cmeraki/elemento huggingface dataset:
Subject: mathematics
Question: Let f(x) be a continuously differentiable function on the interval (0, ∞) such that f(1) = 2 and lim(t^10 f(x) - x^10 f(t) / t^9 - x^9) = 1 for each x > 0. Then, for all x > 0, f(x) is equal to
Choices: (A) 31/11x^10 + 9/11*x^10, (B) 9/11x^10 + 13/11*x^10, (C) -9/11x^10 + 31/11*x^10, (D) 13/11x^10 - 9/11*x^10
Answer: B
Subject: mathematics
Question: A student appears for a quiz consisting of only true-false type questions and answers all the questions. The student knows the answers of some questions and guesses the answers for the remaining questions. Whenever the student knows the answer of a question, he gives the correct answer. Assume that the probability of the student giving the correct answer for a question, given that he has guessed it, is 1/2. Also assume that the probability of the answer for a question being guessed, given that the student's answer is correct, is 1/6. Then the probability that the student knows the answer of a randomly chosen question is
Choices: (A) 1/12, (B) 1/7, (C) 5/7, (D) 5/12
Answer: C
Subject: mathematics
Question: Let π/2 < x < π be such that cot x = -5/√11. Then (sin(11x/2)(sin 6x - cos 6x) + cos(11x/2)(sin 6x + cos 6x)) is equal to
Choices: (A) √11-1/2√3, (B) √11+1/2√3, (C) √11+1/3√2, (D) √11-1/3√2
Answer: (B)
Subject: mathematics
Question: Consider the ellipse x^2/9 + y^2/4 = 1. Let S(p, q) be a point in the first quadrant such that p^2/9 + q^2/4 > 1. Two tangents are drawn from S to the ellipse, of which one meets the ellipse at one end point of the minor axis and the other meets the ellipse at a point T in the fourth quadrant. Let R be the vertex of the ellipse with positive x-coordinate and O be the center of the ellipse. If the area of the triangle ΔORT is 3/2, then which of the following options is correct?
Choices: (A) q = 2, p = 3√3, (B) q = 2, p = 4√3, (C) q =1, p = 5√3, (D) q =1, p = 6√3
Answer: (C)
Subject: mathematics
Question: Let S = {a + b√2 : a, b ∈ ℤ}, T1 = {(-1 + √2)^n : n ∈ ℕ}, and T2 = {(1 + √2)^n : n ∈ ℕ}. Then which of the following statements is (are) TRUE?
Choices: (A) ℤ ⋃ T1 ⋃ T2 ⊆ S, (B) T1 ⋂ (0, 1/2024) = ∅, where ∅ denotes the empty set, (C) T2 ⋂ (2024, ∞) ≠ ∅, (D) For any given a, b ∈ ℤ, cos(π(a + b√2)) + i sin(π(a + b√2)) ∈ ℤ, if and only if b = 0, where i = √-1.
Answer: A,D
Subject: mathematics
Question: Let R^2 denote R×R. Let S = {(a,b,c) : a,b,c ∈ R and ax^2 + 2bxy + cy^2 > 0 for all (x, y) ∈ R^2 − {(0,0)}}. Then which of the following statements is (are) TRUE?
Choices: (A) (2, -2, 6) ∈ S

(B) If (3,b, 1/12) ∈ S, then | 2b | < 1.

(C) For any given (a,b,c) ∈ S, the system of linear equations
ax + by = 1
bx + cy = −1
has a unique solution.

(D) For any given (a,b,c) ∈ S, the system of linear equations
(a + 1)x + by = 0
bx + (c + 1)y = 0
has a unique solution.
Answer: No selected answer
Subject: mathematics
Question: Let R^3 denote the three-dimensional space. Take two points P = (1, 2, 3) and Q = (4, 2, 7). Let dist(X,Y) denote the distance between two points X and Y in R^3. Let S = {X ∈ R^3 : (dist(X, P))^2 − (dist(X, Q))^2 = 50} and T = {Y ∈ R^3 : (dist(Y, Q))^2 − (dist(Y, P))^2 = 50}. Then which of the following statements is (are) TRUE?
Choices: (A) There is a triangle whose area is 1 and all of whose vertices are from S.
(B) There are two distinct points L and M in T such that each point on the line segment LM is also in T.
(C) There are infinitely many rectangles of perimeter 48, two of whose vertices are from S and the other two vertices are from T.
(D) There is a square of perimeter 48, two of whose vertices are from S and the other two vertices are from T.
Answer: No selected answer
Subject: mathematics
Question: Let f(x) = x⁴ + αx³ + βx² + c be a polynomial with real coefficients such that f(1) = −9. Suppose that t√3 is a root of the equation  4x³ + 3αx² + 2βx = 0, where i = √−1. If α₁, α₂, α₃, and α₄ are all the roots of the equation  f(x) = 0, then |α₁|² + |α₂|² + |α₃|² + |α₄|² is equal to _______.
Subject: mathematics
Question: A group of  9 students, s₁, s₂, . . . , s₉, is to be divided to form three teams  X, Y , and Z  of sizes 2, 3, and  4, respectively. Suppose that  s₁  cannot be selected for the team  X, and  s₂  cannot be selected for the team  Y . Then the number of ways to form such teams, is _______.
Subject: mathematics
Question: Let OP = (α−1)i + j + k, OQ = i + (β−1)j + k and OR = i + j + (1/2)k be three vectors, where α, β ∈ R−{0} and O denotes the origin. If (OP×OQ) · OR = 0 and the point (α, β, 2) lies on the plane 3x + 3y - z + l = 0, then the value of l is _______.
Subject: mathematics
Question: Let X be a random variable, and let P(X = x) denote the probability that X takes the value x. Suppose that the points (x, P(X = x)), x = 0,1,2,3,4, lie on a fixed straight line in the xy-plane, and P(X = x) = 0 for all x ∈ R−{0,1,2,3,4}. If the mean of X is 5/2, and the variance of X is α, then the value of 24α is _______.
Subject: mathematics
Question: Let α and β be the distinct roots of the equation x^2 + x - 1 = 0. Consider the set T = {1, α, β}. For a 3×3 matrix M = (aij)i≤3,j≤3, define Ri = ai1 + ai2 + ai3 and Cj = a1j + a2j + a3j for i = 1, 2, 3 and j = 1, 2, 3.
Choices: (P) The number of matrices M = (aij)i≤3,j≤3 with all entries in T such that Ri = Cj = 0 for all i, j, is (1) / (Q) The number of symmetric matrices M = (aij)i≤3,j≤3 with all entries in T such that Cj = 0 for all j, is (2) 12 / (R) Let M = (aij)i≤3,j≤3 be a skew symmetric matrix such that aij ∈ T for i > j. Then the number of elements in the set [[x], [y], [z]]: x, y, z ∈ ℝ, M [y] = [0], [z] [-a23]] is (3) infinite / (S) Let M = (aij)i≤3,j≤3 be a matrix with all entries in T such that Ri = 0 for all i. Then the absolute value of the determinant of M is (4) 6
Answer: (5) 0
Subject: mathematics
Question: Let the straight line y = 2x touch a circle with center (0, α), α > 0, and radius r at a point A. Let B be the point on the circle such that the line segment AB is a diameter of the circle. Let α + r = 5 + √5. Match each entry in List-I to the correct entry in List-II.
Choices: (P) α equals, (Q) r equals, (R) A equals, (S) B equals
Answer: (C) (P) → (4) (Q) → (2) (R) → (5) (S) → (3)
Subject: mathematics
Question: Let y ∈ ℝ be such that the lines L₁: x + 11/1 = y + 21/2 = z + 29/3 and L₂: x + 16/3 = y + 11/2 = z + 4/y intersect. Let R₀ be the point of intersection of L₁ and L₂. Let O = (0, 0, 0), and ṅ denote a unit normal vector to the plane containing both the lines L₁ and L₂. Match each entry in List-I to the correct entry in List-II.
Choices: (P) y equals (Q) A possible choice for ṅ is (R) OR₀ equals (S) A possible value of OR₀ · ṅ is
Answer: (P) → (3), (Q) → (4), (R) → (1), (S) → (2)
Subject: mathematics
Question: Let f : R → R and g : R → R be the functions defined by f(x) = {x | x | sin(1/x), x ≠ 0, 0, x = 0} and g(x) = {1−2x, 0 ≤ x ≤ 1/2, 0, otherwise}. Let a, b, c, d ∈ R. Define the function h : R → R by h(x) = a f(x) + b [g(x) + g(1/2 - x)] + c (x - g(x)) + d g(x), x ∈ R. Match each entry in List-I to the correct entry in List-II.
Choices: List-I 
(P) If a = 0, b = 1, c = 0, and d = 0, then 
(Q) If a = 1, b = 0, c = 0, and d = 0, then 
(R) If a = 0, b = 0, c = 1, and d = 0, then 
(S) If a = 0, b = 0, c = 0, and d = 1, then

List-II
(1) h is one-one.
(2) h is onto.
(3) h is differentiable on R.
(4) the range of h is [0,1].
(5) the range of h is {0,1}.
Answer: (A) (P) → (4) (Q) → (3) (R) → (1) (S) → (2)
Subject: physics
Question: A dimensionless quantity is constructed in terms of electronic charge e, permittivity of free space ε0, Planck's constant h, and speed of light c. If the dimensionless quantity is written as eᵅε0ᵇhᶜcᵈ and n is a non-zero integer, then (α, β, γ, δ) is given by
Choices: (A) (2n, −n, −n, −n), (B) (n, −n, −2n, −n), (C) (n, −n, −n, −2n), (D) (2n, −n, −2n, −2n)
Answer: (C) (n, −n, −n, −2n)
Subject: physics
Question: An infinitely long wire, located on the z-axis, carries a current I along the +z-direction and produces the magnetic field B. The magnitude of the line integral ∫ B⃗ ⋅dℓ⃗  along a straight line from the point (−√3a, a, 0) to (a, a, 0) is given by [μ0 is the magnetic permeability of free space.]
Choices: (A) 7μ0I/24, (B) 7μ0I/12, (C) μ0I/8, (D) μ0I/6
Answer: (C) μ0I/8
Subject: physics
Question: Two beads, each with charge q and mass m, are on a horizontal, frictionless, non-conducting, circular hoop of radius R. One of the beads is glued to the hoop at some point, while the other one performs small oscillations about its equilibrium position along the hoop. The square of the angular frequency of the small oscillations is given by [ε0 is the permittivity of free space.]
Choices: (A) q²/(4πε0R³m), (B) q²/(32πε0R³m), (C) q²/(8πε0R³m), (D) q²/(16πε0R³m)
Answer: (C) q²/(8πε0R³m)
Subject: physics
Question: A block of mass 5 kg moves along the x-direction subject to the force F = (−20x + 10) N, with the value of x in metre. At time t = 0 s, it is at rest at position x = 1 m. The position and momentum of the block at t = (π/4) s are
Choices: (A) −0.5 m, 5 kg m/s, (B) 0.5 m, 0 kg m/s, (C) 0.5 m, −5 kg m/s, (D) −1 m, 5 kg m/s
Answer: (A) −0.5 m, 5 kg m/s
Subject: physics
Question: A particle of mass m is moving in a circular orbit under the influence of the central force F(r) = -kr, corresponding to the potential energy V(r) = kr^2/2, where k is a positive force constant and r is the radial distance from the origin. According to the Bohr's quantization rule, the angular momentum of the particle is given by L = nh, where h = h/(2π), h is the Planck's constant, and n a positive integer. If v and E are the speed and total energy of the particle, respectively, then which of the following expression(s) is(are) correct?
Choices: (A) r^2 = nh √(k/mk) (B) v^2 = nh (k/m) (C) L = (k/mr^2) √(m) (D) E = nh (k/2√m)
Answer: D
Subject: physics
Question: Two uniform strings of mass per unit length μ and 4μ, and length L and 2L, respectively, are joined at point O, and tied at two fixed ends P and Q, as shown in the figure. The strings are under a uniform tension T. If we define the frequency v0 = 1/(2L) √(T/μ), which of the following statement(s) is(are) correct?
Choices: (A) With a node at O, the minimum frequency of vibration of the composite string is v0. (B) With an antinode at O, the minimum frequency of vibration of the composite string is 2v0. (C) When the composite string vibrates at the minimum frequency with a node at O, it has 6 nodes, including the end nodes. (D) No vibrational mode with an antinode at O is possible for the composite string.
Subject: physics
Question: A glass beaker has a solid, plano-convex base of refractive index 1.60, as shown in the figure. The radius of curvature of the convex surface (SPU) is 9 cm, while the planar surface (STU) acts as a mirror. This beaker is filled with a liquid of refractive index n up to the level QPR. If the image of a point object O at a height of h (OT in the figure) is formed onto itself, then, which of the following option(s) is(are) correct?
Choices: (A) For n = 1.42, h = 50 cm. (B) For n = 1.35, h = 36 cm. (C) For n = 1.45, h = 65 cm. (D) For n = 1.48, h = 85 cm.
Subject: physics
Question: The specific heat capacity of a substance is temperature dependent and is given by the formula C = kT, where k is a constant of suitable dimensions in SI units, and T is the absolute temperature. If the heat required to raise the temperature of 1 kg of the substance from −73 °C to 27 °C is nk, the value of n is
Choices: No response choices provided.
Answer: To be entered as a non-negative integer
Subject: physics
Question: A disc of mass M and radius R is free to rotate about its vertical axis as shown in the figure. A battery operated motor of negligible mass is fixed to this disc at a point on its circumference. Another disc of the same mass M and radius R/2 is fixed to the motor's thin shaft. Initially, both the discs are at rest. The motor is switched on so that the smaller disc rotates at a uniform angular speed ω. If the angular speed at which the large disc rotates is ω/n, then the value of n is
Choices: No response choices provided.
Answer: To be entered as a non-negative integer
Subject: physics
Question: A point source S emits unpolarized light uniformly in all directions. At two points A and B, the ratio r = IA/IB of the intensities of light is 2. If a set of two polaroids having 45° angle between their pass-axes is placed just before point B, then the new value of r will be
Choices: No response choices provided.
Answer: To be entered as a non-negative integer
Subject: physics
Question: A source (S) of sound has frequency 240 Hz. When the observer (O) and the source move towards each other at a speed v with respect to the ground (as shown in Case 1 in the figure), the observer measures the frequency of the sound to be 288 Hz. However, when the observer and the source move away from each other at the same speed v with respect to the ground (as shown in Case 2 in the figure), the observer measures the frequency of sound to be n Hz. The value of n is _____.
Choices: No response choices provided
Answer: No answer provided
Subject: physics
Question: Two large, identical water tanks, 1 and 2, kept on the top of a building of height H, are filled with water up to height h in each tank. Both the tanks contain an identical hole of small radius on their sides, close to their bottom. A pipe of the same internal radius as that of the hole is connected to tank 2, and the pipe ends at the ground level. When the water flows from the tanks 1 and 2 through the holes, the times taken to empty the tanks are t1 and t2, respectively. If H = (49/4) h, then the ratio t1/t2 is _____.
Choices: No response choices provided
Answer: No answer provided
Subject: physics
Question: A thin uniform rod of length l, and certain mass is kept on a frictionless horizontal table with a massless string of length L fixed to one end (top view is shown in the figure). The other end of the string is pivoted to a point O. If a horizontal impulse P is imparted to the rod at a distance x = L/n from the mid-point of the rod (see figure), then the rod and string revolve together around the point O, with the rod remaining aligned with the string. In such a case, the value of n is _____.
Choices: No response choices provided
Answer: No answer provided
Subject: physics
Question: One mole of a monatomic ideal gas undergoes the cyclic process J→ K→ L→ M→ J, as shown in the P-T diagram.
Choices: (A) P → 1; Q → 3; R → 5; S → 4
(C) P → 4; Q → 1; R → 2; S → 2
(B) P → 4; Q → 3; R → 5; S → 2
(D) P → 2; Q → 5; R → 3; S → 4
Answer: None
Subject: physics
Question: Four identical thin, square metal sheets, S1, S2, S3 and S4, each of side a are kept parallel to each other with equal distance d (≪ a) between them, as shown in the figure. Let C0 = ε0a^2/d, where ε0 is the permittivity of free space. Match the quantities mentioned in List-I with their values in List-II and choose the correct option.
Choices: List-I: (P) The capacitance between S1 and S4, with S2 and S3 not connected, is; (Q) The capacitance between S1 and S4, with S2 shorted to S3, is; (R) The capacitance between S1 and S4, with S2 shorted to S3, is; (S) The capacitance between S1 and S2, with S3 shorted to S1, and S2 shorted to S4, is. List-II: (1) 3C0; (2) C0/2; (3) C0/3; (4) 2C0/3; (5) 2C0. (A) P → 3; Q → 2; R → 4; S → 5, (B) P → 2; Q → 3; R → 2; S → 1, (C) P → 3; Q → 2; R → 4; S → 1, (D) P → 3; Q → 2; R → 2; S → 5
Answer: (B) P -> 2; Q -> 3; R -> 2; S -> 1
Subject: physics
Question: A light ray is incident on the surface of a sphere of refractive index n at an angle of incidence θ0. The ray partially refracts into the sphere with angle of refraction ϕ0 and then partly reflects from the back surface. The reflected ray then emerges out of the sphere after a partial refraction. The total angle of deviation of the emergent ray with respect to the incident ray is α. Match the quantities mentioned in List-I with their values in List-II and choose the correct option.
Choices: List-I: (P) If n = 2 and α = 180°, then all the possible values of θ0 will be; (Q) If n = √3 and α = 180°, then all the possible values of θ0 will be; (R) If n = √3 and α = 180°, then all the possible values of ϕ0 will be; (S) If n = √2 and θ0 = 45°, then all the possible values of α will be. List-II: (1) 30° and 0°; (2) 60° and 0°; (3) 45° and 0°; (4) 150°; (5) 0°. (A) P → 5; Q → 2; R→ 1; S→ 4, (B) P → 5; Q → 1; R→ 2; S→ 4, (C) P → 3; Q → 2; R→ 1; S→ 4, (D) P → 3; Q → 1; R→ 2; S→ 5
Answer: (B) P -> 5; Q -> 1; R-> 2; S-> 4
Subject: physics
Question: The circuit shown in the figure contains an inductor L, a capacitor C0, a resistor R0 and an ideal battery. The circuit also contains two keys K1 and K2. Initially, both the keys are open and there is no charge on the capacitor. At an instant, key K1 is closed and immediately after this the current in R0 is found to be I1. After a long time, the current attains a steady state value I2. Thereafter, K2 is closed and simultaneously K1 is opened and the voltage across C0 oscillates with amplitude V0 and angular frequency ω0. Match the quantities mentioned in List-I with their values in List-II and choose the correct option.
Choices: (A) P → 1; Q → 3; R → 2; S → 5 (B) P → 1; Q → 2; R → 3; S → 5 (C) P → 1; Q → 3; R → 2; S → 4 (D) P → 2; Q → 5; R → 3; S → 4
Answer: (B) P → 1; Q → 2; R → 3; S → 5
Subject: chemistry
Question: A closed vessel contains 10 g of an ideal gas X at 300 K, which exerts 2 atm pressure. At the same temperature, 80 g of another ideal gas Y is added to it and the pressure becomes 6 atm. The ratio of root mean square velocities of X and Y at 300 K is
Choices: (A) 2√2 : √3, (B) 2√2 : 1, (C) 1 : 2, (D) 2 : 1
Answer: C
Subject: chemistry
Question: At room temperature, disproportionation of an aqueous solution of in situ generated nitrous acid (HNO2) gives the species
Choices: (A) H3O+, NO3- and NO, (B) H3O+, NO3- and NO2, (C) H3O+, NO- and NO2, (D) H3O+, NO3- and N2O
Answer: B
Subject: chemistry
Question: Among the following options, select the option in which each complex in Set-I shows geometrical isomerism and the two complexes in Set-II are ionization isomers of each other. [en = H2NCH2CH2NH2]
Choices: (A) Set-I: [Ni(CO)3] and [PdCl4(PPh3)2] Set-II: [Co(NH3)5Cl]SO4 and [Co(NH3)4(SO4)]Cl, (B) Set-I: [Co(en)(NH3)2Cl2] and [PdCl4(PPh3)2] Set-II: [Co(NH3)6][Cr(CN)6] and [Cr(NH3)6][Co(CN)6], (C) Set-I: [Co(NH3)4(NO2)2] and [Co(en)2Cl2] Set-II: [Co(NH3)5Cl]SO4 and [Co(NH3)4(SO4)]Cl, (D) Set-I: [Cr(NH3)5Cl]Cl2 and [Co(en)(NH3)2Cl2] Set-II: [Cr(H2O)6]Cl3 and [Cr(H2O)4Cl]Cl2·H2O
Answer: (B)
Subject: physics
Question: Among the following, the correct statement(s) for electrons in an atom is(are)
Choices: (A) Uncertainty principle rules out the existence of definite paths for electrons. (B) The energy of an electron in 2s orbital of an atom is lower than the energy of an electron that is infinitely far away from the nucleus. (C) According to Bohr's model, the most negative energy value for an electron is given by n = 1, which corresponds to the most stable orbit. (D) According to Bohr's model, the magnitude of velocity of electrons increases with increase in values of n.
Subject: chemistry
Question: Consider the following volume-temperature (V-T) diagram for the expansion of 5 moles of an ideal monoatomic gas. Considering only P-V work is involved, the total change in enthalpy (in Joule) for the transformation of state in the sequence X->Y->Z is _____.
Choices: The question does not have explicit multiple choice options listed.
Answer: The answer field requires entering a numerical value.
Subject: chemistry
Question: Consider the following reaction, 2H2(g) + 2NO(g) → N2(g) + 2H2O(g) which follows the mechanism given below: 2NO(g) ⇌_k1 N2O2(g) (fast equilibrium) N2O2(g) + H2(g) ⇒_k2 N2O(g) + H2O(g) (slow reaction) N2O(g) + H2(g) ⇒_k3 N2(g) + H2O(g) (fast reaction) The order of the reaction is _______.
Subject: chemistry
Question: Complete reaction of acetaldehyde with excess formaldehyde, upon heating with conc. NaOH solution, gives P and Q. Compound P does not give Tollens' test, whereas Q on acidification gives positive Tollens' test. Treatment of P with excess cyclohexanone in the presence of catalytic amount of p-toluenesulfonic acid (PTSA) gives product R. Sum of the number of methylene groups (-CH2-) and oxygen atoms in R is ______.
Subject: chemistry
Question: Among V(CO)6, Cr(CO)5, Cu(CO)3, Mn(CO)5, Fe(CO)5, [Co(CO)3]3-, [Cr(CO)4]4-, and Ir(CO)3, the total number of species isoelectronic with Ni(CO)4 is ______. [Given, atomic number: V = 23, Cr = 24, Mn = 25, Fe = 26, Co = 27, Ni = 28, Cu = 29, Ir = 77]
Subject: chemistry
Question: In the following reaction sequence, the major product P is formed. 

 i) Hg2+, H3O+ 
 ii) Zn-Hg/HCl 
 iii) H3O+, Δ 

 Glycerol reacts completely with excess P in the presence of an acid catalyst to form Q. Reaction of Q with excess NaOH followed by the treatment with CaCl2 yields Ca-soap R, quantitatively. 

Starting with one mole of Q, the amount of R produced in gram is _____.
Subject: chemistry
Question: Among the following complexes, the total number of diamagnetic species is _____. 

 [Mn(NH3)6]3+, [MnCl6]3-, [FeFe4]2-, [CoF6]3-, [Fe(NH3)6]3+, and [Co(en)3]3+

[Given, atomic number: Mn = 25, Fe = 26, Co = 27; 

en = H2NCH2CH2NH2]
Subject: chemistry
Question: In a conductometric titration, small volume of titrant of higher concentration is added stepwise to a larger volume of titrate of much lower concentration, and the conductance is measured after each addition. Match each entry in List-I with the appropriate entry in List-II and choose the correct option.
Choices: A) P-4, Q-3, R-2, S-5
B) P-2, Q-4, R-3, S-1
C) P-3, Q-4, R-2, S-5
D) P-4, Q-3, R-2, S-1
Subject: chemistry
Question: Based on VSEPR model, match the xenon compounds given in List-I with the corresponding geometries and the number of lone pairs on xenon given in List-II and choose the correct option.
Choices: (A) P-5, Q-2, R-3, S-1
(B) P-5, Q-3, R-2, S-4
(C) P-4, Q-3, R-2, S-1
(D) P-4, Q-2, R-5, S-3
Subject: chemistry
Question: List-I contains various reaction sequences and List-II contains the possible products. Match each entry in List-I with the appropriate entry in List-II and choose the correct option.
Choices: A) P-3, Q-5, R-4, S-1
B) P-3, Q-2, R-4, S-1
C) P-3, Q-5, R-1, S-4
D) P-5, Q-2, R-4, S-1
Answer: (A) P-3, Q-5, R-4, S-1