The value of $\tan ^ { - 1 } \left[ \frac { \sqrt { 1 + x ^ { 2 } } + \sqrt { 1 - x ^ { 2 } } } { \sqrt { 1 + x ^ { 2 } } - \sqrt { 1 - x ^ { 2 } } } \right] , | x | < \frac { 1 } { 2 } , x \neq 0$, is equal to:
(1) $\frac { \pi } { 4 } + \frac { 1 } { 2 } \cos ^ { - 1 } x ^ { 2 }$
(2) $\frac { \pi } { 4 } - \cos ^ { - 1 } x ^ { 2 }$
(3) $\frac { \pi } { 4 } - \frac { 1 } { 2 } \cos ^ { - 1 } x ^ { 2 }$
(4) $\frac { \pi } { 4 } + \cos ^ { - 1 } x ^ { 2 }$

If $\tan A$ and $\tan B$ are the roots of the quadratic equation $3 x ^ { 2 } - 10 x - 25 = 0$, then the value of $3 \sin ^ { 2 } ( A + B ) - 10 \sin ( A + B ) \cos ( A + B ) - 25 \cos ^ { 2 } ( A + B )$ is :
(1) - 25
(2) 10
(3) - 10
(4) 25

If sum of all the solutions of the equation $8 \cos x \cdot \left( \cos \left( \frac { \pi } { 6 } + x \right) \cdot \cos \left( \frac { \pi } { 6 } - x \right) - \frac { 1 } { 2 } \right) = 1$ in $[ 0 , \pi ]$ is $k \pi$, then $k$ is equal to:
(1) $\frac { 20 } { 9 }$
(2) $\frac { 2 } { 3 }$
(3) $\frac { 13 } { 9 }$
(4) $\frac { 8 } { 9 }$

Let $S = \left\{ \theta \in [ - 2 \pi , 2 \pi ] : 2 \cos ^ { 2 } \theta + 3 \sin \theta = 0 \right\}$. Then the sum of the elements of $S$ is:
(1) $\pi$
(2) $\frac { 13 \pi } { 6 }$
(3) $\frac { 5 \pi } { 3 }$
(4) $2 \pi$

If $0 \leq x < \frac{\pi}{2}$, then the number of values of $x$ for which $\sin x - \sin 2x + \sin 3x = 0$, is:
(1) 4
(2) 3
(3) 2
(4) 1

If $x = \sin^{-1}(\sin 10)$ and $y = \cos^{-1}(\cos 10)$, then $y - x$ is equal to:
(1) 10
(2) $\pi$
(3) 0
(4) $7\pi$

The set of all possible values of $\theta$ in the interval $( 0 , \pi )$ for which the points $( 1,2 )$ and $( \sin \theta , \cos \theta )$ lie on the same side of the line $x + y = 1$ is?
(1) $\left( 0 , \frac { \pi } { 2 } \right)$
(2) $\left( \frac { \pi } { 4 } , \frac { 3 \pi } { 4 } \right)$
(3) $\left( 0 , \frac { 3 \pi } { 4 } \right)$
(4) $\left( 0 , \frac { \pi } { 4 } \right)$

The number of solutions of $\sin ^ { 7 } x + \cos ^ { 7 } x = 1 , x \in [ 0,4 \pi ]$ is equal to
(1) 11
(2) 7
(3) 5
(4) 9

The number of solutions of the equation $x + 2 \tan x = \frac { \pi } { 2 }$ in the interval $[ 0,2 \pi ]$ is
(1) 3
(2) 4
(3) 2
(4) 5

All possible values of $\theta \in [ 0,2 \pi ]$ for which $\sin 2 \theta + \tan 2 \theta > 0$ lie in :
(1) $\left( 0 , \frac { \pi } { 2 } \right) \cup \left( \pi , \frac { 3 \pi } { 2 } \right)$
(2) $\left( 0 , \frac { \pi } { 2 } \right) \cup \left( \frac { \pi } { 2 } , \frac { 3 \pi } { 4 } \right) \cup \left( \pi , \frac { 7 \pi } { 6 } \right)$
(3) $\left( 0 , \frac { \pi } { 4 } \right) \cup \left( \frac { \pi } { 2 } , \frac { 3 \pi } { 4 } \right) \cup \left( \pi , \frac { 5 \pi } { 4 } \right) \cup \left( \frac { 3 \pi } { 2 } , \frac { 7 \pi } { 4 } \right)$
(4) $\left( 0 , \frac { \pi } { 4 } \right) \cup \left( \frac { \pi } { 2 } , \frac { 3 \pi } { 4 } \right) \cup \left( \frac { 3 \pi } { 2 } , \frac { 11 \pi } { 6 } \right)$

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

Two poles $A B$ of length $a$ metres and $C D$ of length $a + b ( b \neq a )$ metres are erected at the same horizontal level with bases at $B$ and $D$. If $B D = x$ and $\tan \angle A C B = \frac { 1 } { 2 }$, then: (1) $x ^ { 2 } + 2 ( a + 2 b ) x - b ( a + b ) = 0$ (2) $x ^ { 2 } + 2 ( a + 2 b ) x + a ( a + b ) = 0$ (3) $x ^ { 2 } - 2 a x + b ( a + b ) = 0$ (4) $x ^ { 2 } - 2 a x + a ( a + b ) = 0$

Let in a right angled triangle, the smallest angle be $\theta$. If a triangle formed by taking the reciprocal of its sides is also a right angled triangle, then $\sin \theta$ is equal to:
(1) $\frac { \sqrt { 5 } + 1 } { 4 }$
(2) $\frac { \sqrt { 5 } - 1 } { 2 }$
(3) $\frac { \sqrt { 2 } - 1 } { 2 }$
(4) $\frac { \sqrt { 5 } - 1 } { 4 }$

$\operatorname { cosec } \left[ 2 \cot ^ { - 1 } ( 5 ) + \cos ^ { - 1 } \left( \frac { 4 } { 5 } \right) \right]$ is equal to:
(1) $\frac { 65 } { 56 }$
(2) $\frac { 75 } { 56 }$
(3) $\frac { 65 } { 33 }$
(4) $\frac { 56 } { 33 }$

Let $S _ { k } = \sum _ { r = 1 } ^ { k } \tan ^ { - 1 } \left( \frac { 6 ^ { r } } { 2 ^ { 2 r + 1 } + 3 ^ { 2 r + 1 } } \right)$, then $\lim _ { k \rightarrow \infty } S _ { k }$ is equal to :
(1) $\tan ^ { - 1 } \left( \frac { 3 } { 2 } \right)$
(2) $\frac { \pi } { 2 }$
(3) $\cot ^ { - 1 } \left( \frac { 3 } { 2 } \right)$
(4) $\tan ^ { - 1 } ( 3 )$

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

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 $S = \left\{ \theta \in [ 0,2 \pi ] : 8 ^ { 2 \sin ^ { 2 } \theta } + 8 ^ { 2 \cos ^ { 2 } \theta } = 16 \right\}$. Then $n ( S ) + \sum _ { \theta \in \mathrm { S } } \left( \sec \left( \frac { \pi } { 4 } + 2 \theta \right) \operatorname { cosec } \left( \frac { \pi } { 4 } + 2 \theta \right) \right)$ is equal to:
(1) 0
(2) $- 2$
(3) $- 4$
(4) 12

Let $S = \left\{ \theta \in \left( 0 , \frac { \pi } { 2 } \right) : \sum _ { m = 1 } ^ { 9 } \sec \left( \theta + ( m - 1 ) \frac { \pi } { 6 } \right) \sec \left( \theta + \frac { m \pi } { 6 } \right) = - \frac { 8 } { \sqrt { 3 } } \right\}$. Then
(1) $\mathrm { S } = \left\{ \frac { \pi } { 12 } \right\}$
(2) $S = \left\{ \frac { 2 \pi } { 3 } \right\}$
(3) $\sum _ { \theta \in S } \theta = \frac { \pi } { 2 }$
(4) $\sum _ { \theta \in S } \theta = \frac { 3 \pi } { 4 }$

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

$\lim _ { x \rightarrow \frac { 1 } { \sqrt { 2 } } } \frac { \sin \left( \cos ^ { - 1 } x \right) - x } { 1 - \tan \left( \cos ^ { - 1 } x \right) }$ is equal to
(1) $\frac { 1 } { \sqrt { 2 } }$
(2) $\frac { - 1 } { \sqrt { 2 } }$
(3) $\sqrt { 2 }$
(4) $- 1$

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) $\pi$

Let a vertical tower $A B$ of height $2 h$ stands on a horizontal ground. Let from a point $P$ on the ground a man can see upto height $h$ of the tower with an angle of elevation $2 \alpha$. When from $P$, he moves a distance $d$ in the direction of $\overrightarrow { A P }$, he can see the top $B$ of the tower with an angle of elevation $\alpha$. If $d = \sqrt { 7 } h$, then $\tan \alpha$ is equal to
(1) $\sqrt { 5 } - 2$
(2) $\sqrt { 3 } - 1$
(3) $\sqrt { 7 } - 2$
(4) $\sqrt { 7 } - \sqrt { 3 }$