cmeraki/elemento huggingface dataset:
Subject: physics
Question: Consider a spherical gaseous cloud of mass density ρ(r) in free space where r is the radial distance from its center. The gaseous cloud is made of particles of equal mass m moving in circular orbits about the common center with the same kinetic energy K. The force acting on the particles is their mutual gravitational force. If ρ(r) is constant in time, the particle number density n(r) = ρ(r)/m is [G is universal gravitational constant]
Choices: (A) K/2πr^2m^2G, (B) K/πr^2m^2G, (C) 3K/πr^2m^2G, (D) K/6πr^2m^2G
Answer: (B) K/πr^2m^2G
Subject: physics
Question: A thin spherical insulating shell of radius R carries a uniformly distributed charge such that the potential at its surface is V0. A hole with a small area α4πR^2 (α ≪ 1) is made on the shell without affecting the rest of the shell. Which one of the following statements is correct?
Choices: (A) The potential at the center of the shell is reduced by 2αV0, (B) The magnitude of electric field at the center of the shell is reduced by αV0/2R, (C) The ratio of the potential at the center of the shell to that of the point at 1/2R from center towards the hole will be 1-α/1-2α, (D) The magnitude of electric field at a point, located on a line passing through the hole and shell's center, on a distance 2R from the center of the spherical shell will be reduced by αV0/2R
Answer: (C) The ratio of the potential at the center of the shell to that of the point at 1/2R from center towards the hole will be 1-α/1-2α
Subject: physics
Question: A current carrying wire heats a metal rod. The wire provides a constant power (P) to the rod. The metal rod is enclosed in an insulated container. It is observed that the temperature (T) in the metal rod changes with time (t) as T(t) = T0(1 + βt^2), where β is a constant with appropriate dimension while T0 is a constant with dimension of temperature. The heat capacity of the metal is,
Choices: (A) 4P(T(t)-T0)^3/β^4T03, (B) 4P(T(t)-T0)^4/β^4T03, (C) 4P(T(t)-T0)^2/β^4T03, (D) 4P(T(t)-T0)/β^4T03
Subject: physics
Question: In a radioactive sample, 40K nuclei either decay into stable 40Ca nuclei with decay constant 4.5 x 10^-10 per year or into stable 40Ar nuclei with decay constant 0.5 x 10^-10 per year. Given that in this sample all the stable 40Ca and 40Ar nuclei are produced by the 40K nuclei only. In time t x 10^5 years, if the ratio of the sum of stable 40Ca and 40Ar nuclei to the radioactive 40K nuclei is 99, the value of t will be, [Given: ln 10 = 2.3]
Choices: (A) 1.15, (B) 9.2, (C) 2.3, (D) 4.6
Subject: physics
Question: A cylindrical capillary tube of 0.2 mm radius is made by joining two capillaries T1 and T2 of different materials having water contact angles of 0° and 60°, respectively. The capillary tube is dipped vertically in water in two different configurations, case I and II as shown in figure. Which of the following option(s) is(are) correct? [Surface tension of water = 0.075 N/m, density of water = 1000 kg/m³, take g = 10 m/s²]
Choices: (A) The correction in the height of water column raised in the tube, due to weight of water contained in the meniscus, will be different for both cases. 
(B) For case II, if the capillary joint is 5 cm above the water surface, the height of water column raised in the tube will be 3.75 cm. (Neglect the weight of the water in the meniscus) 
(C) For case I, if the joint is kept at 8 cm above the water surface, the height of water column in the tube will be 7.5 cm. (Neglect the weight of the water in the meniscus) 
(D) For case I, if the capillary joint is 5 cm above the water surface, the height of water column raised in the tube will be more than 8.75 cm. (Neglect the weight of the water in the meniscus)
Subject: physics
Question: A conducting wire of parabolic shape, initially y = x^2, is moving with velocity V = Vot in a non-uniform magnetic field B = B0 (1 + (γ)^β) k, as shown in figure. If V0, B0, L and β are positive constants and Δφ is the potential difference developed between the ends of the wire, then the correct statement(s) is/are:
Choices: (A) |Δφ| = 1/4 B0V0L for β = 0 <br> (B) |Δφ| = 3/5 B0V0L for β = 2 <br> (C) |Δφ| remains the same if the parabolic wire is replaced by a straight wire, y = x initially, of length √(2L) <br> (D) |Δφ| is proportional to the length of the wire projected on the y-axis.
Answer: B
Subject: physics
Question: In the circuit shown, initially there is no charge on capacitors and keys S1 and S2 are open. The values of the capacitors are C1 = 10 μF, C2 = 30 μF and C3 = C4 = 80 μF. Which of the statement(s) is/are correct?
Choices: A) At time t = 0, the key S1 is closed, the instantaneous current in the closed circuit will be 25 mA.

B) If key S1 is kept closed for long time such that capacitors are fully charged, the voltage across the capacitor C1 will be 4 V.

C) The key S1 is kept closed for long time such that capacitors are fully charged. Now key S2 is closed, at this time, the instantaneous current across 30 Ω resistor (between points P and Q) will be 0.2 A (round off to 1* decimal place).

D) If key S1 is kept closed for long time such that capacitors are fully charged, the voltage difference between points P and Q will be 10 V.
Subject: physics
Question: A charged shell of radius R carries a total charge Q. Given Φ as the flux of electric field through a closed cylindrical surface of height h, radius r and with its center same as that of the shell. Here, center of the cylinder is a point on the axis of the cylinder which is equidistant from its top and bottom surfaces. Which of the following option(s) is/are correct? [ε0 is the permittivity of free space]
Choices: A) If h > 2R and r > R then Φ = Q/ε0

B) If h < 8R/5 and r = 3R/5 then Φ = 0

C) If h > 2R and r = 3R/5 then Φ = Q/5ε0

D) If h > 2R and r = 4R/5 then Φ = Q/5ε0
Subject: physics
Question: One mole of a monatomic ideal gas goes through a thermodynamic cycle, as shown in the volume versus temperature (V-T) diagram. The correct statement(s) is/are: [R is the gas constant]
Choices: (A) Work done in this thermodynamic cycle (1 → 2 → 3 → 4 → 1) is |W| = 1/2 RT0, (B) The above thermodynamic cycle exhibits only isochoric and adiabatic processes, (C) The ratio of heat transfer during processes 1 → 2 and 2 → 3 is |Q1→2|/|Q2→3| = 5/3, (D) The ratio of heat transfer during processes 1 → 2 and 3 → 4 is |Q1→2|/|Q3→4| = 1/2
Subject: physics
Question: A thin convex lens is made of two materials with refractive indices n1 and n2, as shown in figure. The radius of curvature of the left and right spherical surfaces are equal. f is the focal length of the lens when n1 = n2 = n. The focal length is f + Δf when n1 = n and n2 = n + Δn. Assuming Δn << (n - 1) and 1 < n < 2, the correct statement(s) is/are.
Choices: A) |f| < |Δn|
B) For n = 1.5, Δn = 10^-3 and f = 20 cm, the value of |Δf| will be 0.02 cm (round off to 2nd decimal place).
C) If Δn < 0 then Δf > 0
D) The relation between Δf and Δn remains unchanged if both the convex surfaces are replaced by concave surfaces of the same radius of curvature.
Subject: physics
Question: Two identical moving coil galvanometers have 10 Ω resistance and full scale deflection at 2 μA current. One of them is converted into a voltmeter of 100 mV full scale reading and the other into an Ammeter of 1mA full scale current using appropriate resistors. These are then used to measure the voltage and current in the Ohm's law experiment with R = 1000 Ω resistor by using an ideal cell. Which of the following statement(s) is/are correct?
Choices: (A) The resistance of the Voltmeter will be 100 kΩ
(B) The resistance of the Ammeter will be 0.02 Ω (round off to 2nd decimal place)
(C) The measured value of R will be 978 Ω < R < 982 Ω
(D) If the ideal cell is replaced by a cell having internal resistance of 5 Ω then the measured value of R will be more than 1000 Ω
Subject: physics
Question: A particle is moved along a path AB-BC-CD-DE-EF-FA, as shown in figure, in presence of a force F⃗ = (αy)iˆ + 2(αx)jˆ N, where x and y are in meter and α = −1 Nm⁻¹. The work done on the particle by this force F⃗ will be ___ Joule.
Subject: physics
Question: A block of weight 100 N is suspended by copper and steel wires of same cross sectional area 0.5 cm² and, length √3 m and 1 m, respectively. Their other ends are fixed on a ceiling as shown in figure. The angles subtended by copper and steel wires with ceiling are 30° and 60°, respectively. If elongation in copper wire is (ΔL₂) and elongation in steel wire is (ΔL₅), then the ratio ΔL₅/ΔL₂ is ___. [Young's modulus for copper and steel are 1×10¹¹ N/m² and 2×10¹¹ N/m², respectively.]
Subject: physics
Question: A train S1, moving with a uniform velocity of 108 km/h, approaches another train S2 standing on a platform. An observer O moves with a uniform velocity of 36 km/h towards S2, as shown in figure. Both the trains are blowing whistles of same frequency 120 Hz. When O is 600 m away from S2 and distance between S1 and S2 is 800 m, the number of beats heard by O is __________. [Speed of the sound = 330 m/s]
Choices: No multiple choice options provided
Answer: <blank>
Subject: physics
Question: A parallel plate capacitor of capacitance C has spacing d between two plates having area A. The region between the plates is filled with N dielectric layers, parallel to its plates, each with thickness δ = (d/N). The dielectric constant of the mth layer is Km = K (1 + m/N). For a very large N (> 10^3), the capacitance C is α (KϵoA/d ln 2). The value of α will be _______. [ϵo is the permittivity of free space]
Choices: No multiple choice options provided
Answer: <blank>
Subject: physics
Question: A liquid at 30°C is poured very slowly into a Calorimeter that is at temperature of 110°C. The boiling temperature of the liquid is 80°C. It is found that the first 5 gm of the liquid completely evaporates. After pouring another 80 gm of the liquid the equilibrium temperature is found to be 50°C. The ratio of the Latent heat of the liquid to its specific heat will be ______°C. [Neglect the heat exchange with surrounding]
Choices: No multiple choice options provided
Answer: <blank>
Subject: physics
Question: A planar structure of length L and width W is made of two different optical media of refractive indices n1 = 1.5 and n2 = 1.44 as shown in figure. If L >> W, a ray entering from end AB will emerge from end CD only if the total internal reflection condition is met inside the structure. For L = 9.6 m, if the incident angle θ is varied, the maximum time taken by a ray to exit the plane CD is t x 10^-9 s, where t is _______. [Speed of light c = 3 x 10^8 m/s]
Choices: No response choices provided
Answer: No answer provided
Subject: chemistry
Question: Calamine, malachite, magneite and cryolite, respectively, are
Choices: (A) ZnSO4, CuCO3, Fe2O3, AlP3 (B) ZnSO4, Cu(OH)2, Fe3O4, Na3AlF6 (C) ZnCO3, CuCO3,Cu(OH)2, Fe3O4, Na3AlF6 (D) ZnCO3, CuCO3, Fe2O3, Na3AlF6
Answer: B
Subject: chemistry
Question: Molar conductivity (Am) of aqueous solution of sodium stearate, which behaves as a strong electrolyte, is recorded at varying concentrations (c) of sodium stearate. Which one of the following plots provides the correct representation of micelle formation in the solution? (critical micelle concentration (CMC) is marked with an arrow in the figures)
Choices: (A) (B) (C) (D)
Answer: B
Subject: chemistry
Question: A tin chloride Q undergoes the following reactions (not balanced) Q + Cl- → X Q + Mg3N2 → Y Q + CuCl2 → Z + CuCl X is a monoanion having pyramidal geometry. Both Y and Z are neutral compounds. Choose the correct option(s)
Choices: (A) The central atom in X is sp3 hybridized (B) There is a coordinate bond in Y (C) The oxidation state of the central atom in Z is +2 (D) The central atom in Z has one lone pair of electrons
Subject: chemistry
Question: Fusion of MnO2 with KOH in presence of O2 produces a salt W. Alkaline solution of W upon electrolytic oxidation yields another salt X. The manganese containing ions present in W and X, respectively, are Y and Z. Correct statement(s) is(are)
Choices: (A) In aqueous acidic solution, Y undergoes disproportionation reaction to give Z and MnO2, (B) Both Y and Z are coloured and have tetrahedral shape, (C) Y is diamagnetic in nature while Z is paramagnetic, (D) In both Y and Z, π-bonding occurs between p-orbitals of oxygen and d-orbitals of manganese
Subject: chemistry
Question: Choose the reaction(s) from the following options, for which the standard enthalpy of reaction is equal to the standard enthalpy of formation.
Choices: (A) 2H2(g) + O2(g) → 2H2O (l), (B) 2C(g) + 3H2(g) → C2H6(g), (C) ½O2(g) → O3(g), (D) ½Na(S) + O2(g) → SO2(g)
Subject: chemistry
Question: Which of the following statement(s) is(are) correct regarding the root mean square speed (urms) and average translational kinetic energy (ϵav) of a molecule in a gas at equilibrium ?
Choices: (A) urms is doubled when its temperature is increased four times, (B) ϵav is doubled when its temperature is increased four times, (C) ϵav at a given temperature does not depend on its molecular mass, (D) urms is inversely proportional to the square root of its molecular mass
Subject: chemistry
Question: Each of the following options contains a set of four molecules. Identify the option(s) where all four molecules possess permanent dipole moment at room temperature.
Choices: (A) BeCl2, CO2, BCl3, CHCl3, (B) NO2, NH3, POCl3, CH3Cl, (C) BF3, O2, SF6, XeF6, (D) SO2, C2H5Cl, H2Se, BrF5
Answer: B
Subject: physics
Question: In the decay sequence, 238U → 234Th → 234Pa → 234U → 230Th, x1, x2, x3 and x4 are particles/radiation emitted by the respective isotopes. The correct option(s) is(are)
Choices: (A) x1 will deflect towards negatively charged plate, (B) x2 is β⁻, (C) x3 is γ-ray, (D) Z is an isotope of uranium
Answer: B,C,D
Subject: chemistry
Question: Which of the following statement(s) is(are) true?
Choices: (A) Monosaccharides cannot be hydrolysed to give polyhydroxy aldehydes and ketones, (B) Oxidation of glucose with bromine water gives glutamic acid, (C) Hydrolysis of sucrose gives dextrorotatory glucose and laevorotatory fructose, (D) The two six-membered cyclic hemiacetal forms of D-(+)-glucose are called anomers
Answer: C,D
Subject: chemistry
Question: For the following reaction, the equilibrium constant Kc at 298 K is 1.6 × 10^17. Fe^2+ (aq) + S^2- (aq) ⇌ FeS (s) When equal volumes of 0.06 M Fe^3 (aq) and 0.2 M S^2- (aq) solutions are mixed, the equilibrium concentration of Fe^2 (aq) is found to be Y × 10^-17 M. The value of Y is_______
Choices: NUMERICAL VALUE
Answer: NUMERICAL VALUE
Subject: chemistry
Question: On dissolving 0.5 g of a non-volatile non-ionic solute to 39 g of benzene, its vapor pressure decreases from 650 mm Hg to 640 mm Hg. The depression of freezing point of benzene (in K) upon addition of the solute is______ (Given data: Molar mass and the molal freezing point depression constant of benzene are 78 g mol^-1 and 5.12 K kg mol^-1, respectively)
Choices: NUMERICAL VALUE
Answer: NUMERICAL VALUE
Subject: chemistry
Question: Consider the kinetic data given in the following table for the reaction A + B + C → Product. [TABLE DATA] The rate of the reaction for [A] = 0.15 mol dm^-3, [B] = 0.25 mol dm^-3 and [C] = 0.15 mol dm^-3 is found to be Y × 10^-5 mol dm^-3 s^-1. The value of Y is______
Choices: MATRIX RESPONSE
Answer: NUMERICAL VALUE
Subject: mathematics
Question: Let S be the set of all complex numbers z satisfying |z - 2 + i| ≥ √5. If the complex number z0 is such that (1/|z0 - 1|) is the maximum of the set {1/|z - 1| : z ∈ S}, then the principal argument of (A - z0 - z̄0)/(z0 - z̄0 + 2i) is
Choices: (A) -(π/2), (B) π/4, (C) π/2, (D) 3π/4
Answer: C
Subject: mathematics
Question: Let M = [sin^4θ  -1 - sin^2θ]/[1 + cos^2θ  cos^4θ] = αI + βM^-1, where α = α(θ) and β = β(θ) are real numbers, and I is the 2 x 2 identity matrix. If α* is the minimum of the set {α(θ): θ ∈ [0,2π)} and β* is the minimum of the set {β(θ): θ ∈ [0,2π)}, then the value of α* + β* is
Choices: (A) -37/16, (B) -31/16, (C) -29/16, (D) -17/16
Answer: C
Subject: mathematics
Question: A line y = mx + 1 intersects the circle (x - 3)^2 + (y + 2)^2 = 25 at the points P and Q. If the midpoint of the line segment PQ has x-coordinate = -2, then which one of the following options is correct?
Choices: (A) -3 ≤ m < -1, (B) 2 ≤ m < 4, (C) 4 ≤ m < 6, (D) 6 ≤ m < 8
Answer: B
Subject: mathematics
Question: The area of the region {(x, y) : xy ≤ 8, 1 ≤ y ≤ x²} is
Choices: (A) 16 log₂ 2 - 14/3, (B) 8 log₂ 2 - 14/3, (C) 16 log₂ 2 - 6, (D) 8 log₂ 2 - 7/3
Subject: mathematics
Question: Let α and β be the roots of x^2 - x - 1 = 0, with α > β. For all positive integers n, define a_n = (α^n - β^n) / (α - β), n ≥ 1, b_1 = 1 and b_n = a_n-1 + a_n+1, n ≥ 2. Then which of the following options is/are correct?
Choices: A) a_1 + a_2 + a_3 + ... + a_n = a_n+2 - 1 for all n ≥ 1
B) Σ a_n = 10/89 from n=1 to infinity
C) b_n = α^n + β^n for all n ≥ 1
D) Σ b_n = 8/89 from n=1 to infinity
Answer: A, B
Subject: mathematics
Question: Let M = [[0, 1, a], [1, 2, 3], [3, b, 1]] and adjM = [[-1, 1, -1], [8, -6, 2], [-5, 3, -1]] where a and b are real numbers. Which of the following options is/are correct?
Choices: (A) a + b = 3
(B) (adjM)^-1 + adjM^-1 = -M
(C) det(adjM^2) = 81
(D) If M [[α], [β], [γ]] = [[1], [2], [3]], then α - β + γ = 3
Answer: Not enough information to determine answer
Subject: mathematics
Question: There are three bags B1, B2 and B3. The bag B1 contains 5 red and 5 green balls, B2 contains 3 red and 5 green balls, and B3 contains 5 red and 3 green balls. Bags B1, B2 and B3 have probabilities 3/10, 3/10 and 4/10 respectively of being chosen. A bag is selected at random and a ball is chosen at random from the bag. Then which of the following options is/are correct?
Choices: (A) Probability that the chosen ball is green, given that the selected bag is B3, equals 3/8
(B) Probability that the chosen ball is green equals 39/80
(C) Probability that the selected bag is B3, given that the chosen ball is green, equals 5/13
(D) Probability that the selected bag is B3 and the chosen ball is green equals 3/10
Answer: Not enough information to determine answer
Subject: mathematics
Question: In a non-right-angled triangle ΔPQR, let p, q, r denote the lengths of the sides opposite to the angles at P, Q, R respectively. The median from R meets the side PQ at S, the perpendicular from P meets the side QR at E, and RS and PE intersect at O. If p = √3, q = 1, and the radius of the circumcircle of the ΔPQR equals 1, then which of the following options is/are correct?
Choices: (A) Length of RS = √7/2 
(B) Area of ΔSOE = √3/12 
(C) Length of OE = 1/6 
(D) Radius of incircle of ΔPQR = √3/2 (2 - √3)
Answer: (C) Length of OE = 1/6
Subject: mathematics
Question: Define the collections {E1, E2, E3, ...} of ellipses and {R1, R2, R3, ...} of rectangles as follows: 
E3: x^2/9 + y^2/4 = 1; 
R3: rectangle of largest area, with sides parallel to the axes, inscribed in E3; 
En: ellipse x^2/a^2 + y^2/b^2 = 1 of largest area inscribed in Rn-1, n > 1; 
Rn: rectangle of largest area, with sides parallel to the axes, inscribed in En, n > 1. 
Then which of the following options is/are correct?
Choices: (A) The eccentricities of E18 and E19 are NOT equal 
(B) ∑ (area of Rn) < 24, for each positive integer N 
    n=1 
(C) The length of latus rectum of E9 is 1/6 
(D) The distance of a focus from the centre in E9 is √5/32
Answer: (B) ∑ (area of Rn) < 24, for each positive integer N
Subject: mathematics
Question: Let Γ denote a curve y = y(x) which is in the first quadrant and let the point (1,0) lie on it. Let the tangent to Γ at a point P intersect the y-axis at Yp. If PYp has length 1 for each point P on Γ, then which of the following options is/are correct?
Choices: (A) y = log₁₁(1 + √(1 - x²)/x) - √(1 - x²); (B) xy' + √(1 - x²) = 0; (C) y = -log₁₁((1 + √(1 - x²))/x) + √(1 - x²); (D) xy' - √(1 - x²) = 0
Answer: A, B, C
Subject: mathematics
Question: Let l₁ and l₂ denote the lines r⃗ = i⃗ + λ (−i⃗ + 2j⃗ + 2k⃗), λ ∈ ℝ and r⃗ = μ (2i⃗ − j⃗ + 2k⃗), μ ∈ ℝ respectively. If l₃ is a line which is perpendicular to both l₁ and l₂ and cuts both of them, then which of the following options describe(s) l₃?
Choices: (A) r⃗ = ½ (4i⃗ + j⃗ + k⃗) + t(2i⃗ + 2j⃗ − k⃗), t ∈ ℝ
(B) r⃗ = ½ (2i⃗ − j⃗ + 2k⃗) + t (2i⃗ + 2j⃗ − k⃗), t ∈ ℝ
(C) r⃗ = ⅓ (2i⃗ + k⃗) + t(2i⃗ + 2j⃗ − k⃗), t ∈ ℝ
(D) r⃗ = t(2i⃗ + 2j⃗ − k⃗), t ∈ ℝ
Subject: mathematics
Question: Let S be the sample space of all 3 × 3 matrices with entries from the set {0, 1}. Let the events E1 and E2 be given by E1 = {A ∈ S : det A = 0} and E2 = {A ∈ S : sum of entries of A is 7}. If a matrix is chosen at random from S, then the conditional probability P(E1|E2) equals
Subject: mathematics
Question: Let the point B be the reflection of the point A(2, 3) with respect to the line 8x − 6y − 23 = 0. Let ΓA and ΓB be circles of radii 2 and 1 with centres A and B respectively. Let T be a common tangent to the circles ΓA and ΓB such that both the circles are on the same side of T. If C is the point of intersection of T and the line passing through A and B, then the length of the line segment AC is
Subject: mathematics
Question: Three lines are given by r = λi, λ ∈ R r = μ (i + j), μ ∈ R and r = v(i + j + k), v ∈ R. Let the lines cut the plane x + y + z = 1 at the points A, B and C respectively. If the area of the triangle ABC is A then the value of (6A)^2 equals___