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_008706 | chemistry | physical_chemistry | equilibrium_degree_dissociation | very_hard | provisional_template_level_label | single_correct_mcq | For A(g) ⇌ 2B(g), the initial concentration of A is 1.99 mol L⁻¹ and initially B is absent. At equilibrium, 39% of A has dissociated at constant volume. Find K_c. | {
"A": "-0.01522",
"B": "1.98478",
"C": "4.98478",
"D": "1.19848"
} | B | 1.98478 | Let α=0.39. At equilibrium [A]=c₀(1−α) and [B]=2c₀α. Thus K_c=[B]²/[A]=4c₀α²/(1−α)=4(1.99)(0.39)²/(1−0.39)=1.98478. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 1.98478, not -0.01522.",
"B": "Correct. The worked calculation gives 1.98478.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 1.98478, not 4.98478.",
"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_001130 | mathematics | calculus | polynomial_differentiation | hard | provisional_template_level_label | single_correct_mcq | Let f(x)=-1x³ +5x² +2x+7. Evaluate f′(-2). | {
"A": "-30",
"B": "-60",
"C": "-24",
"D": "-22.5"
} | A | -30 | Differentiate: f′(x)=3(-1)x²+2(5)x+(2). At x=-2, f′(-2)=3(-1)(-2)²+2(5)(-2)+(2)=-30. | {
"A": "Correct. The worked calculation gives -30.",
"B": "Incorrect. Substituting the stated values into the governing relation gives -30, not -60.",
"C": "Incorrect. Substituting the stated values into the governing relation gives -30, not -24.",
"D": "Incorrect. Substituting the stated values into the govern... | [
"power rule",
"polynomial derivative"
] | 3 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | math_derivative_cubic | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_005390 | physics | electrostatics | capacitor_energy | hard | provisional_template_level_label | single_correct_mcq | A 19 μF capacitor is charged to a potential difference of 22 V. Calculate the energy stored, expressed in μJ. | {
"A": "4600 μJ",
"B": "3448.5 μJ",
"C": "4597 μJ",
"D": "4598 μJ"
} | D | 4598 μJ | U=½CV². Since C=19 μF, U in μJ is ½×19×(22)²=4598 μJ. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 4598 μJ, not 4600 μJ.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 4598 μJ, not 3448.5 μJ.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 4598 μJ, not 459... | [
"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_000756 | mathematics | sequences_and_series | geometric_progression | hard | provisional_template_level_label | single_correct_mcq | A geometric progression has first term 1 and common ratio -3. What is the sum of its first 5 terms? | {
"A": "122",
"B": "-61",
"C": "73.2",
"D": "61"
} | D | 61 | For r≠1, Sₙ=a(rⁿ−1)/(r−1). Thus Sₙ=1(-3^5−1)/(-3−1)=61. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 61, not 122.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 61, not -61.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 61, not 73.2.",
"D": "Correct. The... | [
"geometric progression",
"finite geometric sum"
] | 3 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | math_gp_sum | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_003781 | physics | mechanics | work_energy_theorem | very_hard | provisional_template_level_label | single_correct_mcq | A 8 kg block initially moves at 8 m s⁻¹. A constant net force does 5 N of work over 13 m in the direction of motion. Find its final kinetic energy. | {
"A": "160.5 J",
"B": "321 J",
"C": "33.1 J",
"D": "385.2 J"
} | B | 321 J | By the work–energy theorem, K_f=K_i+W=½mu²+Fs=½(8)(8)²+(5)(13)=321 J. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 321 J, not 160.5 J.",
"B": "Correct. The worked calculation gives 321 J.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 321 J, not 33.1 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 |
openstem_005520 | physics | electrostatics | capacitor_energy | hard | provisional_template_level_label | single_correct_mcq | A 53 μF capacitor is charged to a potential difference of 4 V. Calculate the energy stored, expressed in μJ. | {
"A": "424 μJ",
"B": "425 μJ",
"C": "508.8 μJ",
"D": "43.4 μJ"
} | A | 424 μJ | U=½CV². Since C=53 μF, U in μJ is ½×53×(4)²=424 μJ. | {
"A": "Correct. The worked calculation gives 424 μJ.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 424 μJ, not 425 μJ.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 424 μJ, not 508.8 μJ.",
"D": "Incorrect. Substituting the stated value... | [
"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_004230 | physics | mechanics | inclined_plane_with_friction | very_hard | provisional_template_level_label | single_correct_mcq | A 9 kg block slides down a 37° rough incline. The coefficient of kinetic friction is 0.22. Taking g=10 m s⁻², find the net force down the plane. Use sin37°=0.6 and cos37°=0.8. | {
"A": "76.32 N",
"B": "38.16 N",
"C": "28.62 N",
"D": "30.528 N"
} | B | 38.16 N | Net force=mg(sinθ−μcosθ)=9×10×(0.6−0.22×0.8)=38.16 N. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 38.16 N, not 76.32 N.",
"B": "Correct. The worked calculation gives 38.16 N.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 38.16 N, not 28.62 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_009539 | 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_5N_1X_4. Treat X as the total number of monovalent halogen atoms. | {
"A": "4",
"B": "2",
"C": "3",
"D": "1"
} | B | 2 | Use DBE=(2C+2+N−H−X)/2. Substitution gives (2×5+2+1−5−4)/2=2. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 2, not 4.",
"B": "Correct. The worked calculation gives 2.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 2, not 3.",
"D": "Incorrect. Substituting the stated values into the governing relati... | [
"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_001410 | mathematics | calculus | definite_integral_polynomial | very_hard | provisional_template_level_label | single_correct_mcq | Evaluate the definite integral ∫₀^5 (6x² -1x -2) dx. | {
"A": "225.5",
"B": "182",
"C": "227.5",
"D": "23.75"
} | C | 227.5 | An antiderivative is (6/3)x³+(-1/2)x²+(-2)x. Evaluating between 0 and 5 gives (6/3)(5)³+(-1/2)(5)²+(-2)(5)=227.5. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 227.5, not 225.5.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 227.5, not 182.",
"C": "Correct. The worked calculation gives 227.5.",
"D": "Incorrect. Substituting the stated values into th... | [
"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_006024 | physics | ray_optics | convex_lens_magnification | very_hard | provisional_template_level_label | single_correct_mcq | A real object is placed 34 cm in front of a thin convex lens of focal length 7 cm. Using the Cartesian sign convention, what is the magnitude of the linear magnification? | {
"A": "0.38889",
"B": "1.25926",
"C": "0.25926",
"D": "-1.74074"
} | C | 0.25926 | Using distances as positive magnitudes, 1/f=1/v+1/u gives v=uf/(u−f)=(34)(7)/(34−7)=8.81481 cm. Thus |m|=v/u=8.81481/34=0.25926. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 0.25926, not 0.38889.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 0.25926, not 1.25926.",
"C": "Correct. The worked calculation gives 0.25926.",
"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_005625 | physics | electrostatics | capacitor_energy | hard | provisional_template_level_label | single_correct_mcq | A 35 μF capacitor is charged to a potential difference of 15 V. Calculate the energy stored, expressed in μJ. | {
"A": "3937.5 μJ",
"B": "3936.5 μJ",
"C": "3938.5 μJ",
"D": "7875 μJ"
} | A | 3937.5 μJ | U=½CV². Since C=35 μF, U in μJ is ½×35×(15)²=3937.5 μJ. | {
"A": "Correct. The worked calculation gives 3937.5 μJ.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 3937.5 μJ, not 3936.5 μJ.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 3937.5 μJ, not 3938.5 μJ.",
"D": "Incorrect. Substituting the... | [
"capacitance",
"energy stored in a capacitor"
] | 3 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | physics_capacitor_energy | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_001186 | mathematics | calculus | polynomial_differentiation | hard | provisional_template_level_label | single_correct_mcq | Let f(x)=7x³ -9x² +3x+7. Evaluate f′(0). | {
"A": "1",
"B": "3",
"C": "2",
"D": "-3"
} | B | 3 | Differentiate: f′(x)=3(7)x²+2(-9)x+(3). At x=0, f′(0)=3(7)(0)²+2(-9)(0)+(3)=3. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 3, not 1.",
"B": "Correct. The worked calculation gives 3.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 3, not 2.",
"D": "Incorrect. Substituting the stated values into the governing relati... | [
"power rule",
"polynomial derivative"
] | 3 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | math_derivative_cubic | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_002730 | mathematics | vectors | dot_product | hard | provisional_template_level_label | single_correct_mcq | Given vectors u=(-5, 7, -2) and v=(4, -4, 9), calculate u·v. | {
"A": "-64",
"B": "-66",
"C": "-65",
"D": "-52.8"
} | B | -66 | The dot product is the sum of corresponding component products: (-5)(4)+(7)(-4)+(-2)(9)=-66. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives -66, not -64.",
"B": "Correct. The worked calculation gives -66.",
"C": "Incorrect. Substituting the stated values into the governing relation gives -66, not -65.",
"D": "Incorrect. Substituting the stated values into the govern... | [
"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_008949 | chemistry | physical_chemistry | equilibrium_degree_dissociation | very_hard | provisional_template_level_label | single_correct_mcq | For A(g) ⇌ 2B(g), the initial concentration of A is 1.09 mol L⁻¹ and initially B is absent. At equilibrium, 64% of A has dissociated at constant volume. Find K_c. | {
"A": "7.44107",
"B": "-4.96071",
"C": "9.92142",
"D": "4.96071"
} | D | 4.96071 | Let α=0.64. At equilibrium [A]=c₀(1−α) and [B]=2c₀α. Thus K_c=[B]²/[A]=4c₀α²/(1−α)=4(1.09)(0.64)²/(1−0.64)=4.96071. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 4.96071, not 7.44107.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 4.96071, not -4.96071.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 4.96071, not 9.92... | [
"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_005616 | physics | electrostatics | capacitor_energy | hard | provisional_template_level_label | single_correct_mcq | A 18 μF capacitor is charged to a potential difference of 31 V. Calculate the energy stored, expressed in μJ. | {
"A": "17298 μJ",
"B": "8651 μJ",
"C": "-8649 μJ",
"D": "8649 μJ"
} | D | 8649 μJ | U=½CV². Since C=18 μF, U in μJ is ½×18×(31)²=8649 μJ. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 8649 μJ, not 17298 μJ.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 8649 μJ, not 8651 μJ.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 8649 μJ, not -864... | [
"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_008437 | 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 1.21 mol kg⁻¹. If its van't Hoff factor is 1 and K_b for water is 0.52 K kg mol⁻¹, find the boiling-point elevation. | {
"A": "0.9438 K",
"B": "0.6292 K",
"C": "0.4719 K",
"D": "-0.6292 K"
} | B | 0.6292 K | ΔT_b=iK_bm. Substitution gives ΔT_b=1×0.52×1.21=0.6292 K. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 0.6292 K, not 0.9438 K.",
"B": "Correct. The worked calculation gives 0.6292 K.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 0.6292 K, not 0.4719 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_006520 | physics | oscillations | mass_spring_period_over_pi | hard | provisional_template_level_label | single_correct_mcq | A mass of 19 kg is attached to an ideal spring of force constant 1112 N m⁻¹ and oscillates horizontally without damping. What is the time period divided by π? | {
"A": "0.26143 s",
"B": "-0.26143 s",
"C": "0.19607 s",
"D": "2.26143 s"
} | A | 0.26143 s | For a mass–spring oscillator, T=2π√(m/k). Therefore T/π=2√(19/1112)=0.26143 s. | {
"A": "Correct. The worked calculation gives 0.26143 s.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 0.26143 s, not -0.26143 s.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 0.26143 s, not 0.19607 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_002388 | mathematics | linear_algebra | 2x2_determinant | hard | provisional_template_level_label | single_correct_mcq | Find the determinant of the matrix [[2, 0], [9, 7]]. | {
"A": "14",
"B": "-14",
"C": "15",
"D": "2.4"
} | A | 14 | For a 2×2 matrix [[a,b],[c,d]], det=ad−bc. Here it is (2)(7)−(0)(9)=14. | {
"A": "Correct. The worked calculation gives 14.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 14, not -14.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 14, 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_002120 | mathematics | probability | same_colour_without_replacement | very_hard | provisional_template_level_label | single_correct_mcq | A bag contains 6 red, 10 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": "1.32468",
"B": "0.32468",
"C": "1.03247",
"D": "0.48701"
} | B | 0.32468 | Favourable pairs = C(6,2)+C(10,2)+C(6,2)=75. Total pairs=C(22,2)=231. Therefore P=75/231=0.32468. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 0.32468, not 1.32468.",
"B": "Correct. The worked calculation gives 0.32468.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 0.32468, not 1.03247.",
"D": "Incorrect. Substituting the stated va... | [
"classical probability",
"combinations",
"without replacement"
] | 4 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | math_probability_colours | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_004380 | 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 22 N m about its central axis. Find its angular acceleration. | {
"A": "0.2037 rad s⁻²",
"B": "-0.89815 rad s⁻²",
"C": "0.10185 rad s⁻²",
"D": "-0.10185 rad s⁻²"
} | C | 0.10185 rad s⁻² | For a solid disc, I=½MR²=½(12)(6)²=216 kg m². Using τ=Iα, α=22/216=0.10185 rad s⁻². | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 0.10185 rad s⁻², not 0.2037 rad s⁻².",
"B": "Incorrect. Substituting the stated values into the governing relation gives 0.10185 rad s⁻², not -0.89815 rad s⁻².",
"C": "Correct. The worked calculation gives 0.10185 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_009695 | chemistry | inorganic_chemistry | complex_ion_oxidation_state | hard | provisional_template_level_label | single_correct_mcq | For a hypothetical coordination ion [Mn(OH)_2]^-4, each OH ligand has charge −1. Determine the formal oxidation state of Mn. | {
"A": "-3",
"B": "0.8",
"C": "-5",
"D": "-2"
} | D | -2 | Let the formal oxidation state of Mn be x. Charge balance gives x+(2)(−1)=-4; hence x=-4+2=-2. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives -2, not -3.",
"B": "Incorrect. Substituting the stated values into the governing relation gives -2, not 0.8.",
"C": "Incorrect. Substituting the stated values into the governing relation gives -2, not -5.",
"D": "Correct. The wo... | [
"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_009079 | chemistry | physical_chemistry | nernst_equation_298K | hard | provisional_template_level_label | single_correct_mcq | At 298 K, a galvanic cell has E°cell=0.50 V, transfers 1 electrons, and has reaction quotient Q=10^-1. Using E=E°−(0.0591/n)log₁₀Q, calculate Ecell. | {
"A": "0.5591 V",
"B": "3.5591 V",
"C": "0.83865 V",
"D": "0.41933 V"
} | A | 0.5591 V | Apply the Nernst equation: E=0.50−(0.0591/1)(-1)=0.5591 V. | {
"A": "Correct. The worked calculation gives 0.5591 V.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 0.5591 V, not 3.5591 V.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 0.5591 V, not 0.83865 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_002450 | mathematics | linear_algebra | 2x2_determinant | hard | provisional_template_level_label | single_correct_mcq | Find the determinant of the matrix [[1, -8], [6, -4]]. | {
"A": "46",
"B": "44",
"C": "66",
"D": "5.4"
} | B | 44 | For a 2×2 matrix [[a,b],[c,d]], det=ad−bc. Here it is (1)(-4)−(-8)(6)=44. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 44, not 46.",
"B": "Correct. The worked calculation gives 44.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 44, not 66.",
"D": "Incorrect. Substituting the stated values into the governing r... | [
"determinants",
"matrix operations"
] | 2 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | math_determinant_2x2 | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_005598 | physics | electrostatics | capacitor_energy | hard | provisional_template_level_label | single_correct_mcq | A 26 μF capacitor is charged to a potential difference of 9 V. Calculate the energy stored, expressed in μJ. | {
"A": "1263.6 μJ",
"B": "1053 μJ",
"C": "1579.5 μJ",
"D": "1052 μJ"
} | B | 1053 μJ | U=½CV². Since C=26 μF, U in μJ is ½×26×(9)²=1053 μJ. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 1053 μJ, not 1263.6 μJ.",
"B": "Correct. The worked calculation gives 1053 μJ.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 1053 μJ, not 1579.5 μJ.",
"D": "Incorrect. Substituting the state... | [
"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_006544 | physics | oscillations | mass_spring_period_over_pi | hard | provisional_template_level_label | single_correct_mcq | A mass of 23 kg is attached to an ideal spring of force constant 418 N m⁻¹ and oscillates horizontally without damping. What is the time period divided by π? | {
"A": "0.46914 s",
"B": "0.93829 s",
"C": "3.46914 s",
"D": "0.23457 s"
} | A | 0.46914 s | For a mass–spring oscillator, T=2π√(m/k). Therefore T/π=2√(23/418)=0.46914 s. | {
"A": "Correct. The worked calculation gives 0.46914 s.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 0.46914 s, not 0.93829 s.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 0.46914 s, not 3.46914 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_001954 | mathematics | combinatorics | binomial_coefficient | hard | provisional_template_level_label | single_correct_mcq | How many distinct subsets containing exactly 11 elements can be selected from a set of 81 distinct elements? | {
"A": "12124169174518",
"B": "12124169174520",
"C": "14549003009424",
"D": "12124169174521"
} | B | 12124169174520 | Order does not matter, so the count is C(81,11)=81!/[11!(70)!]=12124169174520. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 12124169174520, not 12124169174518.",
"B": "Correct. The worked calculation gives 12124169174520.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 12124169174520, not 14549003009424.",
"D": "In... | [
"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_009808 | chemistry | inorganic_chemistry | complex_ion_oxidation_state | hard | provisional_template_level_label | single_correct_mcq | For a hypothetical coordination ion [Pd(Cl)_4]^+0, each Cl ligand has charge −1. Determine the formal oxidation state of Pd. | {
"A": "4",
"B": "7",
"C": "3",
"D": "6"
} | A | 4 | Let the formal oxidation state of Pd be x. Charge balance gives x+(4)(−1)=+0; hence x=0+4=+4. | {
"A": "Correct. The worked calculation gives 4.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 4, not 7.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 4, not 3.",
"D": "Incorrect. Substituting the stated values into the governing relati... | [
"coordination compounds",
"oxidation state",
"charge balance"
] | 3 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | chem_complex_oxidation | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_000584 | mathematics | sequences_and_series | arithmetic_progression | hard | provisional_template_level_label | single_correct_mcq | An arithmetic progression has first term -19 and common difference 12. Find the sum of its first 23 terms. | {
"A": "1949.25",
"B": "5198",
"C": "2599",
"D": "260.9"
} | C | 2599 | Use Sₙ = n/2[2a+(n−1)d]. Substituting a=-19, d=12, n=23: Sₙ=23/2[2(-19)+(23−1)(12)]=2599. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 2599, not 1949.25.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 2599, not 5198.",
"C": "Correct. The worked calculation gives 2599.",
"D": "Incorrect. Substituting the stated values into th... | [
"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_006136 | physics | ray_optics | convex_lens_magnification | very_hard | provisional_template_level_label | single_correct_mcq | A real object is placed 15 cm in front of a thin convex lens of focal length 6 cm. Using the Cartesian sign convention, what is the magnitude of the linear magnification? | {
"A": "1.33333",
"B": "-2.33333",
"C": "0.66667",
"D": "-0.33333"
} | C | 0.66667 | Using distances as positive magnitudes, 1/f=1/v+1/u gives v=uf/(u−f)=(15)(6)/(15−6)=10 cm. Thus |m|=v/u=10/15=0.66667. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 0.66667, not 1.33333.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 0.66667, not -2.33333.",
"C": "Correct. The worked calculation gives 0.66667.",
"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_000775 | mathematics | sequences_and_series | geometric_progression | hard | provisional_template_level_label | single_correct_mcq | A geometric progression has first term 12 and common ratio 2. What is the sum of its first 7 terms? | {
"A": "3048",
"B": "1219.2",
"C": "1828.8",
"D": "1524"
} | D | 1524 | For r≠1, Sₙ=a(rⁿ−1)/(r−1). Thus Sₙ=12(2^7−1)/(2−1)=1524. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 1524, not 3048.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 1524, not 1219.2.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 1524, not 1828.8.",
"D": "... | [
"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_003924 | physics | mechanics | work_energy_theorem | very_hard | provisional_template_level_label | single_correct_mcq | A 6 kg block initially moves at 2 m s⁻¹. A constant net force does 27 N of work over 3 m in the direction of motion. Find its final kinetic energy. | {
"A": "91 J",
"B": "10.3 J",
"C": "93 J",
"D": "94 J"
} | C | 93 J | By the work–energy theorem, K_f=K_i+W=½mu²+Fs=½(6)(2)²+(27)(3)=93 J. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 93 J, not 91 J.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 93 J, not 10.3 J.",
"C": "Correct. The worked calculation gives 93 J.",
"D": "Incorrect. Substituting the stated values into the... | [
"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_007820 | 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 17 mmol L⁻¹, what concentration remains after 120 minutes? | {
"A": "0.26562 mmol L⁻¹",
"B": "-0.26562 mmol L⁻¹",
"C": "0.13281 mmol L⁻¹",
"D": "-0.73438 mmol L⁻¹"
} | A | 0.26562 mmol L⁻¹ | 120 minutes corresponds to 6 half-lives. The remaining concentration is [A]₀(1/2)ⁿ=17/2^6=0.26562 mmol L⁻¹. | {
"A": "Correct. The worked calculation gives 0.26562 mmol L⁻¹.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 0.26562 mmol L⁻¹, not -0.26562 mmol L⁻¹.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 0.26562 mmol L⁻¹, not 0.13281 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_007285 | chemistry | physical_chemistry | ideal_gas_volume | hard | provisional_template_level_label | single_correct_mcq | Assuming ideal behaviour, what volume is occupied by 14 mol of gas at 387 K and 2.61 atm? Use R=0.082057 L atm mol⁻¹ K⁻¹. | {
"A": "127.75426 L",
"B": "172.33901 L",
"C": "170.33901 L",
"D": "18.0339 L"
} | C | 170.33901 L | PV=nRT, so V=nRT/P=(14)(0.082057)(387)/(2.61)=170.33901 L. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 170.33901 L, not 127.75426 L.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 170.33901 L, not 172.33901 L.",
"C": "Correct. The worked calculation gives 170.33901 L.",
"D": "Incorrect. Substi... | [
"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_003472 | physics | mechanics | projectile_range | hard | provisional_template_level_label | single_correct_mcq | A projectile is launched from level ground at 50 m s⁻¹ at 40° to the horizontal. Take g=10 m s⁻². Find its horizontal range. | {
"A": "245.20194 m",
"B": "246.20194 m",
"C": "244.20194 m",
"D": "-246.20194 m"
} | B | 246.20194 m | Range R=u²sin(2θ)/g. Substituting u=50, θ=40° and g=10 gives R=(50)²sin(80°)/10=246.20194 m. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 246.20194 m, not 245.20194 m.",
"B": "Correct. The worked calculation gives 246.20194 m.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 246.20194 m, not 244.20194 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_006267 | physics | ray_optics | convex_lens_magnification | very_hard | provisional_template_level_label | single_correct_mcq | A real object is placed 35 cm in front of a thin convex lens of focal length 7 cm. Using the Cartesian sign convention, what is the magnitude of the linear magnification? | {
"A": "1.25",
"B": "0.125",
"C": "0.25",
"D": "2.25"
} | C | 0.25 | Using distances as positive magnitudes, 1/f=1/v+1/u gives v=uf/(u−f)=(35)(7)/(35−7)=8.75 cm. Thus |m|=v/u=8.75/35=0.25. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 0.25, not 1.25.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 0.25, not 0.125.",
"C": "Correct. The worked calculation gives 0.25.",
"D": "Incorrect. Substituting the stated values into the ... | [
"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_001172 | mathematics | calculus | polynomial_differentiation | hard | provisional_template_level_label | single_correct_mcq | Let f(x)=-3x³ +3x² -12x+7. Evaluate f′(1). | {
"A": "-12",
"B": "-22.5",
"C": "-15",
"D": "-17"
} | C | -15 | Differentiate: f′(x)=3(-3)x²+2(3)x+(-12). At x=1, f′(1)=3(-3)(1)²+2(3)(1)+(-12)=-15. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives -15, not -12.",
"B": "Incorrect. Substituting the stated values into the governing relation gives -15, not -22.5.",
"C": "Correct. The worked calculation gives -15.",
"D": "Incorrect. Substituting the stated values into the gove... | [
"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_005366 | physics | electrostatics | capacitor_energy | hard | provisional_template_level_label | single_correct_mcq | A 62 μF capacitor is charged to a potential difference of 22 V. Calculate the energy stored, expressed in μJ. | {
"A": "15004 μJ",
"B": "30008 μJ",
"C": "-15004 μJ",
"D": "15006 μJ"
} | A | 15004 μJ | U=½CV². Since C=62 μF, U in μJ is ½×62×(22)²=15004 μJ. | {
"A": "Correct. The worked calculation gives 15004 μJ.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 15004 μJ, not 30008 μJ.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 15004 μJ, not -15004 μJ.",
"D": "Incorrect. Substituting the sta... | [
"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_007459 | chemistry | physical_chemistry | strong_acid_base_mixture | very_hard | provisional_template_level_label | single_correct_mcq | 67 mL of 0.10 M HCl is mixed with 90 mL of 0.10 M NaOH at 25°C. Assuming complete dissociation and additive volumes, find the final pH. | {
"A": "9.16583",
"B": "-12.16583",
"C": "12.16583",
"D": "11.16583"
} | C | 12.16583 | Excess OH⁻ moles=0.10×(90−67)/1000=0.002300 mol. Total volume=0.157 L, so [OH⁻]=0.014650 M, pOH=−log₁₀[OH⁻], and pH=14−pOH=12.16583. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 12.16583, not 9.16583.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 12.16583, not -12.16583.",
"C": "Correct. The worked calculation gives 12.16583.",
"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_006389 | physics | oscillations | mass_spring_period_over_pi | hard | provisional_template_level_label | single_correct_mcq | A mass of 8 kg is attached to an ideal spring of force constant 642 N m⁻¹ and oscillates horizontally without damping. What is the time period divided by π? | {
"A": "0.44652 s",
"B": "0.11163 s",
"C": "3.22326 s",
"D": "0.22326 s"
} | D | 0.22326 s | For a mass–spring oscillator, T=2π√(m/k). Therefore T/π=2√(8/642)=0.22326 s. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 0.22326 s, not 0.44652 s.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 0.22326 s, not 0.11163 s.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 0.22326 s,... | [
"simple harmonic motion",
"mass–spring oscillator",
"time period"
] | 3 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | physics_shm_period | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_003061 | mathematics | complex_numbers | real_part_of_complex_quotient | very_hard | provisional_template_level_label | single_correct_mcq | Let z=(9-4i)/(10+1i). What is Re(z), the real part of z? | {
"A": "0.63861",
"B": "3.85149",
"C": "0.85149",
"D": "1.08515"
} | C | 0.85149 | Multiply numerator and denominator by the conjugate (10-1i). Then Re(z)=(ac+bd)/(c²+d²)=((9)(10)+(-4)(1))/(10²+(1)²)=0.85149. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 0.85149, not 0.63861.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 0.85149, not 3.85149.",
"C": "Correct. The worked calculation gives 0.85149.",
"D": "Incorrect. Substituting the stated va... | [
"complex conjugate",
"complex division",
"real and imaginary parts"
] | 4 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | math_complex_quotient | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_000692 | mathematics | sequences_and_series | geometric_progression | hard | provisional_template_level_label | single_correct_mcq | A geometric progression has first term 7 and common ratio 4. What is the sum of its first 11 terms? | {
"A": "11744048.4",
"B": "9786707",
"C": "4893353.5",
"D": "7340030.25"
} | B | 9786707 | For r≠1, Sₙ=a(rⁿ−1)/(r−1). Thus Sₙ=7(4^11−1)/(4−1)=9786707. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 9786707, not 11744048.4.",
"B": "Correct. The worked calculation gives 9786707.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 9786707, not 4893353.5.",
"D": "Incorrect. Substituting the stat... | [
"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_008801 | 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 0.39 mol L⁻¹ and initially B is absent. At equilibrium, 79% of A has dissociated at constant volume. Find K_c. | {
"A": "2.31809",
"B": "9.27234",
"C": "1.46362",
"D": "4.63617"
} | D | 4.63617 | Let α=0.79. At equilibrium [A]=c₀(1−α) and [B]=2c₀α. Thus K_c=[B]²/[A]=4c₀α²/(1−α)=4(0.39)(0.79)²/(1−0.79)=4.63617. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 4.63617, not 2.31809.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 4.63617, not 9.27234.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 4.63617, not 1.463... | [
"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_009348 | 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_8H_16. Treat X as the total number of monovalent halogen atoms. | {
"A": "3",
"B": "1",
"C": "1.1",
"D": "-1"
} | B | 1 | Use DBE=(2C+2+N−H−X)/2. Substitution gives (2×8+2+0−16−0)/2=1. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 1, not 3.",
"B": "Correct. The worked calculation gives 1.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 1, not 1.1.",
"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_007226 | chemistry | physical_chemistry | ideal_gas_volume | hard | provisional_template_level_label | single_correct_mcq | Assuming ideal behaviour, what volume is occupied by 12 mol of gas at 329 K and 0.12 atm? Use R=0.082057 L atm mol⁻¹ K⁻¹. | {
"A": "2699.6753 L",
"B": "2024.75648 L",
"C": "4049.51295 L",
"D": "2697.6753 L"
} | A | 2699.6753 L | PV=nRT, so V=nRT/P=(12)(0.082057)(329)/(0.12)=2699.6753 L. | {
"A": "Correct. The worked calculation gives 2699.6753 L.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 2699.6753 L, not 2024.75648 L.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 2699.6753 L, not 4049.51295 L.",
"D": "Incorrect. Subs... | [
"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_002708 | mathematics | vectors | dot_product | hard | provisional_template_level_label | single_correct_mcq | Given vectors u=(4, 6, -3) and v=(-4, -4, 1), calculate u·v. | {
"A": "-44",
"B": "-21.5",
"C": "-43",
"D": "43"
} | C | -43 | The dot product is the sum of corresponding component products: (4)(-4)+(6)(-4)+(-3)(1)=-43. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives -43, not -44.",
"B": "Incorrect. Substituting the stated values into the governing relation gives -43, not -21.5.",
"C": "Correct. The worked calculation gives -43.",
"D": "Incorrect. Substituting the stated values into the gove... | [
"three-dimensional vectors",
"scalar product"
] | 3 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | math_vector_dot | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_002689 | mathematics | vectors | dot_product | hard | provisional_template_level_label | single_correct_mcq | Given vectors u=(1, 3, 6) and v=(-7, -1, -4), calculate u·v. | {
"A": "-25.5",
"B": "-2.4",
"C": "-34",
"D": "-40.8"
} | C | -34 | The dot product is the sum of corresponding component products: (1)(-7)+(3)(-1)+(6)(-4)=-34. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives -34, not -25.5.",
"B": "Incorrect. Substituting the stated values into the governing relation gives -34, not -2.4.",
"C": "Correct. The worked calculation gives -34.",
"D": "Incorrect. Substituting the stated values into the gov... | [
"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_003571 | physics | mechanics | projectile_range | hard | provisional_template_level_label | single_correct_mcq | A projectile is launched from level ground at 108 m s⁻¹ at 20° to the horizontal. Take g=10 m s⁻². Find its horizontal range. | {
"A": "-749.74747 m",
"B": "750.74747 m",
"C": "562.3106 m",
"D": "749.74747 m"
} | D | 749.74747 m | Range R=u²sin(2θ)/g. Substituting u=108, θ=20° and g=10 gives R=(108)²sin(40°)/10=749.74747 m. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 749.74747 m, not -749.74747 m.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 749.74747 m, not 750.74747 m.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 7... | [
"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_007513 | chemistry | physical_chemistry | strong_acid_base_mixture | very_hard | provisional_template_level_label | single_correct_mcq | 77 mL of 0.10 M HCl is mixed with 39 mL of 0.10 M NaOH at 25°C. Assuming complete dissociation and additive volumes, find the final pH. | {
"A": "-1.51533",
"B": "-0.51533",
"C": "0.48467",
"D": "1.48467"
} | D | 1.48467 | Excess H⁺ moles=0.10×(77−39)/1000=0.003800 mol. Total volume=0.116 L, so [H⁺]=0.032759 M and pH=−log₁₀[H⁺]=1.48467. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 1.48467, not -1.51533.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 1.48467, not -0.51533.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 1.48467, not 0.4... | [
"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_006205 | physics | ray_optics | convex_lens_magnification | very_hard | provisional_template_level_label | single_correct_mcq | A real object is placed 56 cm in front of a thin convex lens of focal length 28 cm. Using the Cartesian sign convention, what is the magnitude of the linear magnification? | {
"A": "3",
"B": "1.5",
"C": "-1",
"D": "1"
} | D | 1 | Using distances as positive magnitudes, 1/f=1/v+1/u gives v=uf/(u−f)=(56)(28)/(56−28)=56 cm. Thus |m|=v/u=56/56=1. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 1, not 3.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 1, not 1.5.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 1, not -1.",
"D": "Correct. The worked... | [
"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_006681 | chemistry | stoichiometry | limiting_reagent_water | very_hard | provisional_template_level_label | single_correct_mcq | Hydrogen reacts with oxygen according to 2H₂ + O₂ → 2H₂O. If 44 mol H₂ and 5 mol O₂ are mixed and the reaction goes to completion, how many moles of H₂O form? | {
"A": "10 mol",
"B": "7.5 mol",
"C": "13 mol",
"D": "20 mol"
} | A | 10 mol | From 2 mol H₂ and 1 mol O₂, 2 mol H₂O form. H₂ can produce 44 mol H₂O; O₂ can produce 10 mol H₂O. The limiting condition is O₂; product=10 mol. | {
"A": "Correct. The worked calculation gives 10 mol.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 10 mol, not 7.5 mol.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 10 mol, not 13 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_007949 | 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 71 mmol L⁻¹, what concentration remains after 80 minutes? | {
"A": "5.4375 mmol L⁻¹",
"B": "3.4375 mmol L⁻¹",
"C": "4.4375 mmol L⁻¹",
"D": "6.4375 mmol L⁻¹"
} | C | 4.4375 mmol L⁻¹ | 80 minutes corresponds to 4 half-lives. The remaining concentration is [A]₀(1/2)ⁿ=71/2^4=4.4375 mmol L⁻¹. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 4.4375 mmol L⁻¹, not 5.4375 mmol L⁻¹.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 4.4375 mmol L⁻¹, not 3.4375 mmol L⁻¹.",
"C": "Correct. The worked calculation gives 4.4375 mmol L⁻¹.",
"D"... | [
"first-order kinetics",
"half-life",
"exponential decay"
] | 3 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | chem_first_order_halflife | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_004841 | physics | electricity | series_parallel_circuit | very_hard | provisional_template_level_label | single_correct_mcq | Resistors 14 Ω and 5 Ω are connected in series; this combination is connected in parallel with a 7 Ω resistor across a 12 V ideal battery. Find the total current supplied by the battery. | {
"A": "3.5188 A",
"B": "1.17293 A",
"C": "2.34586 A",
"D": "1.34586 A"
} | C | 2.34586 A | Series branch resistance=14+5=19 Ω. Parallel equivalent R_eq=(19×7)/(19+7)=5.11538 Ω. Total current I=V/R_eq=12/5.11538=2.34586 A. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 2.34586 A, not 3.5188 A.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 2.34586 A, not 1.17293 A.",
"C": "Correct. The worked calculation gives 2.34586 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_003181 | mathematics | complex_numbers | real_part_of_complex_quotient | very_hard | provisional_template_level_label | single_correct_mcq | Let z=(-9-11i)/(9+12i). What is Re(z), the real part of z? | {
"A": "-0.47333",
"B": "-3.94667",
"C": "1.05333",
"D": "-0.94667"
} | D | -0.94667 | Multiply numerator and denominator by the conjugate (9-12i). Then Re(z)=(ac+bd)/(c²+d²)=((-9)(9)+(-11)(12))/(9²+(12)²)=-0.94667. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives -0.94667, not -0.47333.",
"B": "Incorrect. Substituting the stated values into the governing relation gives -0.94667, not -3.94667.",
"C": "Incorrect. Substituting the stated values into the governing relation gives -0.94667, not ... | [
"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_001577 | mathematics | calculus | definite_integral_polynomial | very_hard | provisional_template_level_label | single_correct_mcq | Evaluate the definite integral ∫₀^2 (-4x² +3x -5) dx. | {
"A": "-14.66667",
"B": "-13.66667",
"C": "-7.33333",
"D": "-0.46667"
} | A | -14.66667 | An antiderivative is (-4/3)x³+(3/2)x²+(-5)x. Evaluating between 0 and 2 gives (-4/3)(2)³+(3/2)(2)²+(-5)(2)=-14.66667. | {
"A": "Correct. The worked calculation gives -14.66667.",
"B": "Incorrect. Substituting the stated values into the governing relation gives -14.66667, not -13.66667.",
"C": "Incorrect. Substituting the stated values into the governing relation gives -14.66667, not -7.33333.",
"D": "Incorrect. Substituting the ... | [
"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_001995 | mathematics | combinatorics | binomial_coefficient | hard | provisional_template_level_label | single_correct_mcq | How many distinct subsets containing exactly 3 elements can be selected from a set of 16 distinct elements? | {
"A": "560",
"B": "559",
"C": "448",
"D": "-560"
} | A | 560 | Order does not matter, so the count is C(16,3)=16!/[3!(13)!]=560. | {
"A": "Correct. The worked calculation gives 560.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 560, not 559.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 560, not 448.",
"D": "Incorrect. Substituting the stated values into the govern... | [
"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_007544 | chemistry | physical_chemistry | strong_acid_base_mixture | very_hard | provisional_template_level_label | single_correct_mcq | 64 mL of 0.10 M HCl is mixed with 80 mL of 0.10 M NaOH at 25°C. Assuming complete dissociation and additive volumes, find the final pH. | {
"A": "11.04576",
"B": "2.20458",
"C": "12.04576",
"D": "9.03432"
} | C | 12.04576 | Excess OH⁻ moles=0.10×(80−64)/1000=0.001600 mol. Total volume=0.144 L, so [OH⁻]=0.011111 M, pOH=−log₁₀[OH⁻], and pH=14−pOH=12.04576. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 12.04576, not 11.04576.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 12.04576, not 2.20458.",
"C": "Correct. The worked calculation gives 12.04576.",
"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_006728 | chemistry | stoichiometry | limiting_reagent_water | very_hard | provisional_template_level_label | single_correct_mcq | Hydrogen reacts with oxygen according to 2H₂ + O₂ → 2H₂O. If 18 mol H₂ and 11 mol O₂ are mixed and the reaction goes to completion, how many moles of H₂O form? | {
"A": "21.6 mol",
"B": "-18 mol",
"C": "18 mol",
"D": "27 mol"
} | C | 18 mol | From 2 mol H₂ and 1 mol O₂, 2 mol H₂O form. H₂ can produce 18 mol H₂O; O₂ can produce 22 mol H₂O. The limiting condition is H₂; product=18 mol. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 18 mol, not 21.6 mol.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 18 mol, not -18 mol.",
"C": "Correct. The worked calculation gives 18 mol.",
"D": "Incorrect. Substituting the stated valu... | [
"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_006734 | chemistry | stoichiometry | limiting_reagent_water | very_hard | provisional_template_level_label | single_correct_mcq | Hydrogen reacts with oxygen according to 2H₂ + O₂ → 2H₂O. If 26 mol H₂ and 28 mol O₂ are mixed and the reaction goes to completion, how many moles of H₂O form? | {
"A": "27 mol",
"B": "26 mol",
"C": "28 mol",
"D": "52 mol"
} | B | 26 mol | From 2 mol H₂ and 1 mol O₂, 2 mol H₂O form. H₂ can produce 26 mol H₂O; O₂ can produce 56 mol H₂O. The limiting condition is H₂; product=26 mol. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 26 mol, not 27 mol.",
"B": "Correct. The worked calculation gives 26 mol.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 26 mol, not 28 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_000130 | mathematics | algebra | quadratic_roots | hard | provisional_template_level_label | single_correct_mcq | For the quadratic equation 5x^2 +5x +7=0, let its roots be α and β. What is α²+β²? | {
"A": "-3.8",
"B": "-0.8",
"C": "-1.35",
"D": "-1.8"
} | D | -1.8 | By Vieta's relations, α+β = -5/5 and αβ = 7/5. Then α²+β²=(α+β)²−2αβ = (-5/5)²−2(7/5) = -1.8. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives -1.8, not -3.8.",
"B": "Incorrect. Substituting the stated values into the governing relation gives -1.8, not -0.8.",
"C": "Incorrect. Substituting the stated values into the governing relation gives -1.8, not -1.35.",
"D": "Cor... | [
"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_007311 | chemistry | physical_chemistry | ideal_gas_volume | hard | provisional_template_level_label | single_correct_mcq | Assuming ideal behaviour, what volume is occupied by 1 mol of gas at 382 K and 1.05 atm? Use R=0.082057 L atm mol⁻¹ K⁻¹. | {
"A": "29.85312 L",
"B": "27.85312 L",
"C": "44.77968 L",
"D": "59.70624 L"
} | A | 29.85312 L | PV=nRT, so V=nRT/P=(1)(0.082057)(382)/(1.05)=29.85312 L. | {
"A": "Correct. The worked calculation gives 29.85312 L.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 29.85312 L, not 27.85312 L.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 29.85312 L, not 44.77968 L.",
"D": "Incorrect. Substitutin... | [
"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_005634 | physics | electrostatics | capacitor_energy | hard | provisional_template_level_label | single_correct_mcq | A 38 μF capacitor is charged to a potential difference of 13 V. Calculate the energy stored, expressed in μJ. | {
"A": "3213 μJ",
"B": "2568.8 μJ",
"C": "3853.2 μJ",
"D": "3211 μJ"
} | D | 3211 μJ | U=½CV². Since C=38 μF, U in μJ is ½×38×(13)²=3211 μJ. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 3211 μJ, not 3213 μJ.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 3211 μJ, not 2568.8 μJ.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 3211 μJ, not 385... | [
"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_006000 | 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 3.1 atm at 425 K. What is its pressure at 764 K? | {
"A": "3.57271 atm",
"B": "2.78635 atm",
"C": "5.57271 atm",
"D": "4.17953 atm"
} | C | 5.57271 atm | At constant volume, P/T is constant. P₂=P₁T₂/T₁=(3.1)(764)/(425)=5.57271 atm. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 5.57271 atm, not 3.57271 atm.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 5.57271 atm, not 2.78635 atm.",
"C": "Correct. The worked calculation gives 5.57271 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_002346 | mathematics | linear_algebra | 2x2_determinant | hard | provisional_template_level_label | single_correct_mcq | Find the determinant of the matrix [[3, -4], [3, 8]]. | {
"A": "35",
"B": "72",
"C": "36",
"D": "43.2"
} | C | 36 | For a 2×2 matrix [[a,b],[c,d]], det=ad−bc. Here it is (3)(8)−(-4)(3)=36. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 36, not 35.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 36, not 72.",
"C": "Correct. The worked calculation gives 36.",
"D": "Incorrect. Substituting the stated values into the governing r... | [
"determinants",
"matrix operations"
] | 2 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | math_determinant_2x2 | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_002192 | mathematics | probability | same_colour_without_replacement | very_hard | provisional_template_level_label | single_correct_mcq | A bag contains 8 red, 3 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": "0.35263",
"B": "-0.35263",
"C": "1.03526",
"D": "3.35263"
} | A | 0.35263 | Favourable pairs = C(8,2)+C(3,2)+C(9,2)=67. Total pairs=C(20,2)=190. Therefore P=67/190=0.35263. | {
"A": "Correct. The worked calculation gives 0.35263.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 0.35263, not -0.35263.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 0.35263, not 1.03526.",
"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_000929 | mathematics | sequences_and_series | geometric_progression | hard | provisional_template_level_label | single_correct_mcq | A geometric progression has first term 4 and common ratio 5. What is the sum of its first 6 terms? | {
"A": "31248",
"B": "1563.4",
"C": "15624",
"D": "18748.8"
} | C | 15624 | For r≠1, Sₙ=a(rⁿ−1)/(r−1). Thus Sₙ=4(5^6−1)/(5−1)=15624. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 15624, not 31248.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 15624, not 1563.4.",
"C": "Correct. The worked calculation gives 15624.",
"D": "Incorrect. Substituting the stated values into... | [
"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_000789 | mathematics | sequences_and_series | geometric_progression | hard | provisional_template_level_label | single_correct_mcq | A geometric progression has first term 3 and common ratio 3. What is the sum of its first 9 terms? | {
"A": "29523",
"B": "-29523",
"C": "14761.5",
"D": "29522"
} | A | 29523 | For r≠1, Sₙ=a(rⁿ−1)/(r−1). Thus Sₙ=3(3^9−1)/(3−1)=29523. | {
"A": "Correct. The worked calculation gives 29523.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 29523, not -29523.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 29523, not 14761.5.",
"D": "Incorrect. Substituting the stated values in... | [
"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_006429 | physics | oscillations | mass_spring_period_over_pi | hard | provisional_template_level_label | single_correct_mcq | A mass of 16 kg is attached to an ideal spring of force constant 679 N m⁻¹ and oscillates horizontally without damping. What is the time period divided by π? | {
"A": "-0.69299 s",
"B": "0.30701 s",
"C": "2.30701 s",
"D": "3.30701 s"
} | B | 0.30701 s | For a mass–spring oscillator, T=2π√(m/k). Therefore T/π=2√(16/679)=0.30701 s. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 0.30701 s, not -0.69299 s.",
"B": "Correct. The worked calculation gives 0.30701 s.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 0.30701 s, not 2.30701 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_001614 | mathematics | calculus | definite_integral_polynomial | very_hard | provisional_template_level_label | single_correct_mcq | Evaluate the definite integral ∫₀^2 (4x² +2x +5) dx. | {
"A": "26.66667",
"B": "18.5",
"C": "37",
"D": "24.66667"
} | D | 24.66667 | An antiderivative is (4/3)x³+(2/2)x²+(5)x. Evaluating between 0 and 2 gives (4/3)(2)³+(2/2)(2)²+(5)(2)=24.66667. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 24.66667, not 26.66667.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 24.66667, not 18.5.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 24.66667, not 37."... | [
"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_007946 | 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 90 mmol L⁻¹, what concentration remains after 140 minutes? | {
"A": "-2.29688 mmol L⁻¹",
"B": "3.70312 mmol L⁻¹",
"C": "0.70312 mmol L⁻¹",
"D": "-0.70312 mmol L⁻¹"
} | C | 0.70312 mmol L⁻¹ | 140 minutes corresponds to 7 half-lives. The remaining concentration is [A]₀(1/2)ⁿ=90/2^7=0.70312 mmol L⁻¹. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 0.70312 mmol L⁻¹, not -2.29688 mmol L⁻¹.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 0.70312 mmol L⁻¹, not 3.70312 mmol L⁻¹.",
"C": "Correct. The worked calculation gives 0.70312 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_000842 | mathematics | sequences_and_series | geometric_progression | hard | provisional_template_level_label | single_correct_mcq | A geometric progression has first term 2 and common ratio -2. What is the sum of its first 7 terms? | {
"A": "129",
"B": "9.6",
"C": "103.2",
"D": "86"
} | D | 86 | For r≠1, Sₙ=a(rⁿ−1)/(r−1). Thus Sₙ=2(-2^7−1)/(-2−1)=86. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 86, not 129.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 86, not 9.6.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 86, not 103.2.",
"D": "Correct. Th... | [
"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_002182 | mathematics | probability | same_colour_without_replacement | very_hard | provisional_template_level_label | single_correct_mcq | A bag contains 7 red, 12 blue and 11 green balls. Two balls are drawn uniformly at random without replacement. What is the probability that both have the same colour? | {
"A": "-1.67356",
"B": "0.32644",
"C": "0.65287",
"D": "2.32644"
} | B | 0.32644 | Favourable pairs = C(7,2)+C(12,2)+C(11,2)=142. Total pairs=C(30,2)=435. Therefore P=142/435=0.32644. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 0.32644, not -1.67356.",
"B": "Correct. The worked calculation gives 0.32644.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 0.32644, not 0.65287.",
"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_003933 | physics | mechanics | work_energy_theorem | very_hard | provisional_template_level_label | single_correct_mcq | A 3 kg block initially moves at 4 m s⁻¹. A constant net force does 20 N of work over 14 m in the direction of motion. Find its final kinetic energy. | {
"A": "243.2 J",
"B": "306 J",
"C": "-304 J",
"D": "304 J"
} | D | 304 J | By the work–energy theorem, K_f=K_i+W=½mu²+Fs=½(3)(4)²+(20)(14)=304 J. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 304 J, not 243.2 J.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 304 J, not 306 J.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 304 J, not -304 J.",
"... | [
"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_001050 | mathematics | calculus | polynomial_differentiation | hard | provisional_template_level_label | single_correct_mcq | Let f(x)=4x³ -2x² +7x+7. Evaluate f′(-3). | {
"A": "127",
"B": "125",
"C": "128",
"D": "-127"
} | A | 127 | Differentiate: f′(x)=3(4)x²+2(-2)x+(7). At x=-3, f′(-3)=3(4)(-3)²+2(-2)(-3)+(7)=127. | {
"A": "Correct. The worked calculation gives 127.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 127, not 125.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 127, not 128.",
"D": "Incorrect. Substituting the stated values into the govern... | [
"power rule",
"polynomial derivative"
] | 3 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | math_derivative_cubic | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_000146 | mathematics | algebra | quadratic_roots | hard | provisional_template_level_label | single_correct_mcq | For the quadratic equation 4x^2 +4x -6=0, let its roots be α and β. What is α²+β²? | {
"A": "4",
"B": "6",
"C": "-4",
"D": "2"
} | A | 4 | By Vieta's relations, α+β = -4/4 and αβ = -6/4. Then α²+β²=(α+β)²−2αβ = (-4/4)²−2(-6/4) = 4. | {
"A": "Correct. The worked calculation gives 4.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 4, not 6.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 4, not -4.",
"D": "Incorrect. Substituting the stated values into the governing relat... | [
"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_006599 | physics | oscillations | mass_spring_period_over_pi | hard | provisional_template_level_label | single_correct_mcq | A mass of 9 kg is attached to an ideal spring of force constant 206 N m⁻¹ and oscillates horizontally without damping. What is the time period divided by π? | {
"A": "0.41804 s",
"B": "-0.58196 s",
"C": "0.62706 s",
"D": "1.41804 s"
} | A | 0.41804 s | For a mass–spring oscillator, T=2π√(m/k). Therefore T/π=2√(9/206)=0.41804 s. | {
"A": "Correct. The worked calculation gives 0.41804 s.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 0.41804 s, not -0.58196 s.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 0.41804 s, not 0.62706 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_009270 | chemistry | physical_chemistry | nernst_equation_298K | hard | provisional_template_level_label | single_correct_mcq | At 298 K, a galvanic cell has E°cell=0.73 V, transfers 1 electrons, and has reaction quotient Q=10^-2. Using E=E°−(0.0591/n)log₁₀Q, calculate Ecell. | {
"A": "1.08482 V",
"B": "2.8482 V",
"C": "-1.1518 V",
"D": "0.8482 V"
} | D | 0.8482 V | Apply the Nernst equation: E=0.73−(0.0591/1)(-2)=0.8482 V. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 0.8482 V, not 1.08482 V.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 0.8482 V, not 2.8482 V.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 0.8482 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_006716 | chemistry | stoichiometry | limiting_reagent_water | very_hard | provisional_template_level_label | single_correct_mcq | Hydrogen reacts with oxygen according to 2H₂ + O₂ → 2H₂O. If 7 mol H₂ and 18 mol O₂ are mixed and the reaction goes to completion, how many moles of H₂O form? | {
"A": "5.25 mol",
"B": "5 mol",
"C": "10.5 mol",
"D": "7 mol"
} | D | 7 mol | From 2 mol H₂ and 1 mol O₂, 2 mol H₂O form. H₂ can produce 7 mol H₂O; O₂ can produce 36 mol H₂O. The limiting condition is H₂; product=7 mol. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 7 mol, not 5.25 mol.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 7 mol, not 5 mol.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 7 mol, not 10.5 mol.",
... | [
"balanced chemical equations",
"limiting reagent",
"mole ratios"
] | 4 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | chem_limiting_reagent | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_008643 | 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 0.84 mol kg⁻¹. If its van't Hoff factor is 3 and K_b for water is 1.0 K kg mol⁻¹, find the boiling-point elevation. | {
"A": "2.52 K",
"B": "-0.48 K",
"C": "4.52 K",
"D": "5.52 K"
} | A | 2.52 K | ΔT_b=iK_bm. Substitution gives ΔT_b=3×1.0×0.84=2.52 K. | {
"A": "Correct. The worked calculation gives 2.52 K.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 2.52 K, not -0.48 K.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 2.52 K, not 4.52 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_003242 | mathematics | complex_numbers | real_part_of_complex_quotient | very_hard | provisional_template_level_label | single_correct_mcq | Let z=(15+3i)/(1+10i). What is Re(z), the real part of z? | {
"A": "-2.55446",
"B": "-0.44554",
"C": "0.44554",
"D": "0.22277"
} | C | 0.44554 | Multiply numerator and denominator by the conjugate (1-10i). Then Re(z)=(ac+bd)/(c²+d²)=((15)(1)+(3)(10))/(1²+(10)²)=0.44554. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 0.44554, not -2.55446.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 0.44554, not -0.44554.",
"C": "Correct. The worked calculation gives 0.44554.",
"D": "Incorrect. Substituting the stated ... | [
"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_006625 | physics | oscillations | mass_spring_period_over_pi | hard | provisional_template_level_label | single_correct_mcq | A mass of 24 kg is attached to an ideal spring of force constant 350 N m⁻¹ and oscillates horizontally without damping. What is the time period divided by π? | {
"A": "-0.52372 s",
"B": "0.52372 s",
"C": "0.78558 s",
"D": "1.52372 s"
} | B | 0.52372 s | For a mass–spring oscillator, T=2π√(m/k). Therefore T/π=2√(24/350)=0.52372 s. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 0.52372 s, not -0.52372 s.",
"B": "Correct. The worked calculation gives 0.52372 s.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 0.52372 s, not 0.78558 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_009911 | chemistry | inorganic_chemistry | complex_ion_oxidation_state | hard | provisional_template_level_label | single_correct_mcq | For a hypothetical coordination ion [Cu(Cl)_3]^+3, each Cl ligand has charge −1. Determine the formal oxidation state of Cu. | {
"A": "4.5",
"B": "1.6",
"C": "7",
"D": "6"
} | D | 6 | Let the formal oxidation state of Cu be x. Charge balance gives x+(3)(−1)=+3; hence x=3+3=+6. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 6, not 4.5.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 6, not 1.6.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 6, not 7.",
"D": "Correct. The worke... | [
"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_003144 | mathematics | complex_numbers | real_part_of_complex_quotient | very_hard | provisional_template_level_label | single_correct_mcq | Let z=(9+1i)/(7+6i). What is Re(z), the real part of z? | {
"A": "-1.18824",
"B": "0.40588",
"C": "0.81176",
"D": "0.60882"
} | C | 0.81176 | Multiply numerator and denominator by the conjugate (7-6i). Then Re(z)=(ac+bd)/(c²+d²)=((9)(7)+(1)(6))/(7²+(6)²)=0.81176. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 0.81176, not -1.18824.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 0.81176, not 0.40588.",
"C": "Correct. The worked calculation gives 0.81176.",
"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_001340 | mathematics | calculus | definite_integral_polynomial | very_hard | provisional_template_level_label | single_correct_mcq | Evaluate the definite integral ∫₀^1 (3x² -8x +0) dx. | {
"A": "-1",
"B": "-4.5",
"C": "0.7",
"D": "-3"
} | D | -3 | An antiderivative is (3/3)x³+(-8/2)x²+(0)x. Evaluating between 0 and 1 gives (3/3)(1)³+(-8/2)(1)²+(0)(1)=-3. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives -3, not -1.",
"B": "Incorrect. Substituting the stated values into the governing relation gives -3, not -4.5.",
"C": "Incorrect. Substituting the stated values into the governing relation gives -3, not 0.7.",
"D": "Correct. The ... | [
"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_002412 | mathematics | linear_algebra | 2x2_determinant | hard | provisional_template_level_label | single_correct_mcq | Find the determinant of the matrix [[6, 4], [8, -6]]. | {
"A": "-69",
"B": "-68",
"C": "68",
"D": "-70"
} | B | -68 | For a 2×2 matrix [[a,b],[c,d]], det=ad−bc. Here it is (6)(-6)−(4)(8)=-68. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives -68, not -69.",
"B": "Correct. The worked calculation gives -68.",
"C": "Incorrect. Substituting the stated values into the governing relation gives -68, not 68.",
"D": "Incorrect. Substituting the stated values into the governi... | [
"determinants",
"matrix operations"
] | 2 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | math_determinant_2x2 | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_006953 | 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 2 mol O₂ are mixed and the reaction goes to completion, how many moles of H₂O form? | {
"A": "6 mol",
"B": "3 mol",
"C": "-4 mol",
"D": "4 mol"
} | D | 4 mol | From 2 mol H₂ and 1 mol O₂, 2 mol H₂O form. H₂ can produce 28 mol H₂O; O₂ can produce 4 mol H₂O. The limiting condition is O₂; product=4 mol. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 4 mol, not 6 mol.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 4 mol, not 3 mol.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 4 mol, not -4 mol.",
"D"... | [
"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_004756 | physics | electricity | series_parallel_circuit | very_hard | provisional_template_level_label | single_correct_mcq | Resistors 5 Ω and 6 Ω are connected in series; this combination is connected in parallel with a 13 Ω resistor across a 6 V ideal battery. Find the total current supplied by the battery. | {
"A": "-0.99301 A",
"B": "0.75524 A",
"C": "1.00699 A",
"D": "1.51049 A"
} | C | 1.00699 A | Series branch resistance=5+6=11 Ω. Parallel equivalent R_eq=(11×13)/(11+13)=5.95833 Ω. Total current I=V/R_eq=6/5.95833=1.00699 A. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 1.00699 A, not -0.99301 A.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 1.00699 A, not 0.75524 A.",
"C": "Correct. The worked calculation gives 1.00699 A.",
"D": "Incorrect. Substituting th... | [
"series resistors",
"parallel resistors",
"Ohm's law"
] | 4 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | physics_series_parallel | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_004622 | physics | rotational_mechanics | disc_angular_acceleration | hard | provisional_template_level_label | single_correct_mcq | A uniform solid disc of mass 9 kg and radius 2 m is acted on by a constant torque of 14 N m about its central axis. Find its angular acceleration. | {
"A": "1.16667 rad s⁻²",
"B": "0.38889 rad s⁻²",
"C": "1.55556 rad s⁻²",
"D": "0.77778 rad s⁻²"
} | D | 0.77778 rad s⁻² | For a solid disc, I=½MR²=½(9)(2)²=18 kg m². Using τ=Iα, α=14/18=0.77778 rad s⁻². | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 0.77778 rad s⁻², not 1.16667 rad s⁻².",
"B": "Incorrect. Substituting the stated values into the governing relation gives 0.77778 rad s⁻², not 0.38889 rad s⁻².",
"C": "Incorrect. Substituting the stated values into the governing r... | [
"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_005591 | physics | electrostatics | capacitor_energy | hard | provisional_template_level_label | single_correct_mcq | A 10 μF capacitor is charged to a potential difference of 15 V. Calculate the energy stored, expressed in μJ. | {
"A": "1125 μJ",
"B": "843.75 μJ",
"C": "113.5 μJ",
"D": "1124 μJ"
} | A | 1125 μJ | U=½CV². Since C=10 μF, U in μJ is ½×10×(15)²=1125 μJ. | {
"A": "Correct. The worked calculation gives 1125 μJ.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 1125 μJ, not 843.75 μJ.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 1125 μJ, not 113.5 μJ.",
"D": "Incorrect. Substituting the stated... | [
"capacitance",
"energy stored in a capacitor"
] | 3 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | physics_capacitor_energy | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_006683 | chemistry | stoichiometry | limiting_reagent_water | very_hard | provisional_template_level_label | single_correct_mcq | Hydrogen reacts with oxygen according to 2H₂ + O₂ → 2H₂O. If 50 mol H₂ and 12 mol O₂ are mixed and the reaction goes to completion, how many moles of H₂O form? | {
"A": "24 mol",
"B": "3.4 mol",
"C": "25 mol",
"D": "12 mol"
} | A | 24 mol | From 2 mol H₂ and 1 mol O₂, 2 mol H₂O form. H₂ can produce 50 mol H₂O; O₂ can produce 24 mol H₂O. The limiting condition is O₂; product=24 mol. | {
"A": "Correct. The worked calculation gives 24 mol.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 24 mol, not 3.4 mol.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 24 mol, not 25 mol.",
"D": "Incorrect. Substituting the stated values... | [
"balanced chemical equations",
"limiting reagent",
"mole ratios"
] | 4 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | chem_limiting_reagent | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_009935 | chemistry | inorganic_chemistry | complex_ion_oxidation_state | hard | provisional_template_level_label | single_correct_mcq | For a hypothetical coordination ion [Fe(OH)_3]^+1, each OH ligand has charge −1. Determine the formal oxidation state of Fe. | {
"A": "4",
"B": "-4",
"C": "1",
"D": "2"
} | A | 4 | Let the formal oxidation state of Fe be x. Charge balance gives x+(3)(−1)=+1; hence x=1+3=+4. | {
"A": "Correct. The worked calculation gives 4.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 4, not -4.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 4, not 1.",
"D": "Incorrect. Substituting the stated values into the governing relat... | [
"coordination compounds",
"oxidation state",
"charge balance"
] | 3 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | chem_complex_oxidation | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
openstem_006960 | chemistry | stoichiometry | limiting_reagent_water | very_hard | provisional_template_level_label | single_correct_mcq | Hydrogen reacts with oxygen according to 2H₂ + O₂ → 2H₂O. If 6 mol H₂ and 30 mol O₂ are mixed and the reaction goes to completion, how many moles of H₂O form? | {
"A": "1.6 mol",
"B": "4.5 mol",
"C": "6 mol",
"D": "3 mol"
} | C | 6 mol | From 2 mol H₂ and 1 mol O₂, 2 mol H₂O form. H₂ can produce 6 mol H₂O; O₂ can produce 60 mol H₂O. The limiting condition is H₂; product=6 mol. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 6 mol, not 1.6 mol.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 6 mol, not 4.5 mol.",
"C": "Correct. The worked calculation gives 6 mol.",
"D": "Incorrect. Substituting the stated values i... | [
"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_004728 | physics | electricity | series_parallel_circuit | very_hard | provisional_template_level_label | single_correct_mcq | Resistors 2 Ω and 12 Ω are connected in series; this combination is connected in parallel with a 13 Ω resistor across a 9 V ideal battery. Find the total current supplied by the battery. | {
"A": "1.13352 A",
"B": "-0.66484 A",
"C": "-1.66484 A",
"D": "1.33516 A"
} | D | 1.33516 A | Series branch resistance=2+12=14 Ω. Parallel equivalent R_eq=(14×13)/(14+13)=6.74074 Ω. Total current I=V/R_eq=9/6.74074=1.33516 A. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 1.33516 A, not 1.13352 A.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 1.33516 A, not -0.66484 A.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 1.33516 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_004565 | physics | rotational_mechanics | disc_angular_acceleration | hard | provisional_template_level_label | single_correct_mcq | A uniform solid disc of mass 13 kg and radius 4 m is acted on by a constant torque of 55 N m about its central axis. Find its angular acceleration. | {
"A": "0.52885 rad s⁻²",
"B": "1.05288 rad s⁻²",
"C": "0.39663 rad s⁻²",
"D": "-0.52885 rad s⁻²"
} | A | 0.52885 rad s⁻² | For a solid disc, I=½MR²=½(13)(4)²=104 kg m². Using τ=Iα, α=55/104=0.52885 rad s⁻². | {
"A": "Correct. The worked calculation gives 0.52885 rad s⁻².",
"B": "Incorrect. Substituting the stated values into the governing relation gives 0.52885 rad s⁻², not 1.05288 rad s⁻².",
"C": "Incorrect. Substituting the stated values into the governing relation gives 0.52885 rad s⁻², not 0.39663 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_006282 | physics | ray_optics | convex_lens_magnification | very_hard | provisional_template_level_label | single_correct_mcq | A real object is placed 32 cm in front of a thin convex lens of focal length 21 cm. Using the Cartesian sign convention, what is the magnitude of the linear magnification? | {
"A": "-0.09091",
"B": "1.90909",
"C": "3.81818",
"D": "1.19091"
} | B | 1.90909 | Using distances as positive magnitudes, 1/f=1/v+1/u gives v=uf/(u−f)=(32)(21)/(32−21)=61.09091 cm. Thus |m|=v/u=61.09091/32=1.90909. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 1.90909, not -0.09091.",
"B": "Correct. The worked calculation gives 1.90909.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 1.90909, not 3.81818.",
"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_004781 | physics | electricity | series_parallel_circuit | very_hard | provisional_template_level_label | single_correct_mcq | Resistors 4 Ω and 5 Ω are connected in series; this combination is connected in parallel with a 2 Ω resistor across a 24 V ideal battery. Find the total current supplied by the battery. | {
"A": "-14.66667 A",
"B": "2.46667 A",
"C": "11.66667 A",
"D": "14.66667 A"
} | D | 14.66667 A | Series branch resistance=4+5=9 Ω. Parallel equivalent R_eq=(9×2)/(9+2)=1.63636 Ω. Total current I=V/R_eq=24/1.63636=14.66667 A. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 14.66667 A, not -14.66667 A.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 14.66667 A, not 2.46667 A.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 14.666... | [
"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_001950 | mathematics | combinatorics | binomial_coefficient | hard | provisional_template_level_label | single_correct_mcq | How many distinct subsets containing exactly 12 elements can be selected from a set of 46 distinct elements? | {
"A": "19455308827.5",
"B": "38910617655",
"C": "58365926482.5",
"D": "38910617656"
} | B | 38910617655 | Order does not matter, so the count is C(46,12)=46!/[12!(34)!]=38910617655. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 38910617655, not 19455308827.5.",
"B": "Correct. The worked calculation gives 38910617655.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 38910617655, not 58365926482.5.",
"D": "Incorrect. Su... | [
"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_002307 | mathematics | probability | same_colour_without_replacement | very_hard | provisional_template_level_label | single_correct_mcq | A bag contains 9 red, 12 blue and 6 green balls. Two balls are drawn uniformly at random without replacement. What is the probability that both have the same colour? | {
"A": "0.66667",
"B": "1.03333",
"C": "0.16667",
"D": "0.33333"
} | D | 0.33333 | Favourable pairs = C(9,2)+C(12,2)+C(6,2)=117. Total pairs=C(27,2)=351. Therefore P=117/351=0.33333. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 0.33333, not 0.66667.",
"B": "Incorrect. Substituting the stated values into the governing relation gives 0.33333, not 1.03333.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 0.33333, not 0.166... | [
"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_000179 | mathematics | algebra | quadratic_roots | hard | provisional_template_level_label | single_correct_mcq | For the quadratic equation 3x^2 -4x +8=0, let its roots be α and β. What is α²+β²? | {
"A": "-3.55556",
"B": "-4.55556",
"C": "-0.55556",
"D": "-7.11111"
} | A | -3.55556 | By Vieta's relations, α+β = 4/3 and αβ = 8/3. Then α²+β²=(α+β)²−2αβ = (4/3)²−2(8/3) = -3.55556. | {
"A": "Correct. The worked calculation gives -3.55556.",
"B": "Incorrect. Substituting the stated values into the governing relation gives -3.55556, not -4.55556.",
"C": "Incorrect. Substituting the stated values into the governing relation gives -3.55556, not -0.55556.",
"D": "Incorrect. Substituting the stat... | [
"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_004220 | physics | mechanics | inclined_plane_with_friction | very_hard | provisional_template_level_label | single_correct_mcq | A 26 kg block slides down a 37° rough incline. The coefficient of kinetic friction is 0.24. Taking g=10 m s⁻², find the net force down the plane. Use sin37°=0.6 and cos37°=0.8. | {
"A": "53.04 N",
"B": "106.08 N",
"C": "-106.08 N",
"D": "159.12 N"
} | B | 106.08 N | Net force=mg(sinθ−μcosθ)=26×10×(0.6−0.24×0.8)=106.08 N. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives 106.08 N, not 53.04 N.",
"B": "Correct. The worked calculation gives 106.08 N.",
"C": "Incorrect. Substituting the stated values into the governing relation gives 106.08 N, not -106.08 N.",
"D": "Incorrect. Substituting the stat... | [
"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_002507 | mathematics | linear_algebra | 2x2_determinant | hard | provisional_template_level_label | single_correct_mcq | Find the determinant of the matrix [[7, 4], [6, 1]]. | {
"A": "-20.4",
"B": "-34",
"C": "-0.7",
"D": "-17"
} | D | -17 | For a 2×2 matrix [[a,b],[c,d]], det=ad−bc. Here it is (7)(1)−(4)(6)=-17. | {
"A": "Incorrect. Substituting the stated values into the governing relation gives -17, not -20.4.",
"B": "Incorrect. Substituting the stated values into the governing relation gives -17, not -34.",
"C": "Incorrect. Substituting the stated values into the governing relation gives -17, not -0.7.",
"D": "Correct... | [
"determinants",
"matrix operations"
] | 2 | numeric | en | synthetic_parametric | AxiloLabz deterministic STEM question generator | math_determinant_2x2 | template_formula_generated_structurally_validated_not_expert_reviewed | Answer is computed by the explicit equation in the generator template; this is not an independent expert proof. | Difficulty is approximate and not independently calibrated by JEE examiners; no ultra-hard claim is made in v0.1. | original_synthetic_question | correct_answer_option_matches_generated_numeric_value |
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