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) $\pi - 2\sin^{-1}\frac{\sqrt{3}}{4}$
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 $\_\_\_\_$.
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) $- \frac { 1 } { 2 }$
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) $z < y < x$