If \(\alpha ={ tan }^{ -1 }\left( tan\frac { 5\pi }{ 4 } \right) \) and \(\beta ={ tan }^{ -1 }\left( -tan\frac { 2\pi }{ 3 } \right) \) then ___________

(a)

\(4\alpha =3\beta \quad \)

(b)

\(3\alpha =4\beta \)

(c)

\(\alpha -\beta =\frac { 7\pi }{ 12 } \)

(d)

none

The number of real solutions of the equation \(\sqrt { 1+cos2x } ={ 2sin }^{ -1 }\left( sinx \right) ,-\pi  is ___________

(a)

0

(b)

1

(c)

2

(d)

infinte

If \(\alpha ={ tan }^{ -1 }\left( \frac { \sqrt { 3 } }{ 2y-x } \right) ,\beta ={ tan }^{ -1 }\left( \frac { 2x-y }{ \sqrt { 3y } } \right) \) then \(\alpha -\beta \) __________

(a)

\(\frac { \pi }{ 6 } \)

(b)

\(\frac { \pi }{ 3 } \)

(c)

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

(d)

\(\frac { -\pi }{ 3 } \)

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

(a)

0

(b)

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

(c)

-1

(d)

none

If tan^-1(3) + tan^-1(x) = tan^-1(8) then x = ____________ 

(a)

5

(b)

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

(c)

\(\frac { 5 }{ 14 } \)

(d)

\(\frac { 14 }{ 5 } \)

The value of \({ cos }^{ -1 }\left( \cos\cfrac { 5\pi }{ 3 } \right) +sin^{ -1 }\left( \sin\cfrac{5\pi }{ 3 } \right) \) is ______________ 

(a)

\(\cfrac { \pi }{ 2 } \)

(b)

\(\cfrac { 5\pi }{ 3 } \)

(c)

\(\cfrac { 10\pi }{ 3 } \)

(d)

0

\(sin\left\{ 2{ cos }^{ -1 }\left( \frac { -3 }{ 5 } \right) \right\} =\) __________

(a)

\(\frac { 6 }{ 15 } \)

(b)

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

(c)

\(\frac { 4 }{ 5 } \)

(d)

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

If \(4{ cos }^{ -1 }x+{ sin }^{ -1 }x=\pi \) then x is _____________

(a)

\(\frac { 3 }{ 2 } \)

(b)

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

(c)

\(\frac { \sqrt { 3 } }{ 2 } \)

(d)

\(\frac { 2 }{ \sqrt { 3 } } \)

If \({ tan }^{ -1 }\left( \frac { x+1 }{ x-1 } \right) +{ tan }^{ -1 }\left( \frac { x-1 }{ x } \right) ={ tan }^{ -1 }\left( -7 \right) \) then x is ___________

(a)

0

(b)

-2

(c)

1

(d)

2

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

In a \(\Delta ABC\)  if C is a right angle, then  \({ tan }^{ -1 }\left( \frac { a }{ b+c } \right) +{ tan }^{ -1 }\left( \frac { b }{ c+a } \right) =\) ________

(a)

\(\frac { \pi }{ 3 } \)

(b)

\(\frac { \pi }{ 4 } \)

(c)

\(\frac { 5\pi }{ 2 } \)

(d)

\(\frac { \pi }{ 6 } \)

\(cot\left( \frac { \pi }{ 4 } -{ cot }^{ -1 }3 \right) \)

(a)

7

(b)

6

(c)

5

(d)

none

If tan^-1(cot \(\theta\)) = 2\(\theta\), then\(\theta\) = _____________

(a)

\(\pm 3\)

(b)

\(\pm \frac { \pi }{ 4 } \)

(c)

\(\pm \frac { \pi }{ 6 } \)

(d)

none

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]

The value of tan \(\left( { cos }^{ -1 }\frac { 3 }{ 5 } +{ tan }^{ -1 }\frac { 1 }{ 4 } \right) \) is ______

(a)

\(\frac { 19 }{ 8 } \)

(b)

\(\frac { 8 }{ 19 } \)

(c)

\(\frac { 19 }{ 12 } \)

(d)

\(\frac { 3 }{ 4 } \)