cmeraki/elemento huggingface dataset:
Subject: physics
Question: A large spherical mass M is fixed at one position and two identical point masses m are kept on a line passing through the centre of M (see figure). The point masses are connected by a rigid massless rod of length l and this assembly is free to move along the line connecting them. All three masses interact only through their mutual gravitational interaction. When the point mass nearer to M is at a distance r = 3l from M, the tension in the rod is zero for m = k (M/238). The value of k is
Choices: Single digit integer ranging from 0 to 9
Answer: 7
Subject: physics
Question: The energy of a system as a function of time t is given as E(t) = A^2 exp(-αt), where α = 0.2 s^-1. The measurement of A has an error of 1.25%. If the error in the measurement of time is 1.50%, the percentage error in the value of E(t) at t = 5 s is
Choices: Single digit integer ranging from 0 to 9
Answer: 4
Subject: physics
Question: The densities of two solid spheres A and B of the same radii R vary with radial distance r as ρA(r) = k (R/r) and ρB(r) = k (R/r)^5, respectively, where k is a constant. The moments of inertia of the individual spheres about axes passing through their centres are IA and IB, respectively. If IB = n/10 IA, the value of n is
Choices: Single digit integer ranging from 0 to 9
Answer: 6
Subject: physics
Question: For a radioactive material, its activity A and rate of change of its activity R are defined as A = dN/dt and R = dA/dt, where N(t) is the number of nuclei at time t. Two radioactive sources P (mean life τ) and Q (mean life 2τ) have the same activity at t = 0. Their rates of change of activities at t = 2τ are Rp and Rq, respectively. If Rp = n, then the value of n is Rq
Answer: 2
Subject: physics
Question: Two spheres P and Q of equal radii have densities ρ₁ and ρ₂, respectively. The spheres are connected by a massless string and placed in liquids L₁ and L₂ of densities σ₁ and σ₂ and viscosities η₁ and η₂, respectively. They float in equilibrium with the sphere P in L₁ and sphere Q in L₂ and the string being taut (see figure). If sphere P alone in L₂ has terminal velocity V̅p and Q alone in L₁ has terminal velocity V̅q, then
Choices: (A) |V̅p| = η₁ / |V̅q| η₂, (B) |V̅p| = η₂ / |V̅q| η₁, (C) V̅p⋅V̅q > 0, (D) V̅p⋅V̅q < 0
Answer: (A) and (D)
Subject: physics
Question: In terms of potential difference V, electric current I, permittivity ε₀, permeability μ₀ and speed of light c, the dimensionally correct equation(s) is(are)
Choices: (A) μ₀I² = ε₀V⁴, (B) ε₀I = μ₀V, (C) I = ε₀cV, (D) μ₀cI = ε₀V
Answer: (A) and (C)
Subject: physics
Question: Consider a uniform spherical charge distribution of radius R2 centred at the origin O. In this distribution, a spherical cavity of radius R1, centred at P with distance OP=a=R2-R1 (see figure) is made. If the electric field inside the cavity at position r̄ is Ē(r̄), then the correct statement(s) is(are)
Choices: (A) Ē is uniform, its magnitude is independent of R2 but its direction depends on r̄
(B) Ē is uniform, its magnitude depends on R2 and its direction depends on r̄
(C) Ē is uniform, its magnitude is independent of a but its direction depends on ā
(D) Ē is uniform and both its magnitude and direction depend on ā
Answer: (D)
Subject: physics
Question: In plotting stress versus strain curves for two materials P and Q, a student by mistake puts strain on the y-axis and stress on the x-axis as shown in the figure. Then the correct statement(s) is(are)
Choices: (A) P has more tensile strength than Q
(B) P is more ductile than Q
(C) P is more brittle than Q
(D) The Young's modulus of P is more than that of Q
Answer: (A) and (B)
Subject: physics
Question: A spherical body of radius R consists of a fluid of constant density and is in equilibrium under its own gravity. If P(r) is the pressure at r(r < R), then the correct option(s) is(are)
Choices: (A) P(r = 0) = 0, (B) P(r = 3R/4) / P(r = 2R/3) = 63 / 80, (C) P(r = 3R/5) / P(r = 2R/5) = 16 / 21, (D) P(r = R/2) / P(r = R/3) = 20 / 27
Answer: (B) and (C)
Subject: physics
Question: A parallel plate capacitor having plates of area S and plate separation d, has capacitance C1 in air. When two dielectrics of different relative permittivities (ε1 = 2 and ε2 = 4) are introduced between the two plates as shown in the figure, the capacitance becomes C2. The ratio C2/C1 is
Choices: (A) 6/5, (B) 5/3, (C) 7/5, (D) 7/3
Answer: (D)
Subject: physics
Question: An ideal monoatomic gas is confined in a horizontal cylinder by a spring loaded piston (as shown in the figure). Initially the gas is at temperature T1, pressure P1 and volume V1 and the spring is in its relaxed state. The gas is then heated very slowly to temperature T2, pressure P2 and volume V2. During this process the piston moves out by a distance x. Ignoring the friction between the piston and the cylinder, the correct statement(s) is(are)
Choices: (A) If V2 = 2V1 and T2 = 3T1, then the energy stored in the spring is 1/4 P1V1
(B) If V2 = 2V1 and T2 = 3T1, then the change in internal energy is 3P1V1
(C) If V2 = 3V1 and T2 = 4T1, then the work done by the gas is 7/3 P1V1
(D) If V2 = 3V1 and T2 = 4T1, then the heat supplied to the gas is 17/6 P1V1
Answer: (A), (B) and (C)
Subject: physics
Question: A fission reaction is given by 236U → 140Xe + 94Sr + x + y, where x and y are two particles. Considering 236U to be at rest, the kinetic energies of the products are denoted by Kxe, Ksr, Kx (2 MeV) and Ky (2 MeV), respectively. Let the binding energies per nucleon of 236U, 140Xe and 94Sr be 7.5 MeV, 8.5 MeV and 8.5 MeV, respectively. Considering different conservation laws, the correct option(s) is(are)
Choices: (A) x = n, y = n, Ksr = 129 MeV, Kxe = 86 MeV
(B) x = p, y = e-, Ksr = 129 MeV, Kxe = 86 MeV
(C) x = p, y = n, Ksr = 129 MeV, Kxe = 86 MeV
(D) x = n, y = n, Ksr = 86 MeV, Kxe = 129 MeV
Answer: A
Subject: physics
Question: Consider two different metallic strips (1 and 2) of the same material. Their lengths are the same, widths are w1 and w2 and thicknesses are d1 and d2, respectively. Two points K and M are symmetrically located on the opposite faces parallel to the x-y plane (see figure). V1 and V2 are the potential differences between K and M in strips 1 and 2, respectively. Then, for a given current I flowing through them in a given magnetic field strength B, the correct statement(s) is(are)
Choices: (A) If w1 = w2 and d1 = 2d2, then V2 = 2V1
(B) If w1 = w2 and d1 = 2d2, then V2 = V1
(C) If w1 = 2w2 and d1 = d2, then V2 = 2V1
(D) If w1 = 2w2 and d1 = d2, then V2 = V1
Answer: (A) and (D)
Subject: physics
Question: Consider two different metallic strips (1 and 2) of same dimensions (length l, width w and thickness d) with carrier densities n1 and n2, respectively. Strip 1 is placed in magnetic field B1 and strip 2 is placed in magnetic field B2, both along positive y-directions. Then V1 and V2 are the potential differences developed between K and M in strips 1 and 2, respectively. Assuming that the current I is the same for both the strips, the correct option(s) is(are)
Choices: (A) If B1 = B2 and n1 = 2n2, then V2 = 2V1
(B) If B1 = B2 and n1 = 2n2, then V2 = V1
(C) If B1 = 2B2 and n1 = n2, then V2 = 0.5V1
(D) If B1 = 2B2 and n1 = n2, then V2 = V1
Answer: (A) and (C)
Subject: physics
Question: For two structures namely S1 with n1 = √45/4 and n2 = 3/2, and S2 with n1 = 8/5 and n2 = 7/5 and taking the refractive index of water to be 4/3 and that of air to be 1, the correct option(s) is(are)
Choices: (A) NA of S1 immersed in water is the same as that of S2 immersed in a liquid of refractive index 16/3√15, (B) NA of S1 immersed in liquid of refractive index 6/√15 is the same as that of S2 immersed in water, (C) NA of S1 placed in air is the same as that of S2 immersed in liquid of refractive index 4/√15, (D) NA of S1 placed in air is the same as that of S2 placed in water
Answer: (A) and (C)
Subject: physics
Question: If two structures of same cross-sectional area, but different numerical apertures NA1 and NA2 (NA2 < NA1) are joined longitudinally, the numerical aperture of the combined structure is
Choices: (A) NA1 NA2 / (NA1 + NA2), (B) NA1 + NA2, (C) NA1, (D) NA2
Answer: (D)
Subject: chemistry
Question: Among the complex ions, [Co(NH₃)₂CH₃-CH₂-NH₂)₂Cl₂]+, [CrCl₄(C₂O₄)₂]³⁻, [Fe(H₂O)₄(OH)₂]+, [Fe(NH₃)₆(CN)₆]³+ and [Co(NH₃)₄(H₂O)Cl]²⁺, the number of complex ion(s) that show(s) cis-trans isomerism is
Answer: 6
Subject: chemistry
Question: The molar conductivity of a solution of a weak acid HX (0.01 M) is 10 times smaller than the molar conductivity of a solution of a weak acid HY (0.10 M). If λ°m ≈ λ°c , the difference in their pK​a values, pK​a(HX) - pK​a(HY), is (consider degree of ionization of both acids to be ≪ 1)
Answer: 3
Subject: chemistry
Question: The correct statement(s) regarding, (i) HClO, (ii) HClO2, (iii) HClO3 and (iv) HClO4, is(are)
Choices: (A) The number of Cl=O bonds in (ii) and (iii) together is two (B) The number of lone pairs of electrons on Cl in (ii) and (iii) together is three (C) The hybridization of Cl in (iv) is sp3 (D) Amongst (i) to (iv), the strongest acid is (i)
Answer: (B) and (C)
Subject: chemistry
Question: The pair(s) of ions where BOTH the ions are precipitated upon passing H2S gas in presence of dilute HCl, is(are)
Choices: (A) Ba2+, Zn2+ (B) Bi3+, Fe3+ (C) Cu2+, Pb2+ (D) Hg2+, Bi3+
Answer: (C) and (D)
Subject: chemistry
Question: Under hydrolytic conditions, the compounds used for preparation of linear polymer and for chain termination, respectively, are
Choices: (A) CH3SiCl3 and Si(CH3)4 (B) (CH3)2SiCl2 and (CH3)3SiCl (C) (CH3)2SiCl2 and CH3SiCl3 (D) SiCl4 and (CH3)3SiCl
Answer: (B)
Subject: chemistry
Question: When O2 is adsorbed on a metallic surface, electron transfer occurs from the metal to O2. The TRUE statement(s) regarding this adsorption is(are)
Choices: (A) O2 is physi­sorbed (B) heat is released (C) occupancy of π2p of O2 is increased (D) bond length of O2 is increased
Answer: (B), (C) and (D)
Subject: mathematics
Question: Suppose that the foci of the ellipse (x^2/9) + (y^2/5) = 1 are (f1, 0) and (f2, 0) where f1 > 0 and f2 < 0. Let P1 and P2 be two parabolas with a common vertex at (0, 0) and with foci at (f1, 0) and (2f2, 0), respectively. Let T1 be a tangent to P2 which passes through (2f2, 0) and T2 be a tangent to P2 which passes through (f1, 0). If m1 is the slope of T1 and m2 is the slope of T2, then the value of (1/m1^2 + m2^2) is
Choices: No multiple choice options given
Answer: 4
Subject: mathematics
Question: Let f : R → R be a continuous odd function, which vanishes exactly at one point and f(t) = 1/2. Suppose that F(x)= ∫f(t) dt for all x ∈ [-1, 2] and G(x)= ∫|f(f(t))| dt for all x ∈ [-1, 2]. If lim F(x)/G(x) = 1/14, then the value of f(1/2) is
Choices: No multiple choice options provided
Answer: 7
Subject: mathematics
Question: Suppose that p̂, q̂ and r̂ are three non-coplanar vectors in R³. Let the components of a vector ŝ along p̂, q̂ and r̂ be 4, 3 and 5, respectively. If the components of this vector ŝ along (-p̂+q̂+r̂), (p̂-q̂+r̂) and (-p̂-q̂+r̂) are x, y and z, respectively, then the value of 2x+y+z is
Choices: No multiple choice options provided
Answer: 9
Subject: mathematics
Question: Let S be the set of all non-zero real numbers α such that the quadratic equation αx^2 - x + α = 0 has two distinct real roots x1 and x2 satisfying the inequality |x1 - x2| < 1. Which of the following intervals is(are) a subset(s) of S?
Choices: (A) (-1/2, -1/√5), (B) (-1/√5, 0), (C) (0, 1/√5), (D) (1/√5, 1/2)
Answer: (A) and (D)
Subject: mathematics
Question: If α = 3sin^-1(6/11) and β = 3cos^-1(4/9), where the inverse trigonometric functions take only the principal values, then the correct option(s) is(are)
Choices: (A) cos β > 0, (B) sin β < 0, (C) cos(α + β) > 0, (D) cos α < 0
Answer: (B), (C) and (D)
Subject: mathematics
Question: Let E₁ and E₂ be two ellipses whose centers are at the origin. The major axes of E₁ and E₂ lie along the x-axis and the y-axis, respectively. Let S be the circle x² + (y - 1)² = 2. The straight line x + y = 3 touches the curves S, E₁ and E₂ at P, Q and R, respectively. Suppose that PQ = PR = 2√2/3 . If e₁ and e₂ are the eccentricities of E₁ and E₂, respectively, then the correct expression(s) is(are)
Choices: (A) e₁² + e₂² = 43/40 (B) e₁e₂ = √7/2√10 (C) |e₁² - e₂²| = 5/8 (D) e₁e₂ = √3/4
Answer: (A) and (B)
Subject: mathematics
Question: Consider the hyperbola H : x² - y² = 1 and a circle S with center N(x₂, 0). Suppose that H and S touch each other at a point P(x₁, y₁) with x₁ > 1 and y₁ > 0. The common tangent to H and S at P intersects the x-axis at point M. If (l, m) is the centroid of the triangle ΔPMN, then the correct expression(s) is(are)
Choices: (A) dl/dx₁ = 1 - 1/3x₁ for x₁ > 1 (B) dm/dx₁ = x₁/√(x₁² - 1) for x₁ > 1 (C) dl/dx₁ = 1 + 1/3x₁ for x₁ > 1 (D) dm/dy₁ = 1/3 for y₁ > 0
Answer: (A), (B) and (D)
Subject: mathematics
Question: The option(s) with the values of a and L that satisfy the following equation is(are) [equation 1] = L? [equation 2]
Choices: (A) a = 2, L = e^(4π - 1)/(e^π - 1) (B) a = 2, L = e^(4π + 1)/(e^π + 1) (C) a = 4, L = e^(4π - 1)/(e^π - 1) (D) a = 4, L = e^(4π + 1)/(e^π + 1)
Answer: (A) and (C)
Subject: mathematics
Question: Let f,g:[-1,2] → R be continuous functions which are twice differentiable on the interval (-1,2). Let the values of f and g at the points -1, 0 and 2 be as given in the following table: [table with values]. In each of the intervals (-1,0) and (0,2) the function (f-3g)^' never vanishes. Then the correct statement(s) is(are)
Choices: (A) f'(x)-3g'(x)=0 has exactly three solutions in (-1,0) U (0,2) (B) f'(x)-3g'(x)=0 has exactly one solution in (-1,0) (C) f'(x)-3g'(x)=0 has exactly one solution in (0,2) (D) f'(x)-3g'(x)=0 has exactly two solutions in (-1,0) and exactly two solutions in (0,2)
Answer: (B) and (C)
Subject: mathematics
Question: Let f(x) = 7 tan^8 x + 7 tan^6 x - 3 tan^4 x - 3 tan^2 x for all x ∈ (- π/2, π/2). Then the correct expression(s) is(are)
Choices: (A) ∫^(π/4)_0 x f(x) dx = 1/12, (B) ∫^(π/4)_0 f(x) dx = 0, (C) ∫^(π/4)_0 x f(x) dx = 1/6, (D) ∫^(π/4)_0 f(x) dx = 1
Answer: (A) and (B)
Subject: mathematics
Question: Let f'(x) = 192 x^3 / (2 + sin^4 x) for all x ∈ ℝ with f(1/2) = 0. If m ≤ ∫^1/2_0 f(x) dx ≤ M , then the possible values of m and M are
Choices: (A) m = 13 , M = 24, (B) m = 1/4, M = 1/2, (C) m = -11, M = 0, (D) m = 1, M = 12
Answer: (D)
Subject: mathematics
Question: One of the two boxes, box I and box II, was selected at random and a ball was drawn randomly out of this box. The ball was found to be red. If the probability that this red ball was drawn from box II is 1/3, then the correct option(s) with the possible values of n1, n2, n3 and n4 is(are)
Choices: (A) n1 = 3 , n2 = 3 , n3 = 5 , n4 = 15, (B) n1 = 3 , n2 = 6 , n3 = 10 , n4 = 50, (C) n1 = 8 , n2 = 6 , n3 = 5 , n4 = 20, (D) n1 = 6 , n2 = 12 , n3 = 5 , n4 = 20
Answer: (A) and (B)
Subject: mathematics
Question: A ball is drawn at random from box I and transferred to box II. If the probability of drawing a red ball from box I, after this transfer, is 1/3, then the correct option(s) with the possible values of n1 and n2 is(are)
Choices: (A) n1 = 4 and n2 = 6, (B) n1 = 2 and n2 = 3, (C) n1 = 10 and n2 = 20, (D) n1 = 3 and n2 = 6
Answer: (C) and (D)
Subject: mathematics
Question: The correct statement(s) is(are)
Choices: (A) f'(1) < 0, (B) f(2) < 0, (C) f'(x) ≠ 0 for any x ∈ (1, 3), (D) f'(x) = 0 for some x ∈ (1, 3)
Answer: (A), (B) and (C)
Subject: mathematics
Question: If ∫₁³ xf''(x) dx = -12 and ∫₁³ x³ f'''(x) dx = 40, then the correct expression(s) is(are)
Choices: (A) 9f'(3) + f'(1) - 32 = 0, (B) ∫₁³ f(x) dx = 12, (C) 9f'(3) - f'(1) + 32 = 0, (D) ∫₁³ f(x) dx = -12
Answer: (C) and (D)