Inverse Trigonometric Functions 513 1. cot −1 [ √ 1−sin x+ √ 1+ sin x √ 1−sin x− √ 1 +sin x ] = [MNR 1986] (a) π−x (b) 2 π−x (c) x 2 (d) π− x 2 2. The principal value of sin −1 [ sin ( 2 π 3 ) ] is [IIT 1986] (a) − 2 π 3 (b) 2 π 3 (c) 4 π 3 (d) None of these 3. If θ=tan −1 a,φ=tan −1 b and ab=−1 , then θ−φ= (a) 0 (b) π 4 (c) π 2 (d) None of these 4. If tan (cos −1 x )=sin ( cot −1 1 2 ) , then x = (a) ± 5 3 (b) ± √ 5 3 (c) ± 5 √ 3 (d) None of these 5. If sin −1 x= π 5 for some x ∈ (−1, 1 ) , then the value of cos −1 x is [IIT 1992] (a) 3 π 10 (b) 5 π 10 (c) 7 π 10 (d) 9 π 10 6. If cos −1 p+cos −1 q+cos −1 r=π then p 2 +q 2 + r 2 +2 pqr= [Karnataka CET 2004] (a) 3 (b) 1 (c) 2 (d) –1 7. tan [ π 4 + 1 2 cos −1 a b ] +tan [ π 4 − 1 2 cos −1 a b ] = [MP PET 1999] (a) 2 a b (b) 2 b a (c) a b (d) b a 8. tan −1 [ √ 1+x 2 +√ 1−x 2 √ 1 +x 2 − √ 1−x 2 ] = (a) π 4 + 1 2 cos −1 x 2 (b) π 4 +cos −1 x 2 (c) π 4 + 1 2 cos −1 x (d) π 4 − 1 2 cos −1 x 2 9. The equation sin −1 x−cos −1 x =cos −1 ( √ 3 2 ) has (a) No solution (b) Unique solution (c) Infinite number of solutions (d) None of these 10. ∑ m=1 n tan −1 ( 2 m m 4 +m 2 + 2 ) is equal to (a) tan −1 ( n 2 +n n 2 + n+2 ) (b) tan −1 ( n 2 −n n 2 −n+ 2 ) (c) tan −1 ( n 2 + n+2 n 2 +n ) (d) None of these 11. The number of real solutions of tan −1 √ x ( x +1 )+ sin −1 √ x 2 +x +1= π 2 is [IIT 1999] (a) Zero (b) One (c) Two (d) Infinite 12. The equation 2cos −1 x+sin −1 x= 11 π 6 has [AMU 1999] (a) No solution (b) Only one solution (c) Two solutions (d) Three solutions 13. If tan −1 x+tan −1 y +tan −1 z=π , then x+ y+z is equal to [Kerala (Engg.) 2002] (a) xyz (b) 0 (c) 1 (d) 2xyz 14. 2tan −1 [ √ a−b a +b tan θ 2 ] = [Dhanbad Engg. 1976] (a) cos −1 ( a cosθ +b a+b cos θ ) (b) cos −1 ( a+b cos θ a cosθ +b )
Ifthen[Karnataka CET 2004](a)3(b)1(c)2(d)17. [MP PET 1999]
(a)(b)
(c)(d)8.
(a)(b)
(c) (d)9. The equation has (a)No solution(b)Unique
solution(c)Infinite number of solutions(d)None of these10. is equal
to
(a)(b)
(c)(d) None of these11. The number of real solutions of is [IIT
1999](a)Zero(b)One(c)Two(d)Infinite12. The equation has [AMU
1999](a)No solution(b)Only one solution(c)Two solutions(d)Three
solutions13.
If , then is equal to [Kerala (Engg.)
2002](a)xyz(b)0(c)1(d)2xyz14. [Dhanbad Engg. 1976]
