www.iitscholars.com Time: 3 hours Maximum marks: 315 PART – I 1. The equation of a circle is x2 + y2 = 25. The equation of its chord whose middle point is (1, −2), is given by (A) x + 2y + 5 = 0 (B) 2x – y – 5 = 0 (C) x – 2y – 5 = 0 (D) 2x + y – 5 = 0 2. The length of the tangent drawn from any point on the circle x2 + y2 + 2gx + 2fy + l = 0 to the circle x2 + y2 + 2gx + 2fy + µ = 0 is (l > µ) (A) µ − l (B) µ + l (C) l − µ (D) none of these 3. If the circles x2 + y2 = 9 and x2 + y2 + 2ax + 2y + 1 = 0 touch each other externally, then a = (A) 0 (B) 1 (C) 43 − (D) 43 4. The circles x2 + y2 = 6 and x2 + y2 – 6x + 8 = 0 are given. Then the equation of the circle through their points of intersection and the point (1, 1) is (A) x2 + y2 – 4y + 2 = 0 (B) x2 + y2 – 6x + 4 = 0 (C) x2 + y2 – 3x + 1 = 0 (D) none of these 5. The number of tangents to the circle x2 + y2 – 8x – 6y + 9 = 0 which pass through the point (3, −2) is (A) 2 (B) 1 (C) 0 (D) none of these 6. The tangent to the circle x2 + y2 = 9, which is parallel to y-axis and does not lie in third quadrant, touches the circle at the point (A) (−3, 0) (B) (3, 0) (C) (0, 3) (D) (0, −3) 7. The circles x2 + y2 + 4x + 6y + 3 = 0 and 2(x2 + y2) + 6x + 4y + c = 0 will cut orthogonally if c equals (A) 18 (B) 16 (C) 12 (D) 4 8. The equation of the circle which has a tangent 2x – y – 1 = 0 at (3, 5) on it and with the centre on x + y = 5, is (A) x2 + y2 + 6x – 16y + 28 = 0 (B) x2 + y2 – 6x +16y – 28 = 0 (C) x2 + y2 + 6x + 6y – 28 = 0 (D) x2 + y2 – 8x –2y – 51 = 0 9. Find the equation of the circle passing through the point (−2, 4) and through the points of intersection of the circle x2 + y2 – 2x – 6y + 6 = 0 and the line 3x + 2y -5 = 0 (A) x2 + y2 + 2x – 4y – 4 = 0 (B) x2 + y2 + 4x – 2y – 4 = 0 (C) x2 + y2 – 3x – 4y = 0 (D) x2 + y2 – 4x – 2y = 0 10. The equation of radical axis of the circles 2x2 + 2y2 – 7x = 0 and x2 + y2 – 4y – 7 = 0 is www.iitscholars.com (A) 7x + 8y + 14 = 0 (B) 7x – 8y + 14 = 0 (C) 7x – 8y – 14 = 0 (D) None of these 11. The number of common tangents to two circles x2 + y2 = 4 and x2 + y2 – 8x + 12 = 0 is (A) 1 (B) 2 (C) 3 (D) 4 12. From the origin, chords are drawn to the circle x2 + y2 – 2y = 0. The locus of the middle points of these chords is (A) x2 + y2 – y = 0 (B) x2 + y2 – x = 0 (C) x2 + y2 – 2x = 0 (D) x2 + y2 – x–y=0 13. The equation of circle with origin as centre and passing through the vertices of an equilateral triangle whose median is of length 3a is (A) x2 + y2 = 9a2 (B) x2 + y2 = 16a2 (C) x2 + y2 = 4a2 (D) x2 + y2 = a2 14. The circles x2 + y2 = 9 and x2 + y2 – 12y + 27 = 0 touch each other. The equation of their common tangent is (A) 4y = 9 (B) y = 3 (C) y = −3 (D) x = 3 15. Tangents are drawn from any point on the circle x2 + y2 = a2 to the circle x2 + y2 = b2. If the chord of contact touches the circle x2 + y2 = c2, a > b, then (A) a, b, c are in A. P. (B) a, b, c are in G. P. (C) a, b, c are in H. P. (D) a, c, b are in G.P. 16. If a circle passes through the points where the lines 3kx – 2y – 1 = 0 and 4x – 3y + 2 = 0 meet the coordinate axes then k = (A) 1 (B) −1 (C) 12 (D) 12 − 17. The equation of a circle, touching y-axis at (0, 3) and making intercept of 8 units on the x-axis, is (A) x2 + y2 – 10x + 6y + 9 = 0 (B) x2 + y2 – 10x − 6y + 9 = 0 (C) x2 + y2 + 10x + 6y + 9 = 0 (D) None of these 18. The equation of the diameter of the circle x2 + y2 – 6x + 2y – 8 = 0, which passes through the origin, is (A) x – 3y = 0 (B) x + 3y = 0 (C) 3x – y = 0 (D) none of these 19. A circle has radius 3 and its centre lies on the line y = x – 1. The equation of the circle, if it passes through (7, 3), is (A) x2 + y2 + 8x – 6y + 16 = 0 (B) x2 + y2 − 8x + 6y + 16 = 0 (C) x2 + y2 − 8x – 6y − 16 = 0 (D) x2 + y2 − 8x – 6y + 16 = 0 20. Equation of a circle with centre (4, 3) and touching the circle x2 + y2 = 1 is (A) x2 + y2 – 8x – 6y + 9 = 0 (B) x2 + y2 + 8x + 6y + 9 = 0 (C) x2 + y2 + 8x + 6y – 9 = 0 (D) none of these 21. Consider a circle with its centre lying on the focus of the parabola y2 = 2px such that it touches the directrix of the parabola. Then point(s) of intersection of the circle and the parabola is/are (A) p p , 2 2 (B) p , p 2 − (C) p ,p 2 − (D) p , p 2 − − 22. The curve described parametrically by x = t2 + t + 1, y = t2 – t + 1 represents (A) a pair of straight lines (B) an ellipse (C) a parabola (D) a hyperbola 23. If the line x – 1 = 0 is the directrix of parabola y2 – kx + 8 = 0 then one of the vertices of k is www.iitscholars.com (A) 18 (B) 8 (C) 4 (D) 14 24. If x + y = k is normal to y2 = 12x, then k is (A) 3 (B) 9 (C) −9 (D) −3 25. A curve y = f(x) passes through the point P(1, 1). The normal to the curve at P is a (y – 1) + (x + 1) = 0. If the slope of the tangent at any point on the curve is proportional to the ordinate of the point, then the equation of the curve is (A) ( ) k x 1 y e − = (B) kx y e = (C) ( ) k x 2 y e − = (D) none of these 26. Let f(x) = x4 – x2 + 1. The range of f is (A) R (B) 3 3 , 4 4 − (C) 3 , 4 −¥ (D) 3 , 4 ¥27. A curve passing through the point (1, 1) has the property that the perpendicular distance of the origin from the normal at any point P of the curve is equal to the distance of P from the x-axis. The equation of the curve is (A) x2 + y2 + 2x = 0 (B) x2 + y2 – 2x = 0 (C) x2 + y2 – 2y = 0 (D) None of these 28. The general solution of the differential equation (1 + tan y)(dx – dy) + 2xdy = 0 is (A) x(sin y + cos y ) = sin y + cey (B) x(sin y + cos y ) = sin y + ce−y (C) x(sin x + cos x ) = sin x + cex (D) None of these 29. Solution of differential equation 2 2 tan y.sec xdx tan x.sec dy 0 + = is (A) tan x + tan y = k (B) tan x – tan y = k (C) tan x k tan y = (D) tan x. tan y = k 30. If O be the origin and if the coordinates of any two points Q1 and Q21 be (x1, y1) and (x2, y2) respectively, then OQ1.OQ2 cos Q1OQ2 = (A) x1x2 – y1y1 (B) x1y1 – x2y2 (C) x1x2 + y1y2 (D) x1y1 + x2y2 31. The area of a parallelogram formed by the lines ax by c 0 ± ± = , is (A) 2 cab (B) 2 2c ab (C) 2 c 2ab (D) none of these 32. The area of a triangle is 5. If two of its vertices are (2, 1), (3, −2) and the third vertex lies on the line y = x + 3, then the third vertex is (A) 7 13 , 2 2 − − (B) 7 13 , 2 2 − (C) 7 13 , 2 2 − (D) 7 13 , 2 2 33. The equation of the line which bisects the obtuse angle between the lines x – 2y + 4 = 0 and 4x – 3y + 2 = 0 is (A) ( ) ( ) ( ) 4 5 x 3 2 5 y 2 4 5 0 − − − + − = (B) ( ) ( ) ( ) 4 5 x 3 2 5 y 2 4 5 0 + − + + + = (C) ( ) ( ) ( ) 4 5 x 3 2 5 y 2 4 5 0 + + + + + = (D) none of these 34. Three points are A(6, 3), B(−3, 5), C(4, −2) and P(x, y) is a point, then the ratio of area of DPBC and DABC is (A) x y 2 7 + − (B) x y 2 2 − + (C) x y 2 2 − − (D) None of these www.iitscholars.com 35. /3 /6 dx 1 tan x pp = + (A) 12 p (B) 6p (C) 3p (D) none of these www.iitscholars.com PART – II 36. Which of the following configuration will be responsible for the colourless atom/ion? (A) d0 (B) d10 (C) Both (D) None of these 37. Which one of the following metal ions is coloured? (A) Cu+ (B) Zn2+ (C) Sc3+ (D) V4+ 38. Transition metals (A) exhibit diamagnetism (B) undergo inert pair effect (C) do not form alloys (D) show variable oxidation states 39. Which of the following transition element shows the highest oxidation state? (A) Mn (B) Fe (C) V (D) Sc 40. If carbon is added to the interstitial sites of a iron, then iron becomes (A) softer (B) less tensile (C) less malleable (D) more ductile 41. The catalytic activity of the transition metals and their compounds is due to (A) their chemical reactivity (B) presence of unpaired electron (C) their unfilled d-orbitals (D) their ability to adopt multiple oxidation states and availability of adsorption sites 42. Extraction of silver from its ore involving NaCN, air and an active metal is known as (A) Pattinson’s method (B) Amalgamation method (C) Mc Arthur-forest method (D) Parke’s method 43. Chalcogens are (A) Hydrocarbons (B) Ore forming elements (C) Oxide forming elements (D) Those having ability to catenate 44. Plumbo-solvency refers to (A) Oxidation of lead to lead oxide (B) Oxidation of lead to red lead (C) Dissolution of lead in water containing air (D) Making lead wires by forcing heated metal through a die 45. In Goldschmidt aluminothermic process, thermite mixture contains (A) 3 parts Fe2O3 and 2 part Al (B) 3 parts Al2O3 and 4 parts Al (C) 1 part Fe2O3 and 12 part Al (D) 3 parts Fe2O3 and 1 part Al 46. The pyrometallurgical operations involves the use of (A) High temperature (B) Sulphide ores (C) Electrolysis (D) Complexation 47. The most common element present in the crust of the Earth are (A) Oxygen, Silicon, Aluminum (B) Oxygen, Iron, Magnesium (C) Silicon, Iron, Potassium (D) Oxygen, Iron, Silicon 48. In the cyanide process for extraction of gold and silver from ores, cyanides solution acts as a (A) reducing agent to reduce the gold and silver compounds present in the ores into the metallic states (B) leaching agent to bring the gold and silver into solution as cyanide complexes and thus separate these metals from the ores (C) leaching agent to dissolve all the other constituents of the ores leaving the gold and silver as metals (D) leaching agent to bring the ores into solution 49. The compound [Cr(H2O)6]Cl3 and [Cr(H2O)4Cl2]Cl. 2H2O represent (A) linkage isomerism (B) hydrate isomerism (C) ligand isomerism (D) none of these 50. Which of the following species represent the example of dsp2 hybridization? (A) [Fe(CN)6]3-(B) [Ni(CN)4]2-(C) [Zn(NH3)4]2+ (D) none of these 51. Which is paramagnetic? (A) [Ni(H2O)6]2+ (B) [Fe(CN)6]4-(C) [Ni(CO)4] (D) [Ni(CN)4]2-52. Which one of the following square planar complex will form two geometrical isomers? (A) MA4 (B) MA3B (C) MA2B2 (D) MAB3 www.iitscholars.com 53. IUPAC name of [CoCl(NO2)(en)2]Cl is (A) chloronitro bis (ethylene diammine) cobalt (III) chloride (B) chloronitro bis (ethylene diammine) cobalt (II) chloride (C) chlorobis (ethylene diammine) nitro cobalt (III) chloride (D) bis (ethylene diammine) chloronitro cobalt (III) chloride 54. ‘en’ is an example of a (A) monodentate ligand (B) bidentate ligand (C) tridentate ligand (D) hexadentate ligand 55. Which of the following is non ionizable? (A) [Co(NH3)3Cl3] (B) [Co(NH3)4Cl2]Cl (C) [Co(NH3)5Cl]Cl2 (D) [Co(NH3)6]Cl2 56. Which of the following represent chelating ligand? (A) Cl-(B) H2O (C) OH-(D) DMG 57. Which of the following complexes has a shape different from others (A) [NiCl4]2-(B) Ni(CO)4 (C) [Ni(CN)4]2-(D) [Zn(NH3)4]2+ 58. The IUPAC name of K3[Fe(CN)6] is (A) potassium ferrocyanide (B) potassium ferricyanide (C) potassium hexacyanoferrate (II) (D) potassium hexacyanoferrate (III) 59. In which of the following, the magnetic moment would be maximum (A) [Co(NH3)6]Cl3 (B) K4[Fe(CN)6] (C) K3[Fe(CN)6] (D) [Fe(H2O)6]SO4 60. K2[NiF6] exhibits d2sp3 hybridization. The number of unpaired electrons in this compound is (A) 3 (B) 2 (C) 1 (D) 0 61. Which of the following oxides of chromium is amphoteric in nature? (A) CrO (B) Cr2O3 (C) CrO3 (D) CrO5 62. For Hf4+ the ionic radius is 0.86 A which is almost the same as that of Zr4+. This is due to (A) Hf4+ forming compounds having lower degree of ionic character (B) difference in co-ordination number of Zr4+ and Hf4+ (C) lanthanide contraction (D) diagonal relationship 63. Which of the following is a hexadentate ligand? (A) DMG (Dimethyl glyoxime) (B) en (ethylenediamine) (C) ox (oxalate) (D) EDTA (Ethylenediamine tetraacetate ion) 64. An octahedral MA4B2 type complex gives (A) 3 geometrical isomers (B) 2 geometrical isomers (C) 1 geometrical isomer (D) 4 geometrical isomers 65. The yield of product in the reaction ( ) ( ) ( ) 2 A g 2B g C g QkJ + + Would be higher at (A) low temperature and high pressure (B) high temperature and high pressure (C) low temperature and low pressure (D) high temperature and low pressure 66. The pH of 0.005 molar aqueous solution of sulphuric acid is approximately (A) 0.005 (B) 1 (C) 0.1 (D) 2.0 67. One litre of water contains 10-7 mole of H+ ions. Degree of ionization of water is (A) 1.8 × 10-7 % (B) 0.8 × 10-9% (C) 3.6 × 10-9 % (D) 3.6 × 10-7 % 68. The difference between nth and (n + 1)th Bohr’s radius of H–atom is equal to its (n – 1)th Bohr’s radius. The value of n is (A) 1 (B) 2 (C) 3 (D) 4 69. 100 ml. of gas A effuses in 10 sec. 150 ml of gas B effuses in 20 seconds. If the molecular weight of gas B is 64, what is molecular weight of A? (A) 36 (B) 64 (C) 100 (D) 48 70. In alkaline condition KMnO4 reacts as follows: 2KMnO4 + 2KOH ¾® 2K2MnO4 + H2O + O the equivalent weight of KMnO4 is (A) 52.7 (B) 158 (C) 31.6 (D) 79 www.iitscholars.com PART – III 71. An ideal monatomic gas is taken round the cycle ABCDA as shown in the P-V diagram. The work done during the cycle is (A) PV (B) 2PV (C) 21 PV (D) zero P V P, V P, 2V 2P, 2V 2P, V A B C D 72. If in the figure shown 1 2 T T ¹ , then which of the following is correct? (A) string is massless (B) string is not massless (C) 2 F m a = where a is acceleration of 2 m (D) Acceleration of 1 m ¹ acceleration of 2 m . m1 m2 F (Frictionless) T1 T2 73. If 1 2 m m > and the motion starts from rest, then the amount of work done by any one side of the string in time ‘t’ (A) 2 1 2 1 2 2m m gt m m + (B) ( ) ( ) 1 2 1 2 2 2 2 1 2 m m m m g t m m− + (C) 2 1 2 1 2 m m 1 gt 2 m m − + (D) ( ) 2 2 1 2 2 1 2 m m g t m m + . m2 m1 T T 74. 1/2 mole of helium gas is contained in a container at S.T.P. The heat energy needed to double the pressure of the gas, keeping volume constant, is (heat capacity of the gas is 3 Jg-1K-1): (A) 3276 J (B) 1638 J (C) 819 J (D) 409.5 J 75. 70 calories of heat is required to raise the temperature of 2 mole of an ideal gas from 30 oC to 35 oC at constant pressure. The heat required (in calories) to raise the temperature of the same gas through the same range (30oC to 35 oC) at constant volume will be (A) 30 (B) 50 (C) 70 (D) 90 76. Four curves A, B, C and D are drawn in the figure for a given amount of gas. The curves which represent adiabatic and isothermal changes are: (A) C and D respectively (B) D and C respectively (C) A and B respectively (D) B and A respectively P V A B C D 77. A monatomic ideal gas, initially at temperature T1, is enclosed in a cylinder fitted with a frictionless piston. The gas is allowed to expand adiabatically to a temperature T2 by releasing the piston suddenly. If L1 and L2 are the lengths of the gas column before and after expansion respectively, then T1/T2 is given by (A) 3 /2 21 LL (B) 21 LL www.iitscholars.com (C) 12 LL (D) 3 /2 12 LL 78. Determine the work done by an ideal gas during 1®4®3®2®1. Given P1 = 105 Pa, Po = 3 × 105 Pa, P3 = 4 × 105 Pa & V2 -V1 = 10 litre. (A) 750 J (B) 500 J (C) 250 J (D) 1000 J 2 V P O 4 3 1 79. In order to increase the efficiency of a Carnot engine most effectively, where T1 is the temperature of the source and T2 is the temperature of the sink (A) increase T1 keeping T2 constant (B) increase T2 keeping T1 constant (C) increase both T1 and T2 (D) decrease both T1 and T2 80. Two bodies are at temperature 27°C and 927°C. The heat energy radiated by them will be in the ratio (A) 1 : 256 (B) 1 : 64 (C) 1 : 4 (D) 1 : 16 81. The temperature of a black body increases from T to 2T. The factor by which the rate of emission will increases is (A) 2 (B) 4 (C) 8 (D) 16 82. During adiabatic process pressure (P) versus density (r) equation is (A) P. g r = constant (B) P. −g r = constant (C) 1 P . g +g r = constant (D) 1 P . g g r = = constant 83. According to Newton’s law of cooling, the rate of cooling of a body is proportional to (Dq)n where Dq is the difference of temperature of the body and the surroundings and n is equal to (A) 3 (B) 4 (C) 1 (D) 2 84. A black body at a temperature 77°C radiates heat at a rate of 10 calcm−2s-1. The rate at which this body would radiate heat in units of calcm−2s-1 at 427°C is closest to (A) 40 (B) 160 (C) 200 (D) 400 85. Two rods of equal length and diameter but of thermal conductivities 2 and 3 units respectively are joined in series. The thermal conductivity of the combination is (A) 6 (B) 5 (C) 1 (D) 2.4 86. Two spheres P and Q of same colour having radii 8 cm and 2 cm are maintained at temperature 127°C and 527°C respectively. The ratio of energy radiated by P and Q is (A) 0.054 (B) 0.0034 (C) 1 (D) 2 87. A horizontal cylinder has two sections of unequal cross-sections, in which two pistons can move freely. The pistons are joined by a string. Some gas is trapped between the pistons. If this gas is heated, the pistons will (A) move to the left (B) move to the right (C) remain stationary (D) either (A) or (B) depending on the initial pressure of the gas 88. In the process PV = constant, pressure (P) versus density (r) graph of an ideal gas is (A) a straight line parallel to P-axis (B) a straight line parallel to r-axis (C) a straight line passing through origin (D) a parabola www.iitscholars.com 89. Pressure versus temperature graph of an ideal gas at constant volume V of an ideal gas is shown by the straight line A. Now mass of the gas is doubled and the volume is halved, then the corresponding pressure versus temperature graph will be shown by the line (A) A (B) B (C) C (D) none of these P T A B C 90. For an adiabatic expansion of a perfect gas, the value of PP D is equal to (A) -VV D g (B) -VV D (C) -g VV D (D) -g2 VV D 91. Three rods made of the same material and having the same crosssecctio have been joined as shown in the figure. Each rod is of the same length. The left and right ends are kept at 00C and 900C respectively. The temperature of the junction of the three rods will be (A) 450C (B) 600C (C) 300C (D) 200C 900C 900C 00C 92. A satellite S is moving in an elliptical orbit around the earth. The mass of the satellite is very small compared to the mass of the earth: (A) The acceleration of S is always directed towards the centre of the earth. (B) The angular momentum of S about the centre of the earth changes in direction, but its magnitude remains constant. (C) The total mechanical energy of S varies periodically with time. (D) The linear momentum of S remains constant in magnitude. 93. Two particles of mass m1 and m2 are initially at rest at infinite distance. Find their velocity of approach due to gravitational attraction, when their separation is d: (A) d ) m m ( G 2 2 1 + (B) d 3 ) m m 2 ( G 2 1 + (C) d ) m m 2 ( G 3 2 1 + (D) d ) m m ( G 2 1 + 94. A closed compartment containing liquid is moving with some acceleration in horizontal direction. Neglect the effect of gravity. Then the pressure in the compartment is : (A) same everywhere (B) lower in the front side (C) lower in the rear side (D) lower in the upper side. 95. A vessel contains oil (density 0.8 g cm-3) over mercury (density 13.6 g cm-3). A homogeneous sphere floats with half volume immersed in mercury and the other half in oil. The density of the material of the sphere in g cm-3 is (A) 3.3 (B) 6.4 (C) 7.2 (D) 12.8. 96. A solid shell loses half its weight in water. Relative density of shell is 5.0. What fraction of its volume is hollow ? (A) 35 (B) 25 (C) 15 (D) 45 97. When a force is applied at one end of an elastic wire, it produces a strain e in the wire. If Y is the Young’s modulus of the material of the wire, the amount of energy stored per unit volume of the wire is given by : (A) Y ×e (B) 12 Y ×e (C) 2 Y ×e (D) 2 12 Y ×e . 98. A cube of wood supporting a mass of 200 g just floats in water. When the mass is removed, the cube rises by 2 cm. What is the size of the cube ? (A) 6 cm (B) 8 cm www.iitscholars.com (C) 10 cm (D) 12 cm. 99. A soap bubble of radius r is blown up to form a bubble of radius 2 r under isothermal conditions. If s is the surface tension of soap solution, the energy spent in doing so is : (A) 2 3 r ps (B) 2 6 r ps (C) 2 12 r ps (D) 2 24 r ps . 100. Two springs having spring constants in ratio m : n are connected to a block as shown in figure. What is the respective ratio of energy stored in these when the bock is displaced? (A) 33 nm (B) mn (C) 2 nm (D) 1. 101. A force ( ) ˆ F 3i 4j N = + acts on a particle moving in x-y plane. Starting from origin the particle first goes along x-axis to the point (4,0)m and then parallel to the y-axis to the point (4, 3)m. The total work done by the force on the particle is (A) + 12J (B) – 6J (C) + 24J (D) – 12J. (4, 3)m (0, 0) (4, 0)m x y 102. A particle moves in a straight line with a uniform acceleration a. Initial velocity of the particle is zero. The average velocity of particle in first S distance will be : (A) aS (B) aS 3 (C) aS 2 (D) 1 aS 2 103. A particle is projected from a horizontal plane with 8 2 m/s at some angle. At highest point its velocity is found to be 8 m/s. It range will be (A) 6.4 m (B) 3.2 m (C) 5.0 m (D) 12.8 m 104. Consider a uniform square plate of side a and mass m. The moment of inertia of this plate about an axis perpendicular to its plane and passing through one of its corners is (A) 2 1 ma 12 (B) 2 7 ma 12 (C) 2 2ma 3 (D) 2 5ma 6 105. A block of mass 0.50 kg is moving with a speed of 1 2.00ms− on a smooth surface. It strikes another mass of 1.00 kg and then they move together as a single body. The energy loss during the collision is (A) 1.00 J (B) 0.67 J (C) 0.34 J (D) 0.16 J