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1. 2. 3. 4. 5. 5c 3b a If log , log and log are in a 5c 3b AP, where a, b, c are in GP, then a, b, c are the length of sides of (a) an isosceles triangle (b) an equilateral triangle (c) a scalene triangle (d) None of the above If the pth, qth and rth terms of an AP are in GP then the common ratio of GP is pq (a) rq rq (b) q p pr (c) pq (d) None of the above If a, b, c, d are four numbers such that the first three are in AP while the last three are in HP then (a) bc = ad (b) ac = cd (c) ab = cd (d) None of the above If Sr denotes the sum of the first r terms S Sr 1 of an AP then 3r is equal to S2r S2r 1 (a) 2r-1 (b) 2r+1 (c) 4r+1 (d) 2r+3 If a, b, c are in AP then a 1 , b 1 , c 1 bc 6. ca ab are in (a) AP (b) GP (c) HP (d) None of the above If then the minimum value 2 2 3 of cos + sec 3 is (a) 1 (b) 2 (c) 0 (d) None of the above XIX 7. FT12 Let Sn denote the sum of the cubes of the first n natural numbers and Sn denotes the sum of the first n natural n S numbers. Then r is equal to r 1 S r n(n 1)(n 2) 6 n(n 1) (b) 2 (a) n 2 3n 2 (c) 6 (d) None of the above 8. If a, b, c are in HP then 1 1 is ba bc equal to 2 (a) b 2 (b) ac 1 1 (c) a c (d) Both (a) and (b) 9. The number of real solutions of 1 1 x 2 2 2 is x 4 x 4 (a) 0 (b) 1 (c) 2 (d) infinite 10. The number of real solutions of the equation ex x is (a) 1 (b) 2 (c) 0 (d) None of the above 11. The number of real solutions of the equation log 0.5 x | x | is (a) (b) (c) (d) 1 2 0 None of the above (1) FT12 12. The equation x 1 x 1 4x 1 has No solution One solutions Two solutions More than two solutions The number of solutions of the equation | x | = cos x is (a) One (b) Two (c) Three (d) Zero If 3 x+1 = 6 log23 then x is (a) 3 (b) 2 (c) log32 (d) log23 If ( 2 ) x ( 3 ) x ( 13 ) x / 2 then the number of values of x is (a) 2 (b) 4 (c) 1 (d) None of the above The number of real solutions of the 6x x 2 equations 2 is x2 x 4 (a) Two (b) One (c) Three (d) None of the above The number of real solutions of (a) (b) (c) (d) 13. 14. 15. 16. 17. x 2 4x 2 x 2 9 4x 2 14x 6 is (a) One (b) Two (c) Three (d) None of the above 18. If [x]2 = [x+2], where [x] = the greatest integer less than or equal to x, then x, must be such that (a) x = 2-1 (b) x [2,3) (c) x[-1,0] (d) None of the above XIX 19. The number of solutions of | [x] – 2x| = 4, where [x] is the greatest integer ≤ x, is (a) 2 (b) 4 (c) 1 (d) Infinite 20. The set of real values of x satisfying | x – 1 | ≤ 3 and | x – 1 | ≥ 1 is (a) [2,4] (b) [- (,2] [4,) (c) [-2,0][2,4] (d) None of the above 21. The solution set of x 2 3x 4 1, xR, is x 1 3,+) -1,1)3,+) [-1,1][3,+) None of the above 22. The number of integral solutions of x2 1 is x2 1 2 (a) 4 (b) 5 (c) 3 (d) None of the above (a) (b) (c) (d) 23. The value of the sum 13 (i n 1 n i n 1 ) where I = 1 is (a) i (b) i-1 (c) -i (d) 0 24. If a+i=+i then b+i5 is equal to (a) +i (b) -i (c) -i (d) --i 25. If is a non real cube root of unity then the expression 1- )1- 2)1 4)1+ 8) is equal to (a) 0 (b) 3 (c) 1 (d) 2 (2) FT12 26. If x2-x+1=0 then the value of 2 n 1 is n 1 xn (a) 8 (b) 10 (c) 12 (d) None of the above 1 z 27. If | z | = 1 then is equal to 1 z (a) z (b) z (c) z+ z (d) None of the above z 1 28. = 1 represents z 1 (a) A circle (b) An ellipse (c) A straight line (d) None of the above ax a x 5 32. The value of x 29. If a x a ax a x = 0 then x is x 0 a 3 2a 0 pq pq (a) (b) (c) (d) 30. qp 0 q r is equal to r p r q 0 (a) p + q + r (b) 0 (c) p – q – r (d) - p + q + r 31. The value of the determinant bc ca ab p q r , where a, b, c are the pth, 1 1 1 qth and rth terms of a HP, is ap + bq + cr (a) a + b + (b) p + q + r) (c) 0 (d) None of the above XIX a1x b1y a 2 x b 2 y a 3 x b3 y b1x a1y b 2 x a 2 y b3 x a 3 y b1x a1 b2 x a 2 b3 x a 3 is equal to (a) x 2 y 2 (b) 0 (c) a1a 2a 3x 2 b1b2b3 y2 (d) None of the above 33. If , are non-real numbers satisfying x3 1 0 λ 1 α α λ β 1 β 1 λα β then the is equal to value of (a) 0 (b) 3 (c) 3 + 1 (d) None of the above 1 i, and is a non-real cube root of unity then the value of 34. If 1 2 1 i 2 i 1 1 i 1 i 2 1 is equal to 1 (a) 1 (b) i (c) (d) 0 35. The system of equations ax + 4y + z =0, bx + 3y + z =0, cx + 2y + z = 0 has nontrivial solutions if a, b, c are in (a) AP (b) GP (c) HP (d) None of the above 36. The equations x + y + z = 6, x + 2y + 3z = 10, x + 2y + mz = n give infinite number of values of the triplet x, y, z) if (a) m = 3, n R (b) m = 3, n 10 (c) m = 3, n = 10 (d) None of the above (3) FT12 37. The system of equations 2x + 3y = 8, 7x – 5y + 3 = 0, 4x – 6y + = 0 is solvable if is (a) 6 (b) 8 (c) –8 (d) –6 38. Indefinite Integra If (x ) cot 4 xdx 1 cot 3 x cot x 3 39. 40. 41. 42. and then ( x ) is 2 2 (a) -x (b) x- (c) -x 2 (d) None of the above x x (1 log x )dx is equal to (a) xxlogx+k x (b) e x k (c) x x k (d) None of the above xdx is equal to 1 x4 (a) tan 1 x 2 k 1 (b) tan 1 x 2 k 2 (c) log( 1 x 4 ) k (d) None of the above x5 / 2 dx is 1 x7 2 (a) log( x 7 / 2 x 7 1) c 7 1 x7 1 log 7 c (b) 2 x 1 (c) 2 1 x 7 c (d) None of the above The primitive of the function x | cos x | when x is given by 2 (a) cos x x sin x (b) cos x x sin x (c) xsinx-cosx (d) None of the above XIX 43. e x (1 tan x ) sec xdx is equal to (a) e x sec c (b) e x tan x c (c) e x tan x c (d) None of the above dx is equal to cos x 3 sin x x Π (a) log tan +k 2 3 x (b) log tan +k 2 3 1 x (c) log tan +k 2 2 3 (d) None of the above 44. 45. In the expansion of 2 x 5 2 , the coefficient of x 4 , if it exists, is 5 7 9 11 1 25 2 4! 2 4 5 7 9 11 1 (b) 4! 2 5 7 9 11 4 (a) (c) 2 21 (d) None of the above 46. The coefficient of x 5 in the expansion 1 x2 , x 1, is 1 x (a) –1 (b) 2 (c) 0 (d) –2 47. The coefficient of x n in the expansion of e2 x 3 is of 2n n! e3 2 n (b) n! (c) a p a q (a) (d) None of the above (4) FT12 48. The coefficient of x10 in the expansion of 10 x in ascending powers of x is (a) (b) (c) log e10 10 10 ! 1 10 ! (log 10e)10 10 ! (d) None of the above 49. The constant term in the expansion of 3x 2 x is x2 (a) log e 3 (b) log e 6 log e 3 2 1 3 log e 6 log e 2 2 (d) None of the above 50. If x <1, the coefficient of x 3 in the (a) (b) (c) (d) 1 is e (1 x ) x 17 6 17 6 11 6 None of the above log e 1 x 1 x 2 3 (a) 2 (b) 2 C1 1 2 (d) None of the above (c) XIX 3 4 x x ...to y 3 4 then y y 2 2! 3 y ...to 3! is equal to (a) x (b) x (c) x + 1 (d) None of the above 53. The matrix λ 7 2 is a singular matrix 4 1 2 1 3 2 if is 2 (a) 5 5 (b) 2 (c) 5 (d) None of the above 0 4 1 exists A 2 λ 3 then A 1 1 2 1 i.e., A is invertible) if (a) 4 (b) 8 (c) = 4 (d) None of the above 1 3 is 4 2 3 1 55. The rank of the matrix λ 2 2 18 11 18 (b) = 11 18 (c) = 11 (d) None of the above 1 56. The value of 3 if (a) 51. If x <1, the coefficient of x 2 in the expansion of 2 54. If (c) expansion of 52. If x x 2 is is equal to 40 20 10 80 1 3 10 2 5 70 3 10 2 5 (b) 70 3 10 2 5 (c) 50 (d) None of the above (a) (5) FT12 57. The square root of 2 x 2 x 2 1 is (a) x 1 x 1 (b) x 1 x 1 58. If x 3 2 ,y (b) 0 (c) –1 3 2 3 2 (d) None of the above then the value of x2+xy+y=is (a) 5 (b) 99 (c) 98 (d) None of the above 59. If y =cos-1cosx) then (a) (b) (c) 2 | x | x 2 1 1 2x x 2 1 1 2x x 1 (d) None of the above XIX 2 1 x 2 1 with x 1 x2 x2 (a) dy 5 at x = is dx 4 dy x 1 then is equal to dx x 1 1 63. The derivative of tan 1 respect to tan 1 x is equal to (a) 1 (b) -1 1 (c) 2 (d) None of the above y .....to dy ye 60. If x e then is dx x (a) 1 x 1 (b) x 1 x (c) x (d) None of the above 61. If y tan 1 dy at x =e is dx (a) 1 (c) x 1 x 1 (d) None of the above 3 2 62. If xy.yx = 16 then (b) 1 1 1 x2 (d) None of the above (c) 64. Let x 3 sin x cos x f (x) 6 1 0 p p2 p3 , where p is a d3 constant . Then {f ( x)} at x 0 is dx 3 (a) p (b) p+p2 (c) p+p3 (d) Independent of p 65. If y =at2, x= 2at, where a is a constant, d2y 1 then at x is 2 2 dx 1 (a) 2a (b) 1 (c) 2a (d) None of the above 66. Let A and B be two sets such that A B A . Then A B is equal to (a) (b) B (c) A (d) None of the above (6) FT12 67. Of the members of three athletic teams in a school 21 are in the cricket team, 26 are in the hockey team and 29 are in the football team. Among them, 14 play hockey and cricket, 15 play hockey and football, and 12 play football and cricket. Eight play all the three games. The total number of members in the three athletic team is (a) 43 (b) 76 (c) 49 (d) None of the above 68. The relations “cingruence modulo m” is (a) Reflexive only (b) Transitive only (c) Symmetric only (d) An equivalence relation XIX 69. Let f : R R such that fx) = 1 , 1 x2 xR. Then f is (a) Injective (b) Surjective (c) Bijective (d) None of the above 70. Let R be the relation over the set of integers such that m Rn if and only if m is a multiple of n. Then R is (a) Reflexive (b) Symmetric (c) Transitive (d) An Equivalence relation (7)