JEE_papers/JSON/JEE_2015_1.jsonl

{"id": 7, "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^2\u03bb^2 - d^2, where \u03bb 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", "response_choices": "", "answer": "3", "page_number": 4, "image_available": "yes", "question_subject": "physics"}
{"id": 8, "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", "response_choices": "", "answer": "7", "page_number": 4, "image_available": "yes", "question_subject": "physics"}
{"id": 9, "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:", "response_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)", "page_number": 5, "image_available": "no", "question_subject": "physics"}
{"id": 10, "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)", "response_choices": "(A) M \u221d \u221ac (B) M \u221d \u221aG (C) L \u221d \u221ah (D) L \u221d \u221aG", "answer": "(A), (C) and (D)", "page_number": 5, "image_available": "no", "question_subject": "physics"}
{"id": 11, "question": "Two independent harmonic oscillators of equal mass are oscillating about the origin with angular frequencies \u03c9\u2081 and \u03c9\u2082 and have total energies E\u2081 and E\u2082, 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)", "response_choices": "(A) E\u2081\u03c9\u2081 = E\u2082\u03c9\u2082, (B) \u03c9\u2082 = n^2 * \u03c9\u2081, (C) \u03c9\u2081\u03c9\u2082 = n^2, (D) E\u2081/\u03c9\u2081 = E\u2082/\u03c9\u2082", "answer": "(B) and (D)", "page_number": 6, "image_available": "yes", "question_subject": "physics"}
{"id": 12, "question": "A ring of mass M and radius R is rotating with angular speed \u03c9 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 \u03c9 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", "response_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)", "page_number": 7, "image_available": "yes", "question_subject": "physics"}
{"id": 13, "question": "The figures below depict two situations in which two infinitely long static line charges of constant positive line charge density \u03bb 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)", "response_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", "page_number": 8, "image_available": "yes", "question_subject": "physics"}
{"id": 14, "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", "response_choices": "A. 60 cm B. 70 cm C. 80 cm D. 90 cm", "answer": "B", "page_number": 8, "image_available": "yes", "question_subject": "physics"}
{"id": 15, "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)", "response_choices": "(A) If B is along z, F \u221d (L+R), (B) If B is along x, F=0, (C) If B is along y, F \u221d (L+R), (D) If B is along z, F=0", "answer": "(A), (B) and (C)", "page_number": 9, "image_available": "yes", "question_subject": "physics"}
{"id": 16, "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)", "response_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 \u221a(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/\u221a2", "answer": "(A), (B) and (D)", "page_number": 9, "image_available": "no", "question_subject": "physics"}
{"id": 17, "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\u00d710^-8 \u03a9 m and 1.0\u00d710^-7 \u03a9 m, respectively. The electrical resistance between the two faces P and Q of the composite bar is", "response_choices": "(A) 2475 \u03bc\u03a9/64\t(B) 1875 \u03bc\u03a9/64\t(C) 1875 \u03bc\u03a9/49\t(D) 2475 \u03bc\u03a9/132", "answer": "B", "page_number": 10, "image_available": "yes", "question_subject": "physics"}
{"id": 18, "question": "For photo-electric effect with incident photon wavelength \u03bb, the stopping potential is V0. Identify the correct variation(s) of V0 with \u03bb and 1/\u03bb.", "response_choices": "(A) [Graph A]\t(B) [Graph B]\t(C) [Graph C]\t(D) [Graph D]", "answer": "A and C", "page_number": 10, "image_available": "yes", "question_subject": "physics"}
{"id": 19, "question": "Match the nuclear processes given in column I with the appropriate option(s) in column II.", "response_choices": "Column I: (A) Nuclear fusion, (B) Fission in a nuclear reactor, (C) \u03b2-decay, (D) \u03b3-ray emission\nColumn 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)", "page_number": 11, "image_available": "no", "question_subject": "physics"}
{"id": 20, "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.", "response_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)", "page_number": 12, "image_available": "no", "question_subject": "physics"}
{"id": 21, "question": "The total number of stereoisomers that can exist for M is", "response_choices": "Single digit integer ranging from 0 to 9", "answer": "2", "page_number": 13, "image_available": "yes", "question_subject": "chemistry"}
{"id": 22, "question": "The number of resonance structures for N is", "response_choices": "Single digit integer ranging from 0 to 9", "answer": "9", "page_number": 13, "image_available": "yes", "question_subject": "chemistry"}
{"id": 23, "question": "The total number of lone pairs of electrons in N2O3 is", "response_choices": "Single digit integer ranging from 0 to 9", "answer": "8", "page_number": 13, "image_available": "no", "question_subject": "chemistry"}
{"id": 24, "question": "For the octahedral complexes of Fe3+ in SCN- (thiocyanato-S) and in CN- ligand environments, the difference between the spin-only magnetic moments in Bohr magnetons (when approximated to the nearest integer) is [Atomic number of Fe = 26]", "response_choices": "Single digit integer ranging from 0 to 9", "answer": "4", "page_number": 13, "image_available": "no", "question_subject": "chemistry"}
{"id": 25, "question": "Among the triatomic molecules/ions, BeCl\u2082, N\u2082, N\u2082O, NO\u2082\u207a, O\u2083, SCl\u2082,ICl\u2082\u207b, I\u2083\u207b and XeF\u2082, 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", "response_choices": "No response choices provided", "answer": "4", "page_number": 14, "image_available": "no", "question_subject": "chemistry"}
{"id": 26, "question": "Not considering the electronic spin, the degeneracy of the second excited state (n = 3) of H atom is 9, while the degeneracy of the second excited state of H\u207b is", "response_choices": "No response choices provided", "answer": "3", "page_number": 14, "image_available": "no", "question_subject": "chemistry"}
{"id": 27, "question": "All the energy released from the reaction X \u2192 Y \u039b G\u2070 = -193 kJ mol\u207b\u00b9 is used for oxidizing M\u207a as M\u207a \u2192 M\u00b3\u207a+ 2e\u207b, E\u2070 = - 0.25 V. Under standard conditions, the number of moles of M\u207a oxidized when one mole of X is converted to Y is [F = 96500 C mol\u207b\u00b9]", "response_choices": "No response choices provided", "answer": "4", "page_number": 14, "image_available": "no", "question_subject": "chemistry"}
{"id": 28, "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\u00b0C, the number of chloride(s) in the coordination sphere of the complex is [K\u1d68 of water = 1.86 K kg mol\u207b\u00b9]", "response_choices": "No response choices provided", "answer": "1", "page_number": 14, "image_available": "no", "question_subject": "chemistry"}
{"id": 29, "question": "Compound(s) that on hydrogenation produce(s) optically inactive compound(s) is(are)", "response_choices": "(A) H3C-CH3, (B) H3C-CH3, (C) H3C-CH3, (D) H3C-CH3", "answer": "(B) and (D)", "page_number": 15, "image_available": "yes", "question_subject": "chemistry"}
{"id": 30, "question": "The major product of the following reaction is", "response_choices": "(A) (image), (B) (image), (C) (image), (D) (image)", "answer": "(A)", "page_number": 15, "image_available": "yes", "question_subject": "chemistry"}
{"id": 31, "question": "In the following reaction, the major product is", "response_choices": "(A) H2C=CCH3Br, (B) H3C=CBr, (C) H2C=CHBr, (D) H3C=CHBr", "answer": "D", "page_number": 16, "image_available": "yes", "question_subject": "chemistry"}
{"id": 32, "question": "The structure of D-(+)-glucose is\n\nCHO\nH OH\nHO H\nH OH\nH OH\nCH2OH\n\nThe structure of L-(-)-glucose is", "response_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", "page_number": 16, "image_available": "yes", "question_subject": "chemistry"}
{"id": 33, "question": "The major product of the reaction is H3C CH2 CO2H + CH3 NH2 + CO2H + NaNO2, aqueous HCl at 0\u00b0C", "response_choices": "(A) H3C CH2 NH2 CH3 OH, (B) H3C CO2H CH3 OH, (C) H3C CO2H CH3 OH, (D) H3C NH2 CH3 OH", "answer": "C", "page_number": 17, "image_available": "yes", "question_subject": "chemistry"}
{"id": 34, "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", "response_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", "page_number": 17, "image_available": "no", "question_subject": "chemistry"}
{"id": 35, "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", "response_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", "page_number": 17, "image_available": "no", "question_subject": "chemistry"}
{"id": 36, "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", "response_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", "page_number": 17, "image_available": "no", "question_subject": "chemistry"}
{"id": 37, "question": "The %yield of ammonia as a function of time in the reaction N2(g) + 3H2(g) \u21cc 2NH3(g), \u0394H < 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", "response_choices": "(A) (B) (C) (D)", "answer": "B", "page_number": 18, "image_available": "yes", "question_subject": "chemistry"}
{"id": 38, "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", "response_choices": "(A) 1/2, 1/8 (B) 1, 1/4 (C) 1/2, 1/2 (D) 1, 1/8", "answer": "A", "page_number": 18, "image_available": "no", "question_subject": "chemistry"}
{"id": 39, "question": "Match the anionic species given in Column I that are present in the ore(s) given in Column II.", "response_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)", "page_number": 19, "image_available": "no", "question_subject": "chemistry"}
{"id": 40, "question": "Match the thermodynamic processes given under Column I with the expressions given under Column II.", "response_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 \n\nColumn II: (P) q = 0, (Q) w = 0, (R) \u0394Ssys < 0, (S) \u0394U = 0, (T) \u0394G = 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)", "page_number": 20, "image_available": "no", "question_subject": "chemistry"}
{"id": 41, "question": "The number of distinct solutions of the equation 5/4 cos^2x + cos^4x + sin^4x + cos^6x + sin^6x = 2 in the interval [0, 2\u03c0] is", "response_choices": "Single digit integer from 0 to 9", "answer": "8", "page_number": 21, "image_available": "no", "question_subject": "mathematics"}
{"id": 42, "question": "Let the curve C be the mirror image of the parabola y^2 = 4x with respect to the line x + y + 4 = 0. If A and B are the points of intersection of C with the line y = -5, then the distance between A and B is", "response_choices": "Single digit integer from 0 to 9", "answer": "4", "page_number": 21, "image_available": "no", "question_subject": "mathematics"}
{"id": 43, "question": "The minimum number of times a fair coin needs to be tossed, so that the probability of getting at least two heads is at least 0.96, is", "response_choices": "Single digit integer from 0 to 9", "answer": "8", "page_number": 21, "image_available": "no", "question_subject": "mathematics"}
{"id": 44, "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", "response_choices": "Single digit integer from 0 to 9", "answer": "5", "page_number": 21, "image_available": "no", "question_subject": "mathematics"}
{"id": 45, "question": "If the normals of the parabola y^2 = 4x drawn at the end points of its latus rectum are tangents to the circle (x - 3)^2 + (y + 2)^2 = r^2, then the value of r^2 is", "response_choices": "Single digit integer from 0 to 9", "answer": "2", "page_number": 21, "image_available": "no", "question_subject": "mathematics"}
{"id": 46, "question": "Let f : R -> R be a function defined by f(x) = {[x], x \u2264 2, 0, x > 2} where [x] is the greatest integer less than or equal to x. If I = \u222b(2/1) y(x^3) / (2 + f(x + 1)) dx, then the value of (4I - 1) is", "response_choices": "", "answer": "0", "page_number": 22, "image_available": "no", "question_subject": "mathematics"}
{"id": 47, "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/250\u03c0 is", "response_choices": "", "answer": "4", "page_number": 22, "image_available": "no", "question_subject": "mathematics"}
{"id": 48, "question": "Let F(x) = \u222b(x/\u03c0)^2 2cos^2 t dt for all x \u2208 R and f : [0, 1/2] \u2192 [0, \u221e) be a continuous function. For \u03b1 \u2208 [0, 1/2], if F'(\u03b1) + 2 is the area of the region bounded by x = 0, y = 0, y = f(x) and x = \u03b1, then f(0) is", "response_choices": "", "answer": "3", "page_number": 22, "image_available": "no", "question_subject": "mathematics"}
{"id": 49, "question": "Let X and Y be two arbitrary, 3\u00d73, non-zero, skew-symmetric matrices and Z be an arbitrary 3\u00d73, non-zero, symmetric matrix. Then which of the following matrices is (are) skew symmetric?", "response_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)", "page_number": 23, "image_available": "no", "question_subject": "mathematics"}
{"id": 50, "question": "Which of the following values of \u03b1 satisfy the equation [(1+\u03b1)^2 (1+2\u03b1)^2 (1+3\u03b1)^2]/[(2+\u03b1)^2 (2+2\u03b1)^2 (2+3\u03b1)^2]/[(3+\u03b1)^2 (3+2\u03b1)^2 (3+3\u03b1)^2] = -648\u03b1^2", "response_choices": "(A) -4, (B) 9, (C) -9, (D) 4", "answer": "(B) and (C)", "page_number": 23, "image_available": "no", "question_subject": "mathematics"}
{"id": 51, "question": "In R^3, consider the planes P\u2081 : y = 0 and P\u2082 : x + z = 1. Let P\u2083 be a plane, different from P\u2081 and P\u2082, which passes through the intersection of P\u2081 and P\u2082. If the distance of the point (0, 1, 0) from P\u2082 is 1 and the distance of a point (\u03b1, \u03b2, \u03b3) from P\u2083 is 2, then which of the following relations is (are) true?", "response_choices": "(A) 2\u03b1 + \u03b2 + 2\u03b3 + 2 = 0, (B) 2\u03b1 - \u03b2 + 2\u03b3 + 4 = 0, (C) 2\u03b1 + \u03b2 - 2\u03b3 - 10 = 0, (D) 2\u03b1 - \u03b2 + 2\u03b3 - 8 = 0", "answer": "(B) and (D)", "page_number": 23, "image_available": "no", "question_subject": "mathematics"}
{"id": 52, "question": "In R\u00b3, 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 P\u2081 : x + 2y - z + 1 = 0 and P\u2082 : 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 P\u2081. Which of the following points lie(s) on M?", "response_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)", "page_number": 24, "image_available": "no", "question_subject": "mathematics"}
{"id": 53, "question": "Let P and Q be distinct points on the parabola y\u00b2 = 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 \u0394OPQ is 3\u221a2, then which of the following is (are) the coordinates of P?", "response_choices": "(A) (4, 2\u221a2) (B) (9, 3\u221a2) (C) (1/4, 1/\u221a2) (D) (1, \u221a2)", "answer": "(A) and (D)", "page_number": 24, "image_available": "no", "question_subject": "mathematics"}
{"id": 54, "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?", "response_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)", "page_number": 24, "image_available": "no", "question_subject": "mathematics"}
{"id": 55, "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?", "response_choices": "(A) P = y + x\n(B) P = y - x\n(C) P + Q = 1 - x + y + y' + (y')^2\n(D) P - Q = x + y - y' - (y')^2", "answer": "(B) and (C)", "page_number": 25, "image_available": "no", "question_subject": "mathematics"}
{"id": 56, "question": "Let g : IR -> IR be a differentiable function with g(0) = 0, g'(0) = 0 and g'(1) \u2260 0. Let\n\nf(x) = { x g(x), x \u2260 0\n       { 0,     x = 0\n\nand h(x) = e^|x| for all x \u2208 IR. Let (f \u2218 h)(x) denote f(h(x)) and (h \u2218 f)(x) denote h(f(x)).\nThen which of the following is (are) true?", "response_choices": "(A) f is differentiable at x = 0\n(B) h is differentiable at x = 0\n(C) f \u2218 h is differentiable at x = 0\n(D) h \u2218 f is differentiable at x = 0", "answer": "(A) and (D)", "page_number": 25, "image_available": "no", "question_subject": "mathematics"}
{"id": 57, "question": "Let f(x) = sin(\u03c0/6 - sin(\u03c0/2)sin x) for all x \u2208 \u211d and g(x) = \u03c0/2 sin x for all x \u2208 \u211d. Let (f \u2218 g)(x) denote f(g(x)) and (g \u2218 f)(x) denote g(f(x)). Then which of the following is (are) true?", "response_choices": "(A) Range of f is [-1, 1] (B) Range of f \u2218 g is [-1, 1] (C) lim f(x)/\u03c0 = x\u21920 g(x) 6 (D) There is an x \u2208 \u211d such that (g \u2218 f)(x) = 1", "answer": "(A), (B) and (C)", "page_number": 26, "image_available": "no", "question_subject": "mathematics"}
{"id": 58, "question": "Let \u0394PQR be a triangle. Let a\u20d7 = QR\u20d7, b\u20d7 = RP\u20d7 and c\u20d7 = PQ\u20d7. If |a\u20d7| = 12, |b\u20d7| = 4\u221a3 and b\u20d7, c\u20d7 = 24, then which of the following is (are) true?", "response_choices": "(A) |c\u20d7|^2 / 2 - |a\u20d7| = 12 (B) |c\u20d7|^2 / 2 + |a\u20d7| = 30 (C) |a\u20d7 \u00d7 b\u20d7 + c\u20d7 \u00d7 a\u20d7| = 48\u221a3 (D) a\u20d7.b\u20d7 = -72", "answer": "(A), (C) and (D)", "page_number": 26, "image_available": "no", "question_subject": "mathematics"}
{"id": 1, "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).", "response_choices": "Column I: (A), (B), (C), (D) \nColumn II: (P), (Q), (R), (S), (T)", "answer": "Matrix responses by darkening bubbles", "page_number": 27, "image_available": "yes", "question_subject": "mathematics"}
{"id": 59, "question": "Column I: (A) In R^3, if the magnitude of the projection vector of the vector \u03b1 i + \u03b2 j on \u221a(3i + j) is \u221a3 and if \u03b1 = 2 + \u221a3\u03b2, then possible value(s) of |\u03b1| 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 \u2265 1 } is differentiable for all x \u2208 R. Then possible value(s) of a is (are) (C) Let \u03c9 \u2260 1 be a complex cube root of unity. If (3 - 3\u03c9 + 2\u03c9^2)^n+3 + (2 + 3\u03c9 - 3\u03c9^2)^n+3 + (-3 + 2\u03c9 + 3\u03c9^2)^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", "response_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)", "page_number": 28, "image_available": "no", "question_subject": "mathematics"}
{"id": 60, "question": "In a triangle \u0394XYZ, 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 \u03bb = sin(X - Y) / sin Z, then possible values of n for which cos(n\u03c0\u03bb) = 0 is (are)\n\nIn a triangle \u0394XYZ, 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)\n\nIn R^2, let \u03b2\u221a3 i + j, i + \u221a3 j and \u03b2 i + (1 - \u03b2) 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/\u221a5 , then possible value(s) of | \u03b2 | is (are)\n\nSuppose that F(\u03b1) denotes the area of the region bounded by x = 0, x = 2, y^2 = 4x and y = | \u03b1 x - 1 | + | \u03b1 x - 2 | + \u03b1 x, where \u03b1 \u2208 {0, 1}. Then the value(s) of F(\u03b1) + 8\u221a2/5 when \u03b1 = 0 and \u03b1 = 1, is (are)", "response_choices": "Column I: (A), (B), (C), (D)\nColumn II: (P) 1, (Q) 2, (R) 3, (S) 5, (T) 6", "answer": "(A) Matches to (P), (R) and (S)\n(B) Matches to (P)\n(C) Matches to (P) and (Q)\n(D) Matches to (S) and (T)", "page_number": 29, "image_available": "no", "question_subject": "mathematics"}