1. UPSEEPAST PAPERSMATHEMATICS - UNSOLVED PAPER - 2000

2. SECTION- I Single Correct Answer Type There are five parts in this question. Four choices are given for each part and one of them iscorrect. Indicate you choice of the correct answer for each part in your answer-book bywriting the letter (a), (b), (c) or (d) whichever is appropriate 3. 01 Problem The area of the triangle formed by the tangent at (3, 4) to the circle x2 + y2 = 25 and the co-ordinate axes is :24 a. 25 sq unit b. 0 sq unit625 c. 24sq unit24 d. - sq unit25 4. 02 Problem Which of the following functions has period 2 ? a. y sin 2 t 2 sin 3 t 3 sin5 t 3 4 b. y sin t sin t3 4 c. y = sin t + cos 2t d. none of the above 5. 03 Problem Two bodies of mases m and 4m are moving with equal momentum. The ratio of their kinetic energy is : a. 1 : 4 b. 4 : 1 c. 1 : 1 d. 1 : 2 6. 04 Problem The line y = mx + 1 is a tangent to the parabola y2 = 4x, if : a. m = 1 b. m = 2 c. m = 4 d. m = 3 7. 05 Problem Let D be the middle point of the side BC of a triangle ABC. If the triangle ADC is equilateral, then a2 : b2 : c2 is equal to : a. 1 : 4 : 3 b. 4 : 1 : 3 c. 4 : 3 : 1 d. 3 : 4 : 1 8. 06 Problem Ifi jijij4 7 8k , 2 3 4k and 2 5 7k are the position vectors of the vertices A, B and C respectively of triangle ABC. The position vector of the point where the bisector of angle A meets BC is : a.31 6i 13 18k j b.32 j6i 12 8k c. 13 6 8 9kij d. 236 12 8ki j 9. 07 Problem The projection of the vector ij 2 ki j on the vector 4 4 7k is equal to : a. 199 9 b.19 3 c.1919 d.3 10. 08 Problem Let i j a 2 k , i j i j b 2 k and c 2k be three vectors. A vector in the plane of whose projection is of magnitude2is : 3 a. i j 2 3 3k b. i j 2 3 3k c. i j 2 5k d. 2 5ki j 11. 09 Problem The value of k for which the vectors a and b 2 kji j i are collinear, is : a. 2 b. 1/2 c. 1/3 d. 3 12. 10 Problem The area of a parallelogram whose adjacent sides are determined by the vectorsi j i j a 2 3k and b 3 2 k is equal to : a. 85 sq unit b. 95 sq unit c. 65 sq unit d. 17 5 sq unit 13. 11 Problem A uniform ladder rest in limiting equilibrium with its lower end on a rough horizontal plane and its upper end against a smooth vertical wall, if is an angle of inclination of the ladder to the vertical wall and is the coefficient of friction, then tan is equal to : a. b. 2 3 c. 2 d. + 1 14. 12 Problemx 1 3x 4 3lim The value of x is equal to : 3x 2 a. e-1/3 b. e-2/3 c. e-1 d. e-2 15. 13 Problem If f(x) = (x + 1)cot x be continuous at x = 0, then f(0) is equal to : a. 0 b. -e c. e d. none of these 16. 14 Problemdy If y = (cos x2)2, thenis equal to :dx a. - 4x sin 2x2 b. - x sin x2 c. - 2x sin 2x2 d. - x cos 2x2 17. 15 Problem The value of the derivative of |x - 1| + |x - 3| at x = 2 is : a. 2 b. 1 c. 0 d. -2 18. 16 Problem 1 x2 The derivative of sin-1 2 x with respect to cos 1 2 is equal to : 2 1 x 1 x a. 1 b. -1 c. 2 d. none of these 19. 17 Problem The minimum value of function f(x) = 3x4 8x3 + 121x2 48x + 25 on [0, 3] is equal to : a. 25 b. - 39 c. - 25 d. 39 20. 18 Problem 21 1 is equal to : 1 ex x 2 dxx e a.e 1 2 b. e(e - 1) c. 0 d. none of these 21. 19 Problem The value of 1 1 is :tan1 tan12 3 a. 0 b. 3 c.6 d. 4 22. 20 Problem If 12Pr = 1320, then r is equal to : a. 5 b. 4 c. 3 d. 2 23. 21 Problem A particle is projected vertically upward takes t1 second to reach a height h. If t2 second is the subsequent time to reach the ground, then the maximum height attained is :1 a. 2g (t1 + t2)2 b. 1 g (t1 + t2)24 c. 1 g (t1 + t2)28 d. none of these 24. 22 Problem A cart of 100 kg is free to move on smooth rails and a block of 20 kg is resting on it. Surface of contact between the cart and the block is smooth. A force of 60 N is applied to the cart. Acceleration of 20 kg block in m/s2 is : a. 3 b. 0.6 c. 0.5 d. 0 25. 23 Problem The eccentricity of the ellipse 9x2 + 5y2 30y = 0 is equal to ;1 a. 32 b.3 c. 34 d. none of these 26. 24 Problem The radius of the circumcircle of an isosceles triangle PQR is equal to PQ (=PR), then the angle P is : a. 6 b. 3 c. 22 d. 3 27. 25 Problem If A = {x, y}, then the power set of A is : a. {xx, yy} b. { , x, y} c. { , {x}, {2y}} d. {f, {x}, }{y}, {x, y}} 28. 26 Problem 2 d2y dy 2 d 2y The degree of the differential equation23 x log 2 is : dx dx dx a. 1 b. 2 c. 3 d. none of these 29. 27 Problem The solution of the differential equation x4 dy + x3y + cosec (xy) = 0 is equal to :dx a. 2cos(xy) + x-2 = c b. 2 cos (xy) + y-2 = c c. 2 sin (xy) + x-2 = c d. 2 sin (xy) + y-2 = c 30. 28 Problem Forces of magnitudes 3 and 2 unit acting in the directions ijij 5 3 4k and 3 4 5k respectively act on a particle which is displaced from the points (1, -1, -1) to (3, 3, 1). The work done by the forces is equal to : a. 80 2unit b. 40 2unit57 c.5 2 unit d. 82 unit 31. 29 Problem The rate of increase of bacteria in a certain culture is proportional to the number present. If it double in 5 h, then in 25 h, its number would be : a. 8 times the original b. 16 times the original c. 32 times the original d. 64 times the original 32. 30 Problem 15k If a of magnitude 50 is collinear with the vector , b 6 8 ijand2 makes an acute angle with the positive direction of z-axis, then the vector is equal to : a.24 32 30kij b.ij 24 32 30k c.16 16 15kij d. ij 12 16 30k 33. 31 Problem1 x yz The value of the determinant is equal to :1 y zx1 z xy a. x b. y c. z d. 0 34. 32 Problem If a man and his wife enter in a bus, in which five seats are vacant, then the number of different ways in which they can be seated, is : a. 2 b. 5 c. 20 d. 40 35. 33 Problem The differential equation of the family of curves for which the length of the normal is equal to a constant k, is given by :dy a. y2dx k2 y2 2 b. dy = k2 y2 y dx 2 dy 2 2 c. y =k +y dx 2 d. y dy = k2 + y2dx 36. 34 Problem If is a complex cube root of unity, then the value of 101 99100is a. 1 b. -1 c. 3 d. 0 37. 35 Problem The roots of the equation (q - r)x2 + (r - p) x + (p - q) = 0 are :r p 1 a. ,qr 2pq b.,1q rq r c. ,1p q r p 1 d., pq 2 38. 36 Problem The value of sin 100 + sin 200 + sin 300 + + sin 3600 is equal to a. 0 b. 1 c. 3 d. 2 39. 37 Problem The mean of observations x1, x2, , xn is x , then (x1x- ) + (x2x- ) + .+ (xn - ) equal to : a. (n - 1) x b. n x c. 0 d. none of these 40. 38 Problem If 2a 2b then x is equal to :sin1 2 sin1 2 2 tan1 x,1 a 1 b ab a.1 ab b b.1 ab b c. 1 abab d. 1 ab 41. 39 Problemx 1 y 3 z 5 x 2 y 4 z 6 The point of intersection of the lines3 3 7 and 1 3 5is : a. 1 , 1 , 3 2 22 1 1 3 b. 2 , 2 , 2 1 1 3 c.2 , 2 , 2 1 1 3 2 , 2 , 2 d. 42. 40 Problem The area of the circle and the area of a regular polygon on n sides and of perimeter equal to that of the circle are in the ratio of : a. tan :n n b. cos :n n c. sin :n n cot : n d. n 43. 41 Problem Real part of 1 is equal to :1 cos i sin 1 a. -21 b.2 c. 1 tan2 2 d. 2 44. 42 Problem An insurance company insured 2000 scooter drivers, 4000 car drivers and 6000 truck drives. The probability of an accident involving a scooter driver, car driver and a truck driver is 0.01 , 0.03, 0.15 respectively. One of the insured persons meets with an accident. The probability that he is a scooter driver, is : 1 a. 521 b. 53 c. 251 d. none of these 45. 43 Problem The inradius of the triangle whose sides are 3, 5, 6 is : a.8 7 b. 8 c. 77 d. 8 46. 44 Problem The area of the region bounded by the curve y = x |x |, x axis and the ordinates x = 1, x = -1 is given by : a. 0 sq unit1 b. 3sq unit2 c. 3sq unit d. 1 sq unit 47. 45 Problemn If (1 + x)n = C r 0r x r , then value of C1 + 2C2 + 3C3 + ..+ nCn is equal to : a. n . 2n b. (n + 2)2n c. n . 2n-1 d. n . 2n +1 48. 46 Problem g x .f a g a .f x If f(a) = 2, f(a) = 1; g(a) = -1, g(a) = 2, then limis equal to :x a xa a. - 5 b. 01 c. 5 d. 5 49. 47 Problem P x k If x has binomial distribution with mean np and variance npq, thenisP x k 1 equal to :n k p a. .k 1 qn k 1 p b. k.q n 1 q c..kpn 1 q d. .k 1 p 50. 48 Problemdx The value of the integral x 1 log x 2is equal to : 1 a.c1 x 1 b. c1 log x c. 1c1 log x 1 d.c1 x2 51. 49 Problem Let E1, E2, E3 be three arbitrary events of a sample space S consider the following statements space S consider the following statements which of the following statements are correct ? a. P (only one of them occurs ) = P (E1E 2E3 + 1E2E3 + E2 E 3) b. P (none of them occurs) = P ( E 1 + E 2 + E3) c. P (at least one of them occurs) = P (E1 + E2 + E3) d. P (all the three occur) = P (E1) P (E1) P (E2) where P (E1) E1 and 1 denotescomplement of E1. 52. 50 Problem Consider the system of linear equations a1x + b1y +c1 z + d1 = 0, a2x + b2y + c2z + d2 = 0 and a3x + b3y + c3z + d3 = 0. Let usa1 b1 c1 denote by the determinanta2 b2 c2 , if a, b, c 0, then the value of x ina3 b3 c3 the unique solution of the above equation is : b, c, d a. a, b, c b, c, d b. a, b, c c. a, c, d a, b, c d. a, b, d a, b, c 53. 51 Problem 10 sgn x [x] dx, where [ ] denotes greatest integer function, is equal to :2 a. -12 b. 10 c. 8 d. 12 54. 52 Problem The equations of one of the bisectors of the angles between the straight lines 3x 4y + 7 = 0 and 12x 5y 8 = 0 are : a. 21x + 27y 131 = 0 b. 12x 5y + 7 = 0 c. 4x 3y + 1 = 0 d. none of the above 55. 53 Problem 2n boys are randomly divided into two subgroups containing n boys each. The probability that the two tallest boys are in different group, is :n a. 2n 1 n 1 b. 2n 1 c. 2n 1 4n2 d. none of these 56. 54 Problem If the sides of a triangle are in the ratio 3 : 7 : 8, then R : r is equal to : a. 2 : 7 b. 7 : 2 c. 3 : 7 d. 7 : 3 57. 55 Problem If there are 6 girls and 5 boys who sit in a row, then the probability that no two boys sit together is : 6 !6 ! a.2!11! 7 !5! b.2 !11! c.6 !7! 2!11! d. none of these 58. 56 Problem The latus rectum of the hyperbola 9x2 16y2 18x + 32y 151 = 0 is :] 5 a.2 2 b. 9 c.9 2 d. none of these 59. 57 Problem A tower subtends an angle at a point in the plane of its base and the angle of depression of the foot of the tower at a point b ft just above A is . Then height of the tower is : a. b tan cot b. b cot tan c. b tan tan d. b cot cot 60. 58 Problem x2 y 2 Line x cos + y sin = p will be tangent to the ellipse 2 2 1 , if : ab a. p2 = a2 cos2 + b2 sin2 b. p = 1 c. p2 = a2 + b2 d. none of the above 61. 59 Problem 1 1 1/ x 1 If I1 x 1 t 2 dt and I2 11 t2 dt for x > 0, then : a. I1 = I2 b. I1 > I2 c. I2 > I1 d. None of these 62. 60 Problem A particle possess simultaneously two velocities 10 m/s and 15 m/s in direction inclined at an angle of 600, then its resultant velocity is : a. 15 m/s b. 5 19 m/s c. 25 m/s d. none of these 63. 61 Problem dx 1 x3 1 If x 1 x 3 a log1 x 3 1+ b, then a is equal to :1 a. 3 b. 231 c.3 d. -2 3 64. 62 Problem The angle of projection of a particle when its range on horizontal is 4 3 times the greatest height attained by it is : a. 150 b. 300 c. 450 d. 600 65. 63 Problem The integrala g x vanishes, if : 0 f x f a x dx a. g (x) is odd b. f(x) = f(a - x) c. g(x) = - g (a - x) d. f(a x ) = g(x) 66. 64 Problemtan xis equal to : sin x cos xdx a. 2 tan x c b. 2 cot x c tan x c.c2 d. none of these 67. 65 Problem The equation x2 + k1y2 + k2xy = 0 represents a pair of perpendicular lines, if : a. k1 = -1 b. k1 = 2k2 c. 2k1 = k2 d. none of these 68. 66 Problem If /2sin x dx sin 2 , then the value of satisfying 0 < < is : 3 a.2 b.6 c.56 d. 2 69. 67 Problem The points (-a, -b), (0, 0), (a, b) and (a2, ab) are : a. Collinear b. Vertices of parallelogram c. Vertices of rectangle d. None of the above 70. 68 Problem A bullet fired into a target loses half of its velocity after penetrating 3 cm. The bullet will penetrate further : a. cm b. 1 cm c. 2 cm d. none of these 71. 69 Problem If a point moves with constant acceleration from A to B in the straight line AB has velocities u and v at A and B respectively, then the velocity at C, the mid point AB is :uv a.2 b. u2 v 2 c.u2 v 22 d. none of these 72. 70 Problem A stone is thrown vertically upwards with an initial velocity u from the top of a tower, reaches the ground with a velocity 3u. The height of the tower is : a. 3u2 g4u2 b.g6u2 c. g 9u2 d. g 73. 71 Problem If the focus of a parabola is at (0, - 3) and its directrix is y = 3, then its equation is : a. x2 = - 12y b. x2 = 12y c. y2 = -12x d. y2 = 12 x 74. 72 Problem a 5/2nC If the second term in the expansion 13 a is 14 athen n 3 is equal C2 a1 to : a. 4 b. 3 c. 12 d. 6 75. 73 Problem 1 iz If z = x + iy and , then| | 1 implies that in the complex plane :z i a. z lies on the imaginary axis b. z lies on the real axis c. z lies on the unit circle d. none of the above 76. 74 Problem The radius of a balloon is increasing at the rate of 10 cm/s. At what rate is the surface area of the balloon increasing when the radius is 15 cm ? a. 120 cm2 /s b. 125 c m2 /s c. 1200 cm2 /s d. 1100 cm2 /s 77. 75 Problem If log5 2, log5 (2x - 3) and log5 17x 1 are in AP, then x is equal to : 2 2 a. 3 b. - 3 c. 2 d. - 2 78. 76 Problem If a, is a non-zero vector of modulus a and m is a non-zero scalar, then m a is a unit vector, if : a. m = 11 b. a = 2 m1/2 c. a = - m1 d. a = | m| 79. 77 Problem A circular wire of radius 15 cm is cut and bent so as to lie along the circumference of a loop of radius 120 cm. The angle subtended by it at the centre is ; a. 300 b. 450 c. 600 d. none of these 80. 78 Problem The angle of expenditure on food in a pie-chart for expenditure pattern : food 40%, clothing- 25%, rent 10%, education 10%, medicine 10%, miscellaneous- 5% is equal to a. 720 b. 360 c. 1440 d. none of these 81. 79 Problemlog 1 ax log 1 bx If f(x) = , x 0 and f (x) is continuous at x = 0, then x kx 0 the value of k is equal to : a. log a log b b. a + b c. 1 d. a - b 82. 80 Problem A box contains two white balls, three black balls and four red balls. In how many ways can three ball be drawn from the box, if at least one black ball is to be included in the draw ? a. 64 b. 129 c. 84 d. none of these 83. 81 Problem (4xy ) For which real value of x and y, the equation sec2 is possible ?(x y )2 a. x = y b. x > y c. x < y d. none of these 84. 82 Problem The equation of the sphere with A (2, 3, 5) and B (4, 9, - 3) as the ends of a diameter is : a. x2 + y2 + z2 8x 12y + 2z = 30 b. x2 + y2 + z2 6x 12y - 2z = - 20 c. x2 + y2 - z2 + 6x 12y + 2z = 15 d. x2 + y2 + z2 6x 12y - z = 20 85. 83 Problem The string of a kite is 100 m long and it makes an angle of 600 with the horizontal. What is the height of the kite assuming that there is no slack in the string ? a. 80 m b. 503m c. 403m d. 60 m 86. 84 Problem The circle whose equations are x2 + y2 + c2 = 2ax and x2 + y2 + c 2 2hy = 01 1 1 2 2 a. c 2 a b b. 11 1 2 2 2a b c c.11 12 2 2 b c a d. none of these 87. 85 Problem the value of sin 3000 tan 3300 sec 4200 is equal to :tan1350 sin 2100 sec 3150 a. 12 b. 21 c.3 d. 3 88. 86 Problem (666......6)2 (888......8) is equal to : ndigits n digits4 2 a. 10n 194 b.910n 1 4 c.9 102n 14 2 d. 102n 19 89. 87 Problem If A = P{x : x is a multiple of 4} and B = {x : x is multiple of 6}, then consists of all multiples of : a. 16 b. 12 c. 8 d. 4 90. 88 Problem The ex-radius r1, r2, r3 of a triangle ABC are in HP. Then a, b, c are in : a. AP b. GP c. HP d. a = b = c 91. 89 Problem The points A (2a, 4a), B(2a, 6a) and C ({2 3}a, 5a) are the vertices of : a. An equilateral triangle b. A scalene triangle c. An isosceles triangle d. None of the above 92. 90 Problem sin sin If are in AP then cos cos is equal to : a. cot b. cos c. tan d. Sin 93. 91 Problem If a and b are eccentric angles of the ends of a focal chord of the ellipsex2 y 2 2 1, then tan tanis equal to :a2b 22e 1 a. e 11e b. 1 ee 1 c. e 1 d. none of these 94. 92 Problem On the occasion of Deepawali festival each student of a class sends greeting cards to the others. If there are 20 students in the class, then the total number of greeting cards exchanged by the students is : a. 20C2 b. 220C2 c. 220P2 d. none of these 95. 93 Problem Let S be the set of all straight lines in a plane. A relation R is defined on S by a R b a b, then R is : a. Reflexive but neither symmetric nor transitive b. Symmetric but neither reflexive nor transitive c. Transitive but neither reflexive nor symmetric d. An equivalence relation 96. 94 Problem The eccentricity of the hyperbola whose latus rectum is 8 and conjugate axis is equal to half the distance between the foci, is :4 a. 3 4 b. 3 2 c. 3 d. none of these 97. 95 Problem In a class of 45 students, 22 can speak Hindi only and 12 can speak English only. The number of students, who can speak both Hindi and English is : a. 9 b. 11 c. 23 d. 17 98. 96 Problem There are 18 points in a plane such that no three of them are in the same line except five points which are collinear. The number of triangles formed by these points is : a. 816 b. 806 c. 805 d. 813 99. 97 Problem The area bounded by y = |x -1| and y = 1 is : a. 1 sq unit b. 2 sq unit1 c. sq unit2 d. none of these 100. 98 Problem10 The middle term in the expansion of X 1 is equal to : 2x 105 a.32x 263 b.8105 c. - 32x 2 63 d. -8 101. 99 Problem The equation of the chord of the hyperbola 25x2 16y2 = 400 that is bisected at point (5, 3) is a. 135x 48y = 481 b. 125x 48 y = 481 c. 125 x 4y y = 48 d. none of the above 102. 100 ProblemThe length of the radius of the circle which passes through the point (6, 2) andwhose two diameters are x + y = 6 and x + 2y = 4, is :a. 10b. 2 5c. 6d. 4 103. FOR SOLUTIONS VISIT WWW.VASISTA.NET