cmeraki/elemento huggingface dataset:
Subject: physics
Question: A Young's double slit interference arrangement with slits S1 and S2 is immersed in water (refractive index = 4/3) as shown in the figure. The positions of maxima on the surface of water are given by x^2 = n^2m^2λ^2 - d^2, where λ is the wavelength of light in air (refractive index = 1), 2d is the separation between the slits and m is an integer. The value of n is
Answer: 3
Subject: physics
Question: Consider a concave mirror and a convex lens (refractive index = 1.5) of focal length 10 cm each, separated by a distance of 50 cm in air (refractive index = 1) as shown in the figure. An object is placed at a distance of 15 cm from the mirror. Its erect image formed by this combination has magnification M1. When the set-up is kept in a medium of refractive index 7/6, the magnification becomes M2. The magnitude |M2/M1| is
Answer: 7
Subject: physics
Question: Consider a Vernier callipers in which each 1 cm on the main scale is divided into 8 equal divisions and a screw gauge with 100 divisions on its circular scale. In the Vernier callipers, 5 divisions of the Vernier scale coincide with 4 divisions on the main scale and in the screw gauge, one complete rotation of the circular scale moves it by two divisions on the linear scale. Then:
Choices: (A) If the pitch of the screw gauge is twice the least count of the Vernier callipers, the least count of the screw gauge is 0.01 mm. (B) If the pitch of the screw gauge is twice the least count of the Vernier callipers, the least count of the screw gauge is 0.005 mm. (C) If the least count of the linear scale of the screw gauge is twice the least count of the Vernier callipers, the least count of the screw gauge is 0.01 mm. (D) If the least count of the linear scale of the screw gauge is twice the least count of the Vernier callipers, the least count of the screw gauge is 0.005 mm.
Answer: (B) and (C)
Subject: physics
Question: Planck's constant h, speed of light c and gravitational constant G are used to form a unit of length L and a unit of mass M. Then the correct option(s) is(are)
Choices: (A) M ∝ √c (B) M ∝ √G (C) L ∝ √h (D) L ∝ √G
Answer: (A), (C) and (D)
Subject: physics
Question: Two independent harmonic oscillators of equal mass are oscillating about the origin with angular frequencies ω₁ and ω₂ and have total energies E₁ and E₂, respectively. The variations of their momenta p with positions x are shown in the figures. If a/b = n^2 and a/R = n, then the correct equation(s) is(are)
Choices: (A) E₁ω₁ = E₂ω₂, (B) ω₂ = n^2 * ω₁, (C) ω₁ω₂ = n^2, (D) E₁/ω₁ = E₂/ω₂
Answer: (B) and (D)
Subject: physics
Question: A ring of mass M and radius R is rotating with angular speed ω about a fixed vertical axis passing through its centre O with two point masses each of mass M/5 at rest at O. These masses can move radially outwards along two massless rods fixed on the ring as shown in the figure. At some instant the angular speed of the system is 8/9 ω and one of the masses is at a distance of 3/5 R from O. At this instant the distance of the other mass from O is
Choices: A) 2/3 R, B) 1/3 R, C) 3/5 R, D) 4/5 R
Answer: (C) OR (D) OR (C) and (D)
Subject: physics
Question: The figures below depict two situations in which two infinitely long static line charges of constant positive line charge density λ are kept parallel to each other. In their resulting electric field, point charges q and -q are kept in equilibrium between them. The point charges are confined to move in the x direction only. If they are given a small displacement about their equilibrium positions, then the correct statement(s) is(are)
Choices: A. Both charges execute simple harmonic motion. B. Both charges will continue moving in the direction of their displacement. C. Charge +q executes simple harmonic motion while charge -q continues moving in the direction of its displacement. D. Charge -q executes simple harmonic motion while charge +q continues moving in the direction of its displacement.
Answer: C
Subject: physics
Question: Two identical glass rods S1 and S2 (refractive index = 1.5) have one convex end of radius of curvature 10 cm. They are placed with the curved surfaces at a distance d as shown in the figure, with their axes (shown by the dashed line) aligned. When a point source of light P is placed inside rod S1 on its axis at a distance of 50 cm from the curved face, the light rays emanating from it are found to be parallel to the axis inside S2. The distance d is
Choices: A. 60 cm B. 70 cm C. 80 cm D. 90 cm
Answer: B
Subject: physics
Question: A conductor (shown in the figure) carrying constant current I is kept in the x-y plane in a uniform magnetic field B. If F is the magnitude of the total magnetic force acting on the conductor, then the correct statement(s) is(are)
Choices: (A) If B is along z, F ∝ (L+R), (B) If B is along x, F=0, (C) If B is along y, F ∝ (L+R), (D) If B is along z, F=0
Answer: (A), (B) and (C)
Subject: physics
Question: A container of fixed volume has a mixture of one mole of hydrogen and one mole of helium in equilibrium at temperature T: Assuming the gases are ideal, the correct statement(s) is(are)
Choices: (A) The average energy per mole of the gas mixture is 2RT, (B) The ratio of speed of sound in the gas mixture to that in helium gas is √(5/5), (C) The ratio of the rms speed of helium atoms to that of hydrogen molecules is 1/2, (D) The ratio of the rms speed of helium atoms to that of hydrogen molecules is 1/√2
Answer: (A), (B) and (D)
Subject: physics
Question: In an aluminum (Al) bar of square cross section, a square hole is drilled and is filled with iron (Fe) as shown in the figure. The electrical resistivities of Al and Fe are 2.7×10^-8 Ω m and 1.0×10^-7 Ω m, respectively. The electrical resistance between the two faces P and Q of the composite bar is
Choices: (A) 2475 μΩ/64	(B) 1875 μΩ/64	(C) 1875 μΩ/49	(D) 2475 μΩ/132
Answer: B
Subject: physics
Question: Match the nuclear processes given in column I with the appropriate option(s) in column II.
Choices: Column I: (A) Nuclear fusion, (B) Fission in a nuclear reactor, (C) β-decay, (D) γ-ray emission
Column II: (P) Absorption of thermal neutrons by 235U92, (Q) 27Co nucleus, (R) Energy production in stars via hydrogen conversion to helium, (S) Heavy water, (T) Neutrino emission
Answer: (A) Matches to (R) OR (R) and (T), (B) Matches to (P) and (S), (C) Matches to (Q) and (T), (D) Matches to (R)
Subject: physics
Question: A particle of unit mass is moving along the x-axis under the influence of a force and its total energy is conserved. Four possible forms of the potential energy of the particle are given in column I (a and U0 are constants). Match the potential energies in column I to the corresponding statement(s) in column II.
Choices: Column I: (A) U1(x) = U0 [1 - (x/a)^2]^2/2, (B) U2(x) = U0 (x/a)^2/2, (C) U3(x) = U0 (x/a)^2 exp[-(x/a)^2]/2, (D) U4(x) = U0 [x - 1/(x/a)^3]/2; Column II: (P) The force acting on the particle is zero at x = a, (Q) The force acting on the particle is zero at x = 0, (R) The force acting on the particle is zero at x = -a, (S) The particle experiences an attractive force towards x = 0 in the region |x| < a, (T) The particle with total energy U0/4 can oscillate about the point x = -a
Answer: (A) Matches to (P), (Q), (R) and (T), (B) Matches to (Q) and (S), (C) Matches to (P), (Q), (R) and (S), (D) Matches to (P), (R) and (T)
Subject: chemistry
Question: Among the triatomic molecules/ions, BeCl₂, N₂, N₂O, NO₂⁺, O₃, SCl₂,ICl₂⁻, I₃⁻ and XeF₂, the total number of linear molecule(s)/ion(s) where the hybridization of the central atom does not have contribution from the d-orbital(s) is
Choices: No response choices provided
Answer: 4
Subject: chemistry
Question: All the energy released from the reaction X → Y Λ G⁰ = -193 kJ mol⁻¹ is used for oxidizing M⁺ as M⁺ → M³⁺+ 2e⁻, E⁰ = - 0.25 V. Under standard conditions, the number of moles of M⁺ oxidized when one mole of X is converted to Y is [F = 96500 C mol⁻¹]
Choices: No response choices provided
Answer: 4
Subject: chemistry
Question: If the freezing point of a 0.01 molal aqueous solution of a cobalt(III) chloride-ammonia complex (which behaves as a strong electrolyte) is -0.0588°C, the number of chloride(s) in the coordination sphere of the complex is [Kᵨ of water = 1.86 K kg mol⁻¹]
Choices: No response choices provided
Answer: 1
Subject: chemistry
Question: The structure of D-(+)-glucose is

CHO
H OH
HO H
H OH
H OH
CH2OH

The structure of L-(-)-glucose is
Choices: (A) CHO HO H H OH HO H H CH2OH, (B) CHO H OH HO H H OH HO H CH2OH, (C) CHO HO H HO H H OH HO H CH2OH, (D) CHO HO H HO H H OH CH2OH
Answer: A
Subject: chemistry
Question: The major product of the reaction is H3C CH2 CO2H + CH3 NH2 + CO2H + NaNO2, aqueous HCl at 0°C
Choices: (A) H3C CH2 NH2 CH3 OH, (B) H3C CO2H CH3 OH, (C) H3C CO2H CH3 OH, (D) H3C NH2 CH3 OH
Answer: C
Subject: chemistry
Question: The correct statement(s) about Cr2+ and Mn3+ is(are) [Atomic numbers of Cr = 24 and Mn = 25] (A) Cr2+ is a reducing agent, (B) Mn3+ is an oxidizing agent, (C) Both Cr2+ and Mn3+ exhibit d4 electronic configuration, (D) When Cr2+ is used as a reducing agent, the chromium ion attains d5 electronic configuration
Choices: (A) Cr2+ is a reducing agent, (B) Mn3+ is an oxidizing agent, (C) Both Cr2+ and Mn3+ exhibit d4 electronic configuration, (D) When Cr2+ is used as a reducing agent, the chromium ion attains d5 electronic configuration
Answer: A, B and C
Subject: chemistry
Question: Copper is purified by electrolytic refining of blister copper. The correct statement(s) about this process is (are) (A) Impure Cu strip is used as cathode, (B) Acidified aqueous CuSO4 is used as electrolyte, (C) Pure Cu deposits at cathode, (D) Impurities settle as anode-mud
Choices: (A) Impure Cu strip is used as cathode, (B) Acidified aqueous CuSO4 is used as electrolyte, (C) Pure Cu deposits at cathode, (D) Impurities settle as anode-mud
Answer: B, C and D
Subject: chemistry
Question: Fe3+ is reduced to Fe2+ by using (A) H2O2 in presence of NaOH, (B) Na2O2 in water, (C) H2O2 in presence of H2SO4, (D) Na2O2 in presence of H2SO4
Choices: (A) H2O2 in presence of NaOH, (B) Na2O2 in water, (C) H2O2 in presence of H2SO4, (D) Na2O2 in presence of H2SO4
Answer: A and B
Subject: chemistry
Question: The %yield of ammonia as a function of time in the reaction N2(g) + 3H2(g) ⇌ 2NH3(g), ΔH < 0 at (P, T1) is given below. If this reaction is conducted at (P, T2), with T2 > T1, the %yield of ammonia as a function of time is represented by
Choices: (A) (B) (C) (D)
Answer: B
Subject: chemistry
Question: If the unit cell of a mineral has cubic close packed (ccp) array of oxygen atoms with m fraction of octahedral holes occupied by aluminium ions and n fraction of tetrahedral holes occupied by magnesium ions, m and n, respectively, are
Choices: (A) 1/2, 1/8 (B) 1, 1/4 (C) 1/2, 1/2 (D) 1, 1/8
Answer: A
Subject: chemistry
Question: Match the anionic species given in Column I that are present in the ore(s) given in Column II.
Choices: (A) Carbonate, (B) Sulphide, (C) Hydroxide, (D) Oxide
Answer: (A) Matches to (P), (Q) and (S); (B) Matches to (T); (C) Matches to (Q) and (R); (D) Matches to (R)
Subject: chemistry
Question: Match the thermodynamic processes given under Column I with the expressions given under Column II.
Choices: Column I: (A) Freezing of water at 273 K and 1 atm, (B) Expansion of 1 mol of an ideal gas into a vacuum under isolated conditions, (C) Mixing of equal volumes of two ideal gases at constant temperature and pressure in an isolated container, (D) Reversible heating of H2(g) at 1 atm from 300 K to 600 K, followed by reversible cooling to 300 K at 1 atm 

Column II: (P) q = 0, (Q) w = 0, (R) ΔSsys < 0, (S) ΔU = 0, (T) ΔG = 0
Answer: (A) Matches to (R) and (T), (B) Matches to (P), (Q) and (S), (C) Matches to (P), (Q) and (S), (D) Matches to (P), (Q), (S) and (T)
Subject: mathematics
Question: Let n be the number of ways in which 5 boys and 5 girls can stand in a queue in such a way that all the girls stand consecutively in the queue. Let m be the number of ways in which 5 boys and 5 girls can stand in a queue in such a way that exactly four girls stand consecutively in the queue. Then the value of m/n is
Choices: Single digit integer from 0 to 9
Answer: 5
Subject: mathematics
Question: Let f : R -> R be a function defined by f(x) = {[x], x ≤ 2, 0, x > 2} where [x] is the greatest integer less than or equal to x. If I = ∫(2/1) y(x^3) / (2 + f(x + 1)) dx, then the value of (4I - 1) is
Answer: 0
Subject: mathematics
Question: A cylindrical container is to be made from certain solid material with the following constraints: It has a fixed inner volume of V mm^3, has a 2 mm thick solid wall and is open at the top. The bottom of the container is a solid circular disc of thickness 2 mm and is of radius equal to the outer radius of the container. If the volume of the material used to make the container is minimum when the inner radius of the container is 10 mm, then the value of V/250π is
Answer: 4
Subject: mathematics
Question: Let F(x) = ∫(x/π)^2 2cos^2 t dt for all x ∈ R and f : [0, 1/2] → [0, ∞) be a continuous function. For α ∈ [0, 1/2], if F'(α) + 2 is the area of the region bounded by x = 0, y = 0, y = f(x) and x = α, then f(0) is
Answer: 3
Subject: mathematics
Question: Let X and Y be two arbitrary, 3×3, non-zero, skew-symmetric matrices and Z be an arbitrary 3×3, non-zero, symmetric matrix. Then which of the following matrices is (are) skew symmetric?
Choices: (A) Y^TZ^t - Z^tY^3, (B) X^4 + Y^4, (C) X^tZ^3 - Z^3X^t, (D) X^23 + Y^23
Answer: (C) and (D)
Subject: mathematics
Question: Which of the following values of α satisfy the equation [(1+α)^2 (1+2α)^2 (1+3α)^2]/[(2+α)^2 (2+2α)^2 (2+3α)^2]/[(3+α)^2 (3+2α)^2 (3+3α)^2] = -648α^2
Choices: (A) -4, (B) 9, (C) -9, (D) 4
Answer: (B) and (C)
Subject: mathematics
Question: In R^3, consider the planes P₁ : y = 0 and P₂ : x + z = 1. Let P₃ be a plane, different from P₁ and P₂, which passes through the intersection of P₁ and P₂. If the distance of the point (0, 1, 0) from P₂ is 1 and the distance of a point (α, β, γ) from P₃ is 2, then which of the following relations is (are) true?
Choices: (A) 2α + β + 2γ + 2 = 0, (B) 2α - β + 2γ + 4 = 0, (C) 2α + β - 2γ - 10 = 0, (D) 2α - β + 2γ - 8 = 0
Answer: (B) and (D)
Subject: mathematics
Question: In R³, let L be a straight line passing through the origin. Suppose that all the points on L are at a constant distance from the two planes P₁ : x + 2y - z + 1 = 0 and P₂ : 2x - y + z - 1 = 0. Let M be the locus of the feet of the perpendiculars drawn from the points on L to the plane P₁. Which of the following points lie(s) on M?
Choices: (A) (0, -, 5/6, 2/3) (B) (-, 1/6, 1, 1/6) (C) (-5/6, 0, 1/6) (D) (-1/3, 0, 2/3)
Answer: (A) and (B)
Subject: mathematics
Question: Let P and Q be distinct points on the parabola y² = 2x such that a circle with PQ as diameter passes through the vertex O of the parabola. If P lies in the first quadrant and the area of the triangle ΔOPQ is 3√2, then which of the following is (are) the coordinates of P?
Choices: (A) (4, 2√2) (B) (9, 3√2) (C) (1/4, 1/√2) (D) (1, √2)
Answer: (A) and (D)
Subject: mathematics
Question: Let y(x) be a solution of the differential equation (1 + e^x)y' + ye^x = 1. If y(0) = 2, then which of the following statements is (are) true?
Choices: (A) y(-4) = 0 (B) y(-2) = 0 (C) y(x) has a critical point in the interval (-1, 0) (D) y(x) has no critical point in the interval (-1, 0)
Answer: (A) and (C)
Subject: mathematics
Question: Consider the family of all circles whose centers lie on the straight line y = x. If this family of circles is represented by the differential equation Py' + Qy' + 1 = 0, where P, Q are functions of x, y and y' (here y' = dy/dx, y'' = d^2y/dx^2), then which of the following statements is (are) true?
Choices: (A) P = y + x
(B) P = y - x
(C) P + Q = 1 - x + y + y' + (y')^2
(D) P - Q = x + y - y' - (y')^2
Answer: (B) and (C)
Subject: mathematics
Question: Let g : IR -> IR be a differentiable function with g(0) = 0, g'(0) = 0 and g'(1) ≠ 0. Let

f(x) = { x g(x), x ≠ 0
       { 0,     x = 0

and h(x) = e^|x| for all x ∈ IR. Let (f ∘ h)(x) denote f(h(x)) and (h ∘ f)(x) denote h(f(x)).
Then which of the following is (are) true?
Choices: (A) f is differentiable at x = 0
(B) h is differentiable at x = 0
(C) f ∘ h is differentiable at x = 0
(D) h ∘ f is differentiable at x = 0
Answer: (A) and (D)
Subject: mathematics
Question: Let f(x) = sin(π/6 - sin(π/2)sin x) for all x ∈ ℝ and g(x) = π/2 sin x for all x ∈ ℝ. Let (f ∘ g)(x) denote f(g(x)) and (g ∘ f)(x) denote g(f(x)). Then which of the following is (are) true?
Choices: (A) Range of f is [-1, 1] (B) Range of f ∘ g is [-1, 1] (C) lim f(x)/π = x→0 g(x) 6 (D) There is an x ∈ ℝ such that (g ∘ f)(x) = 1
Answer: (A), (B) and (C)
Subject: mathematics
Question: Let ΔPQR be a triangle. Let a⃗ = QR⃗, b⃗ = RP⃗ and c⃗ = PQ⃗. If |a⃗| = 12, |b⃗| = 4√3 and b⃗, c⃗ = 24, then which of the following is (are) true?
Choices: (A) |c⃗|^2 / 2 - |a⃗| = 12 (B) |c⃗|^2 / 2 + |a⃗| = 30 (C) |a⃗ × b⃗ + c⃗ × a⃗| = 48√3 (D) a⃗.b⃗ = -72
Answer: (A), (C) and (D)
Subject: mathematics
Question: Match the entries in Column I with the entries in Column II. For each entry in Column I, darken the bubbles of all the matching entries. For example, if entry (A) in Column I matches with entries (Q), (R) and (T), then darken these three bubbles in the ORS. Similarly, for entries (B), (C) and (D).
Choices: Column I: (A), (B), (C), (D) 
Column II: (P), (Q), (R), (S), (T)
Answer: Matrix responses by darkening bubbles
Subject: mathematics
Question: Column I: (A) In R^3, if the magnitude of the projection vector of the vector α i + β j on √(3i + j) is √3 and if α = 2 + √3β, then possible value(s) of |α| is (are) (B) Let a and b be real numbers such that the function f(x) = { -3ax^2 - 2, x < 1 , bx + a^2, x ≥ 1 } is differentiable for all x ∈ R. Then possible value(s) of a is (are) (C) Let ω ≠ 1 be a complex cube root of unity. If (3 - 3ω + 2ω^2)^n+3 + (2 + 3ω - 3ω^2)^n+3 + (-3 + 2ω + 3ω^2)^n+3 = 0, then possible value(s) of n is (are) (D) Let the harmonic mean of two positive real numbers a and b be 4. If q is a positive real number such that a, 5, q, b is an arithmetic progression, then the value(s) of |q - a| is (are) Column II: (P) 1 (Q) 2 (R) 3 (S) 4 (T) 5
Choices: (A) Matches to (P) and (Q), (B) Matches to (P) and (Q), (C) Matches to (P), (Q), (S) and (T), (D) Matches to (Q) and (T)
Answer: (A) Matches to (P) and (Q), (B) Matches to (P) and (Q), (C) Matches to (P), (Q), (S) and (T), (D) Matches to (Q) and (T)
Subject: mathematics
Question: In a triangle ΔXYZ, let a, b and c be the lengths of the sides opposite to the angles X, Y and Z, respectively. If 2(a^2 - b^2) = c^2 and λ = sin(X - Y) / sin Z, then possible values of n for which cos(nπλ) = 0 is (are)

In a triangle ΔXYZ, let a, b and c be the lengths of the sides opposite to the angles X, Y and Z, respectively. If 1 + cos 2X - 2 cos 2Y = 2 sin X sin Y, then possible value(s) of c / b is (are)

In R^2, let β√3 i + j, i + √3 j and β i + (1 - β) j be the position vectors of X, Y and Z with respect to the origin O, respectively. If the distance of Z from the bisector of the acute angle of OX with OY is 3/√5 , then possible value(s) of | β | is (are)

Suppose that F(α) denotes the area of the region bounded by x = 0, x = 2, y^2 = 4x and y = | α x - 1 | + | α x - 2 | + α x, where α ∈ {0, 1}. Then the value(s) of F(α) + 8√2/5 when α = 0 and α = 1, is (are)
Choices: Column I: (A), (B), (C), (D)
Column II: (P) 1, (Q) 2, (R) 3, (S) 5, (T) 6
Answer: (A) Matches to (P), (R) and (S)
(B) Matches to (P)
(C) Matches to (P) and (Q)
(D) Matches to (S) and (T)