Let $S$ be the set of all solutions of the equation $\cos^{-1}(2x) - 2\cos^{-1}\left(\sqrt{1 - x^2}\right) = \pi$, $x \in \left[-\frac{1}{2}, \frac{1}{2}\right]$. Then $\sum_{x \in S} 2\sin^{-1}(x^2 - 1)$ is equal to
(1) 0
(2) $\frac{-2\pi}{3}$
(3) $\pi - \sin^{-1}\frac{\sqrt{3}}{4}$
(4) $\pi - 2\sin^{-1}\frac{\sqrt{3}}{4}$

If $\sin x = - \frac { 3 } { 5 }$, where $\pi < x < \frac { 3 \pi } { 2 }$, then $80 \left( \tan ^ { 2 } x - \cos x \right)$ is equal to
(1) 108
(2) 109
(3) 18
(4) 19

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

Given that the inverse trigonometric function assumes principal values only. Let $x , y$ be any two real numbers in $[ - 1,1 ]$ such that $\cos ^ { - 1 } x - \sin ^ { - 1 } y = \alpha , \frac { - \pi } { 2 } \leq \alpha \leq \pi$. Then, the minimum value of $x ^ { 2 } + y ^ { 2 } + 2 x y \sin \alpha$ is
(1) 0
(2) - 1
(3) $\frac { 1 } { 2 }$
(4) $- \frac { 1 } { 2 }$

Let the range of the function $f ( x ) = \frac { 1 } { 2 + \sin 3 x + \cos 3 x } , x \in \mathbb { R }$ be $[ a , b ]$. If $\alpha$ and $\beta$ are respectively the A.M. and the G.M. of $a$ and $b$, then $\frac { \alpha } { \beta }$ is equal to
(1) $\pi$
(2) $\sqrt { \pi }$
(3) 2
(4) $\sqrt { 2 }$

Let the set of all $a \in R$ such that the equation $\cos 2 x + a \sin x = 2 a - 7$ has a solution be $[ p , q ]$ and $r = \tan 9 ^ { \circ } - \tan 27 ^ { \circ } - \frac { 1 } { \cot 63 ^ { \circ } } + \tan 81 ^ { \circ }$, then $p q r$ is equal to $\_\_\_\_$.

Using the principal values of the inverse trigonometric functions, the sum of the maximum and the minimum values of $16 \left( \left( \sec ^ { - 1 } x \right) ^ { 2 } + \left( \operatorname { cosec } ^ { - 1 } x \right) ^ { 2 } \right)$ is:
(1) $24 \pi ^ { 2 }$
(2) $22 \pi ^ { 2 }$
(3) $31 \pi ^ { 2 }$
(4) $18 \pi ^ { 2 }$

Q72. Given that the inverse trigonometric function assumes principal values only. Let $x , y$ be any two real numbers in $[ - 1,1 ]$ such that $\cos ^ { - 1 } x - \sin ^ { - 1 } y = \alpha , \frac { - \pi } { 2 } \leq \alpha \leq \pi$. Then, the minimum value of $x ^ { 2 } + y ^ { 2 } + 2 x y \sin \alpha$ is
(1) 0
(2) - 1
(3) $\frac { 1 } { 2 }$
(4) $- \frac { 1 } { 2 }$

3. For ALL APPLICANTS.

Note that the arguments of all trigonometric functions in this question are given in terms of degrees. You are not expected to differentiate such a function. The notation $\cos ^ { n } x$ means $( \cos x ) ^ { n }$ throughout.
(i) $[ 1$ mark $]$ Without differentiating, write down the maximum value of $\cos \left( 2 x + 30 ^ { \circ } \right)$. [0pt] (ii) [4 marks] Again without differentiating, find the maximum value of
$$\cos \left( 2 x + 30 ^ { \circ } \right) \left( 1 - \cos \left( 2 x + 30 ^ { \circ } \right) \right)$$
(iii) [4 marks] Hence write down the maximum value of
$$\cos ^ { 5 } \left( 2 x + 30 ^ { \circ } \right) \left( 1 - \cos \left( 2 x + 30 ^ { \circ } \right) \right) ^ { 5 }$$
(iv) [6 marks] Find the maximum value of
$$\left( 1 - \cos ^ { 2 } \left( 3 x - 60 ^ { \circ } \right) \right) ^ { 4 } \left( 3 - \cos \left( 150 ^ { \circ } - 3 x \right) \right) ^ { 8 }$$

2. Let $a = \cos(\pi^{2})$. Which of the following options is correct?
(1) $a = -1$
(2) $-1 < a \leq -\frac{1}{2}$
(3) $-\frac{1}{2} < a \leq 0$
(4) $0 < a \leq \frac{1}{2}$
(5) $\frac{1}{2} < a \leq 1$

8. Let $\theta_{1}, \theta_{2}, \theta_{3}, \theta_{4}$ be angles in the first, second, third, and fourth quadrants respectively, all between 0 and $2\pi$. Given that $|\cos \theta_{1}| = |\cos \theta_{2}| = |\cos \theta_{3}| = |\cos \theta_{4}| = \frac{1}{3}$, which of the following options are correct?
(1) $\theta_{1} < \frac{\pi}{4}$
(2) $\theta_{1} + \theta_{2} = \pi$
(3) $\cos \theta_{3} = -\frac{1}{3}$
(4) $\sin \theta_{4} = \frac{2\sqrt{2}}{3}$
(5) $\theta_{4} = \theta_{3} + \frac{\pi}{2}$

On the coordinate plane, a circle with center at the origin $O$ and radius 1 intersects the positive directions of the coordinate axes at points $A$ and $B$ respectively. On the circular arc in the first quadrant, a point $C$ is taken to draw a tangent line to the circle that intersects the two axes at points $D$ and $E$ respectively, as shown in the figure. Let $\angle OEC = \theta$. Select the option that represents $\tan \theta$.
(1) $\overline{OE}$
(2) $\overline{OC}$
(3) $\overline{OD}$
(4) $\overline{CE}$
(5) $\overline{CD}$

Let $\Gamma$ be the function graph of $y = \sin \pi x$ for $0 \leq x \leq 3$. A horizontal line $L : y = k$ intersects $\Gamma$ at three points $P \left( x _ { 1 } , k \right) , Q \left( x _ { 2 } , k \right) , R \left( x _ { 3 } , k \right)$ satisfying $x _ { 1 } < x _ { 2 } < 1 < x _ { 3 }$. Select the correct options.
(1) $k > 0$
(2) $L$ and $\Gamma$ have exactly 3 intersection points
(3) $x _ { 1 } + x _ { 2 } < 1$
(4) If $2 \overline { P Q } = \overline { Q R }$, then $k = \frac { 1 } { 2 }$
(5) The sum of $x$-coordinates of all intersection points of $L$ and $\Gamma$ is greater than 5

On a certain day at a certain location, the duration of daylight (from sunrise to sunset) is exactly 12 hours. The UVI value at that location $x$ hours after sunrise ($0 \leq x \leq 12$) can be expressed by the function $f ( x ) = a \sin ( b x )$ , where $a , b > 0$ . Assume that the UVI value is positive during daylight and 0 during non-daylight hours (i.e., $f ( 0 ) = f ( 12 ) = 0$), and the UVI value 2 hours after sunrise on that day is 4. Find the values of $a$ and $b$.

The diagram shows the graphs of $y = \sin 2 x$ and $y = \cos 2 x$ for $- \frac { \pi } { 2 } < x < \frac { \pi } { 2 }$
Which one of the following is not true?
A $\cos 2 x < \sin 2 x < \tan x$ for some real number $x$ with $- \frac { \pi } { 2 } < x < \frac { \pi } { 2 }$
B $\cos 2 x < \tan x < \sin 2 x$ for some real number $x$ with $- \frac { \pi } { 2 } < x < \frac { \pi } { 2 }$
C $\sin 2 x < \cos 2 x < \tan x$ for some real number $x$ with $- \frac { \pi } { 2 } < x < \frac { \pi } { 2 }$
D $\sin 2 x < \tan x < \cos 2 x$ for some real number $x$ with $- \frac { \pi } { 2 } < x < \frac { \pi } { 2 }$
E $\tan x < \sin 2 x < \cos 2 x$ for some real number $x$ with $- \frac { \pi } { 2 } < x < \frac { \pi } { 2 }$
F $\tan x < \cos 2 x < \sin 2 x$ for some real number $x$ with $- \frac { \pi } { 2 } < x < \frac { \pi } { 2 }$

What is the smallest positive value of $a$ for which the line $x = a$ is a line of symmetry of the graph of $y = \sin \left( 2 x - \frac { 4 \pi } { 3 } \right)$ ?
A $\frac { \pi } { 12 }$
B $\frac { 5 \pi } { 12 }$
C $\quad \frac { 7 \pi } { 12 }$
D $\frac { 11 \pi } { 12 }$
E $\frac { 19 \pi } { 12 }$

The functions $f _ { 1 }$ to $f _ { 5 }$ are defined on the real numbers by
$$\begin{aligned} & \mathrm { f } _ { 1 } ( x ) = \cos x \\ & \mathrm { f } _ { 2 } ( x ) = \sin ( \cos x ) \\ & \mathrm { f } _ { 3 } ( x ) = \cos ( \sin ( \cos x ) ) \\ & \mathrm { f } _ { 4 } ( x ) = \sin ( \cos ( \sin ( \cos x ) ) ) \\ & \mathrm { f } _ { 5 } ( x ) = \cos ( \sin ( \cos ( \sin ( \cos x ) ) ) ) \end{aligned}$$
where all numbers are taken to be in radians. These functions have maximum values $m _ { 1 } , m _ { 2 } , m _ { 3 } , m _ { 4 }$ and $m _ { 5 }$, respectively. Which one of the following statements is true?
A $m _ { 1 } , m _ { 2 } , m _ { 3 } , m _ { 4 }$ and $m _ { 5 }$ are all equal to 1
B $0 < m _ { 5 } < m _ { 4 } < m _ { 3 } < m _ { 2 } < m _ { 1 } = 1$
C $\quad m _ { 1 } = m _ { 3 } = m _ { 5 } = 1$ and $0 < m _ { 2 } = m _ { 4 } < 1$
D $m _ { 1 } = m _ { 3 } = m _ { 5 } = 1$ and $0 < m _ { 4 } < m _ { 2 } < 1$
E $m _ { 1 } = m _ { 3 } = 1$ and $0 < m _ { 2 } = m _ { 4 } < 1$ and $0 < m _ { 5 } < 1$ F $m _ { 1 } = m _ { 3 } = 1$ and $0 < m _ { 4 } < m _ { 2 } < 1$ and $0 < m _ { 5 } < 1$

Let $\mathrm { a } \in \left( \frac { \pi } { 12 } , \frac { \pi } { 6 } \right)$.
$$\begin{aligned} & x = \sin ( 3 a ) \\ & y = \cos ( 3 a ) \\ & z = \tan ( 3 a ) \end{aligned}$$
What is the correct ordering of the numbers?
A) $x < y < z$
B) $x < z < y$
C) $y < x < z$
D) $y < z < x$
E) $z < x < y$

Let $a = \sin(40^{\circ})$
$$\begin{aligned} & b = \sec(40^{\circ}) \\ & c = \tan(40^{\circ}) \end{aligned}$$
Which of the following is the correct ordering of the numbers $a$, $b$ and $c$?
A) $a < b < c$ B) $a < c < b$ C) $b < a < c$ D) $b < c < a$ E) $c < a < b$

Let $x, y$ and $z$ be distinct elements of the set $\left\{\frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}\right\}$ such that
$$\sin x < \tan y < \sec z$$
Which of the following is the correct ordering of $x$, $y$ and $z$?
A) $x < y < z$ B) $y < x < z$ C) $y < z < x$ D) $z < x < y$ E) $z < y < x$