cmeraki/elemento huggingface dataset:
Subject: mathematics
Question: Let S = (0,1) ∪ (1,2) ∪ (3,4) and T = {0,1,2,3}. Then which of the following statements is(are) true?
Choices: (A) There are infinitely many functions from S to T
(B) There are infinitely many strictly increasing functions from S to T
(C) The number of continuous functions from S to T is at most 120
(D) Every continuous function from S to T is differentiable
Subject: mathematics
Question: Let f : (0,1) -> R be the function defined as f(x) = sqrt(n) if x = (1/n + 1/n) where n ∈ N . Let g : (0,1) -> R be a function such that (∫(x to 1) (1-t/sqrt(t)) dt <= g(x) <= 2sqrt(x) for all x ∈ (0,1). Then lim f(x)g(x)x->0+
Choices: (A) does NOT exist
(B) is equal to 1
(C) is equal to 2
(D) is equal to 3
Answer: C
Subject: mathematics
Question: Let Q be the cube with the set of vertices {(x1,x2,x3) ∈ R^3 : x1,x2,x3 ∈ {0,1}}. Let F be the set of all twelve lines containing the diagonals of the six faces of the cube Q. Let S be the set of all four lines containing the main diagonals of the cube Q; for instance, the line passing through the vertices (0,0,0) and (1,1,1) is in S . For lines ℓ1 and ℓ2 , let d(ℓ1,ℓ2) denote the shortest distance between them. Then the maximum value of d(ℓ1,ℓ2), as ℓ1 varies over F and ℓ2 varies over S , is
Choices: (A) 1/sqrt(6)	(B) 1/sqrt(8)	(C) 1/sqrt(3)	(D) 1/sqrt(12)
Answer: B
Subject: mathematics
Question: Let X = {(x, y) ∈ Z×Z : x^2/8 + y^2/20 < 1 and y^2 <= 5x}. Three distinct points P, Q and R are randomly chosen from X . Then the probability that P, Q and R form a triangle whose area is a positive integer, is
Choices: (A) 71/220	(B) 73/220	(C) 79/220	(D) 83/220
Answer: C
Subject: mathematics
Question: Let P be a point on the parabola y^2 = 4ax , where a > 0 . The normal to the parabola at P meets the x-axis at a point Q. The area of the triangle PFQ , where F is the focus of the parabola, is 120. If the slope m of the normal and a are both positive integers, then the pair (a,m) is
Choices: (A) (2,3)	(B) (1,3)	(C) (2,4)	(D) (3,4)
Answer: (D) (3,4)
Subject: mathematics
Question: Let tan^(-1)(x) ∈ (-π/2, π/2), for x ∈ ℝ. Then the number of real solutions of the equation √(1+cos(2x)) = √2 tan^(-1)(tan x) in the set (-3π/2, -π/2) ∪ (-π/2, π/2) ∪ (π/2, 3π/2) is equal to
Choices: No response choices provided.
Answer: 2
Subject: mathematics
Question: Let n ≥ 2 be a natural number and f :[0,1] → ℝ be the function defined by f(x) = n(1-2nx) if 0 ≤ x ≤ 1/2n, 2n(2nx-1) if 1/2n ≤ x ≤ 3/4n, 4n(1-nx) if 3/4n ≤ x ≤ 1/n, n/(n-1) (nx-1) if 1/n ≤ x ≤ 1. If n is such that the area of the region bounded by the curves x = 0 , x = 1 , y = 0 and y = f (x) is 4 , then the maximum value of the function f is
Choices: No response choices provided.
Answer: 2
Subject: mathematics
Question: Let 75⋯57 denote the (r + 2) digit number where the first and the last digits are 7 and the remaining r digits are 5. Consider the sum S = 77 + 757 + 7557 + ⋯ + 75⋯57. If S = 75⋯57 + m/n , where m and n are natural numbers less than 3000, then the value of m + n is
Choices: No response choices provided.
Answer: 1567
Subject: mathematics
Question: Let P be the plane √3x + 2y + 3z = 16 and let S = {αi + βj + γk : α² + β² + γ² = 1 and the distance of (α, β, γ) from the plane P is 7/2}. Let u, v and w be three distinct vectors in S such that |u - v| = |v - w| = |w - u|. Let V be the volume of the parallelepiped determined by vectors u, v and w. Then the value of 80√3/V is
Choices: No multiple choice options provided
Answer: No answer provided
Subject: mathematics
Question: Let α, β and γ be real numbers. Consider the following system of linear equations 
x + 2y + z = 7
x + αz = 11
2x - 3y + βz = γ

Match each entry in List-I to the correct entries in List-II.
Choices: (P) If β = 1/2 (7α - 3) and γ = 28, then the system has 
(Q) If β = 1/2 (7α - 3) and γ ≠ 28, then the system has
(R) If β ≠ 1/2 (7α - 3) where α = 1 and γ ≠ 28, then the system has
(S) If β ≠ 1/2 (7α - 3) where α = 1 and γ = 28, then the system has
Answer: (A) (P) → (3) 	(Q) → (2) 	(R) → (1) 	(S) → (4)
Subject: mathematics
Question: Consider the given data with frequency distribution xi  3  8  11  10  5  4 fi  5  2  3  2  4  4 Match each entry in List-I to the correct entries in List-II.  List-I  (P) The mean of the above data is                (Q) The median of the above data is                (R) The mean deviation about the mean of the above data is   (S) The mean deviation about the median of the above data is     List-II (1) 2.5 (2) 5 (3) 6 (4) 2.7 (5) 2.4  The correct option is: (A) (P) → (3)   (Q) → (2)   (R) → (4)   (S) → (5) (B) (P) → (3)   (Q) → (2)   (R) → (1)   (S) → (5) (C) (P) → (2)   (Q) → (3)   (R) → (4)   (S) → (1) (D) (P) → (3)   (Q) → (3)   (R) → (5)   (S) → (5)
Choices: (A) (P) → (3)   (Q) → (2)   (R) → (4)   (S) → (5), (B) (P) → (3)   (Q) → (2)   (R) → (1)   (S) → (5), (C) (P) → (2)   (Q) → (3)   (R) → (4)   (S) → (1), (D) (P) → (3)   (Q) → (3)   (R) → (5)   (S) → (5)
Answer: (D) (P) → (3)   (Q) → (3)   (R) → (5)   (S) → (5)
Subject: mathematics
Question: Let ℓ1 and ℓ2 be the lines ᾽r = λ(iˆ + jˆ + kˆ) and ᾽r = (jˆ - kˆ) + μ(iˆ + kˆ), respectively. Let X be the set of all the planes H that contain the line ℓ1. For a plane H, let d(H) denote the smallest possible distance between the points of ℓ2 and H. Let H0 be a plane in X for which d(H0) is the maximum value of d(H) as H varies over all planes in X.
Choices: Match each entry in List-I to the correct entries in List-II. 

List-I
(P) The value of d(H0) is
(Q) The distance of the point (0,1,2) from H0 is

(R) The distance of origin from H0 is
(S) The distance of origin from the point of intersection of planes y = z, x = 1 and H0 is 

List-II
(1) √3
(2) 1/√3
(3) 0
(4) √2

(5) 1/√2
Answer: (A) (P) → (2)   (Q) → (4)   (R) → (5)   (S) → (1)
Subject: mathematics
Question: Let z be a complex number satisfying | z |³ + 2z² + 4z̄ - 8 = 0 , where z̄ denotes the complex conjugate of z . Let the imaginary part of z be nonzero.
Choices: Match each entry in List-I to the correct entries in List-II.

List-I
(P) | z |² is equal to
(Q) | z - z̄ |² is equal to
(R) | z |⁴ + | z + z̄ |² is equal to
(S) | z + 1|² is equal to

List-II
(1) 12
(2) 4
(3) 8
(4) 10
(5) 7
Answer: (A) (P) → (1)   (Q) → (3)   (R) → (5)   (S) → (4)
Subject: physics
Question: A slide with a frictionless curved surface, which becomes horizontal at its lower end, is fixed on the terrace of a building of height 3h from the ground, as shown in the figure. A spherical ball of mass m is released on the slide from rest at a height h from the top of the terrace. The ball leaves the slide with a velocity ũ0 = u0x̂ and falls on the ground at a distance d from the building making an angle θ with the horizontal. It bounces off with a velocity ṽ and reaches a maximum height h1. The acceleration due to gravity is g and the coefficient of restitution of the ground is 1/√3. Which of the following statement(s) is(are) correct?
Choices: A) ũ0 = √(2gh)x̂
B) ṽ = √(2gh(x̂ - λ))
C) θ = 60°
D) d/h1 = 2√3
Answer: A
Subject: physics
Question: A plane polarized blue light ray is incident on a prism such that there is no reflection from the surface of the prism. The angle of deviation of the emergent ray is δ = 60° (see Figure-1). The angle of minimum deviation for red light from the same prism is δmin = 30° (see Figure-2). The refractive index of the prism material for blue light is √3. Which of the following statement(s) is(are) correct?
Choices: (A) The blue light is polarized in the plane of incidence.
(B) The angle of the prism is 45°.
(C) The refractive index of the material of the prism for red light is √2.
(D) The angle of refraction for blue light in air at the exit plane of the prism is 60°.
Subject: physics
Question: In a circuit shown in the figure, the capacitor C is initially uncharged and the key K is open. In this condition, a current of 1 A flows through the 1 Ω resistor. The key is closed at time t = t0. Which of the following statement(s) is(are) correct?
Choices: (A) The value of the resistance R is 3 Ω.
(B) For t < t0, the value of current I1 is 2 A.
(C) At t = t0 + 7.2 μs, the current in the capacitor is 0.6 A.
(D) For t → ∞, the charge on the capacitor is 12 μC.
Subject: physics
Question: A bar of mass M = 1.00 kg and length L = 0.20 m is lying on a horizontal frictionless surface. One end of the bar is pivoted at a point about which it is free to rotate. A small mass m = 0.10 kg is moving on the same horizontal surface with 5.00 m s−1 speed on a path perpendicular to the bar. It hits the bar at a distance L/2 from the pivoted end and returns back on the same path with speed v. After this elastic collision, the bar rotates with an angular velocity ω. Which of the following statement is correct?
Choices: (A) ω = 6.98 rad s−1 and v = 4.30 m s−1 
(C) ω = 3.75 rad s−1 and v = 10.0 m s−1
Answer: B
Subject: physics
Question: A container has a base of 50 cm × 5 cm and height 50 cm, as shown in the figure. It has two parallel electrically conducting walls each of area 50 cm × 50 cm. The remaining walls of the container are thin and non-conducting. The container is being filled with a liquid of dielectric constant 3 at a uniform rate of 250 cm3 s−1. What is the value of the capacitance of the container after 10 seconds?
Choices: (A) 27 pF 
(C) 81 pF 
(D) 135 pF
Answer: D
Subject: physics
Question: One mole of an ideal gas expands adiabatically from an initial state (Ti, Vi) to final state (Tf, 5Vi). Another mole of the same gas expands isothermally from a different initial state (TB, VB) to the same final state (Tf, 5Vi). The ratio of the specific heats at constant pressure and constant volume of this ideal gas is γ. What is the ratio TB/Ti?
Choices: (A) 5γ−1, (B) 51−γ, (C) 5γ, (D) 51+γ
Answer: (B) 51−γ
Subject: physics
Question: Two satellites P and Q are moving in different circular orbits around the Earth (radius R). The heights of P and Q from the Earth surface are hP and hQ, respectively, where hP = R/3. The accelerations of P and Q due to Earth's gravity are gP and gQ, respectively. If gP/gQ = 36/25, what is the value of hQ?
Choices: (A) 3R/5, (B) R/6, (C) 6R/5, (D) 5R/6
Answer: (B) R/6
Subject: physics
Question: A Hydrogen-like atom has atomic number Z. Photons emitted in the electronic transitions from level n = 4 to level n = 3 in these atoms are used to perform photoelectric effect experiment on a target metal. The maximum kinetic energy of the photoelectrons generated is 1.95 eV. If the photoelectric threshold wavelength for the target metal is 310 nm, the value of Z is _______.
Subject: physics
Question: An optical arrangement consists of two concave mirrors M1 and M2, and a convex lens L with a common principal axis, as shown in the figure. The focal length of L is 10 cm. The radii of curvature of M1 and M2 are 20 cm and 24 cm, respectively. The distance between L and M2 is 20 cm. A point object S is placed at the mid-point between L and M2 on the axis. When the distance between L and M1 is n/7 cm, one of the images coincides with S. The value of n is _______.
Subject: physics
Question: A closed container contains a homogeneous mixture of two moles of an ideal monatomic gas (γ = 5/3) and one mole of an ideal diatomic gas (γ = 7/5). Here, γ is the ratio of the specific heats at constant pressure and constant volume of an ideal gas. The gas mixture does a work of 66 Joule when heated at constant pressure. The change in its internal energy is _______ Joule.
Choices: Multiple choice options not provided
Answer: No answer provided
Subject: physics
Question: A person of height 1.6 m is walking away from a lamp post of height 4 m along a straight path on the flat ground. The lamp post and the person are always perpendicular to the ground. If the speed of the person is 60 cm s^-1, the speed of the tip of the person's shadow on the ground with respect to the person is ______ cm s^-1.
Choices: Multiple choice options not provided
Answer: No answer provided
Subject: physics
Question: Two point-like objects of masses 20 gm and 30 gm are fixed at the two ends of a rigid massless rod of length 10 cm. This system is suspended vertically from a rigid ceiling using a thin wire attached to its center of mass, as shown in the figure. The resulting torsional pendulum undergoes small oscillations. The torsional constant of the wire is 1.2 × 10^-8 N m rad^-1. The angular frequency of the oscillations in n × 10^-3 rad s^-1. The value of n is _____.
Choices: Multiple choice options not provided
Answer: No answer provided
Subject: physics
Question: List-I shows different radioactive decay processes and List-II provides possible emitted particles. Match each entry in List-I with an appropriate entry from List-II, and choose the correct option.
Choices: (A) P → 4, Q → 3, R → 2, S → 1 (C) P → 5, Q → 3, R → 1, S → 4 (B) P → 4, Q → 1, R → 2, S → 5 (D) P → 5, Q → 1, R → 3, S → 2
Answer: (B) P → 4, Q → 1, R → 2, S → 5
Subject: physics
Question: Match the temperature of a black body given in List-I with an appropriate statement in List-II, and choose the correct option. [Given: Wien's constant as 2.9 × 10^-3 m-K and hc/e = 1.24 × 10^-6 V-m]
Choices: (A) P → 3, Q → 5, R → 2, S → 3 (C) P → 3, Q → 4, R → 2, S → 1 (B) P → 3, Q → 2, R → 4, S → 1 (D) P → 1, Q → 2, R → 5, S → 3
Answer: (B) P → 3, Q → 2, R → 4, S → 1
Subject: physics
Question: A series LCR circuit is connected to a 45 sin(ωt) Volt source. The resonant angular frequency of the circuit is 10^5 rad s^-1 and current amplitude at resonance is I0. When the angular frequency of the source is ω = β × 10^4 rad s^-1, the current amplitude in the circuit is 0.05 I0. If β = 50 mH, match each entry in List-I with an appropriate value from List-II and choose the correct option.
Choices: (P) I0 in mA, (Q) The quality factor of the circuit, (R) The bandwidth of the circuit in rad s^-1, (S) The peak power dissipated at resonance in Watt; (A) P → 2, Q → 3, R → 5, S → 1, (C) P → 4, Q → 5, R → 3, S → 1, (B) P → 3, Q → 1, R → 4, S → 2, (D) P → 4, Q → 2, R → 1, S → 5
Answer: B
Subject: physics
Question: A thin conducting rod MN of mass 20 gm, length 25 cm and resistance 1.0 Ω is held on frictionless, long, perfectly conducting vertical rails as shown in the figure. There is a uniform magnetic field B0 = 4 T directed perpendicular to the plane of the rod-rail arrangement. The rod is released from rest at time t = 0 and it moves down along the rails. Assume air drag is negligible. Match each quantity in List-I with an appropriate value from List-II, and choose the correct option. [Given: The acceleration due to gravity g = 10 m s^-2 and e^-1 = 0.4]
Choices: (P) At t = 0.2 s, the magnitude of the induced emf in Volt, (Q) At t = 0.2 s, the magnitude of the magnetic force in Newton, (R) At t = 0.2 s, the power dissipated as heat in Watt, (S) The magnitude of terminal velocity of the rod in m s^-1; (A) P → 5, Q → 2, R → 3, S → 1, (C) P → 4, Q → 3, R → 1, S → 2, (B) P → 3, Q → 1, R → 4, S → 5, (D) P → 3, Q → 4, R → 2, S → 5
Answer: B
Subject: chemistry
Question: The correct statement(s) related to processes involved in the extraction of metals is(are)
Choices: (A) Roasting of Malachite produces Cuprite, (B) Calcination of Calamine produces Zincite, (C) Copper pyrites is heated with silica in a reverberatory furnace to remove iron, (D) Impure silver is treated with aqueous KCN in the presence of oxygen followed by reduction with zinc metal
Answer: A,B,D
Subject: chemistry
Question: Plotting 1/Λm against c√m for aqueous solutions of a monobasic weak acid (HX) resulted in a straight line with y-axis intercept of P and slope of S. The ratio P/S is
Choices: (A) Ka Λ^0_m 
(B) Ka Λ^0_m/2 
(C) 2 Ka Λ^0_m 
(D) 1 / (Ka Λ^0_m)
Answer: D
Subject: chemistry
Question: On decreasing the pH from 7 to 2, the solubility of a sparingly soluble salt (MX) of a weak acid (HX) increased from 10^-4 mol L^-1 to 10^-3 mol L^-1. The pKa of HX is
Choices: (A) 3, (B) 4, (C) 5, (D) 2
Answer: (D) 2
Subject: chemistry
Question: In the given reaction scheme, P is a phenyl alkyl ether, Q is an aromatic compound; R and S are the major products. The correct statement about S is
Choices: (A) It primarily inhibits noradrenaline degrading enzymes. (B) It inhibits the synthesis of prostaglandin. (C) It is a narcotic drug. (D) It is ortho-acetylbenzoic acid.
Answer: (D) It is ortho-acetylbenzoic acid.
Subject: chemistry
Question: A gas has a compressibility factor of 0.5 and a molar volume of 0.4 dm3 mol−1 at a temperature of 800 K and pressure x atm. If it shows ideal gas behaviour at the same temperature and pressure, the molar volume will be y dm3 mol−1. The value of x/y is ___.
Choices: Integer response
Answer: Integer value to be entered
Subject: chemistry
Question: In a one-litre flask, 6 moles of A undergoes the reaction A (g) ⇌ P (g). The progress of product formation at two temperatures (in Kelvin), T1 and T2, is shown in the figure:
Answer: If T1 = 2T2 and (ΔGᶻ₁ - ΔGᶻ₂) = RT2 ln x, then the value of x is ___.
Subject: chemistry
Question: Match the reactions (in the given stoichiometry of the reactants) in List-I with one of their products given in List-II and choose the correct option.
Choices: (A) P → 2; Q → 3; R → 1; S → 5
(B) P → 3; Q → 5; R → 4; S → 2
(C) P → 5; Q → 2; R → 1; S → 3
(D) P → 2; Q → 3; R → 4; S → 5
Subject: chemistry
Question: The major products obtained from the reactions in List-II are the reactants for the named reactions mentioned in List-I. Match List-I with List-II and choose the correct option.
Choices: (A) P -> 2; Q -> 4; R -> 1; S -> 3, (B) P -> 1; Q -> 3; R -> 5; S -> 2, (C) P -> 3; Q -> 2; R -> 1; S -> 4, (D) P -> 3; Q -> 4; R -> 5; S -> 2
Answer: (B) P -> 1; Q -> 3; R -> 5; S -> 2