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_001947 | mathematics | combinatorics | binomial_coefficient | hard | provisional_template_level_label | single_correct_mcq | How many distinct subsets containing exactly 8 elements can be selected from a set of 52 distinct elements? | {
"A": "752538150",
"B": "752538149",
"C": "752538148",
"D": "602030520"
} | A | 752538150 | Order does not matter, so the count is C(52,8)=52!/[8!(44)!]=752538150. | {
"A": "Correct. The worked calculation gives 752538150.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 752538150, not 752538149.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 752538150, not 752538148.",
"D": "Incorrect. Substituting the... | [
"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_003302 | mathematics | complex_numbers | real_part_of_complex_quotient | very_hard | provisional_template_level_label | single_correct_mcq | Let z=(6-6i)/(11+8i). What is Re(z), the real part of z? | {
"A": "0.0973",
"B": "1.0973",
"C": "0.07297",
"D": "-1.9027"
} | A | 0.0973 | Multiply numerator and denominator by the conjugate (11-8i). Then Re(z)=(ac+bd)/(c²+d²)=((6)(11)+(-6)(8))/(11²+(8)²)=0.0973. | {
"A": "Correct. The worked calculation gives 0.0973.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 0.0973, not 1.0973.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 0.0973, not 0.07297.",
"D": "Incorrect. Substituting the stated values... | [
"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_002959 | mathematics | vectors | dot_product | hard | provisional_template_level_label | single_correct_mcq | Given vectors u=(-7, 3, 5) and v=(-4, -7, -7), calculate u·v. | {
"A": "-27",
"B": "-21",
"C": "-22.4",
"D": "-28"
} | D | -28 | The dot product is the sum of corresponding component products: (-7)(-4)+(3)(-7)+(5)(-7)=-28. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives -28, not -27.",
"B": "Incorrect. Substituting the stated values into the governing relation gives -28, not -21.",
"C": "Incorrect. Substituting the stated values into the governing relation gives -28, not -22.4.",
"D": "Correct.... | [
"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_002593 | mathematics | linear_algebra | 2x2_determinant | hard | provisional_template_level_label | single_correct_mcq | Find the determinant of the matrix [[3, -5], [7, 3]]. | {
"A": "45",
"B": "46",
"C": "33",
"D": "44"
} | D | 44 | For a 2×2 matrix [[a,b],[c,d]], det=ad−bc. Here it is (3)(3)−(-5)(7)=44. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 44, not 45.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 44, not 46.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 44, not 33.",
"D": "Correct. The wor... | [
"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_005866 | 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 2.8 atm at 222 K. What is its pressure at 695 K? | {
"A": "8.76577 atm",
"B": "6.76577 atm",
"C": "-8.76577 atm",
"D": "6.57432 atm"
} | A | 8.76577 atm | At constant volume, P/T is constant. P₂=P₁T₂/T₁=(2.8)(695)/(222)=8.76577 atm. | {
"A": "Correct. The worked calculation gives 8.76577 atm.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 8.76577 atm, not 6.76577 atm.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 8.76577 atm, not -8.76577 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_009444 | 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_5H_2N_3X_1. Treat X as the total number of monovalent halogen atoms. | {
"A": "1.6",
"B": "-6",
"C": "4",
"D": "6"
} | D | 6 | Use DBE=(2C+2+N−H−X)/2. Substitution gives (2×5+2+3−2−1)/2=6. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 6, not 1.6.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 6, not -6.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 6, not 4.",
"D": "Correct. The worked... | [
"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_009892 | chemistry | inorganic_chemistry | complex_ion_oxidation_state | hard | provisional_template_level_label | single_correct_mcq | For a hypothetical coordination ion [Co(CN)_3]^-4, each CN ligand has charge −1. Determine the formal oxidation state of Co. | {
"A": "-0.75",
"B": "-2",
"C": "0",
"D": "-1"
} | D | -1 | Let the formal oxidation state of Co be x. Charge balance gives x+(3)(−1)=-4; hence x=-4+3=-1. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives -1, not -0.75.",
"B": "Incorrect. Substituting the stated values into the governing relation gives -1, not -2.",
"C": "Incorrect. Substituting the stated values into the governing relation gives -1, not 0.",
"D": "Correct. The w... | [
"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_002203 | mathematics | probability | same_colour_without_replacement | very_hard | provisional_template_level_label | single_correct_mcq | A bag contains 6 red, 9 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": "0.5",
"B": "3.33333",
"C": "0.16667",
"D": "0.33333"
} | D | 0.33333 | Favourable pairs = C(6,2)+C(9,2)+C(4,2)=57. Total pairs=C(19,2)=171. Therefore P=57/171=0.33333. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 0.33333, not 0.5.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 0.33333, not 3.33333.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 0.33333, not 0.16667."... | [
"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_008776 | 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 0.15 mol L⁻¹ and initially B is absent. At equilibrium, 23% of A has dissociated at constant volume. Find K_c. | {
"A": "3.04122",
"B": "0.03092",
"C": "0.04122",
"D": "0.06183"
} | C | 0.04122 | Let α=0.23. At equilibrium [A]=c₀(1−α) and [B]=2c₀α. Thus K_c=[B]²/[A]=4c₀α²/(1−α)=4(0.15)(0.23)²/(1−0.23)=0.04122. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 0.04122, not 3.04122.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 0.04122, not 0.03092.",
"C": "Correct. The worked calculation gives 0.04122.",
"D": "Incorrect. Substituting the stated va... | [
"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_009649 | 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_5H_7N_2X_3. Treat X as the total number of monovalent halogen atoms. | {
"A": "2",
"B": "1.5",
"C": "0",
"D": "4"
} | A | 2 | Use DBE=(2C+2+N−H−X)/2. Substitution gives (2×5+2+2−7−3)/2=2. | {
"A": "Correct. The worked calculation gives 2.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 2, not 1.5.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 2, not 0.",
"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_001605 | mathematics | calculus | definite_integral_polynomial | very_hard | provisional_template_level_label | single_correct_mcq | Evaluate the definite integral ∫₀^1 (3x² +5x -5) dx. | {
"A": "0.85",
"B": "1.5",
"C": "-1.5",
"D": "-2.25"
} | C | -1.5 | An antiderivative is (3/3)x³+(5/2)x²+(-5)x. Evaluating between 0 and 1 gives (3/3)(1)³+(5/2)(1)²+(-5)(1)=-1.5. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives -1.5, not 0.85.",
"B": "Incorrect. Substituting the stated values into the governing relation gives -1.5, not 1.5.",
"C": "Correct. The worked calculation gives -1.5.",
"D": "Incorrect. Substituting the stated values into the go... | [
"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_003596 | physics | mechanics | projectile_range | hard | provisional_template_level_label | single_correct_mcq | A projectile is launched from level ground at 73 m s⁻¹ at 40° to the horizontal. Take g=10 m s⁻². Find its horizontal range. | {
"A": "53.48041 m",
"B": "787.20608 m",
"C": "526.80405 m",
"D": "524.80405 m"
} | D | 524.80405 m | Range R=u²sin(2θ)/g. Substituting u=73, θ=40° and g=10 gives R=(73)²sin(80°)/10=524.80405 m. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 524.80405 m, not 53.48041 m.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 524.80405 m, not 787.20608 m.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 524... | [
"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_008229 | 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 14128 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": "23.24166 g",
"B": "21.24166 g",
"C": "24.24166 g",
"D": "34.86249 g"
} | A | 23.24166 g | Cu²⁺+2e⁻→Cu, so z=2. By Faraday's law m=MIt/(zF)=63.5×5×14128/(2×96500)=23.24166 g. | {
"A": "Correct. The worked calculation gives 23.24166 g.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 23.24166 g, not 21.24166 g.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 23.24166 g, not 24.24166 g.",
"D": "Incorrect. Substitutin... | [
"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_002614 | mathematics | linear_algebra | 2x2_determinant | hard | provisional_template_level_label | single_correct_mcq | Find the determinant of the matrix [[3, -4], [1, 0]]. | {
"A": "3",
"B": "4",
"C": "8",
"D": "7"
} | B | 4 | For a 2×2 matrix [[a,b],[c,d]], det=ad−bc. Here it is (3)(0)−(-4)(1)=4. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 4, not 3.",
"B": "Correct. The worked calculation gives 4.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 4, not 8.",
"D": "Incorrect. Substituting the stated values into the governing relati... | [
"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_002648 | mathematics | linear_algebra | 2x2_determinant | hard | provisional_template_level_label | single_correct_mcq | Find the determinant of the matrix [[7, -2], [7, 6]]. | {
"A": "56",
"B": "54",
"C": "55",
"D": "6.6"
} | A | 56 | For a 2×2 matrix [[a,b],[c,d]], det=ad−bc. Here it is (7)(6)−(-2)(7)=56. | {
"A": "Correct. The worked calculation gives 56.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 56, not 54.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 56, not 55.",
"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_007523 | chemistry | physical_chemistry | strong_acid_base_mixture | very_hard | provisional_template_level_label | single_correct_mcq | 34 mL of 0.10 M HCl is mixed with 27 mL of 0.10 M NaOH at 25°C. Assuming complete dissociation and additive volumes, find the final pH. | {
"A": "1.45517",
"B": "-1.05977",
"C": "1.94023",
"D": "-0.05977"
} | C | 1.94023 | Excess H⁺ moles=0.10×(34−27)/1000=0.000700 mol. Total volume=0.061 L, so [H⁺]=0.011475 M and pH=−log₁₀[H⁺]=1.94023. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 1.94023, not 1.45517.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 1.94023, not -1.05977.",
"C": "Correct. The worked calculation gives 1.94023.",
"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_005233 | physics | electrostatics | coulombs_law | hard | provisional_template_level_label | single_correct_mcq | Two point charges of +8 μC and +4 μC are separated by 11 cm in vacuum. Using k=9×10⁹ N m² C⁻², find the magnitude of the electrostatic force. | {
"A": "25.80165 N",
"B": "-23.80165 N",
"C": "23.80165 N",
"D": "3.38017 N"
} | C | 23.80165 N | Convert charges to coulombs and distance to metres. F=k|q₁q₂|/r²=9×10⁹(8×10⁻⁶)(4×10⁻⁶)/(11×10⁻²)²=23.80165 N. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 23.80165 N, not 25.80165 N.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 23.80165 N, not -23.80165 N.",
"C": "Correct. The worked calculation gives 23.80165 N.",
"D": "Incorrect. Substituti... | [
"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_002341 | mathematics | linear_algebra | 2x2_determinant | hard | provisional_template_level_label | single_correct_mcq | Find the determinant of the matrix [[5, 4], [1, 8]]. | {
"A": "36",
"B": "27",
"C": "4.6",
"D": "43.2"
} | A | 36 | For a 2×2 matrix [[a,b],[c,d]], det=ad−bc. Here it is (5)(8)−(4)(1)=36. | {
"A": "Correct. The worked calculation gives 36.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 36, not 27.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 36, not 4.6.",
"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 |
openstem_008147 | 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 13 A for 22667 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": "97.95134 g",
"B": "116.34161 g",
"C": "-96.95134 g",
"D": "96.95134 g"
} | D | 96.95134 g | Cu²⁺+2e⁻→Cu, so z=2. By Faraday's law m=MIt/(zF)=63.5×13×22667/(2×96500)=96.95134 g. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 96.95134 g, not 97.95134 g.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 96.95134 g, not 116.34161 g.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 96.95... | [
"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_003028 | mathematics | complex_numbers | real_part_of_complex_quotient | very_hard | provisional_template_level_label | single_correct_mcq | Let z=(14-12i)/(8+2i). What is Re(z), the real part of z? | {
"A": "2.29412",
"B": "1.94118",
"C": "1.29412",
"D": "-1.70588"
} | C | 1.29412 | Multiply numerator and denominator by the conjugate (8-2i). Then Re(z)=(ac+bd)/(c²+d²)=((14)(8)+(-12)(2))/(8²+(2)²)=1.29412. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 1.29412, not 2.29412.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 1.29412, not 1.94118.",
"C": "Correct. The worked calculation gives 1.29412.",
"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_006542 | 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 1004 N m⁻¹ and oscillates horizontally without damping. What is the time period divided by π? | {
"A": "0.0998 s",
"B": "0.1996 s",
"C": "3.1996 s",
"D": "-2.8004 s"
} | B | 0.1996 s | For a mass–spring oscillator, T=2π√(m/k). Therefore T/π=2√(10/1004)=0.1996 s. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 0.1996 s, not 0.0998 s.",
"B": "Correct. The worked calculation gives 0.1996 s.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 0.1996 s, not 3.1996 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_008304 | 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 7 A for 8622 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": "19.8574 g",
"B": "9.9287 g",
"C": "21.8574 g",
"D": "2.98574 g"
} | A | 19.8574 g | Cu²⁺+2e⁻→Cu, so z=2. By Faraday's law m=MIt/(zF)=63.5×7×8622/(2×96500)=19.8574 g. | {
"A": "Correct. The worked calculation gives 19.8574 g.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 19.8574 g, not 9.9287 g.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 19.8574 g, not 21.8574 g.",
"D": "Incorrect. Substituting the ... | [
"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_000237 | mathematics | algebra | quadratic_roots | hard | provisional_template_level_label | single_correct_mcq | For the quadratic equation 4x^2 +7x -7=0, let its roots be α and β. What is α²+β²? | {
"A": "13.125",
"B": "5.5625",
"C": "6.5625",
"D": "-6.5625"
} | C | 6.5625 | By Vieta's relations, α+β = -7/4 and αβ = -7/4. Then α²+β²=(α+β)²−2αβ = (-7/4)²−2(-7/4) = 6.5625. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 6.5625, not 13.125.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 6.5625, not 5.5625.",
"C": "Correct. The worked calculation gives 6.5625.",
"D": "Incorrect. Substituting the stated values ... | [
"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_000367 | mathematics | sequences_and_series | arithmetic_progression | hard | provisional_template_level_label | single_correct_mcq | An arithmetic progression has first term -7 and common difference 6. Find the sum of its first 12 terms. | {
"A": "468",
"B": "234",
"C": "313",
"D": "312"
} | D | 312 | Use Sₙ = n/2[2a+(n−1)d]. Substituting a=-7, d=6, n=12: Sₙ=12/2[2(-7)+(12−1)(6)]=312. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 312, not 468.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 312, not 234.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 312, not 313.",
"D": "Correct. T... | [
"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_009186 | chemistry | physical_chemistry | nernst_equation_298K | hard | provisional_template_level_label | single_correct_mcq | At 298 K, a galvanic cell has E°cell=1.00 V, transfers 3 electrons, and has reaction quotient Q=10^2. Using E=E°−(0.0591/n)log₁₀Q, calculate Ecell. | {
"A": "1.9606 V",
"B": "-0.0394 V",
"C": "0.9606 V",
"D": "0.4803 V"
} | C | 0.9606 V | Apply the Nernst equation: E=1.00−(0.0591/3)(2)=0.9606 V. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 0.9606 V, not 1.9606 V.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 0.9606 V, not -0.0394 V.",
"C": "Correct. The worked calculation gives 0.9606 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_007649 | chemistry | physical_chemistry | strong_acid_base_mixture | very_hard | provisional_template_level_label | single_correct_mcq | 31 mL of 0.10 M HCl is mixed with 64 mL of 0.10 M NaOH at 25°C. Assuming complete dissociation and additive volumes, find the final pH. | {
"A": "6.2704",
"B": "25.08158",
"C": "10.54079",
"D": "12.54079"
} | D | 12.54079 | Excess OH⁻ moles=0.10×(64−31)/1000=0.003300 mol. Total volume=0.095 L, so [OH⁻]=0.034737 M, pOH=−log₁₀[OH⁻], and pH=14−pOH=12.54079. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 12.54079, not 6.2704.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 12.54079, not 25.08158.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 12.54079, not 10... | [
"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_006491 | physics | oscillations | mass_spring_period_over_pi | hard | provisional_template_level_label | single_correct_mcq | A mass of 18 kg is attached to an ideal spring of force constant 199 N m⁻¹ and oscillates horizontally without damping. What is the time period divided by π? | {
"A": "0.60151 s",
"B": "-2.39849 s",
"C": "1.20301 s",
"D": "-0.60151 s"
} | A | 0.60151 s | For a mass–spring oscillator, T=2π√(m/k). Therefore T/π=2√(18/199)=0.60151 s. | {
"A": "Correct. The worked calculation gives 0.60151 s.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 0.60151 s, not -2.39849 s.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 0.60151 s, not 1.20301 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_005257 | physics | electrostatics | coulombs_law | hard | provisional_template_level_label | single_correct_mcq | Two point charges of +3 μC and +2 μC are separated by 7 cm in vacuum. Using k=9×10⁹ N m² C⁻², find the magnitude of the electrostatic force. | {
"A": "11.02041 N",
"B": "2.10204 N",
"C": "16.53061 N",
"D": "22.04082 N"
} | A | 11.02041 N | Convert charges to coulombs and distance to metres. F=k|q₁q₂|/r²=9×10⁹(3×10⁻⁶)(2×10⁻⁶)/(7×10⁻²)²=11.02041 N. | {
"A": "Correct. The worked calculation gives 11.02041 N.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 11.02041 N, not 2.10204 N.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 11.02041 N, not 16.53061 N.",
"D": "Incorrect. Substituting... | [
"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_005069 | physics | electrostatics | coulombs_law | hard | provisional_template_level_label | single_correct_mcq | Two point charges of +1 μC and +9 μC are separated by 8 cm in vacuum. Using k=9×10⁹ N m² C⁻², find the magnitude of the electrostatic force. | {
"A": "18.98438 N",
"B": "12.65625 N",
"C": "15.65625 N",
"D": "-12.65625 N"
} | B | 12.65625 N | Convert charges to coulombs and distance to metres. F=k|q₁q₂|/r²=9×10⁹(1×10⁻⁶)(9×10⁻⁶)/(8×10⁻²)²=12.65625 N. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 12.65625 N, not 18.98438 N.",
"B": "Correct. The worked calculation gives 12.65625 N.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 12.65625 N, not 15.65625 N.",
"D": "Incorrect. Substitutin... | [
"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_001419 | mathematics | calculus | definite_integral_polynomial | very_hard | provisional_template_level_label | single_correct_mcq | Evaluate the definite integral ∫₀^2 (1x² +0x -3) dx. | {
"A": "-2.33333",
"B": "-3.33333",
"C": "-4.33333",
"D": "-5"
} | B | -3.33333 | An antiderivative is (1/3)x³+(0/2)x²+(-3)x. Evaluating between 0 and 2 gives (1/3)(2)³+(0/2)(2)²+(-3)(2)=-3.33333. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives -3.33333, not -2.33333.",
"B": "Correct. The worked calculation gives -3.33333.",
"C": "Incorrect. Substituting the stated values into the governing relation gives -3.33333, not -4.33333.",
"D": "Incorrect. Substituting the stat... | [
"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_008421 | 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 1.85 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": "3.7 K",
"B": "7.4 K",
"C": "1.74 K",
"D": "8.4 K"
} | B | 7.4 K | ΔT_b=iK_bm. Substitution gives ΔT_b=4×1.0×1.85=7.4 K. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 7.4 K, not 3.7 K.",
"B": "Correct. The worked calculation gives 7.4 K.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 7.4 K, not 1.74 K.",
"D": "Incorrect. Substituting the stated values into... | [
"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_007926 | 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 85 mmol L⁻¹, what concentration remains after 20 minutes? | {
"A": "34 mmol L⁻¹",
"B": "43.5 mmol L⁻¹",
"C": "42.5 mmol L⁻¹",
"D": "85 mmol L⁻¹"
} | C | 42.5 mmol L⁻¹ | 20 minutes corresponds to 1 half-lives. The remaining concentration is [A]₀(1/2)ⁿ=85/2^1=42.5 mmol L⁻¹. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 42.5 mmol L⁻¹, not 34 mmol L⁻¹.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 42.5 mmol L⁻¹, not 43.5 mmol L⁻¹.",
"C": "Correct. The worked calculation gives 42.5 mmol L⁻¹.",
"D": "Incorrect... | [
"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_009670 | chemistry | inorganic_chemistry | complex_ion_oxidation_state | hard | provisional_template_level_label | single_correct_mcq | For a hypothetical coordination ion [Cr(Cl)_4]^-2, each Cl ligand has charge −1. Determine the formal oxidation state of Cr. | {
"A": "1.2",
"B": "0",
"C": "2",
"D": "-1"
} | C | 2 | Let the formal oxidation state of Cr be x. Charge balance gives x+(4)(−1)=-2; hence x=-2+4=+2. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 2, not 1.2.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 2, not 0.",
"C": "Correct. The worked calculation gives 2.",
"D": "Incorrect. Substituting the stated values into the governing rela... | [
"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_001797 | 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 25 distinct elements? | {
"A": "-2042975",
"B": "2451570",
"C": "2042975",
"D": "1634380"
} | C | 2042975 | Order does not matter, so the count is C(25,9)=25!/[9!(16)!]=2042975. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 2042975, not -2042975.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 2042975, not 2451570.",
"C": "Correct. The worked calculation gives 2042975.",
"D": "Incorrect. Substituting the stated v... | [
"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_006472 | physics | oscillations | mass_spring_period_over_pi | hard | provisional_template_level_label | single_correct_mcq | A mass of 2 kg is attached to an ideal spring of force constant 640 N m⁻¹ and oscillates horizontally without damping. What is the time period divided by π? | {
"A": "2.1118 s",
"B": "0.1118 s",
"C": "-0.1118 s",
"D": "1.01118 s"
} | B | 0.1118 s | For a mass–spring oscillator, T=2π√(m/k). Therefore T/π=2√(2/640)=0.1118 s. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 0.1118 s, not 2.1118 s.",
"B": "Correct. The worked calculation gives 0.1118 s.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 0.1118 s, not -0.1118 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_001800 | mathematics | combinatorics | binomial_coefficient | hard | provisional_template_level_label | single_correct_mcq | How many distinct subsets containing exactly 4 elements can be selected from a set of 88 distinct elements? | {
"A": "1748917.5",
"B": "2331890",
"C": "1865512",
"D": "233190"
} | B | 2331890 | Order does not matter, so the count is C(88,4)=88!/[4!(84)!]=2331890. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 2331890, not 1748917.5.",
"B": "Correct. The worked calculation gives 2331890.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 2331890, not 1865512.",
"D": "Incorrect. Substituting the stated ... | [
"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_004509 | physics | rotational_mechanics | disc_angular_acceleration | hard | provisional_template_level_label | single_correct_mcq | A uniform solid disc of mass 4 kg and radius 2 m is acted on by a constant torque of 49 N m about its central axis. Find its angular acceleration. | {
"A": "9.1875 rad s⁻²",
"B": "6.125 rad s⁻²",
"C": "9.125 rad s⁻²",
"D": "5.125 rad s⁻²"
} | B | 6.125 rad s⁻² | For a solid disc, I=½MR²=½(4)(2)²=8 kg m². Using τ=Iα, α=49/8=6.125 rad s⁻². | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 6.125 rad s⁻², not 9.1875 rad s⁻².",
"B": "Correct. The worked calculation gives 6.125 rad s⁻².",
"C": "Incorrect. Substituting the stated values into the governing relation gives 6.125 rad s⁻², not 9.125 rad s⁻².",
"D": "Incorr... | [
"moment of inertia",
"rotational dynamics",
"torque"
] | 4 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | physics_disc_torque | 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_007564 | chemistry | physical_chemistry | strong_acid_base_mixture | very_hard | provisional_template_level_label | single_correct_mcq | 46 mL of 0.10 M HCl is mixed with 78 mL of 0.10 M NaOH at 25°C. Assuming complete dissociation and additive volumes, find the final pH. | {
"A": "12.41173",
"B": "9.41173",
"C": "13.41173",
"D": "18.61759"
} | A | 12.41173 | Excess OH⁻ moles=0.10×(78−46)/1000=0.003200 mol. Total volume=0.124 L, so [OH⁻]=0.025806 M, pOH=−log₁₀[OH⁻], and pH=14−pOH=12.41173. | {
"A": "Correct. The worked calculation gives 12.41173.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 12.41173, not 9.41173.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 12.41173, not 13.41173.",
"D": "Incorrect. Substituting the state... | [
"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_001209 | mathematics | calculus | polynomial_differentiation | hard | provisional_template_level_label | single_correct_mcq | Let f(x)=3x³ -4x² -3x+7. Evaluate f′(-4). | {
"A": "86.5",
"B": "259.5",
"C": "172",
"D": "173"
} | D | 173 | Differentiate: f′(x)=3(3)x²+2(-4)x+(-3). At x=-4, f′(-4)=3(3)(-4)²+2(-4)(-4)+(-3)=173. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 173, not 86.5.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 173, not 259.5.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 173, not 172.",
"D": "Correct... | [
"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_005425 | physics | electrostatics | capacitor_energy | hard | provisional_template_level_label | single_correct_mcq | A 14 μF capacitor is charged to a potential difference of 34 V. Calculate the energy stored, expressed in μJ. | {
"A": "9710.4 μJ",
"B": "16184 μJ",
"C": "8092 μJ",
"D": "8091 μJ"
} | C | 8092 μJ | U=½CV². Since C=14 μF, U in μJ is ½×14×(34)²=8092 μJ. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 8092 μJ, not 9710.4 μJ.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 8092 μJ, not 16184 μJ.",
"C": "Correct. The worked calculation gives 8092 μJ.",
"D": "Incorrect. Substituting the stated... | [
"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_005569 | physics | electrostatics | capacitor_energy | hard | provisional_template_level_label | single_correct_mcq | A 63 μF capacitor is charged to a potential difference of 32 V. Calculate the energy stored, expressed in μJ. | {
"A": "16128 μJ",
"B": "32256 μJ",
"C": "25804.8 μJ",
"D": "32258 μJ"
} | B | 32256 μJ | U=½CV². Since C=63 μF, U in μJ is ½×63×(32)²=32256 μJ. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 32256 μJ, not 16128 μJ.",
"B": "Correct. The worked calculation gives 32256 μJ.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 32256 μJ, not 25804.8 μJ.",
"D": "Incorrect. Substituting the st... | [
"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_002061 | mathematics | probability | same_colour_without_replacement | very_hard | provisional_template_level_label | single_correct_mcq | A bag contains 6 red, 9 blue and 3 green balls. Two balls are drawn uniformly at random without replacement. What is the probability that both have the same colour? | {
"A": "0.35294",
"B": "1.03529",
"C": "2.35294",
"D": "0.26471"
} | A | 0.35294 | Favourable pairs = C(6,2)+C(9,2)+C(3,2)=54. Total pairs=C(18,2)=153. Therefore P=54/153=0.35294. | {
"A": "Correct. The worked calculation gives 0.35294.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 0.35294, not 1.03529.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 0.35294, not 2.35294.",
"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_001943 | mathematics | combinatorics | binomial_coefficient | hard | provisional_template_level_label | single_correct_mcq | How many distinct subsets containing exactly 8 elements can be selected from a set of 78 distinct elements? | {
"A": "46893762630",
"B": "23446881314",
"C": "18757505052",
"D": "23446881315"
} | D | 23446881315 | Order does not matter, so the count is C(78,8)=78!/[8!(70)!]=23446881315. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 23446881315, not 46893762630.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 23446881315, not 23446881314.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 23... | [
"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_001274 | mathematics | calculus | polynomial_differentiation | hard | provisional_template_level_label | single_correct_mcq | Let f(x)=-1x³ -5x² -2x+7. Evaluate f′(-1). | {
"A": "2",
"B": "8",
"C": "5",
"D": "1.5"
} | C | 5 | Differentiate: f′(x)=3(-1)x²+2(-5)x+(-2). At x=-1, f′(-1)=3(-1)(-1)²+2(-5)(-1)+(-2)=5. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 5, not 2.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 5, not 8.",
"C": "Correct. The worked calculation gives 5.",
"D": "Incorrect. Substituting the stated values into the governing relati... | [
"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_001539 | mathematics | calculus | definite_integral_polynomial | very_hard | provisional_template_level_label | single_correct_mcq | Evaluate the definite integral ∫₀^4 (0x² +0x -1) dx. | {
"A": "0.6",
"B": "4",
"C": "-4",
"D": "-6"
} | C | -4 | An antiderivative is (0/3)x³+(0/2)x²+(-1)x. Evaluating between 0 and 4 gives (0/3)(4)³+(0/2)(4)²+(-1)(4)=-4. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives -4, not 0.6.",
"B": "Incorrect. Substituting the stated values into the governing relation gives -4, not 4.",
"C": "Correct. The worked calculation gives -4.",
"D": "Incorrect. Substituting the stated values into the governing r... | [
"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_000892 | mathematics | sequences_and_series | geometric_progression | hard | provisional_template_level_label | single_correct_mcq | A geometric progression has first term 9 and common ratio 2. What is the sum of its first 7 terms? | {
"A": "857.25",
"B": "1143",
"C": "1144",
"D": "115.3"
} | B | 1143 | For r≠1, Sₙ=a(rⁿ−1)/(r−1). Thus Sₙ=9(2^7−1)/(2−1)=1143. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 1143, not 857.25.",
"B": "Correct. The worked calculation gives 1143.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 1143, not 1144.",
"D": "Incorrect. Substituting the stated values into the... | [
"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_003686 | physics | mechanics | work_energy_theorem | very_hard | provisional_template_level_label | single_correct_mcq | A 2 kg block initially moves at 8 m s⁻¹. A constant net force does 14 N of work over 11 m in the direction of motion. Find its final kinetic energy. | {
"A": "219 J",
"B": "220 J",
"C": "261.6 J",
"D": "218 J"
} | D | 218 J | By the work–energy theorem, K_f=K_i+W=½mu²+Fs=½(2)(8)²+(14)(11)=218 J. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 218 J, not 219 J.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 218 J, not 220 J.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 218 J, not 261.6 J.",
"D... | [
"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_009943 | chemistry | inorganic_chemistry | complex_ion_oxidation_state | hard | provisional_template_level_label | single_correct_mcq | For a hypothetical coordination ion [Mn(OH)_2]^-2, each OH ligand has charge −1. Determine the formal oxidation state of Mn. | {
"A": "1",
"B": "3",
"C": "2",
"D": "0"
} | D | 0 | Let the formal oxidation state of Mn be x. Charge balance gives x+(2)(−1)=-2; hence x=-2+2=+0. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 0, not 1.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 0, not 3.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 0, not 2.",
"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_006905 | chemistry | stoichiometry | limiting_reagent_water | very_hard | provisional_template_level_label | single_correct_mcq | Hydrogen reacts with oxygen according to 2H₂ + O₂ → 2H₂O. If 45 mol H₂ and 18 mol O₂ are mixed and the reaction goes to completion, how many moles of H₂O form? | {
"A": "43.2 mol",
"B": "36 mol",
"C": "37 mol",
"D": "54 mol"
} | B | 36 mol | From 2 mol H₂ and 1 mol O₂, 2 mol H₂O form. H₂ can produce 45 mol H₂O; O₂ can produce 36 mol H₂O. The limiting condition is O₂; product=36 mol. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 36 mol, not 43.2 mol.",
"B": "Correct. The worked calculation gives 36 mol.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 36 mol, not 37 mol.",
"D": "Incorrect. Substituting the stated value... | [
"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_007553 | chemistry | physical_chemistry | strong_acid_base_mixture | very_hard | provisional_template_level_label | single_correct_mcq | 41 mL of 0.10 M HCl is mixed with 25 mL of 0.10 M NaOH at 25°C. Assuming complete dissociation and additive volumes, find the final pH. | {
"A": "-0.38458",
"B": "4.61542",
"C": "1.61542",
"D": "2.42314"
} | C | 1.61542 | Excess H⁺ moles=0.10×(41−25)/1000=0.001600 mol. Total volume=0.066 L, so [H⁺]=0.024242 M and pH=−log₁₀[H⁺]=1.61542. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 1.61542, not -0.38458.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 1.61542, not 4.61542.",
"C": "Correct. The worked calculation gives 1.61542.",
"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_008326 | 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 6028 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": "5.93322 g",
"B": "4.93322 g",
"C": "7.93322 g",
"D": "1.79332 g"
} | C | 7.93322 g | Cu²⁺+2e⁻→Cu, so z=2. By Faraday's law m=MIt/(zF)=63.5×4×6028/(2×96500)=7.93322 g. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 7.93322 g, not 5.93322 g.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 7.93322 g, not 4.93322 g.",
"C": "Correct. The worked calculation gives 7.93322 g.",
"D": "Incorrect. Substituting the... | [
"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_008212 | 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 15 A for 20298 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": "150.26304 g",
"B": "99.17536 g",
"C": "101.17536 g",
"D": "100.17536 g"
} | D | 100.17536 g | Cu²⁺+2e⁻→Cu, so z=2. By Faraday's law m=MIt/(zF)=63.5×15×20298/(2×96500)=100.17536 g. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 100.17536 g, not 150.26304 g.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 100.17536 g, not 99.17536 g.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 100... | [
"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_004932 | physics | electricity | series_parallel_circuit | very_hard | provisional_template_level_label | single_correct_mcq | Resistors 14 Ω and 3 Ω are connected in series; this combination is connected in parallel with a 13 Ω resistor across a 36 V ideal battery. Find the total current supplied by the battery. | {
"A": "1.88688 A",
"B": "7.88688 A",
"C": "-4.88688 A",
"D": "4.88688 A"
} | D | 4.88688 A | Series branch resistance=14+3=17 Ω. Parallel equivalent R_eq=(17×13)/(17+13)=7.36667 Ω. Total current I=V/R_eq=36/7.36667=4.88688 A. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 4.88688 A, not 1.88688 A.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 4.88688 A, not 7.88688 A.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 4.88688 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_000232 | mathematics | algebra | quadratic_roots | hard | provisional_template_level_label | single_correct_mcq | For the quadratic equation 7x^2 +18x -12=0, let its roots be α and β. What is α²+β²? | {
"A": "10.04082",
"B": "20.08163",
"C": "2.00408",
"D": "13.04082"
} | A | 10.04082 | By Vieta's relations, α+β = -18/7 and αβ = -12/7. Then α²+β²=(α+β)²−2αβ = (-18/7)²−2(-12/7) = 10.04082. | {
"A": "Correct. The worked calculation gives 10.04082.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 10.04082, not 20.08163.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 10.04082, not 2.00408.",
"D": "Incorrect. Substituting the state... | [
"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_000637 | mathematics | sequences_and_series | arithmetic_progression | hard | provisional_template_level_label | single_correct_mcq | An arithmetic progression has first term 5 and common difference 9. Find the sum of its first 26 terms. | {
"A": "-3055",
"B": "306.5",
"C": "3057",
"D": "3055"
} | D | 3055 | Use Sₙ = n/2[2a+(n−1)d]. Substituting a=5, d=9, n=26: Sₙ=26/2[2(5)+(26−1)(9)]=3055. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 3055, not -3055.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 3055, not 306.5.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 3055, not 3057.",
"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_000136 | mathematics | algebra | quadratic_roots | hard | provisional_template_level_label | single_correct_mcq | For the quadratic equation 4x^2 +14x -9=0, let its roots be α and β. What is α²+β²? | {
"A": "16.75",
"B": "-16.75",
"C": "33.5",
"D": "12.5625"
} | A | 16.75 | By Vieta's relations, α+β = -14/4 and αβ = -9/4. Then α²+β²=(α+β)²−2αβ = (-14/4)²−2(-9/4) = 16.75. | {
"A": "Correct. The worked calculation gives 16.75.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 16.75, not -16.75.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 16.75, not 33.5.",
"D": "Incorrect. Substituting the stated values into ... | [
"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_008254 | 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 8 A for 9407 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": "-24.76039 g",
"B": "37.14059 g",
"C": "22.76039 g",
"D": "24.76039 g"
} | D | 24.76039 g | Cu²⁺+2e⁻→Cu, so z=2. By Faraday's law m=MIt/(zF)=63.5×8×9407/(2×96500)=24.76039 g. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 24.76039 g, not -24.76039 g.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 24.76039 g, not 37.14059 g.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 24.76... | [
"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_009629 | 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_18H_27N_1. Treat X as the total number of monovalent halogen atoms. | {
"A": "1.6",
"B": "9",
"C": "6",
"D": "4"
} | C | 6 | Use DBE=(2C+2+N−H−X)/2. Substitution gives (2×18+2+1−27−0)/2=6. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 6, not 1.6.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 6, not 9.",
"C": "Correct. The worked calculation gives 6.",
"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_009154 | chemistry | physical_chemistry | nernst_equation_298K | hard | provisional_template_level_label | single_correct_mcq | At 298 K, a galvanic cell has E°cell=0.80 V, transfers 2 electrons, and has reaction quotient Q=10^-5. Using E=E°−(0.0591/n)log₁₀Q, calculate Ecell. | {
"A": "1.42163 V",
"B": "1.8955 V",
"C": "-0.05225 V",
"D": "0.94775 V"
} | D | 0.94775 V | Apply the Nernst equation: E=0.80−(0.0591/2)(-5)=0.94775 V. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 0.94775 V, not 1.42163 V.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 0.94775 V, not 1.8955 V.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 0.94775 V, ... | [
"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_009015 | chemistry | physical_chemistry | nernst_equation_298K | hard | provisional_template_level_label | single_correct_mcq | At 298 K, a galvanic cell has E°cell=1.32 V, transfers 2 electrons, and has reaction quotient Q=10^3. Using E=E°−(0.0591/n)log₁₀Q, calculate Ecell. | {
"A": "-1.23135 V",
"B": "3.23135 V",
"C": "1.23135 V",
"D": "1.12313 V"
} | C | 1.23135 V | Apply the Nernst equation: E=1.32−(0.0591/2)(3)=1.23135 V. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 1.23135 V, not -1.23135 V.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 1.23135 V, not 3.23135 V.",
"C": "Correct. The worked calculation gives 1.23135 V.",
"D": "Incorrect. Substituting th... | [
"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_006334 | physics | ray_optics | convex_lens_magnification | very_hard | provisional_template_level_label | single_correct_mcq | A real object is placed 120 cm in front of a thin convex lens of focal length 25 cm. Using the Cartesian sign convention, what is the magnitude of the linear magnification? | {
"A": "1.02632",
"B": "-0.26316",
"C": "0.26316",
"D": "0.13158"
} | C | 0.26316 | Using distances as positive magnitudes, 1/f=1/v+1/u gives v=uf/(u−f)=(120)(25)/(120−25)=31.57895 cm. Thus |m|=v/u=31.57895/120=0.26316. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 0.26316, not 1.02632.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 0.26316, not -0.26316.",
"C": "Correct. The worked calculation gives 0.26316.",
"D": "Incorrect. Substituting the stated v... | [
"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_008073 | 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 13 A for 22296 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": "-95.3645 g",
"B": "95.3645 g",
"C": "47.68225 g",
"D": "93.3645 g"
} | B | 95.3645 g | Cu²⁺+2e⁻→Cu, so z=2. By Faraday's law m=MIt/(zF)=63.5×13×22296/(2×96500)=95.3645 g. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 95.3645 g, not -95.3645 g.",
"B": "Correct. The worked calculation gives 95.3645 g.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 95.3645 g, not 47.68225 g.",
"D": "Incorrect. Substituting t... | [
"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_004683 | 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 10 Ω resistor across a 30 V ideal battery. Find the total current supplied by the battery. | {
"A": "-4.875 A",
"B": "4.875 A",
"C": "3.65625 A",
"D": "2.875 A"
} | B | 4.875 A | Series branch resistance=11+5=16 Ω. Parallel equivalent R_eq=(16×10)/(16+10)=6.15385 Ω. Total current I=V/R_eq=30/6.15385=4.875 A. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 4.875 A, not -4.875 A.",
"B": "Correct. The worked calculation gives 4.875 A.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 4.875 A, not 3.65625 A.",
"D": "Incorrect. Substituting the stated... | [
"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_002716 | mathematics | vectors | dot_product | hard | provisional_template_level_label | single_correct_mcq | Given vectors u=(-4, 1, 8) and v=(0, 0, -2), calculate u·v. | {
"A": "-12.8",
"B": "-19.2",
"C": "-16",
"D": "16"
} | C | -16 | The dot product is the sum of corresponding component products: (-4)(0)+(1)(0)+(8)(-2)=-16. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives -16, not -12.8.",
"B": "Incorrect. Substituting the stated values into the governing relation gives -16, not -19.2.",
"C": "Correct. The worked calculation gives -16.",
"D": "Incorrect. Substituting the stated values into the go... | [
"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_004609 | physics | rotational_mechanics | disc_angular_acceleration | hard | provisional_template_level_label | single_correct_mcq | A uniform solid disc of mass 5 kg and radius 2 m is acted on by a constant torque of 6 N m about its central axis. Find its angular acceleration. | {
"A": "-0.4 rad s⁻²",
"B": "0.6 rad s⁻²",
"C": "-2.4 rad s⁻²",
"D": "-0.6 rad s⁻²"
} | B | 0.6 rad s⁻² | For a solid disc, I=½MR²=½(5)(2)²=10 kg m². Using τ=Iα, α=6/10=0.6 rad s⁻². | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 0.6 rad s⁻², not -0.4 rad s⁻².",
"B": "Correct. The worked calculation gives 0.6 rad s⁻².",
"C": "Incorrect. Substituting the stated values into the governing relation gives 0.6 rad s⁻², not -2.4 rad s⁻².",
"D": "Incorrect. Subs... | [
"moment of inertia",
"rotational dynamics",
"torque"
] | 4 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | physics_disc_torque | 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_003299 | mathematics | complex_numbers | real_part_of_complex_quotient | very_hard | provisional_template_level_label | single_correct_mcq | Let z=(-11+14i)/(10+12i). What is Re(z), the real part of z? | {
"A": "-0.7623",
"B": "-0.2377",
"C": "0.2377",
"D": "3.2377"
} | C | 0.2377 | Multiply numerator and denominator by the conjugate (10-12i). Then Re(z)=(ac+bd)/(c²+d²)=((-11)(10)+(14)(12))/(10²+(12)²)=0.2377. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 0.2377, not -0.7623.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 0.2377, not -0.2377.",
"C": "Correct. The worked calculation gives 0.2377.",
"D": "Incorrect. Substituting the stated value... | [
"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_000697 | mathematics | sequences_and_series | geometric_progression | hard | provisional_template_level_label | single_correct_mcq | A geometric progression has first term 12 and common ratio -3. What is the sum of its first 11 terms? | {
"A": "265722",
"B": "-531444",
"C": "1062888",
"D": "531444"
} | D | 531444 | For r≠1, Sₙ=a(rⁿ−1)/(r−1). Thus Sₙ=12(-3^11−1)/(-3−1)=531444. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 531444, not 265722.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 531444, not -531444.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 531444, not 1062888."... | [
"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_008655 | 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 0.36 mol kg⁻¹. If its van't Hoff factor is 4 and K_b for water is 0.52 K kg mol⁻¹, find the boiling-point elevation. | {
"A": "-0.2512 K",
"B": "-1.2512 K",
"C": "0.7488 K",
"D": "0.3744 K"
} | C | 0.7488 K | ΔT_b=iK_bm. Substitution gives ΔT_b=4×0.52×0.36=0.7488 K. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 0.7488 K, not -0.2512 K.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 0.7488 K, not -1.2512 K.",
"C": "Correct. The worked calculation gives 0.7488 K.",
"D": "Incorrect. Substituting the st... | [
"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_007728 | 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 84 mmol L⁻¹, what concentration remains after 120 minutes? | {
"A": "-1.3125 mmol L⁻¹",
"B": "2.625 mmol L⁻¹",
"C": "-0.6875 mmol L⁻¹",
"D": "1.3125 mmol L⁻¹"
} | D | 1.3125 mmol L⁻¹ | 120 minutes corresponds to 6 half-lives. The remaining concentration is [A]₀(1/2)ⁿ=84/2^6=1.3125 mmol L⁻¹. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 1.3125 mmol L⁻¹, not -1.3125 mmol L⁻¹.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 1.3125 mmol L⁻¹, not 2.625 mmol L⁻¹.",
"C": "Incorrect. Substituting the stated values into the governing r... | [
"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_009503 | 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_11H_4N_2. Treat X as the total number of monovalent halogen atoms. | {
"A": "2.1",
"B": "16.5",
"C": "5.5",
"D": "11"
} | D | 11 | Use DBE=(2C+2+N−H−X)/2. Substitution gives (2×11+2+2−4−0)/2=11. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 11, not 2.1.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 11, not 16.5.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 11, not 5.5.",
"D": "Correct. The... | [
"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_002796 | mathematics | vectors | dot_product | hard | provisional_template_level_label | single_correct_mcq | Given vectors u=(6, 5, -5) and v=(6, -2, 9), calculate u·v. | {
"A": "-15.2",
"B": "-22.8",
"C": "-19",
"D": "-21"
} | C | -19 | The dot product is the sum of corresponding component products: (6)(6)+(5)(-2)+(-5)(9)=-19. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives -19, not -15.2.",
"B": "Incorrect. Substituting the stated values into the governing relation gives -19, not -22.8.",
"C": "Correct. The worked calculation gives -19.",
"D": "Incorrect. Substituting the stated values into the go... | [
"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_003905 | physics | mechanics | work_energy_theorem | very_hard | provisional_template_level_label | single_correct_mcq | A 15 kg block initially moves at 6 m s⁻¹. A constant net force does 16 N of work over 9 m in the direction of motion. Find its final kinetic energy. | {
"A": "414 J",
"B": "310.5 J",
"C": "828 J",
"D": "415 J"
} | A | 414 J | By the work–energy theorem, K_f=K_i+W=½mu²+Fs=½(15)(6)²+(16)(9)=414 J. | {
"A": "Correct. The worked calculation gives 414 J.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 414 J, not 310.5 J.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 414 J, not 828 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_005238 | physics | electrostatics | coulombs_law | hard | provisional_template_level_label | single_correct_mcq | Two point charges of +9 μC and +1 μC are separated by 11 cm in vacuum. Using k=9×10⁹ N m² C⁻², find the magnitude of the electrostatic force. | {
"A": "13.38843 N",
"B": "7.69421 N",
"C": "6.69421 N",
"D": "3.69421 N"
} | C | 6.69421 N | Convert charges to coulombs and distance to metres. F=k|q₁q₂|/r²=9×10⁹(9×10⁻⁶)(1×10⁻⁶)/(11×10⁻²)²=6.69421 N. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 6.69421 N, not 13.38843 N.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 6.69421 N, not 7.69421 N.",
"C": "Correct. The worked calculation gives 6.69421 N.",
"D": "Incorrect. Substituting th... | [
"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_006697 | chemistry | stoichiometry | limiting_reagent_water | very_hard | provisional_template_level_label | single_correct_mcq | Hydrogen reacts with oxygen according to 2H₂ + O₂ → 2H₂O. If 37 mol H₂ and 30 mol O₂ are mixed and the reaction goes to completion, how many moles of H₂O form? | {
"A": "37 mol",
"B": "-37 mol",
"C": "35 mol",
"D": "29.6 mol"
} | A | 37 mol | From 2 mol H₂ and 1 mol O₂, 2 mol H₂O form. H₂ can produce 37 mol H₂O; O₂ can produce 60 mol H₂O. The limiting condition is H₂; product=37 mol. | {
"A": "Correct. The worked calculation gives 37 mol.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 37 mol, not -37 mol.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 37 mol, not 35 mol.",
"D": "Incorrect. Substituting the stated values... | [
"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_007994 | 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 69 mmol L⁻¹, what concentration remains after 80 minutes? | {
"A": "4.3125 mmol L⁻¹",
"B": "-4.3125 mmol L⁻¹",
"C": "2.3125 mmol L⁻¹",
"D": "1.3125 mmol L⁻¹"
} | A | 4.3125 mmol L⁻¹ | 80 minutes corresponds to 4 half-lives. The remaining concentration is [A]₀(1/2)ⁿ=69/2^4=4.3125 mmol L⁻¹. | {
"A": "Correct. The worked calculation gives 4.3125 mmol L⁻¹.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 4.3125 mmol L⁻¹, not -4.3125 mmol L⁻¹.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 4.3125 mmol L⁻¹, not 2.3125 mmol L⁻¹.",
"D... | [
"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_002290 | mathematics | probability | same_colour_without_replacement | very_hard | provisional_template_level_label | single_correct_mcq | A bag contains 9 red, 5 blue and 11 green balls. Two balls are drawn uniformly at random without replacement. What is the probability that both have the same colour? | {
"A": "0.16833",
"B": "3.33667",
"C": "-0.66333",
"D": "0.33667"
} | D | 0.33667 | Favourable pairs = C(9,2)+C(5,2)+C(11,2)=101. Total pairs=C(25,2)=300. Therefore P=101/300=0.33667. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 0.33667, not 0.16833.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 0.33667, not 3.33667.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 0.33667, not -0.66... | [
"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_006938 | chemistry | stoichiometry | limiting_reagent_water | very_hard | provisional_template_level_label | single_correct_mcq | Hydrogen reacts with oxygen according to 2H₂ + O₂ → 2H₂O. If 3 mol H₂ and 24 mol O₂ are mixed and the reaction goes to completion, how many moles of H₂O form? | {
"A": "4.5 mol",
"B": "-3 mol",
"C": "6 mol",
"D": "3 mol"
} | D | 3 mol | From 2 mol H₂ and 1 mol O₂, 2 mol H₂O form. H₂ can produce 3 mol H₂O; O₂ can produce 48 mol H₂O. The limiting condition is H₂; product=3 mol. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 3 mol, not 4.5 mol.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 3 mol, not -3 mol.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 3 mol, not 6 mol.",
"... | [
"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_007854 | 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 81 mmol L⁻¹, what concentration remains after 140 minutes? | {
"A": "0.63281 mmol L⁻¹",
"B": "1.26562 mmol L⁻¹",
"C": "-0.63281 mmol L⁻¹",
"D": "2.63281 mmol L⁻¹"
} | A | 0.63281 mmol L⁻¹ | 140 minutes corresponds to 7 half-lives. The remaining concentration is [A]₀(1/2)ⁿ=81/2^7=0.63281 mmol L⁻¹. | {
"A": "Correct. The worked calculation gives 0.63281 mmol L⁻¹.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 0.63281 mmol L⁻¹, not 1.26562 mmol L⁻¹.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 0.63281 mmol L⁻¹, not -0.63281 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_005611 | physics | electrostatics | capacitor_energy | hard | provisional_template_level_label | single_correct_mcq | A 48 μF capacitor is charged to a potential difference of 16 V. Calculate the energy stored, expressed in μJ. | {
"A": "4608 μJ",
"B": "6144 μJ",
"C": "6142 μJ",
"D": "6143 μJ"
} | B | 6144 μJ | U=½CV². Since C=48 μF, U in μJ is ½×48×(16)²=6144 μJ. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 6144 μJ, not 4608 μJ.",
"B": "Correct. The worked calculation gives 6144 μJ.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 6144 μJ, not 6142 μJ.",
"D": "Incorrect. Substituting the stated va... | [
"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_006185 | physics | ray_optics | convex_lens_magnification | very_hard | provisional_template_level_label | single_correct_mcq | A real object is placed 87 cm in front of a thin convex lens of focal length 23 cm. Using the Cartesian sign convention, what is the magnitude of the linear magnification? | {
"A": "1.35938",
"B": "2.35938",
"C": "0.35938",
"D": "-0.64062"
} | C | 0.35938 | Using distances as positive magnitudes, 1/f=1/v+1/u gives v=uf/(u−f)=(87)(23)/(87−23)=31.26562 cm. Thus |m|=v/u=31.26562/87=0.35938. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 0.35938, not 1.35938.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 0.35938, not 2.35938.",
"C": "Correct. The worked calculation gives 0.35938.",
"D": "Incorrect. Substituting the stated va... | [
"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_001301 | mathematics | calculus | polynomial_differentiation | hard | provisional_template_level_label | single_correct_mcq | Let f(x)=2x³ -10x² -8x+7. Evaluate f′(0). | {
"A": "-8",
"B": "-11",
"C": "8",
"D": "-16"
} | A | -8 | Differentiate: f′(x)=3(2)x²+2(-10)x+(-8). At x=0, f′(0)=3(2)(0)²+2(-10)(0)+(-8)=-8. | {
"A": "Correct. The worked calculation gives -8.",
"B": "Incorrect. Substituting the stated values into the governing relation gives -8, not -11.",
"C": "Incorrect. Substituting the stated values into the governing relation gives -8, not 8.",
"D": "Incorrect. Substituting the stated values into the governing r... | [
"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_006261 | physics | ray_optics | convex_lens_magnification | very_hard | provisional_template_level_label | single_correct_mcq | A real object is placed 30 cm in front of a thin convex lens of focal length 22 cm. Using the Cartesian sign convention, what is the magnitude of the linear magnification? | {
"A": "1.75",
"B": "1.275",
"C": "2.0625",
"D": "2.75"
} | D | 2.75 | Using distances as positive magnitudes, 1/f=1/v+1/u gives v=uf/(u−f)=(30)(22)/(30−22)=82.5 cm. Thus |m|=v/u=82.5/30=2.75. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 2.75, not 1.75.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 2.75, not 1.275.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 2.75, not 2.0625.",
"D": "C... | [
"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_004653 | physics | rotational_mechanics | disc_angular_acceleration | hard | provisional_template_level_label | single_correct_mcq | A uniform solid disc of mass 15 kg and radius 4 m is acted on by a constant torque of 50 N m about its central axis. Find its angular acceleration. | {
"A": "0.83333 rad s⁻²",
"B": "0.41667 rad s⁻²",
"C": "-2.58333 rad s⁻²",
"D": "-0.58333 rad s⁻²"
} | B | 0.41667 rad s⁻² | For a solid disc, I=½MR²=½(15)(4)²=120 kg m². Using τ=Iα, α=50/120=0.41667 rad s⁻². | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 0.41667 rad s⁻², not 0.83333 rad s⁻².",
"B": "Correct. The worked calculation gives 0.41667 rad s⁻².",
"C": "Incorrect. Substituting the stated values into the governing relation gives 0.41667 rad s⁻², not -2.58333 rad s⁻².",
"D... | [
"moment of inertia",
"rotational dynamics",
"torque"
] | 4 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | physics_disc_torque | 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_003678 | physics | mechanics | work_energy_theorem | very_hard | provisional_template_level_label | single_correct_mcq | A 11 kg block initially moves at 4 m s⁻¹. A constant net force does 21 N of work over 2 m in the direction of motion. Find its final kinetic energy. | {
"A": "104 J",
"B": "130 J",
"C": "129 J",
"D": "132 J"
} | B | 130 J | By the work–energy theorem, K_f=K_i+W=½mu²+Fs=½(11)(4)²+(21)(2)=130 J. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 130 J, not 104 J.",
"B": "Correct. The worked calculation gives 130 J.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 130 J, not 129 J.",
"D": "Incorrect. Substituting the stated values into ... | [
"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_006432 | physics | oscillations | mass_spring_period_over_pi | hard | provisional_template_level_label | single_correct_mcq | A mass of 11 kg is attached to an ideal spring of force constant 746 N m⁻¹ and oscillates horizontally without damping. What is the time period divided by π? | {
"A": "3.24286 s",
"B": "1.02429 s",
"C": "0.48572 s",
"D": "0.24286 s"
} | D | 0.24286 s | For a mass–spring oscillator, T=2π√(m/k). Therefore T/π=2√(11/746)=0.24286 s. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 0.24286 s, not 3.24286 s.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 0.24286 s, not 1.02429 s.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 0.24286 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_008364 | 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 0.53 mol kg⁻¹. If its van't Hoff factor is 3 and K_b for water is 1.0 K kg mol⁻¹, find the boiling-point elevation. | {
"A": "1.59 K",
"B": "-0.41 K",
"C": "2.59 K",
"D": "1.159 K"
} | A | 1.59 K | ΔT_b=iK_bm. Substitution gives ΔT_b=3×1.0×0.53=1.59 K. | {
"A": "Correct. The worked calculation gives 1.59 K.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 1.59 K, not -0.41 K.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 1.59 K, not 2.59 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_006570 | 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 307 N m⁻¹ and oscillates horizontally without damping. What is the time period divided by π? | {
"A": "2.44209 s",
"B": "0.44209 s",
"C": "-2.55791 s",
"D": "0.88417 s"
} | B | 0.44209 s | For a mass–spring oscillator, T=2π√(m/k). Therefore T/π=2√(15/307)=0.44209 s. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 0.44209 s, not 2.44209 s.",
"B": "Correct. The worked calculation gives 0.44209 s.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 0.44209 s, not -2.55791 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_000233 | mathematics | algebra | quadratic_roots | hard | provisional_template_level_label | single_correct_mcq | For the quadratic equation 5x^2 -4x -8=0, let its roots be α and β. What is α²+β²? | {
"A": "3.84",
"B": "2.84",
"C": "1.92",
"D": "2.88"
} | A | 3.84 | By Vieta's relations, α+β = 4/5 and αβ = -8/5. Then α²+β²=(α+β)²−2αβ = (4/5)²−2(-8/5) = 3.84. | {
"A": "Correct. The worked calculation gives 3.84.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 3.84, not 2.84.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 3.84, not 1.92.",
"D": "Incorrect. Substituting the stated values into the g... | [
"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_008224 | 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 3256 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": "7.2851 g",
"B": "4.2851 g",
"C": "2.2851 g",
"D": "2.14255 g"
} | B | 4.2851 g | Cu²⁺+2e⁻→Cu, so z=2. By Faraday's law m=MIt/(zF)=63.5×4×3256/(2×96500)=4.2851 g. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 4.2851 g, not 7.2851 g.",
"B": "Correct. The worked calculation gives 4.2851 g.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 4.2851 g, not 2.2851 g.",
"D": "Incorrect. Substituting the stat... | [
"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_007312 | chemistry | physical_chemistry | ideal_gas_volume | hard | provisional_template_level_label | single_correct_mcq | Assuming ideal behaviour, what volume is occupied by 8 mol of gas at 300 K and 0.72 atm? Use R=0.082057 L atm mol⁻¹ K⁻¹. | {
"A": "328.228 L",
"B": "273.52333 L",
"C": "410.285 L",
"D": "205.1425 L"
} | B | 273.52333 L | PV=nRT, so V=nRT/P=(8)(0.082057)(300)/(0.72)=273.52333 L. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 273.52333 L, not 328.228 L.",
"B": "Correct. The worked calculation gives 273.52333 L.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 273.52333 L, not 410.285 L.",
"D": "Incorrect. Substituti... | [
"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_006770 | chemistry | stoichiometry | limiting_reagent_water | very_hard | provisional_template_level_label | single_correct_mcq | Hydrogen reacts with oxygen according to 2H₂ + O₂ → 2H₂O. If 32 mol H₂ and 12 mol O₂ are mixed and the reaction goes to completion, how many moles of H₂O form? | {
"A": "24 mol",
"B": "48 mol",
"C": "25 mol",
"D": "36 mol"
} | A | 24 mol | From 2 mol H₂ and 1 mol O₂, 2 mol H₂O form. H₂ can produce 32 mol H₂O; O₂ can produce 24 mol H₂O. The limiting condition is O₂; product=24 mol. | {
"A": "Correct. The worked calculation gives 24 mol.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 24 mol, not 48 mol.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 24 mol, not 25 mol.",
"D": "Incorrect. Substituting the stated values ... | [
"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_001961 | mathematics | combinatorics | binomial_coefficient | hard | provisional_template_level_label | single_correct_mcq | How many distinct subsets containing exactly 12 elements can be selected from a set of 47 distinct elements? | {
"A": "78377101276.5",
"B": "52251400851",
"C": "104502801702",
"D": "52251400850"
} | B | 52251400851 | Order does not matter, so the count is C(47,12)=47!/[12!(35)!]=52251400851. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 52251400851, not 78377101276.5.",
"B": "Correct. The worked calculation gives 52251400851.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 52251400851, not 104502801702.",
"D": "Incorrect. Sub... | [
"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_005345 | physics | electrostatics | capacitor_energy | hard | provisional_template_level_label | single_correct_mcq | A 55 μF capacitor is charged to a potential difference of 11 V. Calculate the energy stored, expressed in μJ. | {
"A": "3327.5 μJ",
"B": "3328.5 μJ",
"C": "3993 μJ",
"D": "1663.75 μJ"
} | A | 3327.5 μJ | U=½CV². Since C=55 μF, U in μJ is ½×55×(11)²=3327.5 μJ. | {
"A": "Correct. The worked calculation gives 3327.5 μJ.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 3327.5 μJ, not 3328.5 μJ.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 3327.5 μJ, not 3993 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_000105 | mathematics | algebra | quadratic_roots | hard | provisional_template_level_label | single_correct_mcq | For the quadratic equation 6x^2 +9x +1=0, let its roots be α and β. What is α²+β²? | {
"A": "4.91667",
"B": "2.875",
"C": "0.91667",
"D": "1.91667"
} | D | 1.91667 | By Vieta's relations, α+β = -9/6 and αβ = 1/6. Then α²+β²=(α+β)²−2αβ = (-9/6)²−2(1/6) = 1.91667. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 1.91667, not 4.91667.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 1.91667, not 2.875.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 1.91667, not 0.91667... | [
"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_004235 | physics | mechanics | inclined_plane_with_friction | very_hard | provisional_template_level_label | single_correct_mcq | A 46 kg block slides down a 37° rough incline. The coefficient of kinetic friction is 0.30. Taking g=10 m s⁻², find the net force down the plane. Use sin37°=0.6 and cos37°=0.8. | {
"A": "165.6 N",
"B": "167.6 N",
"C": "163.6 N",
"D": "198.72 N"
} | A | 165.6 N | Net force=mg(sinθ−μcosθ)=46×10×(0.6−0.30×0.8)=165.6 N. | {
"A": "Correct. The worked calculation gives 165.6 N.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 165.6 N, not 167.6 N.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 165.6 N, not 163.6 N.",
"D": "Incorrect. Substituting the stated va... | [
"Newton's laws",
"inclined plane",
"kinetic friction"
] | 4 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | physics_incline_friction | 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_008451 | 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 0.65 mol kg⁻¹. If its van't Hoff factor is 1 and K_b for water is 1.0 K kg mol⁻¹, find the boiling-point elevation. | {
"A": "1.3 K",
"B": "0.65 K",
"C": "-0.65 K",
"D": "0.4875 K"
} | B | 0.65 K | ΔT_b=iK_bm. Substitution gives ΔT_b=1×1.0×0.65=0.65 K. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 0.65 K, not 1.3 K.",
"B": "Correct. The worked calculation gives 0.65 K.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 0.65 K, not -0.65 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_008756 | 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 0.95 mol L⁻¹ and initially B is absent. At equilibrium, 20% of A has dissociated at constant volume. Find K_c. | {
"A": "3.19",
"B": "2.19",
"C": "1.19",
"D": "0.19"
} | D | 0.19 | Let α=0.2. At equilibrium [A]=c₀(1−α) and [B]=2c₀α. Thus K_c=[B]²/[A]=4c₀α²/(1−α)=4(0.95)(0.2)²/(1−0.2)=0.19. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 0.19, not 3.19.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 0.19, not 2.19.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 0.19, not 1.19.",
"D": "Corr... | [
"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_005726 | 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 0.1 atm at 307 K. What is its pressure at 495 K? | {
"A": "0.32248 atm",
"B": "-0.16124 atm",
"C": "0.16124 atm",
"D": "-0.83876 atm"
} | C | 0.16124 atm | At constant volume, P/T is constant. P₂=P₁T₂/T₁=(0.1)(495)/(307)=0.16124 atm. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 0.16124 atm, not 0.32248 atm.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 0.16124 atm, not -0.16124 atm.",
"C": "Correct. The worked calculation gives 0.16124 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_006698 | chemistry | stoichiometry | limiting_reagent_water | very_hard | provisional_template_level_label | single_correct_mcq | Hydrogen reacts with oxygen according to 2H₂ + O₂ → 2H₂O. If 3 mol H₂ and 31 mol O₂ are mixed and the reaction goes to completion, how many moles of H₂O form? | {
"A": "3 mol",
"B": "1 mol",
"C": "6 mol",
"D": "2 mol"
} | A | 3 mol | From 2 mol H₂ and 1 mol O₂, 2 mol H₂O form. H₂ can produce 3 mol H₂O; O₂ can produce 62 mol H₂O. The limiting condition is H₂; product=3 mol. | {
"A": "Correct. The worked calculation gives 3 mol.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 3 mol, not 1 mol.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 3 mol, not 6 mol.",
"D": "Incorrect. Substituting the stated values into ... | [
"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_007643 | chemistry | physical_chemistry | strong_acid_base_mixture | very_hard | provisional_template_level_label | single_correct_mcq | 61 mL of 0.10 M HCl is mixed with 74 mL of 0.10 M NaOH at 25°C. Assuming complete dissociation and additive volumes, find the final pH. | {
"A": "5.9918",
"B": "12.98361",
"C": "11.98361",
"D": "13.98361"
} | C | 11.98361 | Excess OH⁻ moles=0.10×(74−61)/1000=0.001300 mol. Total volume=0.135 L, so [OH⁻]=0.009630 M, pOH=−log₁₀[OH⁻], and pH=14−pOH=11.98361. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 11.98361, not 5.9918.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 11.98361, not 12.98361.",
"C": "Correct. The worked calculation gives 11.98361.",
"D": "Incorrect. Substituting the stated... | [
"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 |
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