Tiêu đề 10. INVERS TRIGANOMETRIC FUNCTIONS HARD.pdf ✅

1. All possible values of p and q for which 4 3 cos cos 1 cos 1 1 1 1  + − + − = − − − p p q holds, is (a) 2 1 p = −1, q = (b) 2 1 q  1, p = (c) 2 1 0  p  1,q = (d) None of these 2.        +      − − a b b b a co a 2 1 3 2 1 3 tan 2 1 sec 2 tan 2 1 sec 2 is equal to (a) ( )( ) 2 2 a − b a + b (b) ( )( ) 2 2 a + b a − b (c) ( )( ) 2 2 a + b a + b (d) None of these 3. The value of is equal to (a) 7/4 (b) 11/4 (c)/12 (d) 25/12 4. If       = + − − − 5 3 cos ec x 2 cos 7 cos 1 1 1 then the value of x is (a) 44/117 (b) 125/117 (c) 24/7 (d) 5/3 5. If A = 2 tan–1 (2 2 −1) and B=3sin–1 3 1 + sin–1 5 3 , then (a) B < A < 1050 (b) 1050< B < A (c) B < 1050< A (d) 1050< A < B 6. The value of        − − 6 7 cos sin 1 is- (a) 3 5 (b) 6 7 (c) 3  (d) None of these 7. If         −   − , 2 3 , then the value of tan–1 (cot ) – cot–1 (tan ) + sin–1 (sin ) + cos–1 (cos) is equal to- (a) 2 +  (b)  +  (c) 0 (d)  –  8. If z = sec–1       + x 1 x +sec–1         + y 1 y , where xy> 0, then the value of z (among the given values) is not possible- (a) 6 5 (b) 10 7 (c) 10 9 (d) 3 5 9. If    are the roots of the equation x3 + mx2 + 3x + m = 0, then the general value of tan–1 + tan–1+ tan–1  is: (a) (2n + 1) 2  (b) n (c) 2 n (d) Dependent upon the value of p 10. If x          ,2 2 3 , then value of the expression sin–1 (cos–1 (cos x) + sin–1 (sin x))), is equal to (a) –/2 (b) /2 (c) 0 (d) None of these 11. If x > 0, y > 0 and x > y, then tan–1 (x/y) + tan–1 [(x + y)/ (x – y)] is equal to- (a) –/4 (b) /4 (c) 3/4 (d) None of these 12. The value of sin(2 tan–1 (1/3)) + cos (tan–1 2 2 ) is- (a) 12/13 (b) 13/14 (c) 14/15 (d) None of these 13. If tan–1 x 1 x 1 − + + tan–1 x x −1 = tan–1 (–7) +  then x= (a) 2 (b) 3 (c) 4 (d) None of these 14. Find x satisfying [tan–1 x] + [cot–1 x] = 2, [·] →greatest Integer function (a) (cot 3, cot 2) (b) (cot 3, – cot 1) (c) (cot 3, 0) (d) None of these 15. If f(x) = sin–1 x + cos–1 x + tan–1 x + cot–1 x + sec –1 x, then f(x) lies in the interval- (a) [, 2] (b) (, 2) (c)         2 3 ,          , 2 2 3 (d) None of these 16. The solution of tan–1 2x + tan–1 3x = 4  is- (a) 1/6 (b) –1 (c)       ,–1 6 1 (d) 1⁄2 17. The inequality sin–1 (sin 5) > x2 – 4x holds if- (a) x = 2 – 9 − 2 (b) x = 2 + 9 − 2 (c) x (2 – 9 − 2 , 2 + 9 − 2 ) (d) x > 2 + 9 − 2 18. f sin–1 (x – 3) + cos–1 (x – 1) + tan–1       − 2 2 x x = cos–1 k –  then value of k is (a) 2 1 (b) 2 1 − (c) 1 (d) – 1 19. cosec–1 (cosec x) and cosec (cosec–1x) are equal functions then maximum range of value of x is- (a)       −  − , 1 2         2 1, (b)        − , 0 2         2 0, (c) (–, –1]  [1, ) (d) [–1, 0)  [0, 1) 20. If tan       + + + − − − − 5 1 tan 4 1 tan 3 1 tan 2 1 tan 1 1 1 1 = b a then a + b equals (a & b are in their lowest form) (a) 27 (b) 19 (c) 38 (d) None of these 21. If  = sin–1 (cos (sin–1x)) and  = cos–1 (sin (cos–1 x)), then- (a) tan = cot  (b) tan  = – cot  4 tan ( 1) 2 1 3cos 2 1 2sin 2 1 cos 1 1 1 1 − −          + −       −      − − − − −
(c) tan = tan  (d) tan  = – tan  22. There exists a positive real number x satisfying cos (tan–1 x) = x, the value of cos–1         2 x 2 is- (a) 10  (b) 5  (c) 5 2 (d) 5 4 23. If x & y > 0, x > y then tan–1         y x + tan–1         − + x y x y is equal to (a) –/4 (b) /4 (c) 3/4 (d) None of these 24. If A = 2 tan–1 (2 2 – 1) and B = 3 sin–1       3 1 + sin–1       5 3 , then- (a) A = B (b) A < B (c) A > B (d) None of these 25. Which of the following identities does not hold? (a) sin–1 x = cot–1         − x (1 x ) 2 ; 0 < x  1 (b) sin–1 x = cot–1         − x (1 x ) 2 ; –1  x  0 (c) sin–1 x = cos–1 (1 x ) 2 − ; 0  x  1 (d) sin–1 x =1– sin–1 (–x); – 1  x  1 26. Solution of the equation 3 sin–1         + 2 1 x 2x – 4 cos–1         + − 2 2 1 x 1 x + 2 tan–1         − 2 1 x 2x = 3  is- (a) x = 3 (b) x = 3 1 (c) x = 1 (d) x = 0 27. If sin–1 x + sin–1y + sin–1 z = 2 3 and f(1) = 2, f(p +q) = f(p). f(q)  p, q  R, then f(1) f(2) f(3) x + y + z – f(1) f(2) f(3) x y z x y z + + + + = (a) 0 (b) 1 (c) 2 (d) 3 28. If x1, x2, x3, x4 are the roots of the equation x 4 – x 3 sin 2 + x 2 cos2 – xcos – sin  = 0, then, tan–1 x1 + tan–1 x2 + tan–1 x3 + tan–1 x4is equal to- (a)  (b) /2 – (c)  –  (d) –  29. tan–1               +  − b 3a cos 2 1 4 1 + tan               −  b 3a cos 2 1 – 4 1 is equal to - (a) a 2b (b) 2a b (c) a b 3 2 (d) b a 2 3 30. If n N, and the set of equations cos–1 x + (sin–1 y)2 = 4 n 2  and (sin–1y)2 – cos–1 x = 16 2  is consistent then n can be equal to– (a) 0 (b) 1 (c) 3 (d) 4 31. The value of x for which cos–1 (cos 4) > 3x 2 – 4x, is - (a)         +  − 3 2 6 8 0, (b)         −  − ,0 3 2 6 8 (c) (–2, 2) (d)         −  − +  − 3 2 6 8 , 3 2 6 8 32. If u = cot–1 tan – tan–1 tan , then tan        2 μ – 4 is equal to– (a) tan (b) cot (c) tan  (d) cot  33. m m 2 2m tan 4 2 n m 1 1  + + = − = (a) tan–1 (n2 + n + 1) (b) tan–1 (n2 – n + 1) (c) tan–1 n n 2 n n 2 2 + + + (d) None of these 34. Total number of ordered pairs (x, y) satisfying |y| = cosx and y = sin–1 (sin x) where |x|  3, is equal to- (a) 2 (b) 4 (c) 6 (d) 8 35. = − 1000 i 1 i 1 sin x = 500  then value of  = 1000 i 1 i x (a) 500 (b) 100 (c) 100000 (d) 1000 36. If cos–1 p + cos–1 q + cos–1 r = . Then value of p2 + q2 + r2 + 2pqr + 4 is equal to- (a) 1 (b) 0 (c) 5 (d) 3 37. The number of real solution of tan–1 x(x +1) + sin–1 1 2 x + x + = 2  is– (a) Zero (b) One (c) Two (d) Infinite 38. If cos–1       x 1 = , then tan = (a) x 1 1 2 − (b) x 1 2 + (c) 2 1− x (d) x 1 2 − 39. tan (cos–1 x) is equal to- (a) x 1 x 2 − (b) 2 1 x x + (c) x 1 x 2 + (d) 2 1− x 40. If tan–1 x + tan–1 y + tan–1 z = , then xy 1 + yz 1 + zx 1 = (a) 0 (b) 1 (c) 1/xyz (d) xyz
41. sin       − 5 4 cos 2 1 1 = (a) 10 1 (b) – 10 (c) 10 1 (d) – 10 1 42. tan         −     −  5 4 1 2 tan 1 = (a) 7 17 (b) – 7 17 (c) 17 7 (d) – 17 7 43. If tan–1 x + tan–1 y + tan–1 z =  then x + y + z = (a) xyz (b) 0 (c) 1 (d) 2xyz 44. sin–1 sin 22 + cos–1 cos 33 + tan–1 tan 44 = (a) 55 – 17 (b) 16 – 48 (c) 45 – 18 (d) None 45. cos–1              −  10 9 sin 10 9 cos 2 1 = (a) 20 3 (b) 20 7 (c) 10 7 (d) 20 17 46. cos–1         + − − 4 x x 1 x 1 2 1 2 2 2 = cos–1 2 x – cos–1 x holds if (a) |x|  1 (b) x  R (c) 0  x  1 (d) –1  x  0 47.  = tan–1 (2 tan2) – tan–1       tan  3 1 then tan  = (a) –2 (b) –1 (c) 2/3 (d) 2 48. If 2 1 sin–1       +   5 4cos2 3sin 2 = tan–1 x then x = (a) tan 3 (b) 3 tan  (c) 3 1 tan  (d) 3 cot  49. If the sum of the acute angles tan–1 x and tan–1       2 1 is 450 , then value of x is (a) 2 1 (b) 3 1 (c) 4 1 (d) 5 1 50. The greater of two angles A = 2 tan–1 ( 2 2 –1) and B = 3 sin– 1 (1/3) + sin–1 (3/5) is (a) A (b) B (c) Both are equal (d) None of these 51. The solution to the equation sin (1/5 cos–1 x) = 1 is (a) 3  (b) 6  (c) 4  (d) No solution 52. If cos–1x + cos–1y + cos–1 z = , then- (a) x2 + y2 + x2 + xyz = 0 (b) x2 + y2 + z2 + 2xyz = 0 (c) x 2 + y2 + z2 + xyz = 1 (d) x2 + y2 + z2 + 2xyz = 1 53. If xy + yz + zx =1, than tan–1 x + tan–1 y + tan–1 z is equal to - (a)  (b) /2 (c) 0 (d) None of these 54. If (tan–1x)2 + (cot–1x)2 = 8 5 2  , then x equals- (a) – 1 (b) 1 (c) 0 (d) None of these 55. If A = tan–1         2k – x x 3 and B = tan–1         k 3 2x – k , then the value of A – B is (a) 00 (b) 450 (c) 600 (d) 300 56. If cos–1 2 x + cos–1 3 y = 6  , then value of 4 x 2 – 2 3 xy + 9 y 2 is (a) 4 3 (b) 2 1 (c) 4 1 (d) None of these 57. + = − tan[sec 1 x ] 1 2 (a) x 1 (b) x (c) 2 1 x 1 + (d) 2 1 x x + 58. If sin–1 x = 5  for some x (–1,1), then the value of cos–1 x is (a) 10 3 (b) 10 5 (c) 10 7 (d) 10 9 59. If tan(cos–1 x) = sin       − 2 1 cot 1 , then x = (a) 3 5  (b) 3 5  (c) 3 5  (d) None of these 60. Sin (2sin–1 0.8) = (a) 0.96 (b) 0.48 (c) 0.64 (d) None of these 61. If sin–1x + sin–1y + sin–1 z = 2 3 , then the value of x100 + y100 + z100 – 101 101 101 x y z 9 + + is equal to (a) 0 (b) 3 (c) –3 (d) 9 62. If sec–1x = cosec–1y, then cos–1 x 1 + cos–1 y 1 = (a)  (b) 4  (c) 2 − (d) 2  63. The number of real solutions of tan–1 x(x +1) + sin–1 x x 1 2 + + = 2  is (a) Zero (b) One (c) Two (d) Infinite
64. The value of x for which sin[cot–1 (1 + x)] = cos(tan–1x) is (a) 2 1 (b) 1 (c) 0 (d) – 2 1 65. If 0 < x < 1, then 2 1+ x [{cos(cot–1x) + sin (cot–1x)}2 – 1]1/2 is equal to (a) 2 1 x x + (b) x (c) x 2 1+ x (d) 2 1+ x 66. Sum of maximum and minimum values of (sin–1 x)4 + (cos–1 x)4 is– (a) 128 137 2  (b) 17 4  (c) 16 17   (d) 128 137 4  67. + + +  − − − .......... 33 4 tan 9 2 tan 3 1 tan 1 1 1 = (a)  / 4 (b)  / 2 (c)  (d) None