Let $S$ be the set of those real numbers $x$ for which the identity $$\sum _ { n = 2 } ^ { \infty } \cos ^ { n } x = ( 1 + \cos x ) \cot ^ { 2 } x$$ is valid, and the quantities on both sides are finite. Then
(A) $S$ is the empty set.
(B) $S = \{ x \in \mathbb { R } : x \neq n \pi$ for all $n \in \mathbb { Z } \}$.
(C) $S = \{ x \in \mathbb { R } : x \neq 2 n \pi$ for all $n \in \mathbb { Z } \}$.
(D) $S = \{ x \in \mathbb { R } : x \neq ( 2 n + 1 ) \pi$ for all $n \in \mathbb { Z } \}$.

If $0 < x < 1$, then $$\sqrt { 1 + x ^ { 2 } } \left[ \left\{ x \cos \left( \cot ^ { - 1 } x \right) + \sin \left( \cot ^ { - 1 } x \right) \right\} ^ { 2 } - 1 \right] ^ { \frac { 1 } { 2 } } =$$ (A) $\frac { x } { \sqrt { 1 + x ^ { 2 } } }$
(B) $x$
(C) $x \sqrt { 1 + x ^ { 2 } }$
(D) $\sqrt { 1 + x ^ { 2 } }$

The value of $\cot \left( \sum _ { n = 1 } ^ { 23 } \cot ^ { - 1 } \left( 1 + \sum _ { k = 1 } ^ { n } 2 k \right) \right)$ is
(A) $\frac { 23 } { 25 }$
(B) $\frac { 25 } { 23 }$
(C) $\frac { 23 } { 24 }$
(D) $\frac { 24 } { 23 }$

Match List I with List II and select the correct answer using the code given below the lists:
List I

• [P.] $\left( \frac { 1 } { y ^ { 2 } } \left( \frac { \cos \left( \tan ^ { - 1 } y \right) + y \sin \left( \tan ^ { - 1 } y \right) } { \cot \left( \sin ^ { - 1 } y \right) + \tan \left( \sin ^ { - 1 } y \right) } \right) ^ { 2 } + y ^ { 4 } \right) ^ { 1 / 2 }$ takes value
• [Q.] If $\cos x + \cos y + \cos z = 0 = \sin x + \sin y + \sin z$ then possible value of $\cos \frac { x - y } { 2 }$ is
• [R.] If $\cos \left( \frac { \pi } { 4 } - x \right) \cos 2 x + \sin x \sin 2 x \sec x = \cos x \sin 2 x \sec x + \cos \left( \frac { \pi } { 4 } + x \right) \cos 2 x$ then possible value of $\sec x$ is
• [S.] If $\cot \left( \sin ^ { - 1 } \sqrt { 1 - x ^ { 2 } } \right) = \sin \left( \tan ^ { - 1 } ( x \sqrt { 6 } ) \right) , x \neq 0$, then possible value of $x$ is

List II

1. $\frac { 1 } { 2 } \sqrt { \frac { 5 } { 3 } }$
2. $\sqrt { 2 }$
3. $\frac { 1 } { 2 }$
4. $1$

Codes:

	P	Q	R	S
(A)	4	3	1	2
(B)	4	3	2	1
(C)	3	4	2	1
(D)	3	4	1	2

For any positive integer $n$, define $f _ { n } : ( 0 , \infty ) \rightarrow \mathbb { R }$ as
$$f _ { n } ( x ) = \sum _ { j = 1 } ^ { n } \tan ^ { - 1 } \left( \frac { 1 } { 1 + ( x + j ) ( x + j - 1 ) } \right) \text { for all } x \in ( 0 , \infty )$$
(Here, the inverse trigonometric function $\tan ^ { - 1 } x$ assumes values in $\left( - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right)$.) Then, which of the following statement(s) is (are) TRUE?
(A) $\sum _ { j = 1 } ^ { 5 } \tan ^ { 2 } \left( f _ { j } ( 0 ) \right) = 55$
(B) $\sum _ { j = 1 } ^ { 10 } \left( 1 + f _ { j } ^ { \prime } ( 0 ) \right) \sec ^ { 2 } \left( f _ { j } ( 0 ) \right) = 10$
(C) For any fixed positive integer $n , \lim _ { x \rightarrow \infty } \tan \left( f _ { n } ( x ) \right) = \frac { 1 } { n }$
(D) For any fixed positive integer $n , \lim _ { x \rightarrow \infty } \sec ^ { 2 } \left( f _ { n } ( x ) \right) = 1$

The value of $$\sec^{-1}\left(\frac{1}{4}\sum_{k=0}^{10}\sec\left(\frac{7\pi}{12} + \frac{k\pi}{2}\right)\sec\left(\frac{7\pi}{12} + \frac{(k+1)\pi}{2}\right)\right)$$ in the interval $\left[-\frac{\pi}{4}, \frac{3\pi}{4}\right]$ equals

Considering only the principal values of the inverse trigonometric functions, the value of
$$\tan \left( \sin ^ { - 1 } \left( \frac { 3 } { 5 } \right) - 2 \cos ^ { - 1 } \left( \frac { 2 } { \sqrt { 5 } } \right) \right)$$
is
(A) $\frac { 7 } { 24 }$
(B) $\frac { - 7 } { 24 }$
(C) $\frac { - 5 } { 24 }$
(D) $\frac { 5 } { 24 }$

Let
$$\alpha = \frac { 1 } { \sin 60 ^ { \circ } \sin 61 ^ { \circ } } + \frac { 1 } { \sin 62 ^ { \circ } \sin 63 ^ { \circ } } + \cdots + \frac { 1 } { \sin 118 ^ { \circ } \sin 119 ^ { \circ } } .$$
Then the value of
$$\left( \frac { \operatorname { cosec } 1 ^ { \circ } } { \alpha } \right) ^ { 2 }$$
is $\_\_\_\_$.

A value of $x$ for which $\sin \left( \cot ^ { - 1 } ( 1 + x ) \right) = \cos \left( \tan ^ { - 1 } x \right)$, is :
(1) $- \frac { 1 } { 2 }$
(2) 1
(3) 0
(4) $\frac { 1 } { 2 }$

Let $\tan ^ { - 1 } y = \tan ^ { - 1 } x + \tan ^ { - 1 } \left( \frac { 2 x } { 1 - x ^ { 2 } } \right)$, where $| x | < \frac { 1 } { \sqrt { 3 } }$. Then a value of $y$ is
(1) $\frac { 3 x + x ^ { 3 } } { 1 + 3 x ^ { 2 } }$
(2) $\frac { 3 x - x ^ { 3 } } { 1 - 3 x ^ { 2 } }$
(3) $\frac { 3 x + x ^ { 3 } } { 1 - 3 x ^ { 2 } }$
(4) $\frac { 3 x - x ^ { 3 } } { 1 + 3 x ^ { 2 } }$

The value of $\sin 10 ^ { \circ } \sin 30 ^ { \circ } \sin 50 ^ { \circ } \sin 70 ^ { \circ }$ is:
(1) $\frac { 1 } { 36 }$
(2) $\frac { 1 } { 16 }$
(3) $\frac { 1 } { 18 }$
(4) $\frac { 1 } { 32 }$

$2 \pi - \left( \sin ^ { - 1 } \frac { 4 } { 5 } + \sin ^ { - 1 } \frac { 5 } { 13 } + \sin ^ { - 1 } \frac { 16 } { 65 } \right)$ is equal to:
(1) $\frac { \pi } { 2 }$
(2) $\frac { 5 \pi } { 4 }$
(3) $\frac { 3 \pi } { 2 }$
(4) $\frac { 7 \pi } { 4 }$

If $S$ is the sum of the first 10 terms of the series, $\tan ^ { - 1 } \left( \frac { 1 } { 3 } \right) + \tan ^ { - 1 } \left( \frac { 1 } { 7 } \right) + \tan ^ { - 1 } \left( \frac { 1 } { 13 } \right) + \tan ^ { - 1 } \left( \frac { 1 } { 21 } \right) + \ldots\ldots$. then $\tan ( S )$ is equal to :
(1) $\frac { 5 } { 6 }$
(2) $\frac { 5 } { 11 }$
(3) $- \frac { 5 } { 6 }$
(4) $\frac { 10 } { 11 }$

The value of $\cot \frac { \pi } { 24 }$ is:
(1) $\sqrt { 2 } + \sqrt { 3 } + 2 - \sqrt { 6 }$
(2) $\sqrt { 2 } + \sqrt { 3 } + 2 + \sqrt { 6 }$
(3) $\sqrt { 2 } - \sqrt { 3 } - 2 + \sqrt { 6 }$
(4) $3 \sqrt { 2 } - \sqrt { 3 } - \sqrt { 6 }$

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

$\tan \left( 2 \tan ^ { - 1 } \frac { 1 } { 5 } + \sec ^ { - 1 } \frac { \sqrt { 5 } } { 2 } + 2 \tan ^ { - 1 } \frac { 1 } { 8 } \right)$ is equal to:
(1) 1
(2) 2
(3) $\frac { 1 } { 4 }$
(4) $\frac { 5 } { 4 }$

If the inverse trigonometric functions take principal values, then $\cos ^ { - 1 } \left( \frac { 3 } { 10 } \cos \left( \tan ^ { - 1 } \left( \frac { 4 } { 3 } \right) \right) + \frac { 2 } { 5 } \sin \left( \tan ^ { - 1 } \left( \frac { 4 } { 3 } \right) \right) \right)$ is equal to
(1) 0
(2) $\frac { \pi } { 4 }$
(3) $\frac { \pi } { 3 }$
(4) $\frac { \pi } { 6 }$

$\tan ^ { - 1 } \frac { 1 + \sqrt { 3 } } { 3 + \sqrt { 3 } } + \sec ^ { - 1 } \sqrt { \frac { 8 + 4 \sqrt { 3 } } { 6 + 3 \sqrt { 3 } } } =$
(1) $\frac { \pi } { 4 }$
(2) $\frac { \pi } { 2 }$
(3) $\frac { \pi } { 3 }$
(4) $\frac { \pi } { 6 }$

If the value of $\frac { 3 \cos 36 ^ { \circ } + 5 \sin 18 ^ { \circ } } { 5 \cos 36 ^ { \circ } - 3 \sin 18 ^ { \circ } }$ is $\frac { a \sqrt { 5 } - b } { c }$, where $a , b , c$ are natural numbers and $\operatorname { gcd } ( a , c ) = 1$, then $a + b + c$ is equal to : (1) 40 (2) 52 (3) 50 (4) 54

$\cos \left( \sin ^ { - 1 } \frac { 3 } { 5 } + \sin ^ { - 1 } \frac { 5 } { 13 } + \sin ^ { - 1 } \frac { 33 } { 65 } \right)$ is equal to:
(1) 1
(2) 0
(3) $\frac { 32 } { 65 }$
(4) $\frac { 33 } { 65 }$

The value of $\left( \sin 70 ^ { \circ } \right) \left( \cot 10 ^ { \circ } \cot 70 ^ { \circ } - 1 \right)$ is
(1) $2/3$
(2) 1
(3) 0
(4) $3/2$

If $\alpha > \beta > \gamma > 0$, then the expression $\cot^{-1}\left\{ \beta + \frac{(1 + \beta^{2})}{(\alpha - \beta)} \right\} + \cot^{-1}\left\{ \gamma + \frac{(1 + \gamma^{2})}{(\beta - \gamma)} \right\} + \cot^{-1}\left\{ \alpha + \frac{(1 + \alpha^{2})}{(\gamma - \alpha)} \right\}$ is equal to:
(1) $\pi$
(2) 0
(3) $\frac{\pi}{2} - (\alpha + \beta + \gamma)$
(4) $3\pi$

Q65. If the value of $\frac { 3 \cos 36 ^ { \circ } + 5 \sin 18 ^ { \circ } } { 5 \cos 36 ^ { \circ } - 3 \sin 18 ^ { \circ } }$ is $\frac { a \sqrt { 5 } - b } { c }$, where $a , b , c$ are natural numbers and $\operatorname { gcd } ( a , c ) = 1$, then $a + b + c$ is equal to :
(1) 40
(2) 52
(3) 50
(4) 54

By the princibal of inverse trigonometric function, the value of $\tan \left( 2 \sin ^ { - 1 } \left( \frac { 2 } { \sqrt { 13 } } \right) - 2 \cos ^ { - 1 } \left( \frac { 3 } { \sqrt { 10 } } \right) \right)$ is (A) $\frac { 31 } { 55 }$ (B) $\frac { 33 } { 56 }$ (C) $\frac { 32 } { 59 }$ (D) $\frac { 38 } { 55 }$

The value of $\frac { \sqrt { 3 } \operatorname { cosec } 20 ^ { \circ } - \sec 20 ^ { \circ } } { \cos 20 ^ { \circ } \cos 40 ^ { \circ } \cos 60 ^ { \circ } \cos 80 ^ { \circ } }$ is
(A) 12 (8) 64
(C) 16
(D) 32