| {"id": 1, "question": "The electrostatic energy of Z protons uniformly distributed throughout a spherical nucleus of radius R is given by E=3Z(Z-1)e^2/5(4\u03c0\u03f5\u2080R). The measured masses of the neutron, \u00b9H, \u00b9\u2075N and \u00b9\u2077O are 1.008665 u, 1.007825 u, 15.000109 u and 15.003065 u, respectively. Given that the radii of both the \u00b9\u2075N and \u00b9\u2077O nuclei are same, 1 u = 931.5 MeV/c^2 (c is the speed of light) and e\u00b2/(4\u03c0\u03f5\u2080)=1.44 MeV fm. Assuming that the difference between the binding energies of \u00b9\u2075N and \u00b9\u2077O is purely due to the electrostatic energy, the radius of either of the nuclei is (1 fm = 10^-15 m)", "response_choices": "(A) 2.85 fm (B) 3.03 fm (C) 3.42 fm (D) 3.80 fm", "answer": "C", "page_number": 1, "image_available": "no", "question_subject": "physics"} | |
| {"id": 2, "question": "An accident in a nuclear laboratory resulted in deposition of a certain amount of radioactive material of half-life 18 days inside the laboratory. Tests revealed that the radiation was 64 times more than the permissible level required for safe operation of the laboratory. What is the minimum number of days after which the laboratory can be considered safe for use?", "response_choices": "(A) 64 (B) 90 (C) 108 (D) 120", "answer": "C", "page_number": 1, "image_available": "no", "question_subject": "physics"} | |
| {"id": 3, "question": "A gas is enclosed in a cylinder with a movable frictionless piston. Its initial thermodynamic state at pressure Pi=10^6 Pa and volume Vi=10^(-3) m^3 changes to a final state at Pf=(1/32)\u00d710^6 Pa and Vf=8\u00d710^(-3) m^3 in an adiabatic quasi-static process, such that PV^(gamma)= constant. Consider another thermodynamic process that brings the system from the same initial state to the same final state in two steps: an isobaric expansion at Pi followed by an isochoric (isovolumetric) process at volume Vf. The amount of heat supplied to the system in the two-step process is approximately", "response_choices": "(A) 112 J (B) 294 J (C) 588 J (D) 813 J", "answer": "C", "page_number": 2, "image_available": "no", "question_subject": "physics"} | |
| {"id": 4, "question": "The ends Q and R of two thin wires, PQ and RS, are soldered (joined) together. Initially each of the wires has a length of 1 m at 10 \u00b0C. Now the end P is maintained at 10 \u00b0C, while the end S is heated and maintained at 400 \u00b0C. The system is thermally insulated from its surroundings. If the thermal conductivity of wire PQ is twice that of the wire RS and the coefficient of linear thermal expansion of PQ is 1.2 \u00d7 10^(-5) K^(-1), the change in length of the wire PQ is", "response_choices": "(A) 0.78 mm (B) 0.90 mm (C) 1.56 mm (D) 2.34 mm", "answer": "A", "page_number": 2, "image_available": "no", "question_subject": "physics"} | |
| {"id": 5, "question": "A small object is placed 50 cm to the left of a thin convex lens of focal length 30 cm. A convex spherical mirror of radius of curvature 100 cm is placed to the right of the lens at a distance of 50 cm. The mirror is tilted such that the axis of the mirror is at an angle \u03b8 = 30\u00b0 to the axis of the lens, as shown in the figure. If the origin of the coordinate system is taken to be at the centre of the lens, the coordinates (in cm) of the point (x, y) at which the image is formed are", "response_choices": "(A) (0, 0) (B) (50 - 25\u221a3, 25) (C) (25, 25\u221a3) (D) (125/3, 25/\u221a3)", "answer": "(C)", "page_number": 3, "image_available": "yes", "question_subject": "physics"} | |
| {"id": 6, "question": "There are two Vernier calipers both of which have 1 cm divided into 10 equal divisions on the main scale. The Vernier scale of one of the calipers (C1) has 10 equal divisions that correspond to 9 main scale divisions. The Vernier scale of the other caliper (C2) has 10 equal divisions that correspond to 11 main scale divisions. The readings of the two calipers are shown in the figure. The measured values (in cm) by calipers C1 and C2, respectively, are", "response_choices": "(A) 2.85 and 2.82 (B) 2.87 and 2.83 (C) 2.87 and 2.86 (D) 2.87 and 2.87", "answer": "B", "page_number": 4, "image_available": "yes", "question_subject": "physics"} | |
| {"id": 7, "question": "Two thin circular discs of mass m and 4m, having radii of a and 2a, respectively, are rigidly fixed by a massless, rigid rod of length l = \u221a24a through their centers. This assembly is laid on a firm and flat surface, and set rolling without slipping on the surface so that the angular speed about the axis of the rod is \u03c9. The angular momentum of the entire assembly about the point 'O' is L (see the figure). Which of the following statement(s) is(are) true?", "response_choices": "(A) The center of mass of the assembly rotates about the z-axis with an angular speed of \u03c9/5, (B) The magnitude of angular momentum of center of mass of the assembly about the point O is 81ma\u00b2\u03c9, (C) The magnitude of angular momentum of the assembly about its center of mass is 17ma\u00b2\u03c9/2, (D) The magnitude of the z-component of L is 55ma\u00b2\u03c9", "answer": "(A) OR (A) and (C)", "page_number": 5, "image_available": "yes", "question_subject": "physics"} | |
| {"id": 8, "question": "Light of wavelength \u03bbph falls on a cathode plate inside a vacuum tube as shown in the figure. The work function of the cathode surface is \u03d5 and the anode is a wire mesh of conducting material kept at a distance d from the cathode. A potential difference V is maintained between the electrodes. If the minimum de Broglie wavelength of the electrons passing through the anode is \u03bbc, which of the following statement(s) is(are) true?", "response_choices": "(A) \u03bbc decreases with increase in \u03d5 and \u03bbph\n(B) \u03bbc is approximately halved, if d is doubled\n(C) For large potential difference (V \\gg \u03d5e), \u03bbc is approximately halved if V is made four times\n(D) \u03bbc increases at the same rate as \u03bbph for \u03bbph < hc/\u03d5", "answer": "C", "page_number": 6, "image_available": "yes", "question_subject": "physics"} | |
| {"id": 9, "question": "In an experiment to determine the acceleration due to gravity g, the formula used for the time period of a periodic motion is T' = 2\u03c0\u221a((7(R - r) / 5g). The values of R and r are measured to be (60 \u00b1 1) mm and (10 \u00b1 1) mm, respectively. In five successive measurements, the time period is found to be 0.52 s, 0.56 s, 0.57 s, 0.54 s and 0.59 s. The least count of the watch used for the measurement of time period is 0.01 s. Which of the following statement(s) is(are) true?", "response_choices": "(A) The error in the measurement of r is 10% (B) The error in the measurement of T' is 3.57% (C) The error in the measurement of T' is 2% (D) The error in the determined value of g is 11%", "answer": "(A), (B) and (D)", "page_number": 7, "image_available": "no", "question_subject": "physics"} | |
| {"id": 10, "question": "Consider two identical galvanometers and two identical resistors with resistance R. If the internal resistance of the galvanometers Rc < R/2, which of the following statement(s) about any one of the galvanometers is(are) true?", "response_choices": "(A) The maximum voltage range is obtained when all the components are connected in series (B) The maximum voltage range is obtained when the two resistors and one galvanometer are connected in series, and the second galvanometer is connected in parallel to the first galvanometer (C) The maximum current range is obtained when all the components are connected in parallel (D) The maximum current range is obtained when the two galvanometers are connected in series and the combination is connected in parallel with both the resistors", "answer": "(B) and (C)", "page_number": 7, "image_available": "no", "question_subject": "physics"} | |
| {"id": 11, "question": "In the circuit shown below, the key is pressed at time t = 0. Which of the following statement(s) is(are) true?", "response_choices": "(A) The voltmeter displays - 5 V as soon as the key is pressed, and displays + 5 V after a long time\n(B) The voltmeter will display 0 V at time t = ln 2 seconds\n(C) The current in the ammeter becomes 1/e of the initial value after 1 second\n(D) The current in the ammeter becomes zero after a long time", "answer": "(A), (B), (C) and (D)", "page_number": 8, "image_available": "yes", "question_subject": "physics"} | |
| {"id": 12, "question": "A block with mass M is connected by a massless spring with stiffness constant k to a rigid wall and moves without friction on a horizontal surface. The block oscillates with small amplitude A about an equilibrium position x0. Consider two cases: (i) when the block is at x0; and (ii) when the block is at x = x0 + A. In both the cases, a particle with mass m(< M) is softly placed on the block after which they stick to each other. Which of the following statement(s) is(are) true about the motion after the mass m is placed on the mass M?", "response_choices": "(A) The amplitude of oscillation in the first case changes by a factor of \u221a(M/(m+M)), whereas in the second case it remains unchanged\n(B) The final time period of oscillation in both the cases is same\n(C) The total energy decreases in both the cases\n(D) The instantaneous speed at x0 of the combined masses decreases in both the cases", "answer": "(A), (B) and (D)", "page_number": 8, "image_available": "no", "question_subject": "physics"} | |
| {"id": 13, "question": "While conducting the Young's double slit experiment, a student replaced the two slits with a large opaque plate in the x-y plane containing two small holes that act as two coherent point sources (S1, S2) emitting light of wavelength 600 nm. The student mistakenly placed the screen parallel to the x-z plane (for z > 0) at a distance D = 3 m from the mid-point of S1S2, as shown schematically in the figure. The distance between the sources d = 0.6003 mm. The origin O is at the intersection of the screen and the line joining S1S2. Which of the following is(are) true of the intensity pattern on the screen?", "response_choices": "(A) Straight bright and dark bands parallel to the x-axis\n(B) The region very close to the point O will be dark\n(C) Hyperbolic bright and dark bands with foci symmetrically placed about O in the x-direction\n(D) Semi circular bright and dark bands centered at point O", "answer": "(B) and (D)", "page_number": 9, "image_available": "yes", "question_subject": "physics"} | |
| {"id": 14, "question": "A rigid wire loop of square shape having side of length L and resistance R is moving along the x-axis with a constant velocity v0 in the plane of the paper. At t = 0, the right edge of the loop enters a region of length 3L where there is a uniform magnetic field B0 into the plane of the paper, as shown in the figure. For sufficiently large v0, the loop eventually crosses the region. Let x be the location of the right edge of the loop. Let v(x), I(x) and F(x) represent the velocity of the loop, current in the loop, and force on the loop, respectively, as a function of x. Counter-clockwise current is taken as positive. Which of the following schematic plot(s) is(are) correct? (Ignore gravity)", "response_choices": "A, B, C, D", "answer": "A and B", "page_number": 10, "image_available": "yes", "question_subject": "physics"} | |
| {"id": 1, "question": "This section contains instructions about the format and scoring of the following questions. No specific question text is provided.", "response_choices": "", "answer": "", "page_number": 11, "image_available": "no", "question_subject": "mathematics"} | |
| {"id": 15, "question": "The distance r of the block at time t is", "response_choices": "(A) R/4 (e\u03c9t + e-\u03c9t) (B) R/2 cos \u03c9t (C) R/4 (e2\u03c9t + e-2\u03c9t) (D) R/2 cos 2\u03c9t", "answer": "A", "page_number": 12, "image_available": "yes", "question_subject": "physics"} | |
| {"id": 16, "question": "The net reaction of the disc on the block is", "response_choices": "(A) 1/2 m\u03c92R(e2\u03c9t - e-2\u03c9t) j + mgk (B) 1/2 m\u03c92R(e\u03c9t - e-\u03c9t) j + mgk (C) - m\u03c92R cos \u03c9t j - mgk (D) m\u03c92R sin \u03c9t j - mgk", "answer": "B", "page_number": 12, "image_available": "yes", "question_subject": "physics"} | |
| {"id": 17, "question": "Which one of the following statements is correct?", "response_choices": "(A) The balls will stick to the top plate and remain there\n(B) The balls will bounce back to the bottom plate carrying the same charge they went up with\n(C) The balls will bounce back to the bottom plate carrying the opposite charge they went up with\n(D) The balls will execute simple harmonic motion between the two plates", "answer": "C", "page_number": 13, "image_available": "yes", "question_subject": "physics"} | |
| {"id": 18, "question": "The average current in the steady state registered by the ammeter in the circuit will be", "response_choices": "(A) zero\n(B) proportional to the potential V\u2080\n(C) proportional to V\u2080\u00b9\u2044\u00b2\n(D) proportional to V\u2080\u00b2", "answer": "D", "page_number": 13, "image_available": "yes", "question_subject": "physics"} | |
| {"id": 19, "question": "For the following electrochemical cell at 298 K, Pt(s)|H2(g,1 bar)| H\u207a(aq,1 M)| M\u207a\u207a(aq),M\u207a\u207a(aq)| Pt(s) E_cell = 0.092 V when [M^(+2)]/[M^(+4)] = 10^4. Given : E^0_(M^(+4)/M^(+2)) = 0.151 V;2.303RT/F = 0.059 V The value of x is", "response_choices": "(A) -2 (B) -1 (C) 1 (D) 2", "answer": "D", "page_number": 14, "image_available": "no", "question_subject": "chemistry"} | |
| {"id": 20, "question": "The qualitative sketches I, II and III given below show the variation of surface tension with molar concentration of three different aqueous solutions of KCl, CH3OH and CH3(CH2)11OSO3-Na+ at room temperature. The correct assignment of the sketches is", "response_choices": "(A) I : KCl \n\nII : CH3OH \n\nIII : CH3(CH2)11OSO3-Na+\n\n(B) I : CH3(CH2)11OSO3-Na+ II : CH3OH \n\nIII : KCl\n\n(C) I : KCl \n\nII : CH3(CH2)11OSO3-Na+ III : CH3OH\n\n(D) I : CH3OH \n\nII : KCl \n\nIII : CH3(CH2)11OSO3-Na+", "answer": "D", "page_number": 15, "image_available": "yes", "question_subject": "chemistry"} | |
| {"id": 21, "question": "In the following reaction sequence in aqueous solution, the species X, Y and Z, respectively, are", "response_choices": "(A) [Ag(S2O3)]3-, Ag2S2O3, Ag2S\n\n(B) [Ag(S2O3)3]5-, Ag2SO3, Ag2S\n\n(C) [Ag(SO4)2]3-, Ag2S2O3, Ag\n\n(D) [Ag(SO4)3]3-, Ag2SO4, Ag", "answer": "A", "page_number": 15, "image_available": "no", "question_subject": "chemistry"} | |
| {"id": 22, "question": "The geometries of the ammonia complexes of Ni2+, Pt2+ and Zn2+, respectively, are", "response_choices": "(A) octahedral, square planar and tetrahedral\n(B) square planar, octahedral and tetrahedral\n(C) tetrahedral, square planar and octahedral\n(D) octahedral, tetrahedral and square planar", "answer": "(A)", "page_number": 16, "image_available": "no", "question_subject": "chemistry"} | |
| {"id": 23, "question": "The correct order of acidity for the following compounds is", "response_choices": "(A) I > II > III > IV\n(C) III > IV > II > I\n(B) III > I > II > IV\n(D) I > III > IV > II", "answer": "(A)", "page_number": 16, "image_available": "yes", "question_subject": "chemistry"} | |
| {"id": 24, "question": "The major product of the following reaction sequence is", "response_choices": "(A) \n(B) \n(C) \n(D) ", "answer": "(A)", "page_number": 16, "image_available": "yes", "question_subject": "chemistry"} | |
| {"id": 25, "question": "According to Molecular Orbital Theory,", "response_choices": "(A) C2+ is expected to be diamagnetic\n(B) O2+ is expected to have a longer bond length than O2\n(C) N2+ and N2- have the same bond order\n(D) He2+ has the same energy as two isolated He atoms", "answer": "(A) and (C)", "page_number": 17, "image_available": "no", "question_subject": "chemistry"} | |
| {"id": 26, "question": "Mixture(s) showing positive deviation from Raoult's law at 35 \u00b0C is(are)", "response_choices": "(A) carbon tetrachloride + methanol\n(B) carbon disulphide + acetone\n(C) benzene + toluene\n(D) phenol + aniline", "answer": "(A) and (B)", "page_number": 17, "image_available": "no", "question_subject": "chemistry"} | |
| {"id": 27, "question": "The CORRECT statement(s) for cubic close packed (ccp) three dimensional structure is(are)", "response_choices": "(A) The number of the nearest neighbours of an atom present in the topmost layer is 12\n(B) The efficiency of atom packing is 74%\n(C) The number of octahedral and tetrahedral voids per atom are 1 and 2, respectively\n(D) The unit cell edge length is 2\u221a2 times the radius of the atom", "answer": "(B), (C) and (D)", "page_number": 17, "image_available": "no", "question_subject": "chemistry"} | |
| {"id": 28, "question": "Extraction of copper from copper pyrite (CuFeS\u2082) involves", "response_choices": "(A) crushing followed by concentration of the ore by froth-flotation\n(B) removal of iron as slag\n(C) self-reduction step to produce 'blister copper' following evolution of SO\u2082\n(D) refining of 'blister copper' by carbon reduction", "answer": "(A), (B) and (C)", "page_number": 18, "image_available": "no", "question_subject": "chemistry"} | |
| {"id": 29, "question": "The nitrogen containing compound produced in the reaction of HNO\u2083 with P\u2084O\u2081\u2080", "response_choices": "(A) can also be prepared by reaction of P\u2084 and HNO\u2083\n(B) is diamagnetic\n(C) contains one N-N bond\n(D) reacts with Na metal producing a brown gas", "answer": "(B) and (D)", "page_number": 18, "image_available": "no", "question_subject": "chemistry"} | |
| {"id": 30, "question": "For 'invert sugar', the correct statement(s) is(are)\n(Given: specific rotations of (+)-sucrose, (+)-maltose, L-(-)-glucose and L-(+)-fructose in aqueous solution are +66\u00b0, +140\u00b0, -52\u00b0 and +92\u00b0, respectively)", "response_choices": "(A) 'invert sugar' is prepared by acid catalyzed hydrolysis of maltose\n(B) 'invert sugar' is an equimolar mixture of D-(+)-glucose and D-(-)-fructose\n(C) specific rotation of 'invert sugar' is -20\u00b0\n(D) on reaction with Br\u2082 water, 'invert sugar' forms saccharic acid as one of the products", "answer": "(B) and (C)", "page_number": 18, "image_available": "no", "question_subject": "chemistry"} | |
| {"id": 31, "question": "Reagent(s) which can be used to bring about the following transformation is(are)", "response_choices": "(A) LiAlH4 in (C2H5)2O, (B) BH4 in THF, (C) NaBH4 in C2H5OH, (D) Raney Ni/H2 in THF", "answer": "(C) and (D)", "page_number": 19, "image_available": "yes", "question_subject": "chemistry"} | |
| {"id": 32, "question": "Among the following, reaction(s) which gives(give) tert-butyl benzene as the major product is(are)", "response_choices": "(A), (B), (C), (D)", "answer": "(B), (C) and (D)", "page_number": 19, "image_available": "yes", "question_subject": "chemistry"} | |
| {"id": 33, "question": "The equilibrium constant Kp for this reaction at 298 K, in terms of \u03b2equilibrium, is", "response_choices": "(A) 8\u03b2\u00b2equilibrium/(2 - \u03b2equilibrium) (B) 8\u03b2\u00b2equilibrium/(4 - \u03b2\u00b2equilibrium) (C) 4\u03b2\u00b2equilibrium/(2 - \u03b2equilibrium) (D) 4\u03b2\u00b2equilibrium/(4 - \u03b2\u00b2equilibrium)", "answer": "B", "page_number": 20, "image_available": "no", "question_subject": "chemistry"} | |
| {"id": 34, "question": "The INCORRECT statement among the following, for this reaction, is", "response_choices": "(A) Decrease in the total pressure will result in formation of more moles of gaseous X (B) At the start of the reaction, dissociation of gaseous X\u2082 takes place spontaneously (C) \u03b2equilibrium = 0.7 (D) Kc < 1", "answer": "C", "page_number": 20, "image_available": "no", "question_subject": "chemistry"} | |
| {"id": 35, "question": "The compound R is", "response_choices": "(A) NH2, (B) Br, (C) NHBr, (D) NBr", "answer": "(A)", "page_number": 21, "image_available": "yes", "question_subject": "chemistry"} | |
| {"id": 36, "question": "The compound T is", "response_choices": "(A) glycine, (B) alanine, (C) valine, (D) serine", "answer": "(B)", "page_number": 21, "image_available": "no", "question_subject": "chemistry"} | |
| {"id": 37, "question": "Let P = [1 0 0; 4 1 0; 16 4 1] and I be the identity matrix of order 3. If Q = [qij] is a matrix such that P^50 - Q = I, then q31 + q32 / q21 equals", "response_choices": "(A) 52 (B) 103 (C) 201 (D) 205", "answer": "B", "page_number": 22, "image_available": "no", "question_subject": "mathematics"} | |
| {"id": 38, "question": "Let bi > 1 for i = 1, 2, ..., 101. Suppose log2 b1, log2 b2, ..., log2 b101 are in Arithmetic Progression (A.P.) with the common difference log2 2. Suppose a1, a2, ..., a101 are in A.P. such that a1 = b1 and a51 = b51. If t = b1 + b2 + ... + b51 and s = a1 + a2 + ... + a51, then", "response_choices": "(A) s > t and a101 > b101 (B) s > t and a101 < b101 (C) s < t and a101 > b101 (D) s < t and a101 < b101", "answer": "B", "page_number": 22, "image_available": "no", "question_subject": "mathematics"} | |
| {"id": 39, "question": "The value of \u03a3 (1 / sin(\u03c0/4 + (k - 1)\u03c0/6) * sin(\u03c0/4 + k\u03c0/6)) from k=1 to 3 is equal to", "response_choices": "(A) 3 - \u221a3 (B) 2(3 - \u221a3) (C) 2(\u221a3 - 1) (D) 2(2 + \u221a3)", "answer": "C", "page_number": 22, "image_available": "no", "question_subject": "mathematics"} | |
| {"id": 40, "question": "The value of \u222b(x^2 cos x / 1 + e^(x/2)) dx is equal to", "response_choices": "(A) \u03c0^2/4 - 2, (B) \u03c0^2/4 + 2, (C) x^2 - e^x, (D) x^2 + e^(x/2)", "answer": "A", "page_number": 23, "image_available": "no", "question_subject": "mathematics"} | |
| {"id": 41, "question": "Area of the region {(x, y) \u2208 R^2 : y \u2265 \u221a(x + 3), 5y \u2264 x + 9 \u2264 15} is equal to", "response_choices": "(A) 1/6, (B) 4/3, (C) 3/2, (D) 5/3", "answer": "C", "page_number": 23, "image_available": "yes", "question_subject": "mathematics"} | |
| {"id": 42, "question": "Let P be the image of the point (3, 1, 7) with respect to the plane x - y + z = 3. Then the equation of the plane passing through P and containing the straight line x/2 = y/2 = z/1 is", "response_choices": "(A) x + y - 3z = 0, (B) 3x + z = 0, (C) x - 4y + 7z = 0, (D) 2x - y = 0", "answer": "C", "page_number": 23, "image_available": "no", "question_subject": "mathematics"} | |
| {"id": 43, "question": "Let f(x) = lim [n^(x+n)/(x+n/2)*...(x+n)/n!] / [n!*(x^2+n^2)*(x^2+n^2/4)*...(x^2+n^4/n^2)], for all x > 0. Then", "response_choices": "(A) f(1/2) \u2265 f(1) (B) f(1/3) \u2264 f(2/3) (C) f'(2) \u2264 0 (D) f'(3)/f(3) \u2265 f'(2)/f(2)", "answer": "(B) and (C)", "page_number": 24, "image_available": "yes", "question_subject": "mathematics"} | |
| {"id": 44, "question": "Let a, b \u2208 \u211d and f : \u211d\u2192\u211d be defined by f(x) = a cos(|x^3 - x|) + b|x|sin(|x^3 + x|). Then f is", "response_choices": "(A) differentiable at x = 0 if a = 0 and b = 1 (B) differentiable at x = 1 if a = 1 and b = 0 (C) NOT differentiable at x = 0 if a = 1 and b = 0 (D) NOT differentiable at x = 1 if a = 1 and b = 1", "answer": "(A) and (B)", "page_number": 24, "image_available": "no", "question_subject": "mathematics"} | |
| {"id": 45, "question": "Let f : R \u2192(0, \u221e) and g : R \u2192 R be twice differentiable functions such that f'' and g'' are continuous functions on R. Suppose f''(2)=g(2)=0, f''(2)\u22600 and g'(2)\u22600. If lim(x\u21922) f(x)/g(x) = -1, then", "response_choices": "(A) f has a local minimum at x = 2\n(B) f has a local maximum at x = 2\n(C) f''(2) > f(2)\n(D) f(x) - f''(x) = 0 for at least one x \u2208 R", "answer": "(A) and (D)", "page_number": 25, "image_available": "no", "question_subject": "mathematics"} | |
| {"id": 46, "question": "Let f : [1/2, 2] \u2192 R and g : [-1/2, 2] \u2192 R be functions defined by f(x) = [x^2 - 3] and g(x) = [x] f(x) + [4x - 7] f(x), where [y] denotes the greatest integer less than or equal to y for y \u2208 R. Then", "response_choices": "(A) f is discontinuous exactly at three points in [-1/2, 2]\n(B) f is discontinuous exactly at four points in [-1/2, 2]\n(C) g is NOT differentiable exactly at four points in (-1/2, 2)\n(D) g is NOT differentiable exactly at five points in (-1/2, 2)", "answer": "(B) and (C)", "page_number": 25, "image_available": "no", "question_subject": "mathematics"} | |
| {"id": 47, "question": "Let a, b \u2208 \u211d and a^2 + b^2 \u2260 0. Suppose S = {z \u2208 \u2102 : z = (1 +/- i sqrt(-1)t) / (a +/- ibt), t \u2208 \u211d, t \u2260 0}. If z = x + iy and z \u2208 S, then (x, y) lies on", "response_choices": "(A) the circle with radius 1/2a and centre (1/2a, 0) for a > 0, b \u2260 0\n(B) the circle with radius -1/2a and centre (-1/2a, 0) for a < 0, b \u2260 0\n(C) the x-axis for a \u2260 0, b = 0\n(D) the y-axis for a = 0, b \u2260 0", "answer": "(A), (C) and (D)", "page_number": 26, "image_available": "no", "question_subject": "mathematics"} | |
| {"id": 48, "question": "Let P be the point on the parabola y^2 = 4x which is at the shortest distance from the center S of the circle x^2 + y^2 - 4x - 16y + 64 = 0. Let Q be the point on the circle dividing the line segment SP internally. Then", "response_choices": "(A) SP = 2\u221a5\n(B) SQ : QP = (\u221a5 + 1) : 2\n(C) the x-intercept of the normal to the parabola at P is 6\n(D) the slope of the tangent to the circle at Q is 1/2", "answer": "(A), (C) and (D)", "page_number": 26, "image_available": "no", "question_subject": "mathematics"} | |
| {"id": 49, "question": "Let \u03b1, \u03bb, \u03bc \u2208 \u211d. Consider the system of linear equations \u03b1x + 2y = \u03bb 3x - 2y = \u03bc Which of the following statement(s) is(are) correct?", "response_choices": "(A) If \u03b1 = -3, then the system has infinitely many solutions for all values of \u03bb and \u03bc (B) If \u03b1 \u2260 -3, then the system has a unique solution for all values of \u03bb and \u03bc (C) If \u03bb + \u03bc = 0, then the system has infinitely many solutions for \u03b1 = -3 (D) If \u03bb + \u03bc \u2260 0, then the system has no solution for \u03b1 = -3", "answer": "(B), (C) and (D)", "page_number": 27, "image_available": "no", "question_subject": "mathematics"} | |
| {"id": 50, "question": "Let \u016b = u\u2081i + u\u2082j + u\u2083k be a unit vector in \u211d\u00b3 and \u0175 = (1/\u221a6)(i + j + 2k). Given that there exists a vector v\u0304 in \u211d\u00b3 such that |\u016b \u00d7 v\u0304| = 1 and \u0175 \u00b7 (\u016b \u00d7 v\u0304) = 1. Which of the following statement(s) is(are) correct?", "response_choices": "(A) There is exactly one choice for such v\u0304 (B) There are infinitely many choices for such v\u0304 (C) If \u016b lies in the xy-plane then |u\u2081| = |u\u2082| (D) If \u016b lies in the xz-plane then 2|u\u2081| = |u\u2083|", "answer": "(B) and (C)", "page_number": 27, "image_available": "no", "question_subject": "mathematics"} | |
| {"id": 51, "question": "P(X > Y) is", "response_choices": "A) 1/4, B) 5/12, C) 1/2, D) 7/12", "answer": "B", "page_number": 28, "image_available": "no", "question_subject": "mathematics"} | |
| {"id": 52, "question": "P(X = Y) is", "response_choices": "A) 11/36, B) 1/3, C) 13/36, D) 1/2", "answer": "C", "page_number": 28, "image_available": "no", "question_subject": "mathematics"} | |
| {"id": 53, "question": "The orthocentre of the triangle FiMN is", "response_choices": "(A) (-9/10, 0) (B) (2/3, 0) (C) (-9/10, 0) (D) (2/\u221a3, 0)", "answer": "(A)", "page_number": 29, "image_available": "no", "question_subject": "mathematics"} | |
| {"id": 54, "question": "If the tangents to the ellipse at M and N meet at R and the normal to the parabola at M meets the x-axis at Q, then the ratio of area of the triangle MQR to area of the quadrilateral MFiNF2 is", "response_choices": "(A) 3 : 4 (B) 4 : 5 (C) 5 : 8 (D) 2 : 3", "answer": "(C)", "page_number": 29, "image_available": "yes", "question_subject": "mathematics"} | |