| {"id": 1, "question": "A thin and uniform rod of mass M and length L is held vertical on a floor with large friction. The rod is released from rest so that it falls by rotating about its contact-point with the floor without slipping. Which of the following statement(s) is/are correct, when the rod makes an angle 60\u00b0 with vertical? [g is the acceleration due to gravity]", "response_choices": "(A) The angular speed of the rod will be \u221a(3g/2), (B) The angular acceleration of the rod will be 2g/L, (C) The radial acceleration of the rod's center of mass will be 3g/4, (D) The normal reaction force from the floor on the rod will be Mg/16", "answer": "B, C", "page_number": 1, "image_available": "no", "question_subject": "physics"} | |
| {"id": 2, "question": "A block of mass 2M is attached to a massless spring with spring-constant k. This block is connected to two other blocks of masses M and 2M using two massless pulleys and strings. The accelerations of the blocks are a1, a2 and a3 as shown in the figure. The system is released from rest with the spring in its unstretched state. The maximum extension of the spring is x0. Which of the following option(s) is/are correct? [g is the acceleration due to gravity. Neglect friction]", "response_choices": "A) x0 = 4Mg/k\nB) When spring achieves an extension of x0/2 for the first time, the speed of the block connected to the spring is 3g sqrt(M/5k)\nC) At an extension of x0/4 of the spring, the magnitude of acceleration of the block connected to the spring is 3g/10\nD) a2 - a1 = a1 - a3", "answer": "A, D", "page_number": 2, "image_available": "yes", "question_subject": "physics"} | |
| {"id": 3, "question": "A small particle of mass m moving inside a heavy, hollow and straight tube along the tube axis undergoes elastic collision at two ends. The tube has no friction and it is closed at one end by a flat surface while the other end is fitted with a heavy movable flat piston as shown in figure. When the distance of the piston from closed end is L = L0, the particle speed is v = v0. The piston is moved inward at a very low speed V such that V << dL/v0 , where dL is the infinitesimal displacement of the piston. Which of the following statement(s) is/are correct?", "response_choices": "(A) The rate at which the particle strikes the piston is v/L\n(B) After each collision with the piston, the particle speed increases by 2V\n(C) If the piston moves inward by dL, the particle speed increases by 2 v dL/L\n(D) The particle's kinetic energy increases by a factor of 4 when the piston is moved inward from L0 to 1/2 L0", "answer": "C", "page_number": 3, "image_available": "yes", "question_subject": "physics"} | |
| {"id": 4, "question": "An electric dipole with dipole moment p0/(i + j) is held fixed at the origin O in the presence of an uniform electric field of magnitude E0. If the potential is constant on a circle of radius R centered at the origin as shown in figure, then the correct statement(s) is/are: (e0 is permittivity of free space. R >> dipole size)", "response_choices": "(A) R = ( p0 )^(1/3) (4\u03c0\u03b50E0) (B) Total electric field at point A is BA = \u221a2 E0 (i + j) (C) Total electric field at point B is BB = 0 (D) The magnitude of total electric field on any two points of the circle will be same.", "answer": "A", "page_number": 4, "image_available": "yes", "question_subject": "physics"} | |
| {"id": 5, "question": "A mixture of ideal gas containing 5 moles of monatomic gas and 1 mole of rigid diatomic gas is initially at pressure P0, volume V0, and temperature T0. If the gas mixture is adiabatically compressed to a volume V0/4, then the correct statement(s) is/are, (Given \u03b3^(1/2) = 2.3; \u03b3^(3/2) = 9.2; R is gas constant)", "response_choices": "(A) The work |W| done during the process is 13RT0 (B) The average kinetic energy of the gas mixture after compression is in between 18RT0 and 19RT0 (C) The final pressure of the gas mixture after compression is in between 9P0 and 10P0 (D) Adiabatic constant of the gas mixture is 1.6", "answer": "B", "page_number": 4, "image_available": "no", "question_subject": "physics"} | |
| {"id": 6, "question": "Three glass cylinders of equal height H = 30 cm and same refractive index n = 1.5 are placed on a horizontal surface as shown in figure. Cylinder I has a flat top, cylinder II has a convex top and cylinder III has a concave top. The radii of curvature of the two curved tops are same ( R = 3m). If H2, H2, and H3 are the apparent depths of a point X on the bottom of the three cylinders, respectively, the correct statement(s) is/are:", "response_choices": "A) H2 > H1, B) H3 > H1, C) H2 > H3, D) 0.8 cm < (H2 - H1) < 0.9 cm", "answer": "D", "page_number": 5, "image_available": "yes", "question_subject": "physics"} | |
| {"id": 7, "question": "In a Young's double slit experiment, the slit separation d is 0.3 mm and the screen distance D is 1 m. A parallel beam of light of wavelength 600 nm is incident on the slits at angle \u03b1 as shown in figure. On the screen, the point O is equidistant from the slits and distance P O is 11.0 mm. Which of the following statement(s) is/are correct?", "response_choices": "(A) For \u03b1 = 0.3\u03b1 degree, there will be destructive interference at point O. (B) For \u03b1 = 0, there will be constructive interference at point P. (C) For \u03b1 = 0.3\u03b1 \u03bb degree, there will be destructive interference at point P. (D) Fringe spacing depends on \u03b1.", "answer": "", "page_number": 6, "image_available": "yes", "question_subject": "physics"} | |
| {"id": 8, "question": "A free hydrogen atom after absorbing a photon of wavelength \u03bbin gets excited from the state n = 1 to the state n = 4. Immediately after that the electron jumps to n = m state by emitting a photon of wavelength \u03bbe. Let the change in momentum of atom due to the absorption and the emission are \u0394p\u2090 and \u0394pe, respectively. If \u03bb\u2090/\u03bbe = 1/5, which of the option(s) is/are correct? [Use hc = 1242 eV nm ; 1 nm = 10\u207b\u2079 m, h and c are Planck's constant and speed of light, respectively]", "response_choices": "(A) m = 2 (B) \u03bbe = 418 nm (C) \u0394p\u2090/\u0394pe = 1/2 (D) The ratio of kinetic energy of the electron in the state n = m to the state n = 1 is 1/4", "answer": "", "page_number": 6, "image_available": "no", "question_subject": "physics"} | |
| {"id": 9, "question": "A ball is thrown from ground at an angle \u03b8 with horizontal and with an initial speed u0. For the resulting projectile motion, the magnitude of average velocity of the ball up to the point when it hits the ground for the first time is V1. After hitting the ground, the ball rebounds at the same angle \u03b8 but with a reduced speed of u0/\u03b1. Its motion continues for a long time as shown in figure. If the magnitude of average velocity of the ball for entire duration of motion is 0.8 V1, the value of \u03b1 is __________.", "response_choices": "", "answer": "", "page_number": 7, "image_available": "yes", "question_subject": "physics"} | |
| {"id": 10, "question": "A 10 cm long perfectly conducting wire PQ is moving with a velocity 1 cm/s on a pair of horizontal rails of zero resistance. One side of the rails is connected to an inductor L = 1 mH and a resistance R = 1 \u03a9 as shown in figure. The horizontal rails, L and R lie in the same plane with a uniform magnetic field \ud835\udc35\u20d7 = 1 T perpendicular to the plane. If the key S is closed at certain instant, the current in the circuit after 1 millisecond is x \u00d7 10^-3 A, where the value of x is _______. [Assume the velocity of wire PQ remains constant (1 cm/s) after key S is closed. Given: e^-1 = 0.37, where e is base of the natural logarithm]", "response_choices": "No options provided", "answer": "Blank", "page_number": 8, "image_available": "yes", "question_subject": "physics"} | |
| {"id": 11, "question": "A monochromatic light is incident from air on a refracting surface of a prism of angle 75\u2070 and refractive index n\u2080 = \u221a3. The other refracting surface of the prism is coated by a thin film of material of refractive index n as shown in figure. The light suffers total internal reflection at the coated prism surface for an incidence angle of \u03b8 \u2264 60\u2070. The value of n^2 is _______.", "response_choices": "No options provided", "answer": "Blank", "page_number": 8, "image_available": "yes", "question_subject": "physics"} | |
| {"id": 12, "question": "A perfectly reflecting mirror of mass M mounted on a spring constitutes a spring-mass system of angular frequency \u03a9 such that 4\u03c0M\u03a9/h = 10^24 m^-2 with h as Planck's constant. N photons of wavelength \u03bb = 8\u03c0 x 10^-6 m strike the mirror simultaneously at normal incidence such that the mirror gets displaced by 1 \u03bcm. If the value of N is x \u00d7 10^12, then the value of x is _____. [Consider the spring as massless]", "response_choices": "", "answer": "", "page_number": 9, "image_available": "yes", "question_subject": "physics"} | |
| {"id": 13, "question": "Suppose a \u00b2\u00b2\u2076Ra nucleus at rest and in ground state undergoes \u03b1-decay to a \u00b2\u00b2\u00b2Rn nucleus in its excited state. The kinetic energy of the emitted \u03b1 particle is found to be 4.44 MeV. \u00b2\u00b2\u00b2Rn nucleus then goes to its ground state by \u03b3-decay. The energy of the emitted \u03b3 photon is ____ keV. [Given: atomic mass of \u00b2\u00b2\u2076Ra = 226.005 u, atomic mass of \u00b2\u00b2\u00b2Rn = 222.000 u, atomic mass of \u03b1 particle = 4.000 u, 1 u = 931 MeV/c\u00b2, c is speed of the light]", "response_choices": "", "answer": "", "page_number": 9, "image_available": "no", "question_subject": "physics"} | |
| {"id": 14, "question": "An optical bench has 1.5 m long scale having four equal divisions in each cm. While measuring the focal length of a convex lens, the lens is kept at 75 cm mark of the scale and the object pin is kept at 45 cm mark. The image of the object pin on the other side of the lens overlaps with image pin that is kept at 135 cm mark. In this experiment, the percentage error in the measurement of the focal length of the lens is _____.", "response_choices": "", "answer": "", "page_number": 9, "image_available": "no", "question_subject": "physics"} | |
| {"id": 15, "question": "If the tension in each string is T0, the correct match for the highest fundamental frequency in f0 units will be,", "response_choices": "(A) I \u2192 P, II \u2192 Q, III \u2192 T, IV \u2192 S\n(B) I \u2192 P, II \u2192 R, III \u2192 S, IV \u2192 Q\n(C) I \u2192 Q, II \u2192 S, III \u2192 R, IV \u2192 P\n(D) I \u2192 Q, II \u2192 P, III \u2192 R, IV \u2192 T", "answer": "(B) I \u2192 P, II \u2192 R, III \u2192 S, IV \u2192 Q", "page_number": 11, "image_available": "no", "question_subject": "physics"} | |
| {"id": 16, "question": "The length of the strings 1, 2, 3 and 4 are kept fixed at L0, 3L0/2 , 5L0/4 and 7L0/4 , respectively. Strings 1, 2, 3, and 4 are vibrated at their 1st, 3rd , 5th, and 14th harmonics, respectively such that all the strings have same frequency. The correct match for the tension in the four strings in the units of T0 will be,", "response_choices": "(A) I \u2192 P, II \u2192 R, III \u2192 T, IV \u2192 U\n(B) I \u2192 P, II \u2192 Q, III \u2192 T, IV \u2192 U\n(C) I \u2192 P, II \u2192 Q, III \u2192 R, IV \u2192 T\n(D) I \u2192 T, II \u2192 Q, III \u2192 R, IV \u2192 U", "answer": "(B) I \u2192 P, II \u2192 Q, III \u2192 T, IV \u2192 U", "page_number": 11, "image_available": "no", "question_subject": "physics"} | |
| {"id": 17, "question": "Answer Q.17 and Q.18 by appropriately matching the lists based on the information given in the paragraph", "response_choices": "", "answer": "", "page_number": 12, "image_available": "no", "question_subject": "physics"} | |
| {"id": 18, "question": "In a thermodynamic process on an ideal monatomic gas, the infinitesimal heat absorbed by the gas is given by TdX, where T is the temperature of the system and dX is the infinitesimal change in a thermodynamic quantity X of the system. For a mole of monatomic ideal gas X = (3/2)R ln (Vd/Va) + R ln (Vd/Va). Here, R is gas constant, V is volume of gas, Td and Va are constants.", "response_choices": "", "answer": "", "page_number": 12, "image_available": "no", "question_subject": "physics"} | |
| {"id": 19, "question": "The List-I below gives some quantities involved in a process and List-II gives some possible values of these quantities.", "response_choices": "List-I: (I) Work done by the system in process 1->2->3, (II) Change in internal energy in process 1->2->3, (III) Heat absorbed by the system in process 1->2->3, (IV) Heat absorbed by the system in process 1->2; List-II: (P) 1/2 RTo ln 2, (Q) 1/3 RTo, (R) RTo, (S) 4/3 RTo, (T) 1/3 RTo(3 + ln 2), (U) 5/3 RTo", "answer": "(I) P, (II) Q, (III) R, (IV) S", "page_number": 12, "image_available": "no", "question_subject": "physics"} | |
| {"id": 17, "question": "If the process carried out on one mole of monatomic ideal gas is as shown in figure in the PV-diagram with P0V0 = 4/3 RT0, the correct match is,", "response_choices": "(A) I \u2192 Q, II \u2192 R, III \u2192 P, IV \u2192 U; (B) I \u2192 Q, II \u2192 R, III \u2192 S, IV \u2192 U; (C) I \u2192 S, II \u2192 R, III \u2192 Q, IV \u2192 T; (D) I \u2192 Q, II \u2192 S, III \u2192 R, IV \u2192 U", "answer": "(D) I \u2192 Q, II \u2192 S, III \u2192 R, IV \u2192 U", "page_number": 13, "image_available": "yes", "question_subject": "physics"} | |
| {"id": 18, "question": "If the process on one mole of monatomic ideal gas is as shown in the TV-diagram with P0V0 = 1/3 RT0, the correct match is,", "response_choices": "(A) I \u2192 P, II \u2192 R, III \u2192 T, IV \u2192 S; (B) I \u2192 P, II \u2192 T, III \u2192 Q, IV \u2192 T; (C) I \u2192 P, II \u2192 R, III \u2192 T, IV \u2192 P; (D) I \u2192 S, II \u2192 T, III \u2192 Q, IV \u2192 U", "answer": "(C) I \u2192 P, II \u2192 R, III \u2192 T, IV \u2192 P", "page_number": 13, "image_available": "yes", "question_subject": "physics"} | |
| {"id": 1, "question": "The cyanide process of gold extraction involves leaching out gold from its ore with CN\u207b in the presence of O in water to form R. Subsequently, R is treated with T to obtain Au and Z. Choose the correct option(s)", "response_choices": "(A) O is O\u2082, (B) T is Zn, (C) Z is [Zn(CN)\u2084]\u00b2\u207b, (D) R is [Au(CN)\u2082]\u207b", "answer": "C", "page_number": 14, "image_available": "no", "question_subject": "chemistry"} | |
| {"id": 2, "question": "With reference to aqua regia, choose the correct option(s)", "response_choices": "(A) Aqua regia is prepared by mixing conc. HCl and conc. HNO\u2083 in 3:1 (v/v) ratio, (B) Reaction of gold with aqua regia produces an anion having Au in +3 oxidation state, (C) Reaction of gold with aqua regia produces NO\u2082 in the absence of air, (D) The yellow colour of aqua regia is due to the presence of NOCl and Cl\u2082", "answer": "A,B,D", "page_number": 14, "image_available": "no", "question_subject": "chemistry"} | |
| {"id": 3, "question": "Consider the following reactions (unbalanced)\n\nZn + hot conc. H\u2082SO\u2084 \u2192 G + R + X\nZn + conc. NaOH \u2192 T + Q\nG + H\u2082S + NH\u2084OH \u2192 Z (a precipitate) + X + Y", "response_choices": "(A) Z is dirty white in colour\n\n(B) The oxidation state of Zn in T is +1\n\n(C) R is a V-shaped molecule\n\n(D) Bond order of Q is 1 in its ground state", "answer": "", "page_number": 15, "image_available": "no", "question_subject": "chemistry"} | |
| {"id": 4, "question": "The ground state energy of hydrogen atom is -13.6 eV. Consider an electronic state \u03a8'of He\u207a whose energy, azimuthal quantum number and magnetic quantum number are -3.4 eV, 2 and 0, respectively. Which of the following statement(s) is(are) true for the state \u03a8'?", "response_choices": "(A) It is a 4d state\n\n(B) It has 2 angular nodes\n\n(C) It has 3 radial nodes\n\n(D) The nuclear charge experienced by the electron in this state is less than 2e, where e is the magnitude of the electronic charge", "answer": "", "page_number": 15, "image_available": "no", "question_subject": "chemistry"} | |
| {"id": 5, "question": "Which of the following reactions produce(s) propane as a major product?", "response_choices": "(A)\n\nH\u2083C\u23bb\u23bb\u23bbCOONa NaOH, CaO, \u0394\n\n(B)\n\nH\u2083C\u23bb\u23bb\u23bbCOONa + H\u2082O electrolysis\n\n(C)\n\nH\u2083C\u23bb\u23bb\u23bbCl Zn, dil. HCl\n\n(D)\n\nBr\n| \nH\u2083C\u23bb\u23bb\u23bbBr Zn", "answer": "", "page_number": 15, "image_available": "yes", "question_subject": "chemistry"} | |
| {"id": 6, "question": "Choose the correct option(s) that give(s) an aromatic compound as the major product", "response_choices": "(A) i) alc. KOH, ii) NaNH2, iii) red hot iron tube, 873 K, (B) NaOMe, (C) Br, NaOEt, (D) + Cl2 (excess), UV, 500 K", "answer": "D", "page_number": 16, "image_available": "yes", "question_subject": "chemistry"} | |
| {"id": 7, "question": "Choose the correct option(s) for the following reaction sequence", "response_choices": "A) \n\nMeO Q\n\nMeO S\n\nB) \n\nMeO R\n\nMeO S\n\nC) \n\nMeO Q\n\nMeO R\n\nD) \n\nMeO R\n\nMeO S", "answer": "B", "page_number": 17, "image_available": "yes", "question_subject": "chemistry"} | |
| {"id": 8, "question": "Choose the correct option(s) from the following", "response_choices": "A) Natural rubber is polyisoprene containing trans alkene units\n\nB) Nylon-6 has amide linkages\n\nC) Teflon is prepared by heating tetrafluoroethene in presence of a persulphate catalyst at high pressure\n\nD) Cellulose has only \u03b1-D-glucose units that are joined by glycosidic linkages", "answer": "", "page_number": 17, "image_available": "no", "question_subject": "chemistry"} | |
| {"id": 9, "question": "The amount of water produced (in g) in the oxidation of 1 mole of rhombic sulphur by conc. HNO3 to a compound with the highest oxidation state of sulphur is _____\n(Given data: Molar mass of water = 18 g mol^-1)", "response_choices": "", "answer": "", "page_number": 18, "image_available": "no", "question_subject": "chemistry"} | |
| {"id": 10, "question": "Total number of cis N-Mn-Cl bond angles (that is, Mn-N and Mn-Cl bonds in cis positions) present in a molecule of cis-[Mn(en)2Cl2] complex is _____\n(en = NH2CH2CH2NH2)", "response_choices": "", "answer": "", "page_number": 18, "image_available": "no", "question_subject": "chemistry"} | |
| {"id": 11, "question": "The decomposition reaction 2N2O5(g) \u2192 2N2O4(g) + O2(g) is started in a closed cylinder under isothermal isochoric condition at an initial pressure of 1 atm. After Y \u00d7 10^8 s, the pressure inside the cylinder is found to be 1.45 atm. If the rate constant of the reaction is 5 \u00d7 10^-4 s^-1, assuming ideal gas behavior, the value of Y is _____", "response_choices": "", "answer": "", "page_number": 18, "image_available": "no", "question_subject": "chemistry"} | |
| {"id": 12, "question": "The mole fraction of urea in an aqueous urea solution containing 900 g of water is 0.05. If the density of the solution is 1.2 g cm^-3, the molarity of urea solution is _____\n(Given data: Molar masses of urea and water are 60 g mol^-1 and 18 g mol^-1, respectively)", "response_choices": "", "answer": "", "page_number": 18, "image_available": "no", "question_subject": "chemistry"} | |
| {"id": 13, "question": "Total number of hydroxyl groups present in a molecule of the major product P is ____", "response_choices": "No multiple choice options provided", "answer": "No answer marked", "page_number": 19, "image_available": "yes", "question_subject": "chemistry"} | |
| {"id": 14, "question": "Total number of isomers, considering both structural and stereoisomers, of cyclic ethers with the molecular formula C4H8O is ____", "response_choices": "No multiple choice options provided", "answer": "No answer marked", "page_number": 19, "image_available": "no", "question_subject": "chemistry"} | |
| {"id": 15, "question": "Which of the following options has the correct combination considering List-I and List-II?", "response_choices": "(A) (I), (P) (B) (I), (T) (C) (II), (Q) (D) (II), (R)", "answer": "C", "page_number": 20, "image_available": "no", "question_subject": "physics"} | |
| {"id": 16, "question": "Which of the following options has the correct combination considering List-I and List-II?", "response_choices": "(A) (III), (P) (B) (III), (S) (C) (IV), (Q) (D) (IV), (U)", "answer": "B", "page_number": 20, "image_available": "no", "question_subject": "physics"} | |
| {"id": 17, "question": "Answer Q.17 and Q.18 by appropriately matching the lists based on the information given in the paragraph", "response_choices": "", "answer": "", "page_number": 21, "image_available": "yes", "question_subject": "chemistry"} | |
| {"id": 18, "question": "List-I includes starting materials and reagents of selected chemical reactions. List-II gives structures of compounds that may be formed as intermediate products and/or final products from the reactions of List-I.", "response_choices": "(I) i) DIBAL-H, ii) dil. HCl, iii) NaBH4, iv) conc. H2SO4; (II) i) O3, ii) Zn, H2O, iii) NaBH4, iv) conc. H2SO4; (III) i) KCN, ii) H3O+, \u0394, iii) LiAlH4, iv) conc. H2SO4; (IV) i) LiAlH4, ii) conc. H2SO4; (P) CHO-CO2H; (Q) OH-OH; (R) O; (S) OH-CO2H; (T) CO2H-CO2H; (U) O-O", "answer": "(I) P, (II) Q, (III) R, (IV) S, (V) T", "page_number": 21, "image_available": "yes", "question_subject": "chemistry"} | |
| {"id": 17, "question": "Which of the following options has correct combination considering List-I and List-II?", "response_choices": "(A) (I), (Q), (T), (U) | (B) (II), (P), (S), (T) | (C) (II), (P), (S), (U) | (D) (I), (S), (Q), (R)", "answer": "(B) (II), (P), (S), (T)", "page_number": 22, "image_available": "no", "question_subject": "mathematics"} | |
| {"id": 18, "question": "Which of the following options has correct combination considering List-I and List-II?", "response_choices": "(A) (III), (S), (R) | (B) (IV), (Q), (U) | (C) (III), (T), (U) | (D) (IV), (Q), (R)", "answer": "(B) (IV), (Q), (U)", "page_number": 22, "image_available": "no", "question_subject": "mathematics"} | |
| {"id": 1, "question": "Let P1 = I = [1 0 0; 0 1 0; 0 0 1], P2 = [1 0 0; 0 0 1; 0 1 0], P3 = [0 1 0; 1 0 0; 0 0 1], P4 = [0 1 0; 0 0 1; 1 0 0], P5 = [0 0 1; 1 0 0; 0 1 0], P6 = [0 0 1; 0 1 0; 1 0 0] and X = \u2211(6 k=1) Pk [2 1 3; 1 0 2; 2 2 1]PkT where PkT denotes the transpose of the matrix Pk. Then which of the following options is/are correct? (A) If X [1 1 1]T = \u03b1 [1 1 1]T, then \u03b1 = 30 (B) X is a symmetric matrix (C) The sum of diagonal entries of X is 18 (D) X \u2212 30I is an invertible matrix", "response_choices": "(A) If X [1 1 1]T = \u03b1 [1 1 1]T, then \u03b1 = 30, (B) X is a symmetric matrix, (C) The sum of diagonal entries of X is 18, (D) X \u2212 30I is an invertible matrix", "answer": "", "page_number": 23, "image_available": "no", "question_subject": "mathematics"} | |
| {"id": 2, "question": "Let x \u2208 \u211d and let P = [1 1 1, 0 2 2, 0 0 3], Q = [2 x x, 0 4 0] and R = PQP^(-1). Then which of the following options is/are correct?", "response_choices": "(A) There exists a real number x such that PQ = QP, (B) det R = det [0 4 0] + 8, for all x \u2208 \u211d, (C) For x = 0, if R [a] = 6 [a], then a + b = 5, (D) For x = 1, there exists a unit vector \u03b1\u200bi + \u03b2\u200bj + \u03b3k\u200b\u200b for which R [\u03b2] = [0]", "answer": "(B)", "page_number": 24, "image_available": "yes", "question_subject": "mathematics"} | |
| {"id": 3, "question": "For non-negative integers n, let f(n) = (\u03a3 from k=0 to n) sin((k+1)\u03c0/(n+2)\u03c0) sin((k+2)/(n+2)\u03c0) / (\u03a3 from k=0 to n) sin^2((k+1)\u03c0/(n+2)\u03c0). Assuming cos^(-1)x takes values in [0,\u03c0], which of the following options is/are correct?", "response_choices": "(A) f(4) = \u221a3/2, (B) lim f(n) = 1/2, n->\u221e, (C) If \u03b1 = tan(cos^(-1)f(6)), then \u03b1^2 + 2\u03b1 - 1 = 0, (D) sin(7 cos^(-1)f(5)) = 0", "answer": "(B),(D)", "page_number": 24, "image_available": "no", "question_subject": "mathematics"} | |
| {"id": 4, "question": "Let f: R \u2192 R be a function. We say that f has PROPERTY 1 if lim [f(h) - f(0)]/sqrt(|h|) exists and is finite, and PROPERTY 2 if lim [f(h) - f(0)]/h^2 exists and is finite. Then which of the following options is/are correct?", "response_choices": "(A) f(x) = |x| has PROPERTY 1, (B) f(x) = x^(2/3) has PROPERTY 1, (C) f(x) = x|x| has PROPERTY 2, (D) f(x) = sin x has PROPERTY 2", "answer": "", "page_number": 25, "image_available": "no", "question_subject": "mathematics"} | |
| {"id": 5, "question": "Let f(x) = sin \u03c0x/x^2, x > 0. Let x1 < x2 < x3 < ... < xn < ... be all the points of local maximum of f and y1 < y2 < y3 < ... < yn < ... be all the points of local minimum of f. Then which of the following options is/are correct?", "response_choices": "(A) x1 < y1, (B) xn+1 - xn > 2 for every n, (C) xn \u2208 (2n,2n + 1) for every n, (D) |xn - yn| > 1 for every n", "answer": "", "page_number": 25, "image_available": "no", "question_subject": "mathematics"} | |
| {"id": 6, "question": "For a \u2208 R, |a| > 1, let lim (1 + sqrt(2) + ... + sqrt(n))/(n^(7/3) ((an+1)^2 + (an+2)^2 + ... + (an+n)^2)) = 54. Then the possible value(s) of a is/are", "response_choices": "(A) -9, (B) -6, (C) 7, (D) 8", "answer": "", "page_number": 25, "image_available": "no", "question_subject": "mathematics"} | |
| {"id": 7, "question": "Let f: \u211d \u2192 \u211d be given by f(x) = (x - 1)(x - 2)(x - 5). Define F(x) = \u222b[x to \u221e] f(t) dt, x > 0. Then which of the following options is/are correct?", "response_choices": "(A) F has a local minimum at x = 1 \n(B) F has a local maximum at x = 2 \n(C) F has two local maxima and one local minimum in (0, \u221e) \n(D) F(x) \u2260 0 for all x \u2208 (0, 5)", "answer": "C", "page_number": 26, "image_available": "no", "question_subject": "mathematics"} | |
| {"id": 8, "question": "Three lines l1: r\u20d7 = \u03bbi\u20d7, \u03bb \u2208 \u211d, l2: r\u20d7 = k\u20d7 + \u03bcj\u20d7, \u03bc \u2208 \u211d and l3: r\u20d7 = i\u20d7 + j\u20d7 + \u03bdk\u20d7, \u03bd \u2208 \u211d are given. For which point(s) Q on l2 can we find a point P on l1 and a point R on l3 so that P, Q and R are collinear?", "response_choices": "(A) k\u20d7 - 1/2j\u20d7 \n(B) k\u20d7 \n(C) k\u20d7 + 1/2j\u20d7 \n(D) k\u20d7 + j\u20d7", "answer": "C", "page_number": 26, "image_available": "no", "question_subject": "mathematics"} | |
| {"id": 9, "question": "Suppose det [ (Sum from k=0 to n) k, (Sum from k=0 to n) nCk k^2 ] = 0 [ (Sum from k=0 to n) nCk k, (Sum from k=0 to n) nCk 3k ] holds for some positive integer n. Then (Sum from k=0 to n) nCk / (k + 1) equals _", "response_choices": "", "answer": "", "page_number": 27, "image_available": "yes", "question_subject": "mathematics"} | |
| {"id": 10, "question": "Five persons A, B, C, D and E are seated in a circular arrangement. If each of them is given a hat of one of the three colours red, blue and green, then the number of ways of distributing the hats such that the persons seated in adjacent seats get different coloured hats is _", "response_choices": "", "answer": "", "page_number": 27, "image_available": "no", "question_subject": "mathematics"} | |
| {"id": 11, "question": "Let |X| denote the number of elements in a set X. Let S = {1, 2, 3, 4, 5, 6} be a sample space, where each element is equally likely to occur. If A and B are independent events associated with S, then the number of ordered pairs (A, B) such that 1 \u2264 |B| < |A|, equals _", "response_choices": "", "answer": "", "page_number": 27, "image_available": "no", "question_subject": "mathematics"} | |
| {"id": 12, "question": "The value of sec^-1 (1/4 (sum from k=0 to 10) sec (2\u03c0/12 + k\u03c0/2) + sec (7\u03c0/12 + (k + 1)\u03c0/2)) in the interval [-\u03c0 3\u03c0/4, \u03c0/4] equals ___", "response_choices": "No response choices provided", "answer": "No answer provided", "page_number": 28, "image_available": "yes", "question_subject": "mathematics"} | |
| {"id": 13, "question": "The value of the integral from 0 to \u03c0/2 of (3 \u221a(cos \u03b8)) / ((\u221a(cos \u03b8) + \u221a(sin \u03b8))^2) d\u03b8 equals ___", "response_choices": "No response choices provided", "answer": "No answer provided", "page_number": 28, "image_available": "yes", "question_subject": "mathematics"} | |
| {"id": 14, "question": "Let \u00e2 = 2\u00ee + j - k\u0302 and b\u0302 = \u00ee + 2j + k be two vectors. Consider a vector \u0109 = a\u00e2 + \u03b2b\u0302, \u03b1, \u03b2 \u2208 \u211d. If the projection of \u0109 on the vector (\u00e2 + b\u0302) is 3\u221a2, then the minimum value of (\u0109 - (\u00e2 \u00d7 b\u0302)) \u22c5 \u0109 equals___", "response_choices": "No response choices provided", "answer": "No answer provided", "page_number": 28, "image_available": "no", "question_subject": "mathematics"} | |
| {"id": 15, "question": "Which of the following is the only CORRECT combination?", "response_choices": "A) (I), (P), (R) B) (II), (Q), (T) C) (I), (Q), (U) D) (II), (R), (S)", "answer": "B", "page_number": 30, "image_available": "no", "question_subject": "mathematics"} | |
| {"id": 16, "question": "Which of the following is the only CORRECT combination?", "response_choices": "A) (III), (R), (U) B) (IV), (P), (R), (S) C) (III), (P), (Q), (U) D) (IV), (Q), (T)", "answer": "B", "page_number": 30, "image_available": "no", "question_subject": "mathematics"} | |
| {"id": 17, "question": "Which of the following is the only CORRECT combination?", "response_choices": "(A) (I), (S) (B) (I), (U) (C) (II), (Q) (D) (II), (T)", "answer": "(C) (II), (Q)", "page_number": 31, "image_available": "yes", "question_subject": "mathematics"} | |
| {"id": 18, "question": "Which of the following is the only INCORRECT combination?", "response_choices": "(A) (I), (P) (B) (IV), (U) (C) (III), (R) (D) (IV), (S)", "answer": "(D) (IV), (S)", "page_number": 31, "image_available": "yes", "question_subject": "mathematics"} | |