Title 10. Inverse triganometric functions Medium.pdf ✅

1. The principal value of        −  − 2 3 sin 1 is (a) 3 2 − (b) 3  − (c) 3 4 (d) 8 5 2. − = − sec [sec( 30 )] 1 o (a) – o 60 (b) o − 30 (c) o 30 (d) o 150 3. The principal value of      −  3 5 sin sin 1  is (a) 3 5 (b) 3 5 − (c) 3  − (d) 3 4 4. The principal value of            −  3 2 sin sin 1  is (a) 3 2 − (b) 3 2 (c) 3 4 (d) None of these 5. Considering only the principal values, if             = − − 2 1 tan(cos ) sin cot 1 1 x , then x is equal to (a) 5 1 (b) 5 2 (c) 5 3 (d) 3 5 6. If sin [sin( 600 )] 1 = −  −  , then one of the possible value of  is (a) 3  (b) 2  (c) 3 2 (d) 3 2 − 7. Value of        +     −  − 3 5 sin sin 3 5 cos cos 1  1  is (a) 0 (b) 2  (c) 3 2 (d) 3 10 8. The equation 6 11 2 cos sin 1 1  + = − − x x has (a) No solution (b) Only one solution (c) Two solutions (d) Three solutions 9. If , 3 2 sin sin 1 1  + = − − x y then + = − − x y 1 1 cos cos (a) 3 2 (b) 3  (c) 6  (d)  10. If         − + − − ... 2 4 sin 2 3 1 x x x 2 ... 2 4 cos 4 6 1 2  =         + − + − − x x x for 0 | x |  2, then x equals (a) 2 1 (b) 1 (c) 2 1 − (d) –1 11. If , 2 2 1 sin cot 1 1   =      + − − x then x is (a) 0 (b) 5 1 (c) 5 2 (d) 2 3 12. The value of sin(cot ) 1 x − is (a) 2 3 / 2 (1 + x ) (b) 2 3 / 2 (1 ) − + x (c) 2 1 / 2 (1 + x ) (d) 2 1 / 2 (1 ) − + x 13. The number of real solutions of 2 tan ( 1) sin 1 1 1 2  + + + + = − − x x x x is (a) Zero (b) One (c) Two (d) Infinite 14. The value of                  +     −  − 13 3 cos 5 3 tan sin 1 1 is (a) 17 6 (b) 13 6 (c) 5 13 (d) 6 17 15. + = − − 3 1 tan 2 1 tan 1 1 (a) 0 (b) 4  (c) 2  (d)  16. If sin , 3 2 sin 3 1 sin 1 1 1 x − − − + = then x is equal to (a) 0 (b) 9 5 − 4 2 (c) 9 5 + 4 2 (d) 2  17. cot 3 5 1 sin−1 −1 + is equal to (a) 6  (b) 4  (c) 3  (d) 2  18. If sin , 13 12 cos 5 3 sin 1 1 1 C − − −  =      + then C = (a) 56 65 (b) 65 24 (c) 65 16 (d) 65 56 19. + + = − − − 16 63 tan 5 4 cos 13 12 sin 1 1 1 (a) 0 (b) 2  (c)  (d) 2 3 20. If 3 1 sin 5 4 sin−1 −1  = + and 3 1 cos 5 4 cos−1 −1  = + , then (a)    (b)  =  (c)    (d) None of these 21. If , 3 cos 2 cos 1 1 + = − x − y then − + = 2 2 9x 12 xy cos 4y (a)  2 36 sin (b)  2 36 cos (c)  2 36 tan (d) None of these 22. The number of solutions of 3 sin sin 2 1 1  + = − − x x is (a) 0 (b) 1 (c) 2 (d) Infinitie
23. 239 1 tan 5 1 4 tan−1 −1 − is equal to (a)  (b) 2  (c) 3  (d) 4  24. The formula x x x 1 2 2 1 2 tan 1 1 cos− − = + − holds only for (a) x R (b) | x | 1 (c) x (−1,1] (d) x [1, + ) 25. 2 1 1 1 2 2 tan sin x x x + + − − is independent of x , then (a) x [1, + ) (b) x [−1,1] (c) x (−, −1 ] (d) None of these 26. The equation       = − −1 −1 2 2cos x sin 2x 1 x is valid for all values of x satisfying (a) −1  x  1 (b) 0  x  1 (c) 2 1 0  x  (d) x 1 2 1   27. If 1< x < 2 , then number of solutions of the equation tan–1 (x – 1) + tan–1 x + tan–1 (x + 1) = tan–1 3x, is/are (a) 0 (b) 1 (c) 2 (d) 3 28. If (cot–1 x)2 – 3 (cot–1 x) + 2 > 0, then x lies in (a) (cot 2, cot 1) (b) (– , cot 2)  (cot 1, ) (c) (cot 1, ) (d) (–, cot 1)  (cot 2, ) 29. The value of tan              +     −  − 3 2 tan 5 4 cos 1 1 is: (a) 6/17 (b) 7/16 (c) 16/7 (d) None of these 30. 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 31. Complete solution set of tan2 (sin–1 x) > 1 is- (a)         2 1 –1,–          ,1 2 1 (b)         − 2 1 , 2 1 ~ {0} (c) (–1, 1) ~ {0} (d) None of these 32. The value of cos (2cos–1 x + sin–1 x) at x = 6 1 : (a) 6 37 (b) 6 35 – (c) 6 35 (d) – 6 37 33. If cos–1 2 2 1 x 1– x + = 2tan–1 x then x = (a) x R (b) |x|  1 (c) x  (–1, 1] (d) x  [1, ) 34. If sin cos tan , 1 1 1 x x x − − −  = + − x  0 then the smallest interval in which  lies is (a) 4 3 2      (b) 4 0    (c) 0 4 −    (d) 4 2      35. If  =       +     −  − b y a 1 x 1 cos cos , then − + = 2 2 2 2 cos 2 b y ab xy a x  (a)  2 sin (b)  2 cos (c)  2 tan (d)  2 cot 36. If a, b, c be positive real numbers and the value of ca b a b c bc a a b c ( ) tan ( ) tan 1 1 + + + + + = − −  ab c(a b c) tan 1 + + + − , then tan  is (a) 0 (b) 1 (c) a + b + c (d) None of these 37. 2 tan (cos ) tan (cosec ), 1 1 2 x x − − = then x = (a) 2  (b)  (c) 6  (d) 3  38. The solution set of the equation x x 1 1 sin 2 tan − − = is (a) {1, 2} (b) {−1, 2} (c) {−1, 1, 0} (d) ,0} 2 1 {1, 39.                 + − +        −  − − 2 2 1 2 1 1 1 cos 2 1 sin tan x x x x is equal to (a) 0 (b) 1 (c) 2 (d) 2 1 40. If , 1 2 tan 1 1 cos 1 2 sin 2 1 2 2 1 2 1 x x b b a a − = + − − + − − − then x = (a) a (b) b (c) ab a b − + 1 (d) ab a b + − 1 41.  =       −     −  5 4 1 tan 2 tan 1  (a) 7 17 (b) 7 17 − (c) 17 7 (d) 17 7 − 42. − + = − − − 99 1 tan 70 1 tan 5 1 4 tan 1 1 1 (a) 2  (b) 3  (c) 4  (d) None of these 43. The value of + =              −  − cos(tan 2 2) 3 1 sin 2 tan 1 1 (a) 15 16 (b) 15 14 (c) 15 12 (d) 15 11 44.       + −       + − − b a b 1 a 1 cos 2 1 4 cos tan 2 1 4 tan   equal to
(a) b 2a (b) a 2b (c) b a (d) a b 45. The number of positive integral solutions of the equation 10 3 sin 1 tan cos 1 2 −1 −1 − = + + y y x or tan cot tan 3 −1 −1 −1 x + y = is (a) One (b) Two (c) Zero (d) None 46. If 0  x  1 and sin x cos x tan x −1 −1 −1  = + − , then (a) /2 (b) /4 (c)  = /4 (d) /4 /2 47. If 5 1 x = , the value of cos(cos x 2sin x) −1 −1 + is (a) 25 24 − (b) 25 24 (c) 5 1 − (d) 5 1 48. 2tan (cosec tan 3 tan cot 3) −1 −1 −1 − is equal to (a) /16 (b) /6 (c) /3 (d) /2 49. 2(tan 1 tan 2 tan 3) −1 −1 −1 + + is equal to (a) /4 (b) /2 (c)  (d) 2 50. If 4 x 1 x 1 tan 2 1 = − − − , then (a) x = tan 2 (b) x = tan 4 (c) x = tan(1/4) (d) x = tan 8 51. sec (tan 2) cosec (cot 3) 2 −1 2 −1 + is equal to (a) 1 (b) 5 (c) 10 (d) 15 52. The values of x satisfying tan(sec–1x) = ) 5 1 sin(cos−1 is/are (a) 3 5 (b) 5 3 (c) 3 5 − (d) 5 3 − 53. The number of solution of equation  cot–1 (x–1) + (–1) cot–1x = 2 – 1 (a) 0 (b) 1 (c) 2 (d) 3 54. If x [–1, 0) then cos–1 (2x2 –1) –2 sin–1x equals (a) 2 – (b) 2 3 (c) –2 (d) None 55. If          ,2 2 3 x then value of the expression sin–1 (cos(cos–1 (cosx) + sin–1 (sinx))) equals (a) 2  − (b) 2  (c) 0 (d) None 56. Which of the following is negative (a) cos (tan–1 (tan 4)) (b) sin (cot–1 (cot 4)) (c) tan (cos–1 (cos 5)) (d) cot (sin–1 (sin 4)) 57. If x1, x2, x3, x4 are roots of the equation x4 – x 3 sin 2 + x2 cos 2 – x cos – sin  = 0 then  = − 4 i 1 1 tan xi is equal to (a)  –  (b)  – 2 (c) /2 –  (d) /2 – 2  The inequality log2(x) < sin–1 (sin(5)) holds if x  (a) (0, 25–2 ) (b) (25–2 , ) (c) (22–5 , ) (d) (0, 22–5 ) 59. 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 60. 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 61. 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 62. 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 63. Which one of the following is correct? (a) tan 1 > tan– 1 (b) tan 1 < tan–11 (c) tan 1 = tan–1 1 (d) None of these 64. If tan–1 x + cos–1 2 1 y y + = sin–1 10 3 where x and y are +ve integer, then number of possible pair of (x, y) is- (a) 1 (b) 2 (c) 3 (d) 4 65. The value of tan              +     −  − 3 2 tan 5 4 cos 1 1 is: (a) 6/17 (b) 7/16 (c) 16/7 (d) None of these 66. The value of 'a' for which ax2+sin–1 (x2 – 2x +2) + cos–1 (x2 – 2x + 2) = 0 has a real solution is (a) /2 (b) –  /2 (c) 2/ (d) –2/
67. The value of cos–1 3 2 – cos–1 2 3 6 +1 is equal to - (a) 12  (b) 8  (c) 6  (d) 3  68. If 2 1 < x < 1 then cos–1x+cos–1         + − 2 x 1 x 2 is equal to - (a) 2 cos–1x – 4  (b) 2 cos–1x (c) 4  (d) 0 69. The set values of x for which the equation cos–1 x + cos–1       + − 2 3 3x 2 1 2 x = 3  holds good, is - (a) [0, 1] (b)       2 1 0, (c)       ,1 2 1 (d) {–1, 0, 1} 70. The no. of pair of solutions (x, y) of equation 1+ x2 + 2x sin (cos–1y) = 0 is/are (a) 4 (b) 3 (c) 2 (d) 1 71. Number of solution of equation sin–1 x + n sin–1 (1 –x) = 2 m where n > 0, m  0 is- (a) 3 (b) 1 (c) 2 (d) None of these 72. Set of values of x satisfying cos–1 x > sin–1 x (a)       2 1 0, (b)       2 1 0, (c)       , 1 2 1 (d)       , 1 2 1 73. If , ,  are roots of x3 – mx2 + 3x – m = 0, then general value of tan–1 + tan–1 + tan–1  is- (a) Depends on m (b) Depends on p (c) n (d) (2n + 1) 2  74. Equation of the image of the line x + y = sin–1 (a3 + 1) + cos– 1 (a2 + 1) – tan–1 (a + 1), a  R about y-axis is given by (a) x – y + 4  = 0 (b) x – y = 0 (c) x – y = 4  (d) x – y = 2  75. Which one of the following is correct? (a) tan 1 > tan– 1 (b) tan 1 < tan–11 (c) tan 1 = tan–1 1 (d) None of these 76. If tan–1 x + cos–1 2 1 y y + = sin–1 10 3 where x and y are +ve integer, then number of possible pair of (x, y) is- (a) 1 (b) 2 (c) 3 (d) 4 77. Range of ƒ(x) = sin–1 x + tan–1 x + sec–1 x is- (a)         4 3 , 4 (b)        4 3 , 4 (c)        4 3 , 4 (d) None of these 78. The minimum value of (sin–1 x) 3 + (cos–1 x) 3 is equal to- (a) 32 3  (b) 32 5 3  (c) 32 9 3  (d) 32 11 3  79. If sin–1 x + sin–1 y + sin–1 z = , then x 4 + y4 + z4 + 4x 2y 2 z 2 = k (x 2y 2 + y2 z 2 + z2 x 2 ), where k is equal to - (a) 1 (b) 2 (c) 4 (d) None of these 80. cos (tan–1 x) = (a) 2 1+ x (b) 2 1 x 1 + (c) 1 + x2 (d) None of these 81. cos–1 2 1 + 2 sin–1 2 1 is equal to- (a) 4  (b) 6  (c) 3  (d) 3 2 82. The principal value of sin–1              3 2 sin is- (a) – 3 2 (b) 3 2 (c) 3 4 (d) None of these 83. 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 84. The value of sin cot–1 (tan (cos–1 x)) is equal to- (a) x (b) 2  (c) 1 (d) None of these 85. The solution set of the equation sin–1 x = 2 tan–1 x is- (a) {1, 2} (b) {–1, 2} (c) {–1, 1, 0} (d) {1, 1/2, 0} 86. tan       + − − 3 2 tan 5 4 cos 1 1 = (a) 17 6 (b) 6 17 (c) 16 7 (d) 7 16 87. If cos–1 x + cos–1 y + cos–1 z = , then- (a) x2 + y2 + z2 + xyz = 0 (b) x2 + y2 + z2 + 2xyz = 0 (c) x 2 + y2 + z2 + xyz = 1 (d) x2 + y2 + z2 + 2xyz = 1 88. If sin–1       + 2 1 a 2a + sin–1       + 2 1 b 2a = 2 tan–1x the x = (a) 1 ab a b + − (b) 1 ab b + (c) 1 ab b − (d) 1 ab a b − +