The number of roots of the equation, $( 81 ) ^ { \sin ^ { 2 } x } + ( 81 ) ^ { \cos ^ { 2 } x } = 30$ in the interval $[ 0 , \pi ]$ is equal to :
(1) 3
(2) 4
(3) 8
(4) 2

Given that the inverse trigonometric functions take principal values only. Then, the number of real values of $x$ which satisfy $\sin ^ { - 1 } \left( \frac { 3x } { 5 } \right) + \sin ^ { - 1 } \left( \frac { 4x } { 5 } \right) = \sin ^ { - 1 } x$ is equal to:
(1) 2
(2) 1
(3) 3
(4) 0

The number of solutions of the equation $\sin ^ { - 1 } \left[ x ^ { 2 } + \frac { 1 } { 3 } \right] + \cos ^ { - 1 } \left[ x ^ { 2 } - \frac { 2 } { 3 } \right] = x ^ { 2 }$ for $x \in [ - 1,1 ]$, and $[ x ]$ denotes the greatest integer less than or equal to $x$, is:
(1) 2
(2) 0
(3) 4
(4) Infinite

The sum of possible values of $x$ for $\tan ^ { - 1 } ( x + 1 ) + \cot ^ { - 1 } \left( \frac { 1 } { x - 1 } \right) = \tan ^ { - 1 } \left( \frac { 8 } { 31 } \right)$ is:
(1) $- \frac { 32 } { 4 }$
(2) $- \frac { 31 } { 4 }$
(3) $- \frac { 30 } { 4 }$
(4) $- \frac { 33 } { 4 }$

Let $S$ be the sum of all solutions (in radians) of the equation $\sin ^ { 4 } \theta + \cos ^ { 4 } \theta - \sin \theta \cos \theta = 0$ in $[ 0,4 \pi ]$ then $\frac { 8 S } { \pi }$ is equal to

The number of solutions of the equation $\cos \left( x + \frac { \pi } { 3 } \right) \cos \left( \frac { \pi } { 3 } - x \right) = \frac { 1 } { 4 } \cos ^ { 2 } 2 x , x \in [ - 3 \pi , 3 \pi ]$ is:
(1) 8
(2) 5
(3) 6
(4) 7

The number of solutions of $\cos x = \sin x$, such that $- 4 \pi \leq x \leq 4 \pi$ is
(1) 4
(2) 6
(3) 8
(4) 12

Let $\mathrm { S } = \left\{ \theta \in [ - \pi , \pi ] - \left\{ \pm \frac { \pi } { 2 } \right\} : \sin \theta \tan \theta + \tan \theta = \sin 2 \theta \right\}$. If $T = \sum _ { \theta \in S } \cos 2 \theta$, then $T + n ( S )$ is equal to
(1) $7 + \sqrt { 3 }$
(2) 5
(3) $8 + \sqrt { 3 }$
(4) 9

The number of elements in the set $S = \left\{x \in \mathbb{R} : 2\cos\left(\frac{x^2 + x}{6}\right) = 4^x + 4^{-x}\right\}$ is
(1) 1
(2) 3
(3) 0
(4) infinite

Let $x \times y = x ^ { 2 } + y ^ { 3 }$ and $( x \times 1 ) \times 1 = x \times ( 1 \times 1 )$. Then a value of $2 \sin ^ { - 1 } \left( \frac { x ^ { 4 } + x ^ { 2 } - 2 } { x ^ { 4 } + x ^ { 2 } + 2 } \right)$ is
(1) $\frac { \pi } { 4 }$
(2) $\frac { \pi } { 3 }$
(3) $\frac { \pi } { 6 }$
(4) $\pi$

Considering the principal values of the inverse trigonometric functions, the sum of all the solutions of the equation $\cos^{-1}x - 2\sin^{-1}x = \cos^{-1}(2x)$ is equal to
(1) 0
(2) 1
(3) $\frac{1}{2}$
(4) $-\frac{1}{2}$

The value of $\tan^{-1}\left(\frac{\cos\frac{15\pi}{4} - 1}{\sin\frac{\pi}{4}}\right)$ is equal to
(1) $-\frac{\pi}{4}$
(2) $-\frac{\pi}{8}$
(3) $-\frac{5\pi}{12}$
(4) $-\frac{4\pi}{9}$

Let $S = \left[ - \pi , \frac { \pi } { 2 } \right) - \left\{ - \frac { \pi } { 2 } , - \frac { \pi } { 4 } , - \frac { 3 \pi } { 4 } , \frac { \pi } { 4 } \right\}$. Then the number of elements in the set $A = \{ \theta \in S : \tan \theta ( 1 + \sqrt { 5 } \tan ( 2 \theta ) ) = \sqrt { 5 } - \tan ( 2 \theta ) \}$ is $\_\_\_\_$.

If the solution of the equation $\log_{\cos x} \cot x + 4\log_{\sin x} \tan x = 1, \quad x \in \left(0, \frac{\pi}{2}\right)$ is $\sin^{-1}\frac{\alpha + \sqrt{\beta}}{2}$, where $\alpha, \beta$ are integers, then $\alpha + \beta$ is equal to:
(1) 3
(2) 5
(3) 6
(4) 4

Let $S = \left\{ x \in \left( - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right) : 9 ^ { 1 - \tan ^ { 2 } x } + 9 ^ { \tan ^ { 2 } x } = 10 \right\}$ and $\beta = \sum _ { x \in S } \tan ^ { 2 } \frac { x } { 3 }$, then $\frac { 1 } { 6 } ( \beta - 14 ) ^ { 2 }$ is equal to
(1) 16
(2) 8
(3) 64
(4) 32

Let $S = \{ \theta \in [ 0,2 \pi ) : \tan ( \pi \cos \theta ) + \tan ( \pi \sin \theta ) = 0 \}$, then $\sum _ { \theta \in S } \sin ^ { 2 } \left( \theta + \frac { \pi } { 4 } \right)$ is equal to

If $m$ and $n$ respectively are the numbers of positive and negative value of $\theta$ in the interval $[ - \pi , \pi ]$ that satisfy the equation $\cos 2 \theta \cos \frac { \theta } { 2 } = \cos 3 \theta \cos \frac { 9 \theta } { 2 }$, then $m n$ is equal to $\_\_\_\_$.

The set of all values of $\lambda$ for which the equation $\cos ^ { 2 } 2 x - 2 \sin ^ { 4 } x - 2 \cos ^ { 2 } x = \lambda$ has a solution is: (1) $[ - 2 , - 1 ]$ (2) $\left[ - 2 , - \frac { 3 } { 2 } \right]$ (3) $\left[ - 1 , - \frac { 1 } { 2 } \right]$ (4) $\left[ - \frac { 3 } { 2 } , - 1 \right]$

Let $S = \{\theta \in [0, 2\pi): \tan(\pi\cos\theta) + \tan(\pi\sin\theta) = 0\}$. Then $\sum_{\theta \in S} \sin\left(\theta + \frac{\pi}{4}\right)$ is equal to $\_\_\_\_$.

If the sum of all the solutions of $\tan ^ { - 1 } \left( \frac { 2 x } { 1 - x ^ { 2 } } \right) + \cot ^ { - 1 } \left( \frac { 1 - x ^ { 2 } } { 2 x } \right) = \frac { \pi } { 3 } , - 1 < x < 1 , x \neq 0$, is $\alpha - \frac { 4 } { \sqrt { 3 } }$, then $\alpha$ is equal to $\_\_\_\_$ .

If $\alpha , - \frac { \pi } { 2 } < \alpha < \frac { \pi } { 2 }$ is the solution of $4 \cos \theta + 5 \sin \theta = 1$, then the value of $\tan \alpha$ is
(1) $\frac { 10 - \sqrt { 10 } } { 6 }$
(2) $\frac { 10 - \sqrt { 10 } } { 12 }$
(3) $\frac { \sqrt { 10 } - 10 } { 12 }$
(4) $\frac { \sqrt { 10 } - 10 } { 6 }$

The number of solutions of the equation $4\sin^2 x - 4\cos^3 x + 9 - 4\cos x = 0$; $x \in [-2\pi, 2\pi]$ is:
(1) 1
(2) 3
(3) 2
(4) 0

The sum of the solutions $x \in R$ of the equation $\frac { 3 \cos 2 x + \cos ^ { 3 } 2 x } { \cos ^ { 6 } x - \sin ^ { 6 } x } = x ^ { 3 } - x ^ { 2 } + 6$ is
(1) 0
(2) 1
(3) - 1
(4) 3

Considering only the principal values of inverse trigonometric functions, the number of positive real values of $x$ satisfying $\tan ^ { - 1 } ( \mathrm { x } ) + \tan ^ { - 1 } ( 2 \mathrm { x } ) = \frac { \pi } { 4 }$ is :
(1) More than 2
(2) 1
(3) 2
(4) 0

For $\alpha , \beta , \gamma \neq 0$. If $\sin ^ { - 1 } \alpha + \sin ^ { - 1 } \beta + \sin ^ { - 1 } \gamma = \pi$ and $(\alpha + \beta + \gamma)(\alpha - \gamma + \beta) = 3 \alpha \beta$, then $\gamma$ equals
(1) $\frac { \sqrt { 3 } } { 2 }$
(2) $\frac { 1 } { \sqrt { 2 } }$
(3) $\frac { \sqrt { 3 } - 1 } { 2 \sqrt { 2 } }$
(4) $\sqrt { 3 }$