cmeraki/elemento huggingface dataset:
Subject: mathematics
Question: Let f :[1,∞) → ℝ be a differentiable function such that f(1) = 3 and 3 ∫1x f(t)dt = x f (x) - x3, x ∈ [1, ∞). Let e denote the base of the natural logarithm. Then the value of f(e) is
Choices: (A) e2 + 4/3, (B) log e 4 + e/3, (C) 4e2/3, (D) e2 - 4/3
Answer: (B) log e 4 + e/3
Subject: mathematics
Question: For any y ∈ ℝ, let cot−1(y) ∈ (0, π) and tan−1(y) ∈ (- π/2, π/2). Then the sum of all the solutions of the equation tan−1(6y/9 - y5) + cot−1(9 - y2/6y) = 2π/3 for 0 ≤| y| ≤ 3, is equal to
Choices: (A) 2√3 - 3, (B) 3 - 2√3, (C) 4√3 - 6, (D) 6 - 4√3
Answer: (C) 4√3 - 6
Subject: mathematics
Question: Let the position vectors of the points P, Q, R and S be ⃗a = i⃗ + 2j⃗ - 5k⃗, ⃗b = 3i⃗ + 6j⃗ + 3k⃗, ⃗c = (17/5)i⃗ + (16/5)j⃗ + 7k⃗ and ⃗d = 2i⃗ + j⃗ + k⃗, respectively. Then which of the following statements is true?
Choices: A) The points P, Q, R and S are NOT coplanar
B) (⃗b + 2⃗d)/3 is the position vector of a point which divides PR internally in the ratio 5:4
C) (⃗b + 2⃗d)/3 is the position vector of a point which divides PR externally in the ratio 5:4
D) The square of the magnitude of the vector ⃗b × ⃗d is 95
Subject: mathematics
Question: Let M = (aij), i, j ∈ {1, 2, 3}, be the 3×3 matrix such that aij = 1 if j + 1 is divisible by i, otherwise aij = 0 . Then which of the following statements is(are) true?
Choices: (A) M is invertible
(B) There exists a nonzero column matrix ⎡a1⎤ ⎢a2⎥ such that M ⎢a3⎥ = ⎢⎡a1⎤⎢-a2⎥⎢a3⎥⎥
(C) The set {X ∈ R3 : MX = 0} ≠ {0}, where 0 = ⎢0⎥
(D) The matrix (M - 2I) is invertible, where I is the 3×3 identity matrix
Answer: B,C
Subject: mathematics
Question: Let f : (0,1) -> R be the function defined as f(x) = [4x]( x - 1/4 ) ( x - 1/2 ), where [x] denotes the greatest integer less than or equal to x. Then which of the following statements is(are) true?
Choices: (A) The function f is discontinuous exactly at one point in (0,1) 

(B) There is exactly one point in (0,1) at which the function f is continuous but NOT differentiable 

(C) The function f is NOT differentiable at more than three points in (0,1) 

(D) The minimum value of the function f is -1/512
Answer: D
Subject: mathematics
Question: Let S be the set of all twice differentiable functions f from R to R such that d^2f/dx^2 (x) > 0 for all x ∈ (-1,1). For f ∈ S , let X_f be the number of points x ∈ (-1,1) for which f'(x) = x. Then which of the following statements is(are) true?
Choices: (A) There exists a function f ∈ S such that X_f = 0 

(B) For every function f ∈ S , we have X_f ≤ 2 

(C) There exists a function f ∈ S such that X_f = 2 

(D) There does NOT exist any function f in S such that X_f = 1
Subject: mathematics
Question: For x ∈ ℝ, let tan^(-1)(x) ∈ (-π/2, π/2). Then the minimum value of the function f : ℝ → ℝ defined by f(x) = ∫(x tan^(-1)x) / (e^((x-cos(x))) / (1 + t^2/25)) dt is
Choices: No response choices provided
Answer: Numeric response required
Subject: mathematics
Question: Let X be the set of all five digit numbers formed using 1,2,2,2,4,4,0. For example, 22240 is in X while 02244 and 44422 are not in X. Suppose that each element of X has an equal chance of being chosen. Let p be the conditional probability that an element chosen at random is a multiple of 20 given that it is a multiple of 5. Then the value of 38p is equal to
Choices: No response choices provided
Answer: Numeric response required
Subject: mathematics
Question: Let A1, A2, A3, ..., A8 be the vertices of a regular octagon that lie on a circle of radius 2. Let P be a point on the circle and let PAi denote the distance between the points P and Ai for i = 1, 2, ..., 8. If P varies over the circle, then the maximum value of the product PA1 · PA2 · ... · PA8 , is
Choices: No response choices provided
Answer: Numeric response required
Subject: mathematics
Question: Let C1 be the circle of radius 1 with center at the origin. Let C2 be the circle of radius r with center at the point A = (4, 1), where 1 < r < 3. Two distinct common tangents PQ and ST of C1 and C2 are drawn. The tangent PQ touches C1 at P and C2 at Q. The tangent ST touches C1 at S and C2 at T. Mid points of the line segments PQ and ST are joined to form a line which meets the x-axis at a point B. If AB = √5 , then the value of r2 is
Choices: No multiple choice options provided
Answer: No answer provided
Subject: physics
Question: An electric dipole is formed by two charges +q and −q located in xy-plane at (0,2) mm and (0,−2) mm, respectively, as shown in the figure. The electric potential at point P (100,100) mm due to the dipole is V0. The charges +q and −q are then moved to the points (−1,2) mm and (1,−2) mm, respectively. What is the value of electric potential at P due to the new dipole?
Choices: (A) V0/4, (B) V0/2, (C) V0/√2, (D) 3V0/4
Answer: (C) V0/√2
Subject: physics
Question: Young's modulus of elasticity Y is expressed in terms of three derived quantities, namely, the gravitational constant G, Planck's constant ħ and the speed of light c, as Y = c^α ħ^β G^γ. Which of the following is the correct option?
Choices: (A) α = 7, β = −1, γ = −2, (B) α = −7, β = −1, γ = −2, (C) α = 7, β = −1, γ = 2, (D) α = −7, β = 1, γ = −2
Answer: (C) α = 7, β = −1, γ = 2
Subject: physics
Question: A particle of mass m is moving in the xy-plane such that its velocity at a point (x, y) is given as ⃗v = α(yx̂ + 2xŷ), where α is a non-zero constant. What is the force F⃗ acting on the particle?
Choices: (A) F⃗ = 2mα2(xx̂ + yŷ), (B) F⃗ = mα2(yx̂ + 2xŷ), (C) F⃗ = 2mα2(yx̂ + xŷ), (D) F⃗ = mα2(xx̂ + 2yŷ)
Answer: (B) F⃗ = mα2(yx̂ + 2xŷ)
Subject: physics
Question: An ideal gas is in thermodynamic equilibrium. The number of degrees of freedom of a molecule of the gas is n. The internal energy of one mole of the gas is U₀ and the speed of sound in the gas is v₀. At a fixed temperature and pressure, which of the following is the correct option?
Choices: (A) v₃ < v₆ and U₃ > U₆, (B) v₅ > v₃ and U₃ > U₅, (C) v₅ > v₇ and U₅ < U₇, (D) v₆ < v₇ and U₆ < U₇
Subject: physics
Question: A monochromatic light wave is incident normally on a glass slab of thickness d, as shown in the figure. The refractive index of the slab increases linearly from n1 to n2 over the height h. Which of the following statement(s) is(are) true about the light wave emerging out of the slab?
Choices: (A) It will deflect up by an angle tan^-1 [(n2^2-n1^2)/2h], (B) It will deflect up by an angle tan^-1 [(n2-n1)/d], (C) It will not deflect, (D) The deflection angle depends only on (n2 - n1) and not on the individual values of n1 and n2.
Answer: D
Subject: physics
Question: An annular disk of mass M, inner radius a and outer radius b is placed on a horizontal surface with coefficient of friction μ, as shown in the figure. At some time, an impulse J0 is applied at a height h above the center of the disk. If h = hcrit then the disk rolls without slipping along the x-axis. Which of the following statement(s) is(are) correct?
Choices: (A) For μ ≠ 0 and a → 0, hcrit = b/2, (B) For μ ≠ 0 and a → b, hcrit = b, (C) For h = hcrit, the initial angular velocity does not depend on the inner radius a, (D) For μ = 0 and h = 0, the wheel always slides without rolling.
Answer: D
Subject: physics
Question: The electric field associated with an electromagnetic wave propagating in a dielectric medium is given by E = 30(2x + y)sin [2π (5 × 10^14 t - 10^7/3)] V m^-1. Which of the following option(s) is(are) correct?
Choices: (A) Bx = -2 × 10^-7 sin [2π (5 × 10^14 t - 10^7/3)] Wb m^-2, (B) By = 2 × 10^-7 sin [2π (5 × 10^14 t - 10^7/3)] Wb m^-2, (C) The wave is polarized in the xy-plane with polarization angle 30° with respect to the x-axis, (D) The refractive index of the medium is 2.
Subject: physics
Question: A thin circular coin of mass 5 gm and radius 4/3 cm is initially in a horizontal xy-plane. The coin is tossed vertically up (+z direction) by applying an impulse of √6 x 10^-2 N-s at a distance 2/3 cm from its center. The coin spins about its diameter and moves along the +z direction. By the time the coin reaches back to its initial position, it completes n rotations. The value of n is _____.
Choices: n/a (Non-Multiple Choice)
Answer: 4
Subject: physics
Question: A rectangular conducting loop of length 4 cm and width 2 cm is in the xy-plane, as shown in the figure. It is being moved away from a thin and long conducting wire along the direction 2/3 x + y with a constant speed v. The wire is carrying a steady current I = 10 A in the positive x-direction. A current of 10 μA flows through the loop when it is at a distance d = 4 cm from the wire. If the resistance of the loop is 0.1 Ω, then the value of v is _____ m s^-1.
Choices: n/a (Non-Multiple Choice)
Answer: -10^-6
Subject: physics
Question: An incompressible liquid is kept in a container having a weightless piston with a hole. A capillary tube of inner radius 0.1 mm is dipped vertically into the liquid through the airtight piston hole, as shown in the figure. The air in the container is isothermally compressed from its original volume V0 to 100/101 V0 with the movable piston. Considering air as an ideal gas, the height (h) of the liquid column in the capillary above the liquid level in cm is_______.
Subject: physics
Question: In a radioactive decay process, the activity is defined as A = - dN/dt, where N(t) is the number of radioactive nuclei at time t. Two radioactive sources, S1 and S2 have same activity at time t = 0. At a later time, the activities of S1 and S2 are A1 and A2, respectively. When S1 and S2 have just completed their 3rd and 7th half-lives, respectively, the ratio A1/A2 is __________.
Subject: physics
Question: One mole of an ideal gas undergoes two different cyclic processes I and II, as shown in the P-V diagrams below. In cycle I, processes a, b, c and d are isobaric, isothermal, isobaric and isochoric, respectively. In cycle II, processes a', b', c' and d' are isothermal, isochoric, isobaric and isochoric, respectively. The total work done during cycle I is WI and that during cycle II is WII. The ratio WI/WII is _______.
Choices: No multiple choice options provided
Answer: No answer provided
Subject: chemistry
Question: Consider the following statements related to colloids. 

(I) Lyophobic colloids are not formed by simple mixing of dispersed phase and dispersion medium. 
(II) For emulsions, both the dispersed phase and the dispersion medium are liquid. 
(III) Micelles are produced by dissolving a surfactant in any solvent at any temperature. 
(IV) Tyndall effect can be observed from a colloidal solution with dispersed phase having the same refractive index as that of the dispersion medium.
Choices: (A) (I) and (II)

(B) (II) and (III)

(C) (III) and (IV)

(D) (II) and (IV)
Answer: (B) (II) and (III)
Subject: chemistry
Question: In the following reactions, P, Q, R, and S are the major products.

(i) Mg, dry ether
(ii) H2O                     P

(i) Mg, dry ether
(ii) CO2, dry ether
(iii) H3O+                   Q
(iv) NaOH

(i) Mg, dry ether
(ii) CH3CHO, then H2O
(iii) CrO3                   R

(i) ethanolic NaCN
(ii) H2/ Ni
(iii) CHCI3/KOH, Δ           S
(iv) LiAIH4, then H2O
Choices: (A) P is a primary alcohol with four carbons.
(B) Q undergoes Kolbe's electrolysis to give an eight-carbon product.
(C) R has six carbons and it undergoes Cannizzaro reaction.
(D) S is a primary amine with six carbons.
Answer: (B) Q undergoes Kolbe's electrolysis to give an eight-carbon product.
Subject: chemistry
Question: The complex(es), which can exhibit the type of isomerism shown by [Pt(NH3)3Br3], is(are) [en = H2NCH2CH2NH2]
Choices: (A) [Pt(en)(SCN)2] (B) [Zn(NH3)4Cl2] (C) [Pt(NH3)2Cl4] (D) [Cr(en)3(H2O)(SO4)]
Answer: C
Subject: physics
Question: Atoms of metals x, y, and z form face-centred cubic (fcc) unit cell of edge length Lx, body-centred cubic (bcc) unit cell of edge length Ly, and simple cubic unit cell of edge length Lz, respectively. If rx = (√3)/2 ry : ry = 8/(√3) rx : Mz = 3/2 My and Mz = 3Mx, then the correct statement(s) is(are)
Choices: (A) Packing efficiency of unit cell of x > Packing efficiency of unit cell of y > Packing efficiency of unit cell of z (B) Lx > Lz (C) Lx > Ly (D) Density of x > Density of y
Subject: chemistry
Question: In the following reactions, P, Q, R, and S are the major products.
Choices: (A) P and Q are monomers of polymers dacron and glyptal, respectively. (B) P, Q, and R are dicarboxylic acids. (C) Compounds Q and R are the same. (D) R does not undergo aldol condensation and S does not undergo Cannizzaro reaction.
Subject: chemistry
Question: 50 mL of 0.2 molal urea solution (density = 1.012 g mL^-1 at 300 K) is mixed with 250 mL of a solution containing 0.06 g of urea. Both the solutions were prepared in the same solvent. The osmotic pressure (in Torr) of the resulting solution at 300 K is ___.
Choices: No response choices provided
Answer: No answer provided
Subject: chemistry
Question: The reaction of 4-methyloct-1-ene (P, 2.52 g) with HBr in the presence of (C2H5CO)2O2 gives two isomeric bromides in a 9 : 1 ratio, with a combined yield of 50%. Of these, the entire amount of the primary alkyl bromide was reacted with an appropriate amount of diethylamine followed by treatment with aq. K2CO3 to give a non-ionic product S in 100% yield. The mass (in mg) of S obtained is ___.
Choices: No response choices provided
Answer: No answer provided