1. UPSEEPAST PAPERSMATHEMATICS - UNSOLVED PAPER 2006

2. SECTION 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 If z1, z2 are any two complex numbers, then : a. | z1 z2 | | z1 | | z2 | b. | z1 z2 | | z1 | | z2 | c. | z1 z2 | | z1 | | z2 | d. | z1 z2 | | z1 | | z2 | 4. 02 Problem If z = x + iy is a variable complex number such that arg , z 1 then :z 1 4 a. x2 y2 2x = 1 b. x2 + y2 2x = 1 c. x2 + y2 2y = 1 d. x2 + y2 + 2x = 1 5. 03 Problem If arithmetic mean of tow positive numbers is A, their geometric mean is G and harmonic mean is H, then H is equal to : a. G2 / A b. A2 / G2 c. A / G2 d. G / A2 6. 04 Problem The sun of n terms of two arithmetic series are in the ratio 2n + 3 : 6n + 5, then the ratio of their 13th terms is : a. 53 : 155 b. 27 : 87 c. 29 : 83 d. 31 : 89 7. 05 Problem If , , are the roots of the equation x3 + x + 1 = 0 , then the value of3 3 3 is : a. 0 b. 3 c. - 3 d. - 1 8. 06 Problem0 0 1 If , then A-1 is :A 0 1 01 0 0 a. - A b. A c. 1 d. none of these 9. 07 Problem6 8 54 2 3 If A = 9 7 1 is the sum of symmetric matrix B and skew-symmetric matrix C, then B is :6 6 76 2 5 a. 9 7 10 2 22 5 2 b.2 2 06 6 7 c.6 25 7 510 6 2 d. 2 0 2 22 0 10. 08 Problem1 1 If A = 1 1, then A100 is equal to : a. 2100 A b. 299A c. 100 A d. 299A 11. 09 Problem1 1 1 x 0x If 1 2 2 y 3 , then y is equal to :1 3 1 z 4z0 a. 111 b. 2352 c.112 d. 3 12. 10 Problem If y = cos2 x + sec2 x, then : a. y 2 b. y 1 c. y 2 d. 1 < y < 2 13. 11 Problem1 1 If x 2 cos , then x2is equal to :2 x3 a. sin 3 b. 2 sin 3 c. cos 3 d. 2 cos 3 14. 12 Problem If sin + cosec = 2, the value of sin10 + cosec10 is : a. 2 b. 210 c. 29 d. 10 15. 13 Problem The value of sin357is : sinsinsin16 16 16 16 2 a.16 b. 18 1 c. 16 2 d. 32 16. 14 Problemtan 31 If3 , then the general value of is :tan 31 n a. 312 b. n 712 c.n7 336 d. n12 17. 15 Problemsin B In any triangle ABC, If cos A 2 sin C , then : a. a = b = c b. c = a c. a = b d. b = c 18. 16 Problem In a triangle ABC, if b + c = 2a and A = 600 then ABC is : a. Equilateral b. Right angled c. Isosceles d. Scalene 19. 17 Problem The co-ordinates of the point which divides the join of the points (2, -1, 3) and (4, 3, 1) in the ratio 3 : 4 internally are given by : .2 20 10 a.,,7 7 710 15 2 b. ,, 7 7 720 5 15, , c. 7 7 7 15 20 3, ,7 7 7 d. 20. 18 Problem The area of the triangle ABC, in which a = 1, b = 2, C = 600, is : a. 4 sq unit1 b. 2 sq unit c. 3 sq unit2 d. 3 sq unit 21. 19 Problem In a triangle ABC, b = , c = 1 and = 300 , then the largest angle of the triangle is : a. 600 b. 1350 c. 900 d. 1200 22. 20 Problem If A + B + C = , then sin 2A + sin 2B + sin2C is equal to : a. 4 sin A sin B sic C b. 4 cos A cos B cos C c. 2 cos A cos B cos C d. 2 sin A sin B sin C 23. 21 Problem A flag is standing vertically on a tower of height b. On a point at a distance a from the foot of the tower, the flag and the tower subtend equal angles. The height of the flag is :a2b2 a.a2b2 a2 b2 b. a2 b2 c.a2b2 a2b2 a2 b2 d.a2 b2 24. 22 Problem If are the roots of the equation 6x2 5x + 1 = 0, then the value of tan 1 tan 1 is a. 0 b. /4 c. 1 d. /2 25. 23 Problem The three stainght lines ax + by = c, b x + cy =a and c x + ay =b are collinear, if: a. b+ c=a b. c + a=b c. a+ b + c=0 d. a + b =c 26. 24 Problem The length of perpendicular from the point (a cos , a sin ) upon the straight line y = x tan + c, c > 0, is : a. c b. c sin2 c. c cos2 d. c sec2 27. 25 Problem The equation of the circumcircle of the triangle formed by the lines x = 0, y = 0, 2x + 3y = 5 is : a. 6(x2 + y2) + 5 (3x 2y) = 0 b. x2 + y2 2x 3y + 5 = 0 c. x2 + y2 + 2x 3y - 5 = 0 d. 6(x2 + y2) - 5 (3x + 2y) = 0 28. 26 Problem The differential equation of system of concentric circles with centre (1, 2) is : a. (x - 2) + (y -1) dy = 0 dxdy b. (x - 1) + (y -2)dx =0dy c. (x + 1) dx + (y - 2) = 0dy d. (x + 2) dx+ (y -1) = 0 29. 27 Problem The equation of pair of lines joining origin to the points of intersection of x2 + y2 = 9 and x + y = 3 is : a. x2 + (3 - x)2 = 9 b. xy = 0 c. (3 + y)2 + y2 = 9 d. (x - y)2 = 9 30. 28 Problem The value of , for which the circle x2 + y2 + 2 x + 6y + 1 = 0 intersects the circle x2 + y2 + 4x + 2y = 0 intersects the circle x2 + y2 + 4x + 2y = 0 orthogonally, is : a. 11/8 b. -1 c. -5/4 d. 5/2 31. 29 Problem The value of m, for which the line y = mx + 25 3 is a normal to the conicx2 y2 31, is16 92 a. 3 b.3 c. 32 d. none of these 32. 30 Problem The value of c, for which the line y = 2x + c is a tangent to the circle x2 + y2 = 16, is : a. -16 5 b. 4 5 c. 16 5 d. 20 33. 31 Problem The value of , for which the equation x2 y2 x + y 2 = 0 represents a pair of straight lines, are : a. -3, 1 b. -1, 1 c. 3, -3 d. 3, 1 34. 32 Problem The focus of the parabola x2 + 2y + 6x = 0 is : a. (-3, 4) b. (3, 4) c. (3, -4) d. (-3, -4) 35. 33 Problem The value of m, for which the line y = mx + 2 becomes a tangent to the conic 4x2 9y2 = 36, are ; a. 23 b. 2 2 3 8 c. 9 4 2 d. 3 36. 34 Problem The eccentricity of the conic 4x2 + 16y2 24x 32y = 1 is : a. 12 b. 33 c. 23 d. 4 37. 35 Problem The number of maximum normals which can be drawn from a point to ellipse is : a. 4 b. 2 c. 1 d. 3 38. 36 Problem The equation of line of intersection of planes 4x + 4y 5z = 12, 8x + 12y 31z = 32 can be written as : a. x 1 y 2z 2 3 4 b. x 1y 2 z23 4x y 1z 2 c. 234xy z 2 d.23 4 39. 37 Problem If a line makes angle , , ,with four diagonals of a cube, then the value ofsin2sin2 sin2sin2 is : a. 4/3 b. 8/3 c. 7/3 d. 1 40. 38 Problem The equation of the plane, which makes with co-ordinate axes, a triangle with its centroid , , , is : a.xy z 3 b. x y z 1x y z c. =3 d. x y z =1 41. 39 Problem If the points (1, 1), (-1, -1), (- 3, 3 ) are the vertices of a triangle, then this triangle is : a. Right-angled b. Isosceles c. Equilateral d. None of these 42. 40 Problem A variable plane moves so that sum of the reciprocals of its intercepts on the co-ordinate axes is . Then the plane passes through :1 1 1 , , a. 2 2 2 b. (-1, 1, 1) c. (2, 2, 2) d. (0, 0, 0) 43. 41 Problem The direction cosines l, m, n of two lines are connected by the relations l + m + n = 0, lm = 0, then the angle between them is ; a. /3 b. /4 c. /2 d. 0 44. 42 Problem The value of[a b cab c] is : a. [a b c ] b. 0 c. 2 [a b c ] d.a x (b x c ) 45. 43 Problem The area of the triangle having vertices as i 2j i3k , 3j i3k , 4 7 j 7k , is : a. 36 sq unit b. 0 sq unit c. 39 sq unit d. 11 sq unit 46. 44 Problem The figure formed by the four points i j ik , 2 3,5jj 2k and k j is : a. Trapezium b. Rectangle c. Parallelogram d. None of the above 47. 45 Problem The equation of the plane passing through three non-collinear points a, b, c is : a. r (b x c cxa a x b) 0 b. r (b x c cxa a x b) [a b c] r (a x (b x c )) [a b c ] c. d. r (a b c)0 48. 46 Problem The unit vector perpendicular to and coplanar with is :2 i5 j a.29 b.2 5 i j 1 c.2( i j ) d. ij 49. 47 Problem 2 2(a x b)(a b) is equal to : a. a2 b2 b. a2b2 c. 1 d. 2 a2 b2 50. 48 Problem The domain of the function f(x) = exp ( 5x 3 2x2 ) is : a. [3/2, ) b. [1, 3/2] c. (- , 1] d. (1, 3/2) 51. 49 Problemsin xlim is equal to :x x a. b. 1 c. 0 d. does not exist 52. 50 Problem For the function which of the following is correct : a.lim f ( x )does not exist x0 b.lim f ( x)=1 x0 c. lim f (x) exists but f(x) is not continuous at x = 0x 0 d. f(x) is continuous at x = 0 53. 51 Problemx(fo fo...of )(x) If f ( x), thenis equal to : x1 19 times x a.x 1 19 b.x x 1 c.19 x x 1 d. x 54. 52 Problem A function f is defined by f(x) = 2 + (x - 1)2/3 in [0, 2]. What of the following is not correct ? a. f is not derivable in (0, 2) b. f is continuous in [0, 2] c. f(0) = f(2) d. Rolles theoren is ture in [0, 2] 55. 53 Problem 2x 1 If f(x) =x 5(x 5) , then f-1 (x) is equal to : x 51 a.,x2x 12 b. 5x 1 ,x 22 x c.5x 1,x2 2 xx 5 1 d. ,x 2x 1 2 56. 54 Problem d11x2 1 is equal to :tan dx x 1 a.1 x2 x2 b.2 1 x 2 ( 1 x 2 1) c. 2 1 x21 d. 2 1 x2 57. 55 Problemx is equal to :1 cos d1 2tan dx x1 cos2 a. -1/4 b. 1/4 c. -1/2 d. 1/2 58. 56 Problem The maximum value of x1/x is : a. 1/ee b. e c. e1/e d. 1/e 59. 57 Problem The function f defined by f(x) = 4x4 2x + 1 is increasing for : a. x < 1 b. x > 0 c. x < 1/2 d. x > 1/2 60. 58 Problem A particle moves in a straight line so that s = t , then its acceleration is proportional to : a. (veloctiy)3 b. velocity c. (velocity)2 d. (velocity)3/2 61. 59 Problem32 x 2 (log x )2 dx is equal to : a. 8x4(log x)2 + c b. x4{8(log x)2 4(log x} + c c. x4{8(log x)2 4 log x} + c d. x3 {(log x)2 2 log x} + c 62. 60 Problemcos x1 x is equal to : e dxsin x1 e x cos x a. c 1 sin x e x sin x c b.1 sin xex c c. 1 sin x e x cos xc d.1 sin x 63. 61 Problem If f(x) dx = g(x) + c, then f-1 (x) dx is equal to : a. x f-1 (x) + c b. f(g-1 (x)) + c c. xf-1(x) g (f-1 (x)) + c d. g-1(x) + c 64. 62 Problem The value of 2dx is :1 x(1 x 4 )1 17 a. log4 321 32 b. log4 1717 c. log 2 1 17 log d.42 65. 63 Problem The value of the integral b x dxis : a x a b x a.1 b. 2(b - a) c. /2 d. b a 66. 64 Problem The area bounded by y = log x, x-axis and ordinates x = 1, x = 2 is : a. (log 2)2 b. log 2/e c. log 4/e d. log 4 67. 65 Problem The area of the segment of a circle of radius a subtending an angle of 2 at the centre is : 1a2 sin2 a.2 1 2 b.a sin 2 2 1 c. a2 2 sin2 d. a2 68. 66 Problem dy2yx1 The solution of the differential equation is : dx 1 x2 (1 x 2 )2 a. y(1 + x2) = x + tan-1 x y b. 1 x2 = c + tan-1 x c. y log (1 + x2) = c + tan-1 x d. y (1 + x2) = c + sin-1 x 69. 67 Problem The solution of the differential equation xdy y dx = x2 y2 dx is :2 a. x + xy 2 = cx2 b. y - x2y 2 = cx c. x - x2 y 2 = cx d. y + x2y 2 = cx2 70. 68 Problem dy The solution of the differential equationex y x 2e yis : dx a. y = ex-y x2e-y + c1 b. ey - ex = 3 x3 + c 1 c. ex + ey = x3 + c 3 d. ex ey =1 x3 + c 3 71. 69 Problem If A and B are 2 x 2 matrices, then which of the following is true ? a. (A + B)2 = A2 + B2 + 2AB b. (A - B)2 = A2 + B2 2AB c. (A - B) (A + B) = A2 + AB BA B2 d. (A + B) (A - B) = A2 B2 72. 70 Problem If M and N are any two events. The probability. That exactly one of them occurs, is : a. P(M) + P(N) P(M N) b. P(M) + P(N) + P(M N) c. P(M) + P(N) d. P(M) + P(N) 2P(M N) 73. 71 Problem If four dice are thrown together. Probability that the sum of the number appearing on them is 13, is : a.35324 5 b. 216 11 c. 21611 d. 432 74. 72 Problem If is the angle between two regression lines with correlation coefficient , then :2 a. sin 12 b. sin 12 c. sin 12sin 1 d. 75. 73 Problem The value of .0 , where 0.0 stands for the number 0.0373737 ., is : a. 37/1000 b. 37/990 c. 1/37 d. 1/27 76. 74 Problem 2 a b a If is an imaginary root of unity, then the value of 2 is : b c b 2 c a c a. a3 + b3 + c3 b. a2b b2c c. 0 d. a3 + b3 + c3 3abc 77. 75 Problem If A = {x, y}, then the power set of A is ; {xy, yx} { , x, y} { , {x}, {2y}} { , {x}, {y}, {x, y}} 78. FOR SOLUTIONS VISIT WWW.VASISTA.NET