The value of sin^-1 (cos x), \(0\le x\le\pi\) is

(a)

\(\pi-x\)

(b)

\(x-\frac{\pi}{2}\)

(c)

\(\frac{\pi}{2}-x\)

(d)

\(x-\pi\)

If \(\cot ^{-1} x=\frac{2 \pi}{5}\) for some x \(\in\) R, the value of tan^-1 x is

(a)

\(-\frac{\pi}{10}\)

(b)

\(\frac{\pi}{5}\)

(c)

\(\frac{\pi}{10}\)

(d)

\(-\frac{\pi}{5}\)

If \(x = \frac{1}{5}\), the value of cos (cos^-1x+2sin^-1x) is

(a)

\(-\sqrt { \frac { 24 }{ 25 } } \)

(b)

\(\sqrt { \frac { 24 }{ 25 } } \)

(c)

\(\frac{1}{5}\)

(d)

\(-\frac{1}{5}\)

\(\sin ^{-1}\left(\tan \frac{\pi}{4}\right)-\sin ^{-1}\left(\sqrt{\frac{3}{x}}\right)=\frac{\pi}{6}\). Then x is a root of the equation

(a)

x²−x−6 = 0

(b)

x²−x−12 = 0

(c)

x²+x−12 = 0

(d)

x²+x−6 = 0

If \(\sin ^{-1} \frac{x}{5}+\operatorname{cosec}^{-1} \frac{5}{4}=\frac{\pi}{2}\), then the value of x is

(a)

4

(b)

5

(c)

2

(d)

3

If \({ tan }^{ -1 }\left\{ \cfrac { \sqrt { 1+{ x }^{ 2 } } -\sqrt { 1-{ x }^{ 2 } } }{ \sqrt { 1+{ x }^{ 2 } } +\sqrt { 1-{ x }^{ 2 } } } \right\} =\alpha \) then x² = _____________

(a)

\(sin2\alpha \)

(b)

\(sin\alpha \)

(c)

\(cos2\alpha \)

(d)

\(cos\alpha \)

The number of solutions of the equation \({ tan }^{ -1 }2x+{ tan }^{ -1 }3x=\frac { \pi }{ 4 } \) ____________

(a)

2

(b)

3

(c)

1

(d)

none

\({ tan }^{ -1 }\left( \frac { 1 }{ 4 } \right) +{ tan }^{ -1 }\left( \frac { 2 }{ 11 } \right) \) = ____________

(a)

0

(b)

\(\frac { 1 }{ 2 } \)

(c)

-1

(d)

none

If \({ cos }^{ -1 }x>x>{ sin }^{ -1 }x\) then _________

(a)

\(\cfrac { 1 }{ \sqrt { 2 } }

(b)

\(0\le x<\frac { 1 }{ \sqrt { 2 } } \)

(c)

\(-1\le x<\frac { 1 }{ \sqrt { 2 } } \)

(d)

x>0

The domain of cos^-1(x² - 4) is______

(a)

[3, 5]

(b)

[-1, 1]

(c)

\(\left[ -\sqrt { 5 } ,-\sqrt { 3 } \right] \cup \left[ \sqrt { 3 } ,\sqrt { 5 } \right] \)

(d)

[0, 1]