics_capacitor_energy | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_006362 | physics | oscillations | mass_spring_period_over_pi | hard | provisional_template_level_label | single_correct_mcq | A mass of 10 kg is attached to an ideal spring of force constant 1144 N m⁻¹ and oscillates horizontally without damping. What is the time period divided by π? | {
"A": "2.18699 s",
"B": "0.09349 s",
"C": "0.14024 s",
"D": "0.18699 s"
} | D | 0.18699 s | For a mass–spring oscillator, T=2π√(m/k). Therefore T/π=2√(10/1144)=0.18699 s. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 0.18699 s, not 2.18699 s.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 0.18699 s, not 0.09349 s.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 0.18699 s,... | [
"simple harmonic motion",
"mass–spring oscillator",
"time period"
] | 3 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | physics_shm_period | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_007209 | chemistry | physical_chemistry | ideal_gas_volume | hard | provisional_template_level_label | single_correct_mcq | Assuming ideal behaviour, what volume is occupied by 4 mol of gas at 634 K and 0.27 atm? Use R=0.082057 L atm mol⁻¹ K⁻¹. | {
"A": "770.72797 L",
"B": "924.87356 L",
"C": "1156.09196 L",
"D": "616.58238 L"
} | A | 770.72797 L | PV=nRT, so V=nRT/P=(4)(0.082057)(634)/(0.27)=770.72797 L. | {
"A": "Correct. The worked calculation gives 770.72797 L.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 770.72797 L, not 924.87356 L.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 770.72797 L, not 1156.09196 L.",
"D": "Incorrect. Subst... | [
"ideal gas equation",
"unit consistency"
] | 3 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | chem_ideal_gas | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_001741 | mathematics | combinatorics | binomial_coefficient | hard | provisional_template_level_label | single_correct_mcq | How many distinct subsets containing exactly 10 elements can be selected from a set of 57 distinct elements? | {
"A": "43183019880",
"B": "-43183019880",
"C": "32387264910",
"D": "43183019881"
} | A | 43183019880 | Order does not matter, so the count is C(57,10)=57!/[10!(47)!]=43183019880. | {
"A": "Correct. The worked calculation gives 43183019880.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 43183019880, not -43183019880.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 43183019880, not 32387264910.",
"D": "Incorrect. Subst... | [
"combinations",
"factorials"
] | 2 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | math_combinations | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_008298 | chemistry | physical_chemistry | faradays_law_copper | very_hard | provisional_template_level_label | single_correct_mcq | An aqueous Cu²⁺ solution is electrolysed using a current of 5 A for 10730 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. | {
"A": "2.76517 g",
"B": "17.65168 g",
"C": "16.65168 g",
"D": "15.65168 g"
} | B | 17.65168 g | Cu²⁺+2e⁻→Cu, so z=2. By Faraday's law m=MIt/(zF)=63.5×5×10730/(2×96500)=17.65168 g. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 17.65168 g, not 2.76517 g.",
"B": "Correct. The worked calculation gives 17.65168 g.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 17.65168 g, not 16.65168 g.",
"D": "Incorrect. Substituting... | [
"electrolysis",
"Faraday's laws",
"redox stoichiometry"
] | 4 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | chem_faraday_copper | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_009981 | chemistry | inorganic_chemistry | complex_ion_oxidation_state | hard | provisional_template_level_label | single_correct_mcq | For a hypothetical coordination ion [Pd(Cl)_2]^-4, each Cl ligand has charge −1. Determine the formal oxidation state of Pd. | {
"A": "-1.5",
"B": "0.8",
"C": "-4",
"D": "-2"
} | D | -2 | Let the formal oxidation state of Pd be x. Charge balance gives x+(2)(−1)=-4; hence x=-4+2=-2. | {
"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 0.8.",
"C": "Incorrect. Substituting the stated values into the governing relation gives -2, not -4.",
"D": "Correct. The ... | [
"coordination compounds",
"oxidation state",
"charge balance"
] | 3 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | chem_complex_oxidation | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_009272 | chemistry | physical_chemistry | nernst_equation_298K | hard | provisional_template_level_label | single_correct_mcq | At 298 K, a galvanic cell has E°cell=1.09 V, transfers 3 electrons, and has reaction quotient Q=10^2. Using E=E°−(0.0591/n)log₁₀Q, calculate Ecell. | {
"A": "1.0506 V",
"B": "0.5253 V",
"C": "-0.9494 V",
"D": "-1.0506 V"
} | A | 1.0506 V | Apply the Nernst equation: E=1.09−(0.0591/3)(2)=1.0506 V. | {
"A": "Correct. The worked calculation gives 1.0506 V.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 1.0506 V, not 0.5253 V.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 1.0506 V, not -0.9494 V.",
"D": "Incorrect. Substituting the sta... | [
"electrochemical cells",
"Nernst equation",
"logarithms"
] | 3 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | chem_nernst | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_005984 | physics | thermodynamics | ideal_gas_isochoric_pressure | hard | provisional_template_level_label | single_correct_mcq | A fixed amount of ideal gas is kept in a rigid container. Its pressure is 3.5 atm at 330 K. What is its pressure at 794 K? | {
"A": "4.21061 atm",
"B": "8.42121 atm",
"C": "11.42121 atm",
"D": "10.42121 atm"
} | B | 8.42121 atm | At constant volume, P/T is constant. P₂=P₁T₂/T₁=(3.5)(794)/(330)=8.42121 atm. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 8.42121 atm, not 4.21061 atm.",
"B": "Correct. The worked calculation gives 8.42121 atm.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 8.42121 atm, not 11.42121 atm.",
"D": "Incorrect. Subst... | [
"ideal gas law",
"isochoric process",
"absolute temperature"
] | 3 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | physics_isochoric_gas | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_009540 | chemistry | organic_chemistry | degree_of_unsaturation | very_hard | provisional_template_level_label | single_correct_mcq | Assuming a neutral closed-shell formula, calculate the degree of unsaturation (DBE) for C_16H_25N_3X_4. Treat X as the total number of monovalent halogen atoms. | {
"A": "4",
"B": "1.4",
"C": "2",
"D": "6"
} | A | 4 | Use DBE=(2C+2+N−H−X)/2. Substitution gives (2×16+2+3−25−4)/2=4. | {
"A": "Correct. The worked calculation gives 4.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 4, not 1.4.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 4, not 2.",
"D": "Incorrect. Substituting the stated values into the governing rela... | [
"molecular formula",
"degree of unsaturation",
"organic structure"
] | 3 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | chem_dbe | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_005632 | physics | electrostatics | capacitor_energy | hard | provisional_template_level_label | single_correct_mcq | A 74 μF capacitor is charged to a potential difference of 19 V. Calculate the energy stored, expressed in μJ. | {
"A": "-13357 μJ",
"B": "13357 μJ",
"C": "10685.6 μJ",
"D": "13358 μJ"
} | B | 13357 μJ | U=½CV². Since C=74 μF, U in μJ is ½×74×(19)²=13357 μJ. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 13357 μJ, not -13357 μJ.",
"B": "Correct. The worked calculation gives 13357 μJ.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 13357 μJ, not 10685.6 μJ.",
"D": "Incorrect. Substituting the s... | [
"capacitance",
"energy stored in a capacitor"
] | 3 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | physics_capacitor_energy | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_008908 | chemistry | physical_chemistry | equilibrium_degree_dissociation | very_hard | provisional_template_level_label | single_correct_mcq | For A(g) ⇌ 2B(g), the initial concentration of A is 1.85 mol L⁻¹ and initially B is absent. At equilibrium, 45% of A has dissociated at constant volume. Find K_c. | {
"A": "0.72455",
"B": "-2.72455",
"C": "2.04341",
"D": "2.72455"
} | D | 2.72455 | Let α=0.45. At equilibrium [A]=c₀(1−α) and [B]=2c₀α. Thus K_c=[B]²/[A]=4c₀α²/(1−α)=4(1.85)(0.45)²/(1−0.45)=2.72455. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 2.72455, not 0.72455.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 2.72455, not -2.72455.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 2.72455, not 2.04... | [
"chemical equilibrium",
"equilibrium constant",
"degree of dissociation"
] | 4 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | chem_equilibrium_alpha | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_003551 | physics | mechanics | projectile_range | hard | provisional_template_level_label | single_correct_mcq | A projectile is launched from level ground at 80 m s⁻¹ at 65° to the horizontal. Take g=10 m s⁻². Find its horizontal range. | {
"A": "490.26844 m",
"B": "367.70133 m",
"C": "50.02684 m",
"D": "980.53689 m"
} | A | 490.26844 m | Range R=u²sin(2θ)/g. Substituting u=80, θ=65° and g=10 gives R=(80)²sin(130°)/10=490.26844 m. | {
"A": "Correct. The worked calculation gives 490.26844 m.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 490.26844 m, not 367.70133 m.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 490.26844 m, not 50.02684 m.",
"D": "Incorrect. Substit... | [
"projectile motion",
"range equation",
"trigonometry"
] | 4 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | physics_projectile_range | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_004904 | physics | electricity | series_parallel_circuit | very_hard | provisional_template_level_label | single_correct_mcq | Resistors 9 Ω and 3 Ω are connected in series; this combination is connected in parallel with a 2 Ω resistor across a 24 V ideal battery. Find the total current supplied by the battery. | {
"A": "28 A",
"B": "-14 A",
"C": "11 A",
"D": "14 A"
} | D | 14 A | Series branch resistance=9+3=12 Ω. Parallel equivalent R_eq=(12×2)/(12+2)=1.71429 Ω. Total current I=V/R_eq=24/1.71429=14 A. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 14 A, not 28 A.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 14 A, not -14 A.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 14 A, not 11 A.",
"D": "Cor... | [
"series resistors",
"parallel resistors",
"Ohm's law"
] | 4 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | physics_series_parallel | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_000537 | mathematics | sequences_and_series | arithmetic_progression | hard | provisional_template_level_label | single_correct_mcq | An arithmetic progression has first term -13 and common difference 14. Find the sum of its first 32 terms. | {
"A": "13056",
"B": "653.8",
"C": "3264",
"D": "6528"
} | D | 6528 | Use Sₙ = n/2[2a+(n−1)d]. Substituting a=-13, d=14, n=32: Sₙ=32/2[2(-13)+(32−1)(14)]=6528. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 6528, not 13056.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 6528, not 653.8.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 6528, not 3264.",
"D": "Co... | [
"arithmetic progression",
"finite series"
] | 3 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | math_ap_sum | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_009167 | chemistry | physical_chemistry | nernst_equation_298K | hard | provisional_template_level_label | single_correct_mcq | At 298 K, a galvanic cell has E°cell=1.42 V, transfers 2 electrons, and has reaction quotient Q=10^4. Using E=E°−(0.0591/n)log₁₀Q, calculate Ecell. | {
"A": "4.3018 V",
"B": "1.13018 V",
"C": "1.9527 V",
"D": "1.3018 V"
} | D | 1.3018 V | Apply the Nernst equation: E=1.42−(0.0591/2)(4)=1.3018 V. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 1.3018 V, not 4.3018 V.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 1.3018 V, not 1.13018 V.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 1.3018 V, not... | [
"electrochemical cells",
"Nernst equation",
"logarithms"
] | 3 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | chem_nernst | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_005326 | physics | electrostatics | coulombs_law | hard | provisional_template_level_label | single_correct_mcq | Two point charges of +3 μC and +2 μC are separated by 2 cm in vacuum. Using k=9×10⁹ N m² C⁻², find the magnitude of the electrostatic force. | {
"A": "134 N",
"B": "67.5 N",
"C": "135 N",
"D": "202.5 N"
} | C | 135 N | Convert charges to coulombs and distance to metres. F=k|q₁q₂|/r²=9×10⁹(3×10⁻⁶)(2×10⁻⁶)/(2×10⁻²)²=135 N. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 135 N, not 134 N.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 135 N, not 67.5 N.",
"C": "Correct. The worked calculation gives 135 N.",
"D": "Incorrect. Substituting the stated values into... | [
"Coulomb's law",
"SI unit conversion",
"inverse-square law"
] | 4 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | physics_coulomb | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_006260 | physics | ray_optics | convex_lens_magnification | very_hard | provisional_template_level_label | single_correct_mcq | A real object is placed 41 cm in front of a thin convex lens of focal length 9 cm. Using the Cartesian sign convention, what is the magnitude of the linear magnification? | {
"A": "-2.71875",
"B": "3.28125",
"C": "-1.71875",
"D": "0.28125"
} | D | 0.28125 | Using distances as positive magnitudes, 1/f=1/v+1/u gives v=uf/(u−f)=(41)(9)/(41−9)=11.53125 cm. Thus |m|=v/u=11.53125/41=0.28125. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 0.28125, not -2.71875.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 0.28125, not 3.28125.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 0.28125, not -1.7... | [
"thin lens equation",
"linear magnification",
"real image"
] | 4 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | physics_lens_magnification | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_002965 | mathematics | vectors | dot_product | hard | provisional_template_level_label | single_correct_mcq | Given vectors u=(4, 0, 4) and v=(-3, -1, 9), calculate u·v. | {
"A": "19.2",
"B": "24",
"C": "28.8",
"D": "12"
} | B | 24 | The dot product is the sum of corresponding component products: (4)(-3)+(0)(-1)+(4)(9)=24. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 24, not 19.2.",
"B": "Correct. The worked calculation gives 24.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 24, not 28.8.",
"D": "Incorrect. Substituting the stated values into the governi... | [
"three-dimensional vectors",
"scalar product"
] | 3 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | math_vector_dot | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_002934 | mathematics | vectors | dot_product | hard | provisional_template_level_label | single_correct_mcq | Given vectors u=(-3, 2, 3) and v=(9, 1, -4), calculate u·v. | {
"A": "-37",
"B": "-18.5",
"C": "-35",
"D": "-27.75"
} | A | -37 | The dot product is the sum of corresponding component products: (-3)(9)+(2)(1)+(3)(-4)=-37. | {
"A": "Correct. The worked calculation gives -37.",
"B": "Incorrect. Substituting the stated values into the governing relation gives -37, not -18.5.",
"C": "Incorrect. Substituting the stated values into the governing relation gives -37, not -35.",
"D": "Incorrect. Substituting the stated values into the gove... | [
"three-dimensional vectors",
"scalar product"
] | 3 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | math_vector_dot | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_008917 | chemistry | physical_chemistry | equilibrium_degree_dissociation | very_hard | provisional_template_level_label | single_correct_mcq | For A(g) ⇌ 2B(g), the initial concentration of A is 1.81 mol L⁻¹ and initially B is absent. At equilibrium, 23% of A has dissociated at constant volume. Find K_c. | {
"A": "2.4974",
"B": "-2.5026",
"C": "0.37305",
"D": "0.4974"
} | D | 0.4974 | Let α=0.23. At equilibrium [A]=c₀(1−α) and [B]=2c₀α. Thus K_c=[B]²/[A]=4c₀α²/(1−α)=4(1.81)(0.23)²/(1−0.23)=0.4974. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 0.4974, not 2.4974.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 0.4974, not -2.5026.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 0.4974, not 0.37305."... | [
"chemical equilibrium",
"equilibrium constant",
"degree of dissociation"
] | 4 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | chem_equilibrium_alpha | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_004975 | physics | electricity | series_parallel_circuit | very_hard | provisional_template_level_label | single_correct_mcq | Resistors 7 Ω and 9 Ω are connected in series; this combination is connected in parallel with a 2 Ω resistor across a 9 V ideal battery. Find the total current supplied by the battery. | {
"A": "3.0625 A",
"B": "5.0625 A",
"C": "-5.0625 A",
"D": "2.0625 A"
} | B | 5.0625 A | Series branch resistance=7+9=16 Ω. Parallel equivalent R_eq=(16×2)/(16+2)=1.77778 Ω. Total current I=V/R_eq=9/1.77778=5.0625 A. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 5.0625 A, not 3.0625 A.",
"B": "Correct. The worked calculation gives 5.0625 A.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 5.0625 A, not -5.0625 A.",
"D": "Incorrect. Substituting the sta... | [
"series resistors",
"parallel resistors",
"Ohm's law"
] | 4 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | physics_series_parallel | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_007437 | chemistry | physical_chemistry | strong_acid_base_mixture | very_hard | provisional_template_level_label | single_correct_mcq | 55 mL of 0.10 M HCl is mixed with 42 mL of 0.10 M NaOH at 25°C. Assuming complete dissociation and additive volumes, find the final pH. | {
"A": "1.87283",
"B": "-0.12717",
"C": "0.93641",
"D": "3.74566"
} | A | 1.87283 | Excess H⁺ moles=0.10×(55−42)/1000=0.001300 mol. Total volume=0.097 L, so [H⁺]=0.013402 M and pH=−log₁₀[H⁺]=1.87283. | {
"A": "Correct. The worked calculation gives 1.87283.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 1.87283, not -0.12717.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 1.87283, not 0.93641.",
"D": "Incorrect. Substituting the stated v... | [
"strong acid–strong base neutralization",
"molarity",
"pH and pOH"
] | 5 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | chem_acid_base_pH | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_005740 | physics | thermodynamics | ideal_gas_isochoric_pressure | hard | provisional_template_level_label | single_correct_mcq | A fixed amount of ideal gas is kept in a rigid container. Its pressure is 3.5 atm at 180 K. What is its pressure at 626 K? | {
"A": "24.34444 atm",
"B": "6.08611 atm",
"C": "12.17222 atm",
"D": "14.17222 atm"
} | C | 12.17222 atm | At constant volume, P/T is constant. P₂=P₁T₂/T₁=(3.5)(626)/(180)=12.17222 atm. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 12.17222 atm, not 24.34444 atm.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 12.17222 atm, not 6.08611 atm.",
"C": "Correct. The worked calculation gives 12.17222 atm.",
"D": "Incorrect. Su... | [
"ideal gas law",
"isochoric process",
"absolute temperature"
] | 3 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | physics_isochoric_gas | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_009993 | chemistry | inorganic_chemistry | complex_ion_oxidation_state | hard | provisional_template_level_label | single_correct_mcq | For a hypothetical coordination ion [Cr(OH)_6]^-2, each OH ligand has charge −1. Determine the formal oxidation state of Cr. | {
"A": "4",
"B": "3",
"C": "1",
"D": "2"
} | A | 4 | Let the formal oxidation state of Cr be x. Charge balance gives x+(6)(−1)=-2; hence x=-2+6=+4. | {
"A": "Correct. The worked calculation gives 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 1.",
"D": "Incorrect. Substituting the stated values into the governing relati... | [
"coordination compounds",
"oxidation state",
"charge balance"
] | 3 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | chem_complex_oxidation | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_006629 | physics | oscillations | mass_spring_period_over_pi | hard | provisional_template_level_label | single_correct_mcq | A mass of 6 kg is attached to an ideal spring of force constant 1140 N m⁻¹ and oscillates horizontally without damping. What is the time period divided by π? | {
"A": "0.1451 s",
"B": "3.1451 s",
"C": "1.1451 s",
"D": "0.29019 s"
} | A | 0.1451 s | For a mass–spring oscillator, T=2π√(m/k). Therefore T/π=2√(6/1140)=0.1451 s. | {
"A": "Correct. The worked calculation gives 0.1451 s.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 0.1451 s, not 3.1451 s.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 0.1451 s, not 1.1451 s.",
"D": "Incorrect. Substituting the stat... | [
"simple harmonic motion",
"mass–spring oscillator",
"time period"
] | 3 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | physics_shm_period | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_009126 | chemistry | physical_chemistry | nernst_equation_298K | hard | provisional_template_level_label | single_correct_mcq | At 298 K, a galvanic cell has E°cell=0.96 V, transfers 3 electrons, and has reaction quotient Q=10^0. Using E=E°−(0.0591/n)log₁₀Q, calculate Ecell. | {
"A": "0.48 V",
"B": "0.96 V",
"C": "-2.04 V",
"D": "-0.96 V"
} | B | 0.96 V | Apply the Nernst equation: E=0.96−(0.0591/3)(0)=0.96 V. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 0.96 V, not 0.48 V.",
"B": "Correct. The worked calculation gives 0.96 V.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 0.96 V, not -2.04 V.",
"D": "Incorrect. Substituting the stated values... | [
"electrochemical cells",
"Nernst equation",
"logarithms"
] | 3 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | chem_nernst | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_002947 | mathematics | vectors | dot_product | hard | provisional_template_level_label | single_correct_mcq | Given vectors u=(6, 2, -6) and v=(3, 8, 1), calculate u·v. | {
"A": "28",
"B": "29",
"C": "14",
"D": "27"
} | A | 28 | The dot product is the sum of corresponding component products: (6)(3)+(2)(8)+(-6)(1)=28. | {
"A": "Correct. The worked calculation gives 28.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 28, not 29.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 28, not 14.",
"D": "Incorrect. Substituting the stated values into the governing r... | [
"three-dimensional vectors",
"scalar product"
] | 3 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | math_vector_dot | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_006359 | physics | oscillations | mass_spring_period_over_pi | hard | provisional_template_level_label | single_correct_mcq | A mass of 6 kg is attached to an ideal spring of force constant 1108 N m⁻¹ and oscillates horizontally without damping. What is the time period divided by π? | {
"A": "1.01472 s",
"B": "-1.85282 s",
"C": "-2.85282 s",
"D": "0.14718 s"
} | D | 0.14718 s | For a mass–spring oscillator, T=2π√(m/k). Therefore T/π=2√(6/1108)=0.14718 s. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 0.14718 s, not 1.01472 s.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 0.14718 s, not -1.85282 s.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 0.14718 s... | [
"simple harmonic motion",
"mass–spring oscillator",
"time period"
] | 3 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | physics_shm_period | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_008104 | chemistry | physical_chemistry | faradays_law_copper | very_hard | provisional_template_level_label | single_correct_mcq | An aqueous Cu²⁺ solution is electrolysed using a current of 4 A for 16297 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. | {
"A": "19.44787 g",
"B": "32.1718 g",
"C": "21.44787 g",
"D": "3.14479 g"
} | C | 21.44787 g | Cu²⁺+2e⁻→Cu, so z=2. By Faraday's law m=MIt/(zF)=63.5×4×16297/(2×96500)=21.44787 g. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 21.44787 g, not 19.44787 g.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 21.44787 g, not 32.1718 g.",
"C": "Correct. The worked calculation gives 21.44787 g.",
"D": "Incorrect. Substituting... | [
"electrolysis",
"Faraday's laws",
"redox stoichiometry"
] | 4 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | chem_faraday_copper | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_008588 | chemistry | physical_chemistry | boiling_point_elevation | hard | provisional_template_level_label | single_correct_mcq | A non-volatile solute is dissolved in water to produce a solution of molality 1.13 mol kg⁻¹. If its van't Hoff factor is 2 and K_b for water is 1.0 K kg mol⁻¹, find the boiling-point elevation. | {
"A": "0.26 K",
"B": "3.26 K",
"C": "2.26 K",
"D": "1.26 K"
} | C | 2.26 K | ΔT_b=iK_bm. Substitution gives ΔT_b=2×1.0×1.13=2.26 K. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 2.26 K, not 0.26 K.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 2.26 K, not 3.26 K.",
"C": "Correct. The worked calculation gives 2.26 K.",
"D": "Incorrect. Substituting the stated values ... | [
"colligative properties",
"van't Hoff factor",
"molality"
] | 3 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | chem_boiling_elevation | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_008440 | chemistry | physical_chemistry | boiling_point_elevation | hard | provisional_template_level_label | single_correct_mcq | A non-volatile solute is dissolved in water to produce a solution of molality 0.1 mol kg⁻¹. If its van't Hoff factor is 1 and K_b for water is 0.512 K kg mol⁻¹, find the boiling-point elevation. | {
"A": "2.0512 K",
"B": "-0.9488 K",
"C": "0.0512 K",
"D": "0.1024 K"
} | C | 0.0512 K | ΔT_b=iK_bm. Substitution gives ΔT_b=1×0.512×0.1=0.0512 K. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 0.0512 K, not 2.0512 K.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 0.0512 K, not -0.9488 K.",
"C": "Correct. The worked calculation gives 0.0512 K.",
"D": "Incorrect. Substituting the sta... | [
"colligative properties",
"van't Hoff factor",
"molality"
] | 3 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | chem_boiling_elevation | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_007944 | chemistry | physical_chemistry | first_order_half_life | hard | provisional_template_level_label | single_correct_mcq | A first-order reaction has a half-life of 20 minutes. If the initial reactant concentration is 73 mmol L⁻¹, what concentration remains after 160 minutes? | {
"A": "1.28516 mmol L⁻¹",
"B": "0.28516 mmol L⁻¹",
"C": "0.14258 mmol L⁻¹",
"D": "1.02852 mmol L⁻¹"
} | B | 0.28516 mmol L⁻¹ | 160 minutes corresponds to 8 half-lives. The remaining concentration is [A]₀(1/2)ⁿ=73/2^8=0.28516 mmol L⁻¹. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 0.28516 mmol L⁻¹, not 1.28516 mmol L⁻¹.",
"B": "Correct. The worked calculation gives 0.28516 mmol L⁻¹.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 0.28516 mmol L⁻¹, not 0.14258 mmol L⁻¹.",
... | [
"first-order kinetics",
"half-life",
"exponential decay"
] | 3 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | chem_first_order_halflife | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_009150 | chemistry | physical_chemistry | nernst_equation_298K | hard | provisional_template_level_label | single_correct_mcq | At 298 K, a galvanic cell has E°cell=0.95 V, transfers 1 electrons, and has reaction quotient Q=10^3. Using E=E°−(0.0591/n)log₁₀Q, calculate Ecell. | {
"A": "0.38635 V",
"B": "0.7727 V",
"C": "1.15905 V",
"D": "1.07727 V"
} | B | 0.7727 V | Apply the Nernst equation: E=0.95−(0.0591/1)(3)=0.7727 V. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 0.7727 V, not 0.38635 V.",
"B": "Correct. The worked calculation gives 0.7727 V.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 0.7727 V, not 1.15905 V.",
"D": "Incorrect. Substituting the st... | [
"electrochemical cells",
"Nernst equation",
"logarithms"
] | 3 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | chem_nernst | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_000118 | mathematics | algebra | quadratic_roots | hard | provisional_template_level_label | single_correct_mcq | For the quadratic equation 7x^2 -11x -9=0, let its roots be α and β. What is α²+β²? | {
"A": "5.04082",
"B": "3.04082",
"C": "-5.04082",
"D": "1.50408"
} | A | 5.04082 | By Vieta's relations, α+β = 11/7 and αβ = -9/7. Then α²+β²=(α+β)²−2αβ = (11/7)²−2(-9/7) = 5.04082. | {
"A": "Correct. The worked calculation gives 5.04082.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 5.04082, not 3.04082.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 5.04082, not -5.04082.",
"D": "Incorrect. Substituting the stated v... | [
"Vieta's formulas",
"symmetric polynomial expressions"
] | 3 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | math_quadratic_vieta | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_004695 | physics | electricity | series_parallel_circuit | very_hard | provisional_template_level_label | single_correct_mcq | Resistors 5 Ω and 9 Ω are connected in series; this combination is connected in parallel with a 11 Ω resistor across a 12 V ideal battery. Find the total current supplied by the battery. | {
"A": "2.94805 A",
"B": "0.94805 A",
"C": "1.19481 A",
"D": "1.94805 A"
} | D | 1.94805 A | Series branch resistance=5+9=14 Ω. Parallel equivalent R_eq=(14×11)/(14+11)=6.16 Ω. Total current I=V/R_eq=12/6.16=1.94805 A. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 1.94805 A, not 2.94805 A.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 1.94805 A, not 0.94805 A.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 1.94805 A,... | [
"series resistors",
"parallel resistors",
"Ohm's law"
] | 4 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | physics_series_parallel | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_009924 | chemistry | inorganic_chemistry | complex_ion_oxidation_state | hard | provisional_template_level_label | single_correct_mcq | For a hypothetical coordination ion [Co(OH)_2]^+1, each OH ligand has charge −1. Determine the formal oxidation state of Co. | {
"A": "1",
"B": "3",
"C": "-3",
"D": "1.5"
} | B | 3 | Let the formal oxidation state of Co be x. Charge balance gives x+(2)(−1)=+1; hence x=1+2=+3. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 3, not 1.",
"B": "Correct. The worked calculation gives 3.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 3, not -3.",
"D": "Incorrect. Substituting the stated values into the governing relat... | [
"coordination compounds",
"oxidation state",
"charge balance"
] | 3 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | chem_complex_oxidation | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_006456 | physics | oscillations | mass_spring_period_over_pi | hard | provisional_template_level_label | single_correct_mcq | A mass of 8 kg is attached to an ideal spring of force constant 54 N m⁻¹ and oscillates horizontally without damping. What is the time period divided by π? | {
"A": "1.07698 s",
"B": "0.7698 s",
"C": "-0.2302 s",
"D": "0.57735 s"
} | B | 0.7698 s | For a mass–spring oscillator, T=2π√(m/k). Therefore T/π=2√(8/54)=0.7698 s. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 0.7698 s, not 1.07698 s.",
"B": "Correct. The worked calculation gives 0.7698 s.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 0.7698 s, not -0.2302 s.",
"D": "Incorrect. Substituting the st... | [
"simple harmonic motion",
"mass–spring oscillator",
"time period"
] | 3 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | physics_shm_period | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_002041 | mathematics | probability | same_colour_without_replacement | very_hard | provisional_template_level_label | single_correct_mcq | A bag contains 11 red, 6 blue and 9 green balls. Two balls are drawn uniformly at random without replacement. What is the probability that both have the same colour? | {
"A": "0.16308",
"B": "0.32615",
"C": "0.65231",
"D": "-2.67385"
} | B | 0.32615 | Favourable pairs = C(11,2)+C(6,2)+C(9,2)=106. Total pairs=C(26,2)=325. Therefore P=106/325=0.32615. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 0.32615, not 0.16308.",
"B": "Correct. The worked calculation gives 0.32615.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 0.32615, not 0.65231.",
"D": "Incorrect. Substituting the stated va... | [
"classical probability",
"combinations",
"without replacement"
] | 4 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | math_probability_colours | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_006626 | physics | oscillations | mass_spring_period_over_pi | hard | provisional_template_level_label | single_correct_mcq | A mass of 25 kg is attached to an ideal spring of force constant 299 N m⁻¹ and oscillates horizontally without damping. What is the time period divided by π? | {
"A": "-1.42169 s",
"B": "0.57831 s",
"C": "3.57831 s",
"D": "0.43374 s"
} | B | 0.57831 s | For a mass–spring oscillator, T=2π√(m/k). Therefore T/π=2√(25/299)=0.57831 s. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 0.57831 s, not -1.42169 s.",
"B": "Correct. The worked calculation gives 0.57831 s.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 0.57831 s, not 3.57831 s.",
"D": "Incorrect. Substituting th... | [
"simple harmonic motion",
"mass–spring oscillator",
"time period"
] | 3 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | physics_shm_period | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_005347 | physics | electrostatics | capacitor_energy | hard | provisional_template_level_label | single_correct_mcq | A 48 μF capacitor is charged to a potential difference of 20 V. Calculate the energy stored, expressed in μJ. | {
"A": "9600 μJ",
"B": "14400 μJ",
"C": "9601 μJ",
"D": "9602 μJ"
} | A | 9600 μJ | U=½CV². Since C=48 μF, U in μJ is ½×48×(20)²=9600 μJ. | {
"A": "Correct. The worked calculation gives 9600 μJ.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 9600 μJ, not 14400 μJ.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 9600 μJ, not 9601 μJ.",
"D": "Incorrect. Substituting the stated v... | [
"capacitance",
"energy stored in a capacitor"
] | 3 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | physics_capacitor_energy | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_004969 | physics | electricity | series_parallel_circuit | very_hard | provisional_template_level_label | single_correct_mcq | Resistors 11 Ω and 5 Ω are connected in series; this combination is connected in parallel with a 11 Ω resistor across a 12 V ideal battery. Find the total current supplied by the battery. | {
"A": "-1.84091 A",
"B": "3.84091 A",
"C": "1.84091 A",
"D": "0.84091 A"
} | C | 1.84091 A | Series branch resistance=11+5=16 Ω. Parallel equivalent R_eq=(16×11)/(16+11)=6.51852 Ω. Total current I=V/R_eq=12/6.51852=1.84091 A. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 1.84091 A, not -1.84091 A.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 1.84091 A, not 3.84091 A.",
"C": "Correct. The worked calculation gives 1.84091 A.",
"D": "Incorrect. Substituting th... | [
"series resistors",
"parallel resistors",
"Ohm's law"
] | 4 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | physics_series_parallel | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_000799 | mathematics | sequences_and_series | geometric_progression | hard | provisional_template_level_label | single_correct_mcq | A geometric progression has first term 3 and common ratio 4. What is the sum of its first 10 terms? | {
"A": "1048575",
"B": "1048577",
"C": "-1048575",
"D": "1258290"
} | A | 1048575 | For r≠1, Sₙ=a(rⁿ−1)/(r−1). Thus Sₙ=3(4^10−1)/(4−1)=1048575. | {
"A": "Correct. The worked calculation gives 1048575.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 1048575, not 1048577.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 1048575, not -1048575.",
"D": "Incorrect. Substituting the stated v... | [
"geometric progression",
"finite geometric sum"
] | 3 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | math_gp_sum | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_006186 | physics | ray_optics | convex_lens_magnification | very_hard | provisional_template_level_label | single_correct_mcq | A real object is placed 21 cm in front of a thin convex lens of focal length 7 cm. Using the Cartesian sign convention, what is the magnitude of the linear magnification? | {
"A": "-0.5",
"B": "3.5",
"C": "0.25",
"D": "0.5"
} | D | 0.5 | Using distances as positive magnitudes, 1/f=1/v+1/u gives v=uf/(u−f)=(21)(7)/(21−7)=10.5 cm. Thus |m|=v/u=10.5/21=0.5. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 0.5, not -0.5.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 0.5, not 3.5.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 0.5, not 0.25.",
"D": "Correct.... | [
"thin lens equation",
"linear magnification",
"real image"
] | 4 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | physics_lens_magnification | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_001019 | mathematics | calculus | polynomial_differentiation | hard | provisional_template_level_label | single_correct_mcq | Let f(x)=-1x³ +5x² +3x+7. Evaluate f′(-1). | {
"A": "-9",
"B": "-11",
"C": "0",
"D": "-10"
} | D | -10 | Differentiate: f′(x)=3(-1)x²+2(5)x+(3). At x=-1, f′(-1)=3(-1)(-1)²+2(5)(-1)+(3)=-10. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives -10, not -9.",
"B": "Incorrect. Substituting the stated values into the governing relation gives -10, not -11.",
"C": "Incorrect. Substituting the stated values into the governing relation gives -10, not 0.",
"D": "Correct. The ... | [
"power rule",
"polynomial derivative"
] | 3 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | math_derivative_cubic | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_006477 | physics | oscillations | mass_spring_period_over_pi | hard | provisional_template_level_label | single_correct_mcq | A mass of 13 kg is attached to an ideal spring of force constant 764 N m⁻¹ and oscillates horizontally without damping. What is the time period divided by π? | {
"A": "-1.73911 s",
"B": "1.02609 s",
"C": "0.26089 s",
"D": "0.52178 s"
} | C | 0.26089 s | For a mass–spring oscillator, T=2π√(m/k). Therefore T/π=2√(13/764)=0.26089 s. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 0.26089 s, not -1.73911 s.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 0.26089 s, not 1.02609 s.",
"C": "Correct. The worked calculation gives 0.26089 s.",
"D": "Incorrect. Substituting th... | [
"simple harmonic motion",
"mass–spring oscillator",
"time period"
] | 3 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | physics_shm_period | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_004730 | physics | electricity | series_parallel_circuit | very_hard | provisional_template_level_label | single_correct_mcq | Resistors 13 Ω and 9 Ω are connected in series; this combination is connected in parallel with a 3 Ω resistor across a 12 V ideal battery. Find the total current supplied by the battery. | {
"A": "2.54545 A",
"B": "7.54545 A",
"C": "9.09091 A",
"D": "4.54545 A"
} | D | 4.54545 A | Series branch resistance=13+9=22 Ω. Parallel equivalent R_eq=(22×3)/(22+3)=2.64 Ω. Total current I=V/R_eq=12/2.64=4.54545 A. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 4.54545 A, not 2.54545 A.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 4.54545 A, not 7.54545 A.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 4.54545 A,... | [
"series resistors",
"parallel resistors",
"Ohm's law"
] | 4 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | physics_series_parallel | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_008952 | chemistry | physical_chemistry | equilibrium_degree_dissociation | very_hard | provisional_template_level_label | single_correct_mcq | For A(g) ⇌ 2B(g), the initial concentration of A is 0.16 mol L⁻¹ and initially B is absent. At equilibrium, 53% of A has dissociated at constant volume. Find K_c. | {
"A": "0.19125",
"B": "0.3825",
"C": "2.3825",
"D": "-0.6175"
} | B | 0.3825 | Let α=0.53. At equilibrium [A]=c₀(1−α) and [B]=2c₀α. Thus K_c=[B]²/[A]=4c₀α²/(1−α)=4(0.16)(0.53)²/(1−0.53)=0.3825. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 0.3825, not 0.19125.",
"B": "Correct. The worked calculation gives 0.3825.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 0.3825, not 2.3825.",
"D": "Incorrect. Substituting the stated values... | [
"chemical equilibrium",
"equilibrium constant",
"degree of dissociation"
] | 4 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | chem_equilibrium_alpha | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_007983 | chemistry | physical_chemistry | first_order_half_life | hard | provisional_template_level_label | single_correct_mcq | A first-order reaction has a half-life of 20 minutes. If the initial reactant concentration is 15 mmol L⁻¹, what concentration remains after 120 minutes? | {
"A": "0.17578 mmol L⁻¹",
"B": "0.23438 mmol L⁻¹",
"C": "0.46875 mmol L⁻¹",
"D": "0.11719 mmol L⁻¹"
} | B | 0.23438 mmol L⁻¹ | 120 minutes corresponds to 6 half-lives. The remaining concentration is [A]₀(1/2)ⁿ=15/2^6=0.23438 mmol L⁻¹. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 0.23438 mmol L⁻¹, not 0.17578 mmol L⁻¹.",
"B": "Correct. The worked calculation gives 0.23438 mmol L⁻¹.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 0.23438 mmol L⁻¹, not 0.46875 mmol L⁻¹.",
... | [
"first-order kinetics",
"half-life",
"exponential decay"
] | 3 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | chem_first_order_halflife | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_003262 | mathematics | complex_numbers | real_part_of_complex_quotient | very_hard | provisional_template_level_label | single_correct_mcq | Let z=(7+14i)/(10+1i). What is Re(z), the real part of z? | {
"A": "0.41584",
"B": "1.83168",
"C": "3.83168",
"D": "0.83168"
} | D | 0.83168 | Multiply numerator and denominator by the conjugate (10-1i). Then Re(z)=(ac+bd)/(c²+d²)=((7)(10)+(14)(1))/(10²+(1)²)=0.83168. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 0.83168, not 0.41584.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 0.83168, not 1.83168.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 0.83168, not 3.831... | [
"complex conjugate",
"complex division",
"real and imaginary parts"
] | 4 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | math_complex_quotient | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_000142 | mathematics | algebra | quadratic_roots | hard | provisional_template_level_label | single_correct_mcq | For the quadratic equation 2x^2 +16x -4=0, let its roots be α and β. What is α²+β²? | {
"A": "66",
"B": "68",
"C": "136",
"D": "67"
} | B | 68 | By Vieta's relations, α+β = -16/2 and αβ = -4/2. Then α²+β²=(α+β)²−2αβ = (-16/2)²−2(-4/2) = 68. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 68, not 66.",
"B": "Correct. The worked calculation gives 68.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 68, not 136.",
"D": "Incorrect. Substituting the stated values into the governing ... | [
"Vieta's formulas",
"symmetric polynomial expressions"
] | 3 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | math_quadratic_vieta | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_009218 | chemistry | physical_chemistry | nernst_equation_298K | hard | provisional_template_level_label | single_correct_mcq | At 298 K, a galvanic cell has E°cell=1.10 V, transfers 1 electrons, and has reaction quotient Q=10^2. Using E=E°−(0.0591/n)log₁₀Q, calculate Ecell. | {
"A": "-0.9818 V",
"B": "-1.0182 V",
"C": "-0.0182 V",
"D": "0.9818 V"
} | D | 0.9818 V | Apply the Nernst equation: E=1.10−(0.0591/1)(2)=0.9818 V. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 0.9818 V, not -0.9818 V.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 0.9818 V, not -1.0182 V.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 0.9818 V, no... | [
"electrochemical cells",
"Nernst equation",
"logarithms"
] | 3 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | chem_nernst | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_002636 | mathematics | linear_algebra | 2x2_determinant | hard | provisional_template_level_label | single_correct_mcq | Find the determinant of the matrix [[9, -2], [5, 8]]. | {
"A": "81",
"B": "61.5",
"C": "82",
"D": "164"
} | C | 82 | For a 2×2 matrix [[a,b],[c,d]], det=ad−bc. Here it is (9)(8)−(-2)(5)=82. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 82, not 81.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 82, not 61.5.",
"C": "Correct. The worked calculation gives 82.",
"D": "Incorrect. Substituting the stated values into the governing... | [
"determinants",
"matrix operations"
] | 2 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | math_determinant_2x2 | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
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