The value of $\cot \left( \sum _ { n = 1 } ^ { 50 } \tan ^ { - 1 } \left( \frac { 1 } { 1 + n + n ^ { 2 } } \right) \right)$ is
(1) $\frac { 25 } { 26 }$
(2) $\frac { 50 } { 51 }$
(3) $\frac { 26 } { 25 }$
(4) $\frac { 52 } { 51 }$

For $k \in \mathbb { R }$, let the solutions of the equation $\cos \left( \sin ^ { - 1 } \left( x \cot \left( \tan ^ { - 1 } \left( \cos \left( \sin ^ { - 1 } x \right) \right) \right) \right) \right) = k , 0 < | x | < \frac { 1 } { \sqrt { 2 } }$ be $\alpha$ and $\beta$, where the inverse trigonometric functions take only principal values. If the solutions of the equation $x ^ { 2 } - b x - 5 = 0$ are $\frac { 1 } { \alpha ^ { 2 } } + \frac { 1 } { \beta ^ { 2 } }$ and $\frac { \alpha } { \beta }$, then $\frac { b } { k ^ { 2 } }$ 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 $\sin^{-1}\frac{\alpha}{17} + \cos^{-1}\frac{4}{5} - \tan^{-1}\frac{77}{36} = 0$, $0 < \alpha < 13$, then $\sin^{-1}(\sin\alpha) + \cos^{-1}(\cos\alpha)$ is equal to
(1) $\pi$
(2) 16
(3) 0
(4) $16 - 5\pi$

Let $y = f(x)$ represent a parabola with focus $\left(-\frac{1}{2}, 0\right)$ and directrix $y = -\frac{1}{2}$. Then $S = \left\{x \in \mathbb{R} : \tan^{-1}\sqrt{f(x)} + \sin^{-1}\sqrt{f(x)+1} = \frac{\pi}{2}\right\}$:
(1) contains exactly two elements
(2) contains exactly one element
(3) is an infinite set
(4) is an empty set

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) $\sqrt { 3 }$

If $a = \sin^{-1}(\sin 5)$ and $b = \cos^{-1}(\cos 5)$, then $a^2 + b^2$ is equal to
(1) $4\pi^2 + 25$
(2) $8\pi^2 - 40\pi + 50$
(3) $4\pi^2 - 20\pi + 50$
(4) 25

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

Let the inverse trigonometric functions take principal values. The number of real solutions of the equation $2 \sin ^ { - 1 } x + 3 \cos ^ { - 1 } x = \frac { 2 \pi } { 5 }$, is $\_\_\_\_$

If for some $\alpha, \beta$; $\alpha \leq \beta$, $\alpha + \beta = 8$ and $\sec^2(\tan^{-1}\alpha) + \operatorname{cosec}^2(\cot^{-1}\beta) = 36$, then $\alpha^2 + \beta$ is $\underline{\hspace{2cm}}$.

Let $\mathrm{S} = \{x: \cos^{-1}x = \pi + \sin^{-1}x + \sin^{-1}(2x+1)\}$. Then $\sum_{x \in \mathrm{S}}(2x-1)^2$ is equal to \_\_\_\_ .

Q87. For $n \in \mathrm {~N}$, if $\cot ^ { - 1 } 3 + \cot ^ { - 1 } 4 + \cot ^ { - 1 } 5 + \cot ^ { - 1 } n = \frac { \pi } { 4 }$, then $n$ is equal to $\_\_\_\_$

Q86. Let the inverse trigonometric functions take principal values. The number of real solutions of the equation $2 \sin ^ { - 1 } x + 3 \cos ^ { - 1 } x = \frac { 2 \pi } { 5 }$, is $\_\_\_\_$

The number of values of $x$ satisfying $\tan ^ { - 1 } ( 4 x ) + \tan ^ { - 1 } ( 6 x ) = \frac { \pi } { 6 }$ and $\mathrm { x } \in \left[ - \frac { 1 } { 2 \sqrt { 6 } } , \frac { 1 } { 2 \sqrt { 6 } } \right]$ is
(A) 0
(B) 1
(C) 2
(D) 3

Let $\mathrm { k } = \tan \left( \frac { \pi } { 4 } + \frac { 1 } { 2 } \cos ^ { - 1 } \frac { 2 } { 3 } \right) + \tan \left( \frac { 1 } { 2 } \sin ^ { - 1 } \frac { 2 } { 3 } \right)$.
Then number of solutions of the equation $\sin ^ { - 1 } ( k x - 1 ) = \sin ^ { - 1 } x - \cos ^ { - 1 } x$