When $\tan \frac { \theta } { 2 } = \frac { \sqrt { 2 } } { 2 }$, what is the value of $\sec \theta$? (where $0 < \theta < \frac { \pi } { 2 }$ ) [3 points]
(1) 3
(2) $\frac { 10 } { 3 }$
(3) $\frac { 11 } { 3 }$
(4) 4
(5) $\frac { 13 } { 3 }$

When $\tan \theta = 5$, find the value of $\sec ^ { 2 } \theta$. [3 points]

For $\theta$ satisfying $\pi < \theta < \frac { 3 } { 2 } \pi$ and $\tan \theta - \frac { 6 } { \tan \theta } = 1$, what is the value of $\sin \theta + \cos \theta$? [3 points]
(1) $- \frac { 2 \sqrt { 10 } } { 5 }$
(2) $- \frac { \sqrt { 10 } } { 5 }$
(3) 0
(4) $\frac { \sqrt { 10 } } { 5 }$
(5) $\frac { 2 \sqrt { 10 } } { 5 }$

If $\tan \theta < 0$ and $\cos \left( \frac { \pi } { 2 } + \theta \right) = \frac { \sqrt { 5 } } { 5 }$, what is the value of $\cos \theta$? [3 points]
(1) $- \frac { 2 \sqrt { 5 } } { 5 }$
(2) $- \frac { \sqrt { 5 } } { 5 }$
(3) 0
(4) $\frac { \sqrt { 5 } } { 5 }$
(5) $\frac { 2 \sqrt { 5 } } { 5 }$

When $\cos\left(\frac{\pi}{2} + \theta\right) = -\frac{1}{5}$, what is the value of $\frac{\sin\theta}{1 - \cos^{2}\theta}$? [3 points]
(1) $-5$
(2) $-\sqrt{5}$
(3) $0$
(4) $\sqrt{5}$
(5) $5$

The value of $$\sin ^ { - 1 } \cot \left[ \sin ^ { - 1 } \left\{ \frac { 1 } { 2 } \left( 1 - \sqrt { \frac { 5 } { 6 } } \right) \right\} + \cos ^ { - 1 } \sqrt { \frac { 2 } { 3 } } + \sec ^ { - 1 } \sqrt { \frac { 8 } { 3 } } \right]$$ is
(A) 0
(B) $\pi / 6$
(C) $\pi / 4$
(D) $\pi / 2$

Let $n \geq 3$ be an integer. Assume that inside a big circle, exactly $n$ small circles of radius $r$ can be drawn so that each small circle touches the big circle and also touches both its adjacent small circles. Then, the radius of the big circle is:
(a) $r \operatorname { cosec } \frac { \pi } { n }$
(b) $r \left( 1 + \operatorname { cosec } \frac { 2 \pi } { n } \right)$
(c) $r \left( 1 + \operatorname { cosec } \frac { \pi } { 2 n } \right)$
(d) $r \left( 1 + \operatorname { cosec } \frac { \pi } { n } \right)$.

Let $n \geq 3$ be an integer. Assume that inside a big circle, exactly $n$ small circles of radius $r$ can be drawn so that each small circle touches the big circle and also touches both its adjacent small circles. Then, the radius of the big circle is:
(a) $r \operatorname { cosec } \frac { \pi } { n }$
(b) $r \left( 1 + \operatorname { cosec } \frac { 2 \pi } { n } \right)$
(c) $r \left( 1 + \operatorname { cosec } \frac { \pi } { 2 n } \right)$
(d) $r \left( 1 + \operatorname { cosec } \frac { \pi } { n } \right)$.

The value of $$\sin ^ { - 1 } \cot \left[ \sin ^ { - 1 } \left\{ \frac { 1 } { 2 } \left( 1 - \sqrt { \frac { 5 } { 6 } } \right) \right\} + \cos ^ { - 1 } \sqrt { \frac { 2 } { 3 } } + \sec ^ { - 1 } \sqrt { \frac { 8 } { 3 } } \right]$$ is
(A) 0
(B) $\pi/6$
(C) $\pi/4$
(D) $\pi/2$

The value of $$\sin ^ { - 1 } \cot \left[ \sin ^ { - 1 } \left\{ \frac { 1 } { 2 } \left( 1 - \sqrt { \frac { 5 } { 6 } } \right) \right\} + \cos ^ { - 1 } \sqrt { \frac { 2 } { 3 } } + \sec ^ { - 1 } \sqrt { \frac { 8 } { 3 } } \right]$$ is
(A) 0
(B) $\pi / 6$
(C) $\pi / 4$
(D) $\pi / 2$

If $\sin \left( \tan ^ { - 1 } ( x ) \right) = \cot \left( \sin ^ { - 1 } \left( \sqrt { \frac { 13 } { 17 } } \right) \right)$ then $x$ is
(A) $\frac { 4 } { 17 }$
(B) $\frac { 2 } { 3 }$
(C) $\sqrt { \frac { 17 ^ { 2 } - 13 ^ { 2 } } { 17 ^ { 2 } + 13 ^ { 2 } } }$
(D) $\sqrt { \frac { 17 ^ { 2 } - 13 ^ { 2 } } { 17 \times 13 } }$.

The value of $\sin ^ { - 1 } \cot \left[ \sin ^ { - 1 } \left\{ \frac { 1 } { 2 } \left( 1 - \sqrt { \frac { 5 } { 6 } } \right) \right\} + \cos ^ { - 1 } \sqrt { \frac { 2 } { 3 } } + \sec ^ { - 1 } \sqrt { \frac { 8 } { 3 } } \right]$ is
(a) 0 .
(B) $\pi / 6$.
(C) $\pi / 4$.
(D) $\pi / 2$.

The number of solutions of the equation $\sin ^ { - 1 } x = 2 \tan ^ { - 1 } x$ is
(a) 1 .
(B) 2 .
(C) 3 .
(D) 5 .

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) $\sqrt { 1 + x ^ { 2 } }$

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 $n , \lim _ { x \rightarrow \infty } \sec ^ { 2 } \left( f _ { n } ( x ) \right) = 1$

For non-negative integers $n$, let $$f(n) = \frac{\sum_{k=0}^{n} \sin\left(\frac{k+1}{n+2}\pi\right)\sin\left(\frac{k+2}{n+2}\pi\right)}{\sum_{k=0}^{n} \sin^2\left(\frac{k+1}{n+2}\pi\right)}$$
Assuming $\cos^{-1}x$ takes values in $[0, \pi]$, which of the following options is/are correct?
(A) $f(4) = \frac{\sqrt{3}}{2}$
(B) $\lim_{n\rightarrow\infty} f(n) = \frac{1}{2}$
(C) If $\alpha = \tan(\cos^{-1}f(6))$, then $\alpha^2 + 2\alpha - 1 = 0$
(D) $\sin(7\cos^{-1}f(5)) = 0$

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
$$\frac { 3 } { 2 } \cos ^ { - 1 } \sqrt { \frac { 2 } { 2 + \pi ^ { 2 } } } + \frac { 1 } { 4 } \sin ^ { - 1 } \frac { 2 \sqrt { 2 } \pi } { 2 + \pi ^ { 2 } } + \tan ^ { - 1 } \frac { \sqrt { 2 } } { \pi }$$
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 $x \in ( 0,1 )$. The set of all $x$ such that $\sin ^ { -1 } x > \cos ^ { -1 } x$, is the interval:
(1) $\left( \frac { 1 } { 2 } , \frac { 1 } { \sqrt { 2 } } \right)$
(2) $\left( \frac { 1 } { \sqrt { 2 } } , 1 \right)$
(3) $( 0,1 )$
(4) $\left( 0 , \frac { \sqrt { 3 } } { 2 } \right)$

Let $f _ { k } ( x ) = \frac { 1 } { k } \left( \sin ^ { k } x + \cos ^ { k } x \right)$ where $x \in R$ and $k \geq 1$. Then $f _ { 4 } ( x ) - f _ { 6 } ( x )$ equals
(1) $\frac { 1 } { 4 }$
(2) $\frac { 1 } { 12 }$
(3) $\frac { 1 } { 6 }$
(4) $\frac { 1 } { 3 }$

If $\operatorname { cosec } \theta = \frac { \mathrm { p } + \mathrm { q } } { \mathrm { p } - \mathrm { q } } ( \mathrm { p } \neq \mathrm { q } , \mathrm { p } \neq 0 )$, then $\left| \cot \left( \frac { \pi } { 4 } + \frac { \theta } { 2 } \right) \right|$ is equals to:
(1) $p q$
(2) $\sqrt { \frac { p } { q } }$
(3) $\sqrt { \frac { q } { p } }$
(4) $\sqrt { \mathrm { pq } }$

If an angle $A$ of a $\triangle A B C$ satisfies $5 \cos A + 3 = 0$, then the roots of the quadratic equation $9 x ^ { 2 } + 27 x + 20 = 0$ are
(1) $\sec A , \cot A$
(2) $\sec A , \tan A$
(3) $\tan A , \cos A$
(4) $\sin A , \sec A$

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