Let $a _ { 0 } = \frac { 1 } { 2 }$ and $a _ { n }$ be defined inductively by
$$a _ { n } = \sqrt { \frac { 1 + a _ { n - 1 } } { 2 } } , n \geq 1 .$$
(a) Show that for $n = 0,1,2 , \ldots$,
$$a _ { n } = \cos \theta _ { n } \text { for some } 0 < \theta _ { n } < \frac { \pi } { 2 } ,$$
and determine $\theta _ { n }$.
(b) Using (a) or otherwise, calculate
$$\lim _ { n \rightarrow \infty } 4 ^ { n } \left( 1 - a _ { n } \right)$$

The value of $$\sum _ { k = 0 } ^ { 202 } ( - 1 ) ^ { k } \binom { 202 } { k } \cos \left( \frac { k \pi } { 3 } \right)$$ equals
(A) $\quad \sin \left( \frac { 202 } { 3 } \pi \right)$.
(B) $- \sin \left( \frac { 202 } { 3 } \pi \right)$.
(C) $\quad \cos \left( \frac { 202 } { 3 } \pi \right)$.
(D) $\cos ^ { 202 } \left( \frac { \pi } { 3 } \right)$.

Let $\frac { \tan ( \alpha - \beta + \gamma ) } { \tan ( \alpha + \beta - \gamma ) } = \frac { \tan \beta } { \tan \gamma }$. Then
(A) $\sin ( \beta - \gamma ) = \sin ( \alpha - \beta )$.
(B) $\sin ( \alpha - \gamma ) = \sin ( \beta - \gamma )$.
(C) $\sin ( \beta - \gamma ) = 0$.
(D) $\sin 2 \alpha + \sin 2 \beta + \sin 2 \gamma = 0$

Let $\frac { \tan 3 \theta } { \tan \theta } = k$. Then
(A) $k \in ( 1 / 3,3 )$
(B) $k \notin ( 1 / 3,3 )$
(C) $\frac { \sin 3 \theta } { \sin \theta } = \frac { 2 k } { k - 1 }$.
(D) $\frac { \sin 3 \theta } { \sin \theta } > \frac { 2 k } { k - 1 }$

The value of $\sum _ { k = 1 } ^ { 13 } \frac { 1 } { \sin \left( \frac { \pi } { 4 } + \frac { ( k - 1 ) \pi } { 6 } \right) \sin \left( \frac { \pi } { 4 } + \frac { k \pi } { 6 } \right) }$ is equal to
(A) $3 - \sqrt { 3 }$
(B) $2 ( 3 - \sqrt { 3 } )$
(C) $2 ( \sqrt { 3 } - 1 )$
(D) $2 ( 2 + \sqrt { 3 } )$

Let $\alpha$ and $\beta$ be nonzero real numbers such that $2 ( \cos \beta - \cos \alpha ) + \cos \alpha \cos \beta = 1$. Then which of the following is/are true?
[A] $\tan \left( \frac { \alpha } { 2 } \right) + \sqrt { 3 } \tan \left( \frac { \beta } { 2 } \right) = 0$
[B] $\sqrt { 3 } \tan \left( \frac { \alpha } { 2 } \right) + \tan \left( \frac { \beta } { 2 } \right) = 0$
[C] $\tan \left( \frac { \alpha } { 2 } \right) - \sqrt { 3 } \tan \left( \frac { \beta } { 2 } \right) = 0$
[D] $\sqrt { 3 } \tan \left( \frac { \alpha } { 2 } \right) - \tan \left( \frac { \beta } { 2 } \right) = 0$

If the triangle $P Q R$ varies, then the minimum value of
$$\cos ( P + Q ) + \cos ( Q + R ) + \cos ( R + P )$$
is
[A] $- \frac { 5 } { 3 }$
[B] $- \frac { 3 } { 2 }$
[C] $\frac { 3 } { 2 }$
[D] $\frac { 5 } { 3 }$

Let $a , b , c$ be three non-zero real numbers such that the equation $$\sqrt { 3 } a \cos x + 2 b \sin x = c , x \in \left[ - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right]$$ has two distinct real roots $\alpha$ and $\beta$ with $\alpha + \beta = \frac { \pi } { 3 }$. Then, the value of $\frac { b } { a }$ is $\_\_\_\_$.

Let $\alpha$ and $\beta$ be real numbers such that $- \frac { \pi } { 4 } < \beta < 0 < \alpha < \frac { \pi } { 4 }$. If $\sin ( \alpha + \beta ) = \frac { 1 } { 3 }$ and $\cos ( \alpha - \beta ) = \frac { 2 } { 3 }$, then the greatest integer less than or equal to
$$\left( \frac { \sin \alpha } { \cos \beta } + \frac { \cos \beta } { \sin \alpha } + \frac { \cos \alpha } { \sin \beta } + \frac { \sin \beta } { \cos \alpha } \right) ^ { 2 }$$
is $\_\_\_\_$ .

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 $\frac { \pi } { 2 } < x < \pi$ be such that $\cot x = \frac { - 5 } { \sqrt { 11 } }$. Then
$$\left( \sin \frac { 11 x } { 2 } \right) ( \sin 6 x - \cos 6 x ) + \left( \cos \frac { 11 x } { 2 } \right) ( \sin 6 x + \cos 6 x )$$
is equal to
(A) $\frac { \sqrt { 11 } - 1 } { 2 \sqrt { 3 } }$
(B) $\frac { \sqrt { 11 } + 1 } { 2 \sqrt { 3 } }$
(C) $\frac { \sqrt { 11 } + 1 } { 3 \sqrt { 2 } }$
(D) $\frac { \sqrt { 11 } - 1 } { 3 \sqrt { 2 } }$

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 $\_\_\_\_$.

Suppose $\theta$ and $\phi ( \neq 0 )$ are such that $\sec ( \theta + \phi )$, $\sec \theta$ and $\sec ( \theta - \phi )$ are in A.P. If $\cos \theta = k \cos \left( \frac { \phi } { 2 } \right)$ for some $k$, then $k$ is equal to
(1) $\pm \sqrt { 2 }$
(2) $\pm 1$
(3) $\pm \frac { 1 } { \sqrt { 2 } }$
(4) $\pm 2$

The value of $\cos 255 ^ { \circ } + \sin 195 ^ { \circ }$ is
(1) $\frac { \sqrt { 3 } - 1 } { 2 \sqrt { 2 } }$
(2) $\frac { \sqrt { 3 } - 1 } { \sqrt { 2 } }$
(3) $- \frac { \sqrt { 3 } - 1 } { \sqrt { 2 } }$
(4) $\frac { \sqrt { 3 } + 1 } { \sqrt { 2 } }$

If $\tan\left(\frac{\pi}{4} + \frac{\theta}{2}\right) = \tan^3\left(\frac{\pi}{4} + \frac{\alpha}{2}\right)$, then $\sin\theta = $:
(1) $\frac{\sin\alpha(3 + \sin^2\alpha)}{1 + 3\sin^2\alpha}$
(2) $\frac{\sin\alpha(3 - \sin^2\alpha)}{1 + 3\sin^2\alpha}$
(3) $\frac{\sin\alpha(3 + \cos^2\alpha)}{1 + 3\cos^2\alpha}$
(4) $\frac{\sin\alpha(3 - \cos^2\alpha)}{1 + 3\cos^2\alpha}$

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) $\frac { 3 x - x ^ { 3 } } { 1 + 3 x ^ { 2 } }$

If $A > 0 , B > 0$ and $A + B = \frac { \pi } { 6 }$, then the minimum positive value of $( \tan A + \tan B )$ is :
(1) $\sqrt { 3 } - \sqrt { 2 }$
(2) $4 - 2 \sqrt { 3 }$
(3) $\frac { 2 } { \sqrt { 3 } }$
(4) $2 - \sqrt { 3 }$

The value of $\cos \frac { \pi } { 2 ^ { 2 } } \cdot \cos \frac { \pi } { 2 ^ { 3 } } \cdot \ldots \cdot \cos \frac { \pi } { 2 ^ { 10 } } \cdot \sin \frac { \pi } { 2 ^ { 10 } }$ is:
(1) $\frac { 1 } { 1024 }$
(2) $\frac { 1 } { 512 }$
(3) $\frac { 1 } { 2 }$
(4) $\frac { 1 } { 256 }$

If $\cos(\alpha + \beta) = \frac{3}{5}$, $\sin(\alpha - \beta) = \frac{5}{13}$ and $0 < \alpha, \beta < \frac{\pi}{4}$, then $\tan 2\alpha$ is equal to:
(1) $\frac{21}{16}$
(2) $\frac{63}{52}$
(3) $\frac{33}{52}$
(4) $\frac{63}{16}$

If $\alpha = \cos^{-1}\frac{3}{5}$, $\beta = \tan^{-1}\frac{1}{3}$, where $0 < \alpha, \beta < \frac{\pi}{2}$, then $\alpha - \beta$ is equal to
(1) $\tan^{-1}\frac{9}{14}$
(2) $\cos^{-1}\frac{9}{5\sqrt{10}}$
(3) $\sin^{-1}\frac{9}{5\sqrt{10}}$
(4) $\tan^{-1}\frac{9}{5\sqrt{10}}$

The value of $\cos ^ { 3 } \left( \frac { \pi } { 8 } \right) \cdot \cos \left( \frac { 3 \pi } { 8 } \right) + \sin ^ { 3 } \left( \frac { \pi } { 8 } \right) \cdot \sin \left( \frac { 3 \pi } { 8 } \right)$ is:
(1) $\frac { 1 } { \sqrt { 2 } }$
(2) $\frac { 1 } { 2 \sqrt { 2 } }$
(3) $\frac { 1 } { 2 }$
(4) $\frac { 1 } { 4 }$

If $L = \sin^2\left(\frac{\pi}{16}\right) - \sin^2\left(\frac{\pi}{8}\right)$ and $M = \cos^2\left(\frac{\pi}{16}\right) - \sin^2\left(\frac{\pi}{8}\right)$
(1) $L = -\frac{1}{2\sqrt{2}} + \frac{1}{2}\cos\frac{\pi}{8}$
(2) $L = \frac{1}{4\sqrt{2}} - \frac{1}{4}\cos\frac{\pi}{8}$
(3) $M = \frac{1}{4\sqrt{2}} + \frac{1}{4}\cos\frac{\pi}{8}$
(4) $M = \frac{1}{2\sqrt{2}} + \frac{1}{2}\cos\frac{\pi}{8}$

$2 \pi - \left( \sin ^ { - 1 } \frac { 4 } { 5 } + \sin ^ { - 1 } \frac { 5 } { 13 } + \sin ^ { - 1 } \frac { 16 } { 65 } \right)$ 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) $\frac { 10 } { 11 }$

If $e ^ { \cos ^ { 2 } x + \cos ^ { 4 } x + \cos ^ { 6 } x + \ldots \infty \log _ { e } 2 }$ satisfies the equation $t ^ { 2 } - 9 t + 8 = 0$, then the value of $\frac { 2 \sin x } { \sin x + \sqrt { 3 } \cos x }$, where $0 < x < \frac { \pi } { 2 }$, is equal to
(1) $\frac { 3 } { 2 }$
(2) $\frac { 1 } { 2 }$
(3) $\sqrt { 3 }$
(4) $2 \sqrt { 3 }$