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_002556 | mathematics | linear_algebra | 2x2_determinant | hard | provisional_template_level_label | single_correct_mcq | Find the determinant of the matrix [[3, -1], [5, 4]]. | {
"A": "17",
"B": "8.5",
"C": "15",
"D": "2.7"
} | A | 17 | For a 2×2 matrix [[a,b],[c,d]], det=ad−bc. Here it is (3)(4)−(-1)(5)=17. | {
"A": "Correct. The worked calculation gives 17.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 17, not 8.5.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 17, not 15.",
"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_000980 | mathematics | sequences_and_series | geometric_progression | hard | provisional_template_level_label | single_correct_mcq | A geometric progression has first term 2 and common ratio 5. What is the sum of its first 5 terms? | {
"A": "1249.6",
"B": "781",
"C": "2343",
"D": "1562"
} | D | 1562 | For r≠1, Sₙ=a(rⁿ−1)/(r−1). Thus Sₙ=2(5^5−1)/(5−1)=1562. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 1562, not 1249.6.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 1562, not 781.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 1562, not 2343.",
"D": "Cor... | [
"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_006247 | physics | ray_optics | convex_lens_magnification | very_hard | provisional_template_level_label | single_correct_mcq | A real object is placed 38 cm in front of a thin convex lens of focal length 19 cm. Using the Cartesian sign convention, what is the magnitude of the linear magnification? | {
"A": "2",
"B": "4",
"C": "1",
"D": "1.5"
} | C | 1 | Using distances as positive magnitudes, 1/f=1/v+1/u gives v=uf/(u−f)=(38)(19)/(38−19)=38 cm. Thus |m|=v/u=38/38=1. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 1, not 2.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 1, not 4.",
"C": "Correct. The worked calculation gives 1.",
"D": "Incorrect. Substituting the stated values into the governing relati... | [
"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_006110 | physics | ray_optics | convex_lens_magnification | very_hard | provisional_template_level_label | single_correct_mcq | A real object is placed 92 cm in front of a thin convex lens of focal length 27 cm. Using the Cartesian sign convention, what is the magnitude of the linear magnification? | {
"A": "-0.58462",
"B": "0.41538",
"C": "2.41538",
"D": "0.83077"
} | B | 0.41538 | Using distances as positive magnitudes, 1/f=1/v+1/u gives v=uf/(u−f)=(92)(27)/(92−27)=38.21538 cm. Thus |m|=v/u=38.21538/92=0.41538. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 0.41538, not -0.58462.",
"B": "Correct. The worked calculation gives 0.41538.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 0.41538, not 2.41538.",
"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_006147 | physics | ray_optics | convex_lens_magnification | very_hard | provisional_template_level_label | single_correct_mcq | A real object is placed 82 cm in front of a thin convex lens of focal length 20 cm. Using the Cartesian sign convention, what is the magnitude of the linear magnification? | {
"A": "-1.67742",
"B": "0.16129",
"C": "1.32258",
"D": "0.32258"
} | D | 0.32258 | Using distances as positive magnitudes, 1/f=1/v+1/u gives v=uf/(u−f)=(82)(20)/(82−20)=26.45161 cm. Thus |m|=v/u=26.45161/82=0.32258. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 0.32258, not -1.67742.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 0.32258, not 0.16129.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 0.32258, not 1.32... | [
"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_006007 | physics | ray_optics | convex_lens_magnification | very_hard | provisional_template_level_label | single_correct_mcq | A real object is placed 71 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": "1.44898",
"B": "0.44898",
"C": "0.33673",
"D": "2.44898"
} | B | 0.44898 | Using distances as positive magnitudes, 1/f=1/v+1/u gives v=uf/(u−f)=(71)(22)/(71−22)=31.87755 cm. Thus |m|=v/u=31.87755/71=0.44898. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 0.44898, not 1.44898.",
"B": "Correct. The worked calculation gives 0.44898.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 0.44898, not 0.33673.",
"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_004319 | physics | mechanics | inclined_plane_with_friction | very_hard | provisional_template_level_label | single_correct_mcq | A 4 kg block slides down a 37° rough incline. The coefficient of kinetic friction is 0.36. Taking g=10 m s⁻², find the net force down the plane. Use sin37°=0.6 and cos37°=0.8. | {
"A": "13.48 N",
"B": "12.48 N",
"C": "10.48 N",
"D": "-12.48 N"
} | B | 12.48 N | Net force=mg(sinθ−μcosθ)=4×10×(0.6−0.36×0.8)=12.48 N. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 12.48 N, not 13.48 N.",
"B": "Correct. The worked calculation gives 12.48 N.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 12.48 N, not 10.48 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_005070 | physics | electrostatics | coulombs_law | hard | provisional_template_level_label | single_correct_mcq | Two point charges of +1 μC and +3 μC are separated by 8 cm in vacuum. Using k=9×10⁹ N m² C⁻², find the magnitude of the electrostatic force. | {
"A": "2.21875 N",
"B": "7.21875 N",
"C": "5.21875 N",
"D": "4.21875 N"
} | D | 4.21875 N | Convert charges to coulombs and distance to metres. F=k|q₁q₂|/r²=9×10⁹(1×10⁻⁶)(3×10⁻⁶)/(8×10⁻²)²=4.21875 N. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 4.21875 N, not 2.21875 N.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 4.21875 N, not 7.21875 N.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 4.21875 N,... | [
"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_009973 | chemistry | inorganic_chemistry | complex_ion_oxidation_state | hard | provisional_template_level_label | single_correct_mcq | For a hypothetical coordination ion [Pd(CN)_4]^-4, each CN ligand has charge −1. Determine the formal oxidation state of Pd. | {
"A": "1",
"B": "-2",
"C": "2",
"D": "0"
} | D | 0 | Let the formal oxidation state of Pd be x. Charge balance gives x+(4)(−1)=-4; hence x=-4+4=+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 -2.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 0, not 2.",
"D": "Correct. The worked c... | [
"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_004417 | physics | rotational_mechanics | disc_angular_acceleration | hard | provisional_template_level_label | single_correct_mcq | A uniform solid disc of mass 5 kg and radius 3 m is acted on by a constant torque of 45 N m about its central axis. Find its angular acceleration. | {
"A": "3 rad s⁻²",
"B": "2 rad s⁻²",
"C": "0 rad s⁻²",
"D": "4 rad s⁻²"
} | B | 2 rad s⁻² | For a solid disc, I=½MR²=½(5)(3)²=22.5 kg m². Using τ=Iα, α=45/22.5=2 rad s⁻². | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 2 rad s⁻², not 3 rad s⁻².",
"B": "Correct. The worked calculation gives 2 rad s⁻².",
"C": "Incorrect. Substituting the stated values into the governing relation gives 2 rad s⁻², not 0 rad s⁻².",
"D": "Incorrect. Substituting the... | [
"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_000161 | mathematics | algebra | quadratic_roots | hard | provisional_template_level_label | single_correct_mcq | For the quadratic equation 7x^2 -9x -1=0, let its roots be α and β. What is α²+β²? | {
"A": "2.90816",
"B": "1.93878",
"C": "1.19388",
"D": "-1.93878"
} | B | 1.93878 | By Vieta's relations, α+β = 9/7 and αβ = -1/7. Then α²+β²=(α+β)²−2αβ = (9/7)²−2(-1/7) = 1.93878. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 1.93878, not 2.90816.",
"B": "Correct. The worked calculation gives 1.93878.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 1.93878, not 1.19388.",
"D": "Incorrect. Substituting the stated va... | [
"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_001839 | mathematics | combinatorics | binomial_coefficient | hard | provisional_template_level_label | single_correct_mcq | How many distinct subsets containing exactly 7 elements can be selected from a set of 72 distinct elements? | {
"A": "1767731644.8",
"B": "1473109704",
"C": "736554852",
"D": "147310971.4"
} | B | 1473109704 | Order does not matter, so the count is C(72,7)=72!/[7!(65)!]=1473109704. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 1473109704, not 1767731644.8.",
"B": "Correct. The worked calculation gives 1473109704.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 1473109704, not 736554852.",
"D": "Incorrect. Substituti... | [
"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_009478 | 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_6H_15N_1. Treat X as the total number of monovalent halogen atoms. | {
"A": "-2",
"B": "0",
"C": "3",
"D": "1"
} | B | 0 | Use DBE=(2C+2+N−H−X)/2. Substitution gives (2×6+2+1−15−0)/2=0. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 0, not -2.",
"B": "Correct. The worked calculation gives 0.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 0, not 3.",
"D": "Incorrect. Substituting the stated values into the governing relat... | [
"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_009706 | chemistry | inorganic_chemistry | complex_ion_oxidation_state | hard | provisional_template_level_label | single_correct_mcq | For a hypothetical coordination ion [Co(CN)_6]^+1, each CN ligand has charge −1. Determine the formal oxidation state of Co. | {
"A": "10.5",
"B": "9",
"C": "7",
"D": "10"
} | C | 7 | Let the formal oxidation state of Co be x. Charge balance gives x+(6)(−1)=+1; hence x=1+6=+7. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 7, not 10.5.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 7, not 9.",
"C": "Correct. The worked calculation gives 7.",
"D": "Incorrect. Substituting the stated values into the governing rel... | [
"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_007244 | chemistry | physical_chemistry | ideal_gas_volume | hard | provisional_template_level_label | single_correct_mcq | Assuming ideal behaviour, what volume is occupied by 19 mol of gas at 642 K and 0.58 atm? Use R=0.082057 L atm mol⁻¹ K⁻¹. | {
"A": "1294.3077 L",
"B": "1725.7436 L",
"C": "-1725.7436 L",
"D": "3451.48719 L"
} | B | 1725.7436 L | PV=nRT, so V=nRT/P=(19)(0.082057)(642)/(0.58)=1725.7436 L. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 1725.7436 L, not 1294.3077 L.",
"B": "Correct. The worked calculation gives 1725.7436 L.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 1725.7436 L, not -1725.7436 L.",
"D": "Incorrect. Subst... | [
"ideal gas equation",
"unit consistency"
] | 3 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | chem_ideal_gas | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_009165 | chemistry | physical_chemistry | nernst_equation_298K | hard | provisional_template_level_label | single_correct_mcq | At 298 K, a galvanic cell has E°cell=0.76 V, transfers 1 electrons, and has reaction quotient Q=10^-3. Using E=E°−(0.0591/n)log₁₀Q, calculate Ecell. | {
"A": "2.9373 V",
"B": "-1.0627 V",
"C": "-0.0627 V",
"D": "0.9373 V"
} | D | 0.9373 V | Apply the Nernst equation: E=0.76−(0.0591/1)(-3)=0.9373 V. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 0.9373 V, not 2.9373 V.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 0.9373 V, not -1.0627 V.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 0.9373 V, not... | [
"electrochemical cells",
"Nernst equation",
"logarithms"
] | 3 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | chem_nernst | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_005000 | physics | electricity | series_parallel_circuit | very_hard | provisional_template_level_label | single_correct_mcq | Resistors 11 Ω and 11 Ω are connected in series; this combination is connected in parallel with a 11 Ω resistor across a 24 V ideal battery. Find the total current supplied by the battery. | {
"A": "3.27273 A",
"B": "0.27273 A",
"C": "5.27273 A",
"D": "4.90909 A"
} | A | 3.27273 A | Series branch resistance=11+11=22 Ω. Parallel equivalent R_eq=(22×11)/(22+11)=7.33333 Ω. Total current I=V/R_eq=24/7.33333=3.27273 A. | {
"A": "Correct. The worked calculation gives 3.27273 A.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 3.27273 A, not 0.27273 A.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 3.27273 A, not 5.27273 A.",
"D": "Incorrect. Substituting the... | [
"series resistors",
"parallel resistors",
"Ohm's law"
] | 4 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | physics_series_parallel | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_000101 | mathematics | algebra | quadratic_roots | hard | provisional_template_level_label | single_correct_mcq | For the quadratic equation 6x^2 +1x +3=0, let its roots be α and β. What is α²+β²? | {
"A": "0.97222",
"B": "-0.97222",
"C": "-1.97222",
"D": "0.90278"
} | B | -0.97222 | By Vieta's relations, α+β = -1/6 and αβ = 3/6. Then α²+β²=(α+β)²−2αβ = (-1/6)²−2(3/6) = -0.97222. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives -0.97222, not 0.97222.",
"B": "Correct. The worked calculation gives -0.97222.",
"C": "Incorrect. Substituting the stated values into the governing relation gives -0.97222, not -1.97222.",
"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_002128 | mathematics | probability | same_colour_without_replacement | very_hard | provisional_template_level_label | single_correct_mcq | A bag contains 4 red, 10 blue and 9 green balls. Two balls are drawn uniformly at random without replacement. What is the probability that both have the same colour? | {
"A": "0.68775",
"B": "-0.65613",
"C": "0.34387",
"D": "-2.65613"
} | C | 0.34387 | Favourable pairs = C(4,2)+C(10,2)+C(9,2)=87. Total pairs=C(23,2)=253. Therefore P=87/253=0.34387. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 0.34387, not 0.68775.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 0.34387, not -0.65613.",
"C": "Correct. The worked calculation gives 0.34387.",
"D": "Incorrect. Substituting the stated v... | [
"classical probability",
"combinations",
"without replacement"
] | 4 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | math_probability_colours | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_005777 | 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 0.2 atm at 206 K. What is its pressure at 791 K? | {
"A": "1.53592 atm",
"B": "-2.23204 atm",
"C": "0.38398 atm",
"D": "0.76796 atm"
} | D | 0.76796 atm | At constant volume, P/T is constant. P₂=P₁T₂/T₁=(0.2)(791)/(206)=0.76796 atm. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 0.76796 atm, not 1.53592 atm.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 0.76796 atm, not -2.23204 atm.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 0... | [
"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_005606 | physics | electrostatics | capacitor_energy | hard | provisional_template_level_label | single_correct_mcq | A 75 μF capacitor is charged to a potential difference of 19 V. Calculate the energy stored, expressed in μJ. | {
"A": "13537.5 μJ",
"B": "10830 μJ",
"C": "20306.25 μJ",
"D": "13539.5 μJ"
} | A | 13537.5 μJ | U=½CV². Since C=75 μF, U in μJ is ½×75×(19)²=13537.5 μJ. | {
"A": "Correct. The worked calculation gives 13537.5 μJ.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 13537.5 μJ, not 10830 μJ.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 13537.5 μJ, not 20306.25 μJ.",
"D": "Incorrect. Substituting... | [
"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_000559 | mathematics | sequences_and_series | arithmetic_progression | hard | provisional_template_level_label | single_correct_mcq | An arithmetic progression has first term 11 and common difference 12. Find the sum of its first 16 terms. | {
"A": "-1616",
"B": "1616",
"C": "1212",
"D": "1939.2"
} | B | 1616 | Use Sₙ = n/2[2a+(n−1)d]. Substituting a=11, d=12, n=16: Sₙ=16/2[2(11)+(16−1)(12)]=1616. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 1616, not -1616.",
"B": "Correct. The worked calculation gives 1616.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 1616, not 1212.",
"D": "Incorrect. Substituting the stated values into the ... | [
"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_007001 | chemistry | stoichiometry | limiting_reagent_water | very_hard | provisional_template_level_label | single_correct_mcq | Hydrogen reacts with oxygen according to 2H₂ + O₂ → 2H₂O. If 34 mol H₂ and 17 mol O₂ are mixed and the reaction goes to completion, how many moles of H₂O form? | {
"A": "25.5 mol",
"B": "17 mol",
"C": "34 mol",
"D": "32 mol"
} | C | 34 mol | From 2 mol H₂ and 1 mol O₂, 2 mol H₂O form. H₂ can produce 34 mol H₂O; O₂ can produce 34 mol H₂O. The limiting condition is neither; stoichiometric; product=34 mol. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 34 mol, not 25.5 mol.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 34 mol, not 17 mol.",
"C": "Correct. The worked calculation gives 34 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_007711 | 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 94 mmol L⁻¹, what concentration remains after 40 minutes? | {
"A": "17.625 mmol L⁻¹",
"B": "23.5 mmol L⁻¹",
"C": "22.5 mmol L⁻¹",
"D": "35.25 mmol L⁻¹"
} | B | 23.5 mmol L⁻¹ | 40 minutes corresponds to 2 half-lives. The remaining concentration is [A]₀(1/2)ⁿ=94/2^2=23.5 mmol L⁻¹. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 23.5 mmol L⁻¹, not 17.625 mmol L⁻¹.",
"B": "Correct. The worked calculation gives 23.5 mmol L⁻¹.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 23.5 mmol L⁻¹, not 22.5 mmol L⁻¹.",
"D": "Incor... | [
"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_002943 | mathematics | vectors | dot_product | hard | provisional_template_level_label | single_correct_mcq | Given vectors u=(6, -2, -3) and v=(8, -1, 3), calculate u·v. | {
"A": "39",
"B": "42",
"C": "41",
"D": "82"
} | C | 41 | The dot product is the sum of corresponding component products: (6)(8)+(-2)(-1)+(-3)(3)=41. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 41, not 39.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 41, not 42.",
"C": "Correct. The worked calculation gives 41.",
"D": "Incorrect. Substituting the stated values into the governing r... | [
"three-dimensional vectors",
"scalar product"
] | 3 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | math_vector_dot | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_003493 | physics | mechanics | projectile_range | hard | provisional_template_level_label | single_correct_mcq | A projectile is launched from level ground at 64 m s⁻¹ at 40° to the horizontal. Take g=10 m s⁻². Find its horizontal range. | {
"A": "484.05271 m",
"B": "403.37726 m",
"C": "302.53294 m",
"D": "404.37726 m"
} | B | 403.37726 m | Range R=u²sin(2θ)/g. Substituting u=64, θ=40° and g=10 gives R=(64)²sin(80°)/10=403.37726 m. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 403.37726 m, not 484.05271 m.",
"B": "Correct. The worked calculation gives 403.37726 m.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 403.37726 m, not 302.53294 m.",
"D": "Incorrect. Substi... | [
"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_000008 | mathematics | algebra | quadratic_roots | hard | provisional_template_level_label | single_correct_mcq | For the quadratic equation 1x^2 +16x -12=0, let its roots be α and β. What is α²+β²? | {
"A": "420",
"B": "278",
"C": "280",
"D": "140"
} | C | 280 | By Vieta's relations, α+β = -16/1 and αβ = -12/1. Then α²+β²=(α+β)²−2αβ = (-16/1)²−2(-12/1) = 280. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 280, not 420.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 280, not 278.",
"C": "Correct. The worked calculation gives 280.",
"D": "Incorrect. Substituting the stated values into the govern... | [
"Vieta's formulas",
"symmetric polynomial expressions"
] | 3 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | math_quadratic_vieta | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_004302 | physics | mechanics | inclined_plane_with_friction | very_hard | provisional_template_level_label | single_correct_mcq | A 32 kg block slides down a 37° rough incline. The coefficient of kinetic friction is 0.45. Taking g=10 m s⁻², find the net force down the plane. Use sin37°=0.6 and cos37°=0.8. | {
"A": "76.8 N",
"B": "115.2 N",
"C": "77.8 N",
"D": "153.6 N"
} | A | 76.8 N | Net force=mg(sinθ−μcosθ)=32×10×(0.6−0.45×0.8)=76.8 N. | {
"A": "Correct. The worked calculation gives 76.8 N.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 76.8 N, not 115.2 N.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 76.8 N, not 77.8 N.",
"D": "Incorrect. Substituting the stated values... | [
"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_005162 | physics | electrostatics | coulombs_law | hard | provisional_template_level_label | single_correct_mcq | Two point charges of +3 μC and +1 μC are separated by 9 cm in vacuum. Using k=9×10⁹ N m² C⁻², find the magnitude of the electrostatic force. | {
"A": "1.33333 N",
"B": "3.33333 N",
"C": "6.66667 N",
"D": "2.5 N"
} | B | 3.33333 N | Convert charges to coulombs and distance to metres. F=k|q₁q₂|/r²=9×10⁹(3×10⁻⁶)(1×10⁻⁶)/(9×10⁻²)²=3.33333 N. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 3.33333 N, not 1.33333 N.",
"B": "Correct. The worked calculation gives 3.33333 N.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 3.33333 N, not 6.66667 N.",
"D": "Incorrect. Substituting the... | [
"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_004483 | physics | rotational_mechanics | disc_angular_acceleration | hard | provisional_template_level_label | single_correct_mcq | A uniform solid disc of mass 4 kg and radius 1 m is acted on by a constant torque of 63 N m about its central axis. Find its angular acceleration. | {
"A": "37.8 rad s⁻²",
"B": "32.5 rad s⁻²",
"C": "31.5 rad s⁻²",
"D": "33.5 rad s⁻²"
} | C | 31.5 rad s⁻² | For a solid disc, I=½MR²=½(4)(1)²=2 kg m². Using τ=Iα, α=63/2=31.5 rad s⁻². | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 31.5 rad s⁻², not 37.8 rad s⁻².",
"B": "Incorrect. Substituting the stated values into the governing relation gives 31.5 rad s⁻², not 32.5 rad s⁻².",
"C": "Correct. The worked calculation gives 31.5 rad s⁻².",
"D": "Incorrect. S... | [
"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_008498 | 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 1.17 mol kg⁻¹. If its van't Hoff factor is 1 and K_b for water is 0.6 K kg mol⁻¹, find the boiling-point elevation. | {
"A": "2.702 K",
"B": "-1.298 K",
"C": "0.702 K",
"D": "1.404 K"
} | C | 0.702 K | ΔT_b=iK_bm. Substitution gives ΔT_b=1×0.6×1.17=0.702 K. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 0.702 K, not 2.702 K.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 0.702 K, not -1.298 K.",
"C": "Correct. The worked calculation gives 0.702 K.",
"D": "Incorrect. Substituting the stated v... | [
"colligative properties",
"van't Hoff factor",
"molality"
] | 3 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | chem_boiling_elevation | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_001252 | mathematics | calculus | polynomial_differentiation | hard | provisional_template_level_label | single_correct_mcq | Let f(x)=6x³ -7x² +2x+7. Evaluate f′(-4). | {
"A": "415.2",
"B": "348",
"C": "692",
"D": "346"
} | D | 346 | Differentiate: f′(x)=3(6)x²+2(-7)x+(2). At x=-4, f′(-4)=3(6)(-4)²+2(-7)(-4)+(2)=346. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 346, not 415.2.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 346, not 348.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 346, not 692.",
"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_008466 | 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 1.68 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": "1.4944 K",
"B": "5.4944 K",
"C": "3.4944 K",
"D": "5.2416 K"
} | C | 3.4944 K | ΔT_b=iK_bm. Substitution gives ΔT_b=4×0.52×1.68=3.4944 K. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 3.4944 K, not 1.4944 K.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 3.4944 K, not 5.4944 K.",
"C": "Correct. The worked calculation gives 3.4944 K.",
"D": "Incorrect. Substituting the stat... | [
"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_004636 | physics | rotational_mechanics | disc_angular_acceleration | hard | provisional_template_level_label | single_correct_mcq | A uniform solid disc of mass 12 kg and radius 6 m is acted on by a constant torque of 21 N m about its central axis. Find its angular acceleration. | {
"A": "3.09722 rad s⁻²",
"B": "-0.90278 rad s⁻²",
"C": "1.00972 rad s⁻²",
"D": "0.09722 rad s⁻²"
} | D | 0.09722 rad s⁻² | For a solid disc, I=½MR²=½(12)(6)²=216 kg m². Using τ=Iα, α=21/216=0.09722 rad s⁻². | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 0.09722 rad s⁻², not 3.09722 rad s⁻².",
"B": "Incorrect. Substituting the stated values into the governing relation gives 0.09722 rad s⁻², not -0.90278 rad s⁻².",
"C": "Incorrect. Substituting the stated values into the governing ... | [
"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_007837 | 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 60 mmol L⁻¹, what concentration remains after 160 minutes? | {
"A": "0.23438 mmol L⁻¹",
"B": "-0.23438 mmol L⁻¹",
"C": "-2.76562 mmol L⁻¹",
"D": "0.17578 mmol L⁻¹"
} | A | 0.23438 mmol L⁻¹ | 160 minutes corresponds to 8 half-lives. The remaining concentration is [A]₀(1/2)ⁿ=60/2^8=0.23438 mmol L⁻¹. | {
"A": "Correct. The worked calculation gives 0.23438 mmol L⁻¹.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 0.23438 mmol L⁻¹, not -0.23438 mmol L⁻¹.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 0.23438 mmol L⁻¹, not -2.76562 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_005738 | physics | thermodynamics | ideal_gas_isochoric_pressure | hard | provisional_template_level_label | single_correct_mcq | A fixed amount of ideal gas is kept in a rigid container. Its pressure is 3.7 atm at 278 K. What is its pressure at 488 K? | {
"A": "4.49496 atm",
"B": "6.49496 atm",
"C": "8.49496 atm",
"D": "9.49496 atm"
} | B | 6.49496 atm | At constant volume, P/T is constant. P₂=P₁T₂/T₁=(3.7)(488)/(278)=6.49496 atm. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 6.49496 atm, not 4.49496 atm.",
"B": "Correct. The worked calculation gives 6.49496 atm.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 6.49496 atm, not 8.49496 atm.",
"D": "Incorrect. Substi... | [
"ideal gas law",
"isochoric process",
"absolute temperature"
] | 3 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | physics_isochoric_gas | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_009508 | 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_10H_5N_1X_2. Treat X as the total number of monovalent halogen atoms. | {
"A": "1.8",
"B": "8",
"C": "4",
"D": "5"
} | B | 8 | Use DBE=(2C+2+N−H−X)/2. Substitution gives (2×10+2+1−5−2)/2=8. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 8, not 1.8.",
"B": "Correct. The worked calculation gives 8.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 8, not 4.",
"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_000227 | mathematics | algebra | quadratic_roots | hard | provisional_template_level_label | single_correct_mcq | For the quadratic equation 3x^2 -7x +11=0, let its roots be α and β. What is α²+β²? | {
"A": "-3.77778",
"B": "-4.88889",
"C": "0.81111",
"D": "-1.88889"
} | D | -1.88889 | By Vieta's relations, α+β = 7/3 and αβ = 11/3. Then α²+β²=(α+β)²−2αβ = (7/3)²−2(11/3) = -1.88889. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives -1.88889, not -3.77778.",
"B": "Incorrect. Substituting the stated values into the governing relation gives -1.88889, not -4.88889.",
"C": "Incorrect. Substituting the stated values into the governing relation gives -1.88889, not ... | [
"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_000268 | mathematics | algebra | quadratic_roots | hard | provisional_template_level_label | single_correct_mcq | For the quadratic equation 3x^2 -15x -9=0, let its roots be α and β. What is α²+β²? | {
"A": "32",
"B": "31",
"C": "4.1",
"D": "62"
} | B | 31 | By Vieta's relations, α+β = 15/3 and αβ = -9/3. Then α²+β²=(α+β)²−2αβ = (15/3)²−2(-9/3) = 31. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 31, not 32.",
"B": "Correct. The worked calculation gives 31.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 31, not 4.1.",
"D": "Incorrect. Substituting the stated values into the governing ... | [
"Vieta's formulas",
"symmetric polynomial expressions"
] | 3 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | math_quadratic_vieta | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_004463 | physics | rotational_mechanics | disc_angular_acceleration | hard | provisional_template_level_label | single_correct_mcq | A uniform solid disc of mass 6 kg and radius 4 m is acted on by a constant torque of 27 N m about its central axis. Find its angular acceleration. | {
"A": "-1.4375 rad s⁻²",
"B": "3.5625 rad s⁻²",
"C": "1.125 rad s⁻²",
"D": "0.5625 rad s⁻²"
} | D | 0.5625 rad s⁻² | For a solid disc, I=½MR²=½(6)(4)²=48 kg m². Using τ=Iα, α=27/48=0.5625 rad s⁻². | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 0.5625 rad s⁻², not -1.4375 rad s⁻².",
"B": "Incorrect. Substituting the stated values into the governing relation gives 0.5625 rad s⁻², not 3.5625 rad s⁻².",
"C": "Incorrect. Substituting the stated values into the governing rela... | [
"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_001999 | 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 81 distinct elements? | {
"A": "1663738",
"B": "1663740",
"C": "1996488",
"D": "1663742"
} | B | 1663740 | Order does not matter, so the count is C(81,4)=81!/[4!(77)!]=1663740. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 1663740, not 1663738.",
"B": "Correct. The worked calculation gives 1663740.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 1663740, not 1996488.",
"D": "Incorrect. Substituting the stated va... | [
"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_001462 | mathematics | calculus | definite_integral_polynomial | very_hard | provisional_template_level_label | single_correct_mcq | Evaluate the definite integral ∫₀^8 (-1x² +3x +5) dx. | {
"A": "-41.6",
"B": "-36.66667",
"C": "-34.66667",
"D": "-33.66667"
} | C | -34.66667 | An antiderivative is (-1/3)x³+(3/2)x²+(5)x. Evaluating between 0 and 8 gives (-1/3)(8)³+(3/2)(8)²+(5)(8)=-34.66667. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives -34.66667, not -41.6.",
"B": "Incorrect. Substituting the stated values into the governing relation gives -34.66667, not -36.66667.",
"C": "Correct. The worked calculation gives -34.66667.",
"D": "Incorrect. Substituting the sta... | [
"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_004309 | physics | mechanics | inclined_plane_with_friction | very_hard | provisional_template_level_label | single_correct_mcq | A 22 kg block slides down a 37° rough incline. The coefficient of kinetic friction is 0.42. Taking g=10 m s⁻², find the net force down the plane. Use sin37°=0.6 and cos37°=0.8. | {
"A": "58.08 N",
"B": "29.04 N",
"C": "57.08 N",
"D": "69.696 N"
} | A | 58.08 N | Net force=mg(sinθ−μcosθ)=22×10×(0.6−0.42×0.8)=58.08 N. | {
"A": "Correct. The worked calculation gives 58.08 N.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 58.08 N, not 29.04 N.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 58.08 N, not 57.08 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_006675 | chemistry | stoichiometry | limiting_reagent_water | very_hard | provisional_template_level_label | single_correct_mcq | Hydrogen reacts with oxygen according to 2H₂ + O₂ → 2H₂O. If 41 mol H₂ and 30 mol O₂ are mixed and the reaction goes to completion, how many moles of H₂O form? | {
"A": "61.5 mol",
"B": "32.8 mol",
"C": "41 mol",
"D": "20.5 mol"
} | C | 41 mol | From 2 mol H₂ and 1 mol O₂, 2 mol H₂O form. H₂ can produce 41 mol H₂O; O₂ can produce 60 mol H₂O. The limiting condition is H₂; product=41 mol. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 41 mol, not 61.5 mol.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 41 mol, not 32.8 mol.",
"C": "Correct. The worked calculation gives 41 mol.",
"D": "Incorrect. Substituting the stated val... | [
"balanced chemical equations",
"limiting reagent",
"mole ratios"
] | 4 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | chem_limiting_reagent | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_009736 | chemistry | inorganic_chemistry | complex_ion_oxidation_state | hard | provisional_template_level_label | single_correct_mcq | For a hypothetical coordination ion [Pd(Cl)_6]^-4, each Cl ligand has charge −1. Determine the formal oxidation state of Pd. | {
"A": "0",
"B": "1.5",
"C": "2",
"D": "4"
} | C | 2 | Let the formal oxidation state of Pd be x. Charge balance gives x+(6)(−1)=-4; hence x=-4+6=+2. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 2, not 0.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 2, not 1.5.",
"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_008910 | 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 0.93 mol L⁻¹ and initially B is absent. At equilibrium, 19% of A has dissociated at constant volume. Find K_c. | {
"A": "0.12434",
"B": "-2.83421",
"C": "0.16579",
"D": "2.16579"
} | C | 0.16579 | Let α=0.19. At equilibrium [A]=c₀(1−α) and [B]=2c₀α. Thus K_c=[B]²/[A]=4c₀α²/(1−α)=4(0.93)(0.19)²/(1−0.19)=0.16579. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 0.16579, not 0.12434.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 0.16579, not -2.83421.",
"C": "Correct. The worked calculation gives 0.16579.",
"D": "Incorrect. Substituting the stated v... | [
"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_000315 | mathematics | algebra | quadratic_roots | hard | provisional_template_level_label | single_correct_mcq | For the quadratic equation 4x^2 -10x -12=0, let its roots be α and β. What is α²+β²? | {
"A": "10.25",
"B": "14.25",
"C": "12.25",
"D": "24.5"
} | C | 12.25 | By Vieta's relations, α+β = 10/4 and αβ = -12/4. Then α²+β²=(α+β)²−2αβ = (10/4)²−2(-12/4) = 12.25. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 12.25, not 10.25.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 12.25, not 14.25.",
"C": "Correct. The worked calculation gives 12.25.",
"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_003320 | mathematics | complex_numbers | real_part_of_complex_quotient | very_hard | provisional_template_level_label | single_correct_mcq | Let z=(3+15i)/(2+9i). What is Re(z), the real part of z? | {
"A": "3.31765",
"B": "-1.34118",
"C": "1.65882",
"D": "0.65882"
} | C | 1.65882 | Multiply numerator and denominator by the conjugate (2-9i). Then Re(z)=(ac+bd)/(c²+d²)=((3)(2)+(15)(9))/(2²+(9)²)=1.65882. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 1.65882, not 3.31765.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 1.65882, not -1.34118.",
"C": "Correct. The worked calculation gives 1.65882.",
"D": "Incorrect. Substituting the stated v... | [
"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_008767 | 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 0.41 mol L⁻¹ and initially B is absent. At equilibrium, 29% of A has dissociated at constant volume. Find K_c. | {
"A": "0.19426",
"B": "1.01943",
"C": "-0.19426",
"D": "3.19426"
} | A | 0.19426 | Let α=0.29. At equilibrium [A]=c₀(1−α) and [B]=2c₀α. Thus K_c=[B]²/[A]=4c₀α²/(1−α)=4(0.41)(0.29)²/(1−0.29)=0.19426. | {
"A": "Correct. The worked calculation gives 0.19426.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 0.19426, not 1.01943.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 0.19426, not -0.19426.",
"D": "Incorrect. Substituting the stated v... | [
"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_003585 | physics | mechanics | projectile_range | hard | provisional_template_level_label | single_correct_mcq | A projectile is launched from level ground at 114 m s⁻¹ at 18° to the horizontal. Take g=10 m s⁻². Find its horizontal range. | {
"A": "77.38857 m",
"B": "763.88571 m",
"C": "-763.88571 m",
"D": "916.66286 m"
} | B | 763.88571 m | Range R=u²sin(2θ)/g. Substituting u=114, θ=18° and g=10 gives R=(114)²sin(36°)/10=763.88571 m. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 763.88571 m, not 77.38857 m.",
"B": "Correct. The worked calculation gives 763.88571 m.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 763.88571 m, not -763.88571 m.",
"D": "Incorrect. Substi... | [
"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_001869 | 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 48 distinct elements? | {
"A": "1677106641",
"B": "2515659960",
"C": "-1677106640",
"D": "1677106640"
} | D | 1677106640 | Order does not matter, so the count is C(48,9)=48!/[9!(39)!]=1677106640. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 1677106640, not 1677106641.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 1677106640, not 2515659960.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 167710... | [
"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_004388 | physics | rotational_mechanics | disc_angular_acceleration | hard | provisional_template_level_label | single_correct_mcq | A uniform solid disc of mass 12 kg and radius 5 m is acted on by a constant torque of 46 N m about its central axis. Find its angular acceleration. | {
"A": "0.30667 rad s⁻²",
"B": "-1.69333 rad s⁻²",
"C": "2.30667 rad s⁻²",
"D": "0.23 rad s⁻²"
} | A | 0.30667 rad s⁻² | For a solid disc, I=½MR²=½(12)(5)²=150 kg m². Using τ=Iα, α=46/150=0.30667 rad s⁻². | {
"A": "Correct. The worked calculation gives 0.30667 rad s⁻².",
"B": "Incorrect. Substituting the stated values into the governing relation gives 0.30667 rad s⁻², not -1.69333 rad s⁻².",
"C": "Incorrect. Substituting the stated values into the governing relation gives 0.30667 rad s⁻², not 2.30667 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_000862 | mathematics | sequences_and_series | geometric_progression | hard | provisional_template_level_label | single_correct_mcq | A geometric progression has first term 4 and common ratio 4. What is the sum of its first 6 terms? | {
"A": "2730",
"B": "5462",
"C": "547",
"D": "5460"
} | D | 5460 | For r≠1, Sₙ=a(rⁿ−1)/(r−1). Thus Sₙ=4(4^6−1)/(4−1)=5460. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 5460, not 2730.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 5460, not 5462.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 5460, not 547.",
"D": "Corre... | [
"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_006793 | chemistry | stoichiometry | limiting_reagent_water | very_hard | provisional_template_level_label | single_correct_mcq | Hydrogen reacts with oxygen according to 2H₂ + O₂ → 2H₂O. If 28 mol H₂ and 15 mol O₂ are mixed and the reaction goes to completion, how many moles of H₂O form? | {
"A": "-28 mol",
"B": "28 mol",
"C": "29 mol",
"D": "26 mol"
} | B | 28 mol | From 2 mol H₂ and 1 mol O₂, 2 mol H₂O form. H₂ can produce 28 mol H₂O; O₂ can produce 30 mol H₂O. The limiting condition is H₂; product=28 mol. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 28 mol, not -28 mol.",
"B": "Correct. The worked calculation gives 28 mol.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 28 mol, not 29 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_007522 | chemistry | physical_chemistry | strong_acid_base_mixture | very_hard | provisional_template_level_label | single_correct_mcq | 11 mL of 0.10 M HCl is mixed with 43 mL of 0.10 M NaOH at 25°C. Assuming complete dissociation and additive volumes, find the final pH. | {
"A": "6.38638",
"B": "12.77276",
"C": "-12.77276",
"D": "10.77276"
} | B | 12.77276 | Excess OH⁻ moles=0.10×(43−11)/1000=0.003200 mol. Total volume=0.054 L, so [OH⁻]=0.059259 M, pOH=−log₁₀[OH⁻], and pH=14−pOH=12.77276. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 12.77276, not 6.38638.",
"B": "Correct. The worked calculation gives 12.77276.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 12.77276, not -12.77276.",
"D": "Incorrect. Substituting the stat... | [
"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_004284 | physics | mechanics | inclined_plane_with_friction | very_hard | provisional_template_level_label | single_correct_mcq | A 8 kg block slides down a 37° rough incline. The coefficient of kinetic friction is 0.34. Taking g=10 m s⁻², find the net force down the plane. Use sin37°=0.6 and cos37°=0.8. | {
"A": "13.12 N",
"B": "39.36 N",
"C": "26.24 N",
"D": "-26.24 N"
} | C | 26.24 N | Net force=mg(sinθ−μcosθ)=8×10×(0.6−0.34×0.8)=26.24 N. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 26.24 N, not 13.12 N.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 26.24 N, not 39.36 N.",
"C": "Correct. The worked calculation gives 26.24 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_008596 | 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 0.62 mol kg⁻¹. If its van't Hoff factor is 2 and K_b for water is 1.0 K kg mol⁻¹, find the boiling-point elevation. | {
"A": "1.24 K",
"B": "2.24 K",
"C": "2.48 K",
"D": "0.24 K"
} | A | 1.24 K | ΔT_b=iK_bm. Substitution gives ΔT_b=2×1.0×0.62=1.24 K. | {
"A": "Correct. The worked calculation gives 1.24 K.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 1.24 K, not 2.24 K.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 1.24 K, not 2.48 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_002289 | mathematics | probability | same_colour_without_replacement | very_hard | provisional_template_level_label | single_correct_mcq | A bag contains 7 red, 8 blue and 12 green balls. Two balls are drawn uniformly at random without replacement. What is the probability that both have the same colour? | {
"A": "-0.32764",
"B": "0.65527",
"C": "2.32764",
"D": "0.32764"
} | D | 0.32764 | Favourable pairs = C(7,2)+C(8,2)+C(12,2)=115. Total pairs=C(27,2)=351. Therefore P=115/351=0.32764. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 0.32764, not -0.32764.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 0.32764, not 0.65527.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 0.32764, not 2.32... | [
"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_001493 | mathematics | calculus | definite_integral_polynomial | very_hard | provisional_template_level_label | single_correct_mcq | Evaluate the definite integral ∫₀^7 (-3x² +5x -3) dx. | {
"A": "-23.15",
"B": "-120.75",
"C": "-241.5",
"D": "-362.25"
} | C | -241.5 | An antiderivative is (-3/3)x³+(5/2)x²+(-3)x. Evaluating between 0 and 7 gives (-3/3)(7)³+(5/2)(7)²+(-3)(7)=-241.5. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives -241.5, not -23.15.",
"B": "Incorrect. Substituting the stated values into the governing relation gives -241.5, not -120.75.",
"C": "Correct. The worked calculation gives -241.5.",
"D": "Incorrect. Substituting the stated values... | [
"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_006945 | chemistry | stoichiometry | limiting_reagent_water | very_hard | provisional_template_level_label | single_correct_mcq | Hydrogen reacts with oxygen according to 2H₂ + O₂ → 2H₂O. If 15 mol H₂ and 16 mol O₂ are mixed and the reaction goes to completion, how many moles of H₂O form? | {
"A": "15 mol",
"B": "14 mol",
"C": "22.5 mol",
"D": "2.5 mol"
} | A | 15 mol | From 2 mol H₂ and 1 mol O₂, 2 mol H₂O form. H₂ can produce 15 mol H₂O; O₂ can produce 32 mol H₂O. The limiting condition is H₂; product=15 mol. | {
"A": "Correct. The worked calculation gives 15 mol.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 15 mol, not 14 mol.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 15 mol, not 22.5 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_007525 | chemistry | physical_chemistry | strong_acid_base_mixture | very_hard | provisional_template_level_label | single_correct_mcq | 86 mL of 0.10 M HCl is mixed with 96 mL of 0.10 M NaOH at 25°C. Assuming complete dissociation and additive volumes, find the final pH. | {
"A": "12.73993",
"B": "23.47986",
"C": "11.73993",
"D": "9.73993"
} | C | 11.73993 | Excess OH⁻ moles=0.10×(96−86)/1000=0.001000 mol. Total volume=0.182 L, so [OH⁻]=0.005495 M, pOH=−log₁₀[OH⁻], and pH=14−pOH=11.73993. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 11.73993, not 12.73993.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 11.73993, not 23.47986.",
"C": "Correct. The worked calculation gives 11.73993.",
"D": "Incorrect. Substituting the stat... | [
"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_004050 | physics | mechanics | inclined_plane_with_friction | very_hard | provisional_template_level_label | single_correct_mcq | A 14 kg block slides down a 37° rough incline. The coefficient of kinetic friction is 0.33. Taking g=10 m s⁻², find the net force down the plane. Use sin37°=0.6 and cos37°=0.8. | {
"A": "70.56 N",
"B": "94.08 N",
"C": "47.04 N",
"D": "5.704 N"
} | C | 47.04 N | Net force=mg(sinθ−μcosθ)=14×10×(0.6−0.33×0.8)=47.04 N. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 47.04 N, not 70.56 N.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 47.04 N, not 94.08 N.",
"C": "Correct. The worked calculation gives 47.04 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_002111 | mathematics | probability | same_colour_without_replacement | very_hard | provisional_template_level_label | single_correct_mcq | A bag contains 10 red, 11 blue and 6 green balls. Two balls are drawn uniformly at random without replacement. What is the probability that both have the same colour? | {
"A": "0.32764",
"B": "-1.67236",
"C": "1.03276",
"D": "0.65527"
} | A | 0.32764 | Favourable pairs = C(10,2)+C(11,2)+C(6,2)=115. Total pairs=C(27,2)=351. Therefore P=115/351=0.32764. | {
"A": "Correct. The worked calculation gives 0.32764.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 0.32764, not -1.67236.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 0.32764, not 1.03276.",
"D": "Incorrect. Substituting the stated v... | [
"classical probability",
"combinations",
"without replacement"
] | 4 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | math_probability_colours | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_008012 | 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 12 A for 16737 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": "79.29696 g",
"B": "64.0808 g",
"C": "66.0808 g",
"D": "65.0808 g"
} | C | 66.0808 g | Cu²⁺+2e⁻→Cu, so z=2. By Faraday's law m=MIt/(zF)=63.5×12×16737/(2×96500)=66.0808 g. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 66.0808 g, not 79.29696 g.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 66.0808 g, not 64.0808 g.",
"C": "Correct. The worked calculation gives 66.0808 g.",
"D": "Incorrect. Substituting th... | [
"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_001058 | mathematics | calculus | polynomial_differentiation | hard | provisional_template_level_label | single_correct_mcq | Let f(x)=2x³ +7x² +9x+7. Evaluate f′(-2). | {
"A": "3",
"B": "5",
"C": "-5",
"D": "7"
} | B | 5 | Differentiate: f′(x)=3(2)x²+2(7)x+(9). At x=-2, f′(-2)=3(2)(-2)²+2(7)(-2)+(9)=5. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 5, not 3.",
"B": "Correct. The worked calculation gives 5.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 5, not -5.",
"D": "Incorrect. Substituting the stated values into the governing relat... | [
"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_005308 | physics | electrostatics | coulombs_law | hard | provisional_template_level_label | single_correct_mcq | Two point charges of +7 μC and +9 μC are separated by 10 cm in vacuum. Using k=9×10⁹ N m² C⁻², find the magnitude of the electrostatic force. | {
"A": "57.7 N",
"B": "55.7 N",
"C": "56.7 N",
"D": "42.525 N"
} | C | 56.7 N | Convert charges to coulombs and distance to metres. F=k|q₁q₂|/r²=9×10⁹(7×10⁻⁶)(9×10⁻⁶)/(10×10⁻²)²=56.7 N. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 56.7 N, not 57.7 N.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 56.7 N, not 55.7 N.",
"C": "Correct. The worked calculation gives 56.7 N.",
"D": "Incorrect. Substituting the stated values ... | [
"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_000538 | mathematics | sequences_and_series | arithmetic_progression | hard | provisional_template_level_label | single_correct_mcq | An arithmetic progression has first term -6 and common difference 11. Find the sum of its first 17 terms. | {
"A": "1045.5",
"B": "1396",
"C": "-1394",
"D": "1394"
} | D | 1394 | Use Sₙ = n/2[2a+(n−1)d]. Substituting a=-6, d=11, n=17: Sₙ=17/2[2(-6)+(17−1)(11)]=1394. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 1394, not 1045.5.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 1394, not 1396.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 1394, not -1394.",
"D": "C... | [
"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_006644 | physics | oscillations | mass_spring_period_over_pi | hard | provisional_template_level_label | single_correct_mcq | A mass of 12 kg is attached to an ideal spring of force constant 790 N m⁻¹ and oscillates horizontally without damping. What is the time period divided by π? | {
"A": "0.24649 s",
"B": "0.12325 s",
"C": "-1.75351 s",
"D": "-0.75351 s"
} | A | 0.24649 s | For a mass–spring oscillator, T=2π√(m/k). Therefore T/π=2√(12/790)=0.24649 s. | {
"A": "Correct. The worked calculation gives 0.24649 s.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 0.24649 s, not 0.12325 s.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 0.24649 s, not -1.75351 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_005870 | 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 2.7 atm at 441 K. What is its pressure at 584 K? | {
"A": "-3.57551 atm",
"B": "4.57551 atm",
"C": "3.57551 atm",
"D": "5.57551 atm"
} | C | 3.57551 atm | At constant volume, P/T is constant. P₂=P₁T₂/T₁=(2.7)(584)/(441)=3.57551 atm. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 3.57551 atm, not -3.57551 atm.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 3.57551 atm, not 4.57551 atm.",
"C": "Correct. The worked calculation gives 3.57551 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_000371 | mathematics | sequences_and_series | arithmetic_progression | hard | provisional_template_level_label | single_correct_mcq | An arithmetic progression has first term 17 and common difference 9. Find the sum of its first 21 terms. | {
"A": "3370.5",
"B": "4494",
"C": "2247",
"D": "2249"
} | C | 2247 | Use Sₙ = n/2[2a+(n−1)d]. Substituting a=17, d=9, n=21: Sₙ=21/2[2(17)+(21−1)(9)]=2247. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 2247, not 3370.5.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 2247, not 4494.",
"C": "Correct. The worked calculation gives 2247.",
"D": "Incorrect. Substituting the stated values into the... | [
"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_003463 | physics | mechanics | projectile_range | hard | provisional_template_level_label | single_correct_mcq | A projectile is launched from level ground at 46 m s⁻¹ at 15° to the horizontal. Take g=10 m s⁻². Find its horizontal range. | {
"A": "104.8 m",
"B": "52.9 m",
"C": "106.8 m",
"D": "105.8 m"
} | D | 105.8 m | Range R=u²sin(2θ)/g. Substituting u=46, θ=15° and g=10 gives R=(46)²sin(30°)/10=105.8 m. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 105.8 m, not 104.8 m.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 105.8 m, not 52.9 m.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 105.8 m, not 106.8 ... | [
"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_004394 | physics | rotational_mechanics | disc_angular_acceleration | hard | provisional_template_level_label | single_correct_mcq | A uniform solid disc of mass 3 kg and radius 1 m is acted on by a constant torque of 93 N m about its central axis. Find its angular acceleration. | {
"A": "46.5 rad s⁻²",
"B": "62 rad s⁻²",
"C": "61 rad s⁻²",
"D": "63 rad s⁻²"
} | B | 62 rad s⁻² | For a solid disc, I=½MR²=½(3)(1)²=1.5 kg m². Using τ=Iα, α=93/1.5=62 rad s⁻². | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 62 rad s⁻², not 46.5 rad s⁻².",
"B": "Correct. The worked calculation gives 62 rad s⁻².",
"C": "Incorrect. Substituting the stated values into the governing relation gives 62 rad s⁻², not 61 rad s⁻².",
"D": "Incorrect. Substitut... | [
"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_005180 | physics | electrostatics | coulombs_law | hard | provisional_template_level_label | single_correct_mcq | Two point charges of +5 μC and +2 μC are separated by 12 cm in vacuum. Using k=9×10⁹ N m² C⁻², find the magnitude of the electrostatic force. | {
"A": "12.5 N",
"B": "3.25 N",
"C": "6.25 N",
"D": "1.625 N"
} | C | 6.25 N | Convert charges to coulombs and distance to metres. F=k|q₁q₂|/r²=9×10⁹(5×10⁻⁶)(2×10⁻⁶)/(12×10⁻²)²=6.25 N. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 6.25 N, not 12.5 N.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 6.25 N, not 3.25 N.",
"C": "Correct. The worked calculation gives 6.25 N.",
"D": "Incorrect. Substituting the stated values ... | [
"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_005041 | physics | electrostatics | coulombs_law | hard | provisional_template_level_label | single_correct_mcq | Two point charges of +1 μC and +2 μC are separated by 5 cm in vacuum. Using k=9×10⁹ N m² C⁻², find the magnitude of the electrostatic force. | {
"A": "5.4 N",
"B": "7.2 N",
"C": "6.2 N",
"D": "5.2 N"
} | B | 7.2 N | Convert charges to coulombs and distance to metres. F=k|q₁q₂|/r²=9×10⁹(1×10⁻⁶)(2×10⁻⁶)/(5×10⁻²)²=7.2 N. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 7.2 N, not 5.4 N.",
"B": "Correct. The worked calculation gives 7.2 N.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 7.2 N, not 6.2 N.",
"D": "Incorrect. Substituting the stated values into ... | [
"Coulomb's law",
"SI unit conversion",
"inverse-square law"
] | 4 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | physics_coulomb | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_009066 | chemistry | physical_chemistry | nernst_equation_298K | hard | provisional_template_level_label | single_correct_mcq | At 298 K, a galvanic cell has E°cell=1.09 V, transfers 2 electrons, and has reaction quotient Q=10^4. Using E=E°−(0.0591/n)log₁₀Q, calculate Ecell. | {
"A": "1.9718 V",
"B": "3.9718 V",
"C": "-2.0282 V",
"D": "0.9718 V"
} | D | 0.9718 V | Apply the Nernst equation: E=1.09−(0.0591/2)(4)=0.9718 V. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 0.9718 V, not 1.9718 V.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 0.9718 V, not 3.9718 V.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 0.9718 V, not ... | [
"electrochemical cells",
"Nernst equation",
"logarithms"
] | 3 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | chem_nernst | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_003911 | physics | mechanics | work_energy_theorem | very_hard | provisional_template_level_label | single_correct_mcq | A 11 kg block initially moves at 9 m s⁻¹. A constant net force does 17 N of work over 5 m in the direction of motion. Find its final kinetic energy. | {
"A": "530.5 J",
"B": "795.75 J",
"C": "265.25 J",
"D": "532.5 J"
} | A | 530.5 J | By the work–energy theorem, K_f=K_i+W=½mu²+Fs=½(11)(9)²+(17)(5)=530.5 J. | {
"A": "Correct. The worked calculation gives 530.5 J.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 530.5 J, not 795.75 J.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 530.5 J, not 265.25 J.",
"D": "Incorrect. Substituting the stated ... | [
"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_006497 | physics | oscillations | mass_spring_period_over_pi | hard | provisional_template_level_label | single_correct_mcq | A mass of 6 kg is attached to an ideal spring of force constant 689 N m⁻¹ and oscillates horizontally without damping. What is the time period divided by π? | {
"A": "2.18664 s",
"B": "0.37327 s",
"C": "0.18664 s",
"D": "-1.81336 s"
} | C | 0.18664 s | For a mass–spring oscillator, T=2π√(m/k). Therefore T/π=2√(6/689)=0.18664 s. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 0.18664 s, not 2.18664 s.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 0.18664 s, not 0.37327 s.",
"C": "Correct. The worked calculation gives 0.18664 s.",
"D": "Incorrect. Substituting the... | [
"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_001071 | mathematics | calculus | polynomial_differentiation | hard | provisional_template_level_label | single_correct_mcq | Let f(x)=2x³ -10x² +12x+7. Evaluate f′(0). | {
"A": "12",
"B": "9",
"C": "24",
"D": "6"
} | A | 12 | Differentiate: f′(x)=3(2)x²+2(-10)x+(12). At x=0, f′(0)=3(2)(0)²+2(-10)(0)+(12)=12. | {
"A": "Correct. The worked calculation gives 12.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 12, not 9.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 12, not 24.",
"D": "Incorrect. Substituting the stated values into the governing re... | [
"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_000631 | mathematics | sequences_and_series | arithmetic_progression | hard | provisional_template_level_label | single_correct_mcq | An arithmetic progression has first term -14 and common difference 6. Find the sum of its first 10 terms. | {
"A": "132",
"B": "130",
"C": "-130",
"D": "260"
} | B | 130 | Use Sₙ = n/2[2a+(n−1)d]. Substituting a=-14, d=6, n=10: Sₙ=10/2[2(-14)+(10−1)(6)]=130. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 130, not 132.",
"B": "Correct. The worked calculation gives 130.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 130, not -130.",
"D": "Incorrect. Substituting the stated values into the gover... | [
"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_002293 | mathematics | probability | same_colour_without_replacement | very_hard | provisional_template_level_label | single_correct_mcq | A bag contains 10 red, 3 blue and 11 green balls. Two balls are drawn uniformly at random without replacement. What is the probability that both have the same colour? | {
"A": "0.37319",
"B": "-0.62681",
"C": "0.18659",
"D": "1.37319"
} | A | 0.37319 | Favourable pairs = C(10,2)+C(3,2)+C(11,2)=103. Total pairs=C(24,2)=276. Therefore P=103/276=0.37319. | {
"A": "Correct. The worked calculation gives 0.37319.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 0.37319, not -0.62681.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 0.37319, not 0.18659.",
"D": "Incorrect. Substituting the stated v... | [
"classical probability",
"combinations",
"without replacement"
] | 4 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | math_probability_colours | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_008785 | 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 0.6 mol L⁻¹ and initially B is absent. At equilibrium, 15% of A has dissociated at constant volume. Find K_c. | {
"A": "0.06353",
"B": "0.03176",
"C": "1.06353",
"D": "-2.93647"
} | A | 0.06353 | Let α=0.15. At equilibrium [A]=c₀(1−α) and [B]=2c₀α. Thus K_c=[B]²/[A]=4c₀α²/(1−α)=4(0.6)(0.15)²/(1−0.15)=0.06353. | {
"A": "Correct. The worked calculation gives 0.06353.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 0.06353, not 0.03176.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 0.06353, not 1.06353.",
"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_009246 | chemistry | physical_chemistry | nernst_equation_298K | hard | provisional_template_level_label | single_correct_mcq | At 298 K, a galvanic cell has E°cell=0.87 V, transfers 2 electrons, and has reaction quotient Q=10^-4. Using E=E°−(0.0591/n)log₁₀Q, calculate Ecell. | {
"A": "0.9882 V",
"B": "-1.0118 V",
"C": "1.9764 V",
"D": "0.4941 V"
} | A | 0.9882 V | Apply the Nernst equation: E=0.87−(0.0591/2)(-4)=0.9882 V. | {
"A": "Correct. The worked calculation gives 0.9882 V.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 0.9882 V, not -1.0118 V.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 0.9882 V, not 1.9764 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_006082 | physics | ray_optics | convex_lens_magnification | very_hard | provisional_template_level_label | single_correct_mcq | A real object is placed 57 cm in front of a thin convex lens of focal length 14 cm. Using the Cartesian sign convention, what is the magnitude of the linear magnification? | {
"A": "0.65116",
"B": "0.32558",
"C": "1.32558",
"D": "3.32558"
} | B | 0.32558 | Using distances as positive magnitudes, 1/f=1/v+1/u gives v=uf/(u−f)=(57)(14)/(57−14)=18.55814 cm. Thus |m|=v/u=18.55814/57=0.32558. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 0.32558, not 0.65116.",
"B": "Correct. The worked calculation gives 0.32558.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 0.32558, not 1.32558.",
"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_005822 | physics | thermodynamics | ideal_gas_isochoric_pressure | hard | provisional_template_level_label | single_correct_mcq | A fixed amount of ideal gas is kept in a rigid container. Its pressure is 4.1 atm at 432 K. What is its pressure at 719 K? | {
"A": "4.82384 atm",
"B": "7.82384 atm",
"C": "10.23576 atm",
"D": "6.82384 atm"
} | D | 6.82384 atm | At constant volume, P/T is constant. P₂=P₁T₂/T₁=(4.1)(719)/(432)=6.82384 atm. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 6.82384 atm, not 4.82384 atm.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 6.82384 atm, not 7.82384 atm.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 6.... | [
"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_002628 | mathematics | linear_algebra | 2x2_determinant | hard | provisional_template_level_label | single_correct_mcq | Find the determinant of the matrix [[6, 6], [8, 3]]. | {
"A": "-30",
"B": "-24",
"C": "-15",
"D": "-45"
} | A | -30 | For a 2×2 matrix [[a,b],[c,d]], det=ad−bc. Here it is (6)(3)−(6)(8)=-30. | {
"A": "Correct. The worked calculation gives -30.",
"B": "Incorrect. Substituting the stated values into the governing relation gives -30, not -24.",
"C": "Incorrect. Substituting the stated values into the governing relation gives -30, not -15.",
"D": "Incorrect. Substituting the stated values into the govern... | [
"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_009389 | 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_4X_2. Treat X as the total number of monovalent halogen atoms. | {
"A": "6",
"B": "3",
"C": "1.3",
"D": "1"
} | B | 3 | Use DBE=(2C+2+N−H−X)/2. Substitution gives (2×5+2+0−4−2)/2=3. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 3, not 6.",
"B": "Correct. The worked calculation gives 3.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 3, not 1.3.",
"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_008003 | 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 1 A for 21212 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": "4.97908 g",
"B": "6.97908 g",
"C": "5.23431 g",
"D": "1.69791 g"
} | B | 6.97908 g | Cu²⁺+2e⁻→Cu, so z=2. By Faraday's law m=MIt/(zF)=63.5×1×21212/(2×96500)=6.97908 g. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 6.97908 g, not 4.97908 g.",
"B": "Correct. The worked calculation gives 6.97908 g.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 6.97908 g, not 5.23431 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_008446 | 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 0.85 mol kg⁻¹. If its van't Hoff factor is 4 and K_b for water is 0.6 K kg mol⁻¹, find the boiling-point elevation. | {
"A": "1.04 K",
"B": "4.08 K",
"C": "1.53 K",
"D": "2.04 K"
} | D | 2.04 K | ΔT_b=iK_bm. Substitution gives ΔT_b=4×0.6×0.85=2.04 K. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 2.04 K, not 1.04 K.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 2.04 K, not 4.08 K.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 2.04 K, not 1.53 K.",
... | [
"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_003844 | physics | mechanics | work_energy_theorem | very_hard | provisional_template_level_label | single_correct_mcq | A 6 kg block initially moves at 0 m s⁻¹. A constant net force does 20 N of work over 13 m in the direction of motion. Find its final kinetic energy. | {
"A": "-260 J",
"B": "390 J",
"C": "208 J",
"D": "260 J"
} | D | 260 J | By the work–energy theorem, K_f=K_i+W=½mu²+Fs=½(6)(0)²+(20)(13)=260 J. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 260 J, not -260 J.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 260 J, not 390 J.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 260 J, not 208 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_005413 | physics | electrostatics | capacitor_energy | hard | provisional_template_level_label | single_correct_mcq | A 44 μF capacitor is charged to a potential difference of 3 V. Calculate the energy stored, expressed in μJ. | {
"A": "20.8 μJ",
"B": "99 μJ",
"C": "198 μJ",
"D": "158.4 μJ"
} | C | 198 μJ | U=½CV². Since C=44 μF, U in μJ is ½×44×(3)²=198 μJ. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 198 μJ, not 20.8 μJ.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 198 μJ, not 99 μJ.",
"C": "Correct. The worked calculation gives 198 μJ.",
"D": "Incorrect. Substituting the stated values ... | [
"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_005518 | physics | electrostatics | capacitor_energy | hard | provisional_template_level_label | single_correct_mcq | A 46 μF capacitor is charged to a potential difference of 7 V. Calculate the energy stored, expressed in μJ. | {
"A": "1127 μJ",
"B": "1125 μJ",
"C": "2254 μJ",
"D": "-1127 μJ"
} | A | 1127 μJ | U=½CV². Since C=46 μF, U in μJ is ½×46×(7)²=1127 μJ. | {
"A": "Correct. The worked calculation gives 1127 μJ.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 1127 μJ, not 1125 μJ.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 1127 μJ, not 2254 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_000241 | mathematics | algebra | quadratic_roots | hard | provisional_template_level_label | single_correct_mcq | For the quadratic equation 7x^2 +13x +3=0, let its roots be α and β. What is α²+β²? | {
"A": "1.29592",
"B": "2.59184",
"C": "3.88776",
"D": "1.25918"
} | B | 2.59184 | By Vieta's relations, α+β = -13/7 and αβ = 3/7. Then α²+β²=(α+β)²−2αβ = (-13/7)²−2(3/7) = 2.59184. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 2.59184, not 1.29592.",
"B": "Correct. The worked calculation gives 2.59184.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 2.59184, not 3.88776.",
"D": "Incorrect. Substituting the stated va... | [
"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_008135 | 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 9 A for 8343 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": "49.40958 g",
"B": "19.76383 g",
"C": "24.70479 g",
"D": "12.3524 g"
} | C | 24.70479 g | Cu²⁺+2e⁻→Cu, so z=2. By Faraday's law m=MIt/(zF)=63.5×9×8343/(2×96500)=24.70479 g. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 24.70479 g, not 49.40958 g.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 24.70479 g, not 19.76383 g.",
"C": "Correct. The worked calculation gives 24.70479 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_004846 | physics | electricity | series_parallel_circuit | very_hard | provisional_template_level_label | single_correct_mcq | Resistors 9 Ω and 11 Ω are connected in series; this combination is connected in parallel with a 18 Ω resistor across a 24 V ideal battery. Find the total current supplied by the battery. | {
"A": "5.06667 A",
"B": "-0.46667 A",
"C": "1.26667 A",
"D": "2.53333 A"
} | D | 2.53333 A | Series branch resistance=9+11=20 Ω. Parallel equivalent R_eq=(20×18)/(20+18)=9.47368 Ω. Total current I=V/R_eq=24/9.47368=2.53333 A. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 2.53333 A, not 5.06667 A.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 2.53333 A, not -0.46667 A.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 2.53333 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_008484 | 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 0.9 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": "1.404 K",
"B": "1.1872 K",
"C": "1.872 K",
"D": "-0.128 K"
} | C | 1.872 K | ΔT_b=iK_bm. Substitution gives ΔT_b=4×0.52×0.9=1.872 K. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 1.872 K, not 1.404 K.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 1.872 K, not 1.1872 K.",
"C": "Correct. The worked calculation gives 1.872 K.",
"D": "Incorrect. Substituting the stated v... | [
"colligative properties",
"van't Hoff factor",
"molality"
] | 3 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | chem_boiling_elevation | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_007504 | chemistry | physical_chemistry | strong_acid_base_mixture | very_hard | provisional_template_level_label | single_correct_mcq | 71 mL of 0.10 M HCl is mixed with 98 mL of 0.10 M NaOH at 25°C. Assuming complete dissociation and additive volumes, find the final pH. | {
"A": "15.20348",
"B": "12.20348",
"C": "9.15261",
"D": "13.20348"
} | B | 12.20348 | Excess OH⁻ moles=0.10×(98−71)/1000=0.002700 mol. Total volume=0.169 L, so [OH⁻]=0.015976 M, pOH=−log₁₀[OH⁻], and pH=14−pOH=12.20348. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 12.20348, not 15.20348.",
"B": "Correct. The worked calculation gives 12.20348.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 12.20348, not 9.15261.",
"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_000804 | mathematics | sequences_and_series | geometric_progression | hard | provisional_template_level_label | single_correct_mcq | A geometric progression has first term 1 and common ratio 4. What is the sum of its first 11 terms? | {
"A": "1398101",
"B": "1398103",
"C": "699050.5",
"D": "1398100"
} | A | 1398101 | For r≠1, Sₙ=a(rⁿ−1)/(r−1). Thus Sₙ=1(4^11−1)/(4−1)=1398101. | {
"A": "Correct. The worked calculation gives 1398101.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 1398101, not 1398103.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 1398101, not 699050.5.",
"D": "Incorrect. Substituting the stated v... | [
"geometric progression",
"finite geometric sum"
] | 3 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | math_gp_sum | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_004562 | physics | rotational_mechanics | disc_angular_acceleration | hard | provisional_template_level_label | single_correct_mcq | A uniform solid disc of mass 4 kg and radius 5 m is acted on by a constant torque of 71 N m about its central axis. Find its angular acceleration. | {
"A": "1.42 rad s⁻²",
"B": "3.42 rad s⁻²",
"C": "2.42 rad s⁻²",
"D": "4.42 rad s⁻²"
} | A | 1.42 rad s⁻² | For a solid disc, I=½MR²=½(4)(5)²=50 kg m². Using τ=Iα, α=71/50=1.42 rad s⁻². | {
"A": "Correct. The worked calculation gives 1.42 rad s⁻².",
"B": "Incorrect. Substituting the stated values into the governing relation gives 1.42 rad s⁻², not 3.42 rad s⁻².",
"C": "Incorrect. Substituting the stated values into the governing relation gives 1.42 rad s⁻², not 2.42 rad s⁻².",
"D": "Incorrect. S... | [
"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_008408 | 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 1.3 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": "2.028 K",
"B": "2.704 K",
"C": "1.352 K",
"D": "4.704 K"
} | B | 2.704 K | ΔT_b=iK_bm. Substitution gives ΔT_b=4×0.52×1.3=2.704 K. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 2.704 K, not 2.028 K.",
"B": "Correct. The worked calculation gives 2.704 K.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 2.704 K, not 1.352 K.",
"D": "Incorrect. Substituting the stated va... | [
"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_003782 | physics | mechanics | work_energy_theorem | very_hard | provisional_template_level_label | single_correct_mcq | A 4 kg block initially moves at 2 m s⁻¹. A constant net force does 17 N of work over 12 m in the direction of motion. Find its final kinetic energy. | {
"A": "169.6 J",
"B": "22.2 J",
"C": "212 J",
"D": "213 J"
} | C | 212 J | By the work–energy theorem, K_f=K_i+W=½mu²+Fs=½(4)(2)²+(17)(12)=212 J. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 212 J, not 169.6 J.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 212 J, not 22.2 J.",
"C": "Correct. The worked calculation gives 212 J.",
"D": "Incorrect. Substituting the stated values in... | [
"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 |
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