The number of solutions of $\sin ^ { 2 } x + \left( 2 + 2 x - x ^ { 2 } \right) \sin x - 3 ( x - 1 ) ^ { 2 } = 0$, where $- \pi \leq x \leq \pi$, is $\_\_\_\_$

Let the inverse trigonometric functions take principal values. The number of real solutions of the equation $2 \sin ^ { - 1 } x + 3 \cos ^ { - 1 } x = \frac { 2 \pi } { 5 }$, is $\_\_\_\_$

Let $\mathrm{S} = \{x: \cos^{-1}x = \pi + \sin^{-1}x + \sin^{-1}(2x+1)\}$. Then $\sum_{x \in \mathrm{S}}(2x-1)^2$ is equal to \_\_\_\_ .

Q64. Let $| \cos \theta \cos ( 60 - \theta ) \cos ( 60 + \theta ) | \leq \frac { 1 } { 8 } , \theta \epsilon [ 0,2 \pi ]$. Then, the sum of all $\theta \epsilon [ 0,2 \pi ]$, where $\cos 3 \theta$ attains its maximum value, is :
(1) $15 \pi$
(2) $18 \pi$
(3) $6 \pi$
(4) $9 \pi$

Q83. The number of solutions of $\sin ^ { 2 } x + \left( 2 + 2 x - x ^ { 2 } \right) \sin x - 3 ( x - 1 ) ^ { 2 } = 0$, where $- \pi \leq x \leq \pi$, is $\_\_\_\_$

Q86. Let the inverse trigonometric functions take principal values. The number of real solutions of the equation $2 \sin ^ { - 1 } x + 3 \cos ^ { - 1 } x = \frac { 2 \pi } { 5 }$, is $\_\_\_\_$

Q87. For $n \in \mathrm {~N}$, if $\cot ^ { - 1 } 3 + \cot ^ { - 1 } 4 + \cot ^ { - 1 } 5 + \cot ^ { - 1 } n = \frac { \pi } { 4 }$, then $n$ is equal to $\_\_\_\_$

The number of values of $x$ satisfying $\tan ^ { - 1 } ( 4 x ) + \tan ^ { - 1 } ( 6 x ) = \frac { \pi } { 6 }$ and $\mathrm { x } \in \left[ - \frac { 1 } { 2 \sqrt { 6 } } , \frac { 1 } { 2 \sqrt { 6 } } \right]$ is
(A) 0
(B) 1
(C) 2
(D) 3

Let $\mathrm { k } = \tan \left( \frac { \pi } { 4 } + \frac { 1 } { 2 } \cos ^ { - 1 } \frac { 2 } { 3 } \right) + \tan \left( \frac { 1 } { 2 } \sin ^ { - 1 } \frac { 2 } { 3 } \right)$.
Then number of solutions of the equation $\sin ^ { - 1 } ( k x - 1 ) = \sin ^ { - 1 } x - \cos ^ { - 1 } x$

Let $f ( x ) = 4 \sqrt { 3 } e ^ { - x } \cos x + 6 e ^ { - x }$.
(1) Let $a$ and $b$ ($a < b$) be the values of $x$ satisfying $f ( x ) = 0$ on $0 \leqq x < 2 \pi$. Then,
$$a = \frac { \mathbf{A} } { \mathbf{B} } \pi , \quad b = \frac { \mathbf{C} } { \mathbf{D} } \pi$$
(2) The values of the constants $p$ and $q$ satisfying
$$\frac { d } { d x } \left( p e ^ { - x } \cos x + q e ^ { - x } \sin x \right) = e ^ { - x } \cos x$$
are given by
$$p = \frac { \mathbf { E F } } { \mathbf { G } } , \quad q = \frac { \mathbf { H } } { \mathbf { I } } .$$
(3) Using the values of $a$ and $b$ obtained in (1), we set $A = e ^ { - a }$ and $B = e ^ { - b }$. When we calculate the value of $\int _ { a } ^ { b } f ( x ) d x$, we obtain
$$\int _ { a } ^ { b } f ( x ) d x = ( \mathbf { J } - \sqrt { \mathbf{J} } \mathbf { K } ) A - ( \mathbf { L } + \sqrt { \mathbf{L} } ) B .$$

Consider the following two equations in $x$
$$\sin 2x + a\cos x = 0 \tag{1}$$ $$\cos 2x + a\sin x = -2 \tag{2}$$
over the interval $-\frac{\pi}{2} < x < \frac{\pi}{2}$, where $a > 0$.
Let $a = \sqrt{2}$. Then the value of $x$ which satisfies (1) is
$$x = \frac{\mathbf{AB}}{\mathbf{A}}$$
However, at this $x$ the value of the left side of (2) is $\mathbf{DE}$, and so equation (2) does not hold. Hence, when $a = \sqrt{2}$, (1) and (2) have no common solution.
Now, let us find a value of $a$ such that (1) and (2) have a common solution, and also the common solution $x$.
First, from (1) we have
$$\sin x = \frac{\mathbf{FG}}{\mathbf{H}}a, \quad \cos 2x = \mathbf{I} - \frac{a^2}{\mathbf{J}}.$$
When we substitute these into (2), we obtain
$$a^2 = \mathbf{K}.$$
Thus $a = \sqrt{\mathbf{K}}$, and the common solution is
$$x = \frac{\mathbf{LM}}{\mathbf{N}}$$

3. For APPLICANTS IN $\left\{ \begin{array} { l } \text { MATHEMATICS } \\ \text { MATHEMATICS \& STATISTICS } \\ \text { MATHEMATICS \& PHILOSOPHY } \\ \text { MATHEMATICS \& COMPUTER SCIENCE } \end{array} \right\}$ ONLY.

Computer Science applicants should turn to page 14. [0pt] [In this question, you may assume that the derivative of $\sin x$ is $\cos x$.] [Figure]
(i) In the diagram above $O A$ and $O C$ are of length 1 and subtend an angle $x$ at $O$. The angle $B A O$ is a right angle and the circular arc from $A$ to $C$, centred at $O$, is also drawn.
By consideration of various areas in the above diagram, show, for $0 < x < \pi / 2$, that
$$x \cos x < \sin x < x .$$
(ii) Sketch, on the axes provided on the opposite page, the graph of
$$y = \frac { \sin x } { x } , \quad 0 < x < 4 \pi$$
Justify your value that $y$ takes as $x$ becomes small. [0pt] [You do not need to determine the coordinates of the turning points.]
(iii) Drawn below is a graph of $y = \sin x$. Sketch on the same axes the line $y = c x$ where $c > 0$ is such that the equation $\sin x = c x$ has exactly 5 solutions. [Figure]
(iv) Draw the line $y = c$ on the axes on the opposite page.
(v) If $X$ is the largest of the five solutions of the equation $\sin x = c x$, explain why $\tan X = X$. [Figure]

9. Which of the following equations have real solutions?
(1) $x^{3} + x - 1 = 0$
(2) $2^{x} + 2^{-x} = 0$
(3) $\log_{2} x + \log_{x} 2 = 1$
(4) $\sin x + \cos 2x = 3$
(5) $4 \sin x + 3 \cos x = \frac{9}{2}$

China's Tiger Hill Tower, Pearl Tower, and Italy's Leaning Tower of Pisa are three famous leaning towers with tower heights of 48, 19, and 57 meters respectively, and offset distances of 2.3, 2.3, and 4 meters respectively. Their tilt angles are denoted as $\theta_1{}^{\circ}, \theta_2{}^{\circ}$, and $\theta_3{}^{\circ}$ respectively. Compare the size relationship of $\theta_1, \theta_2$, and $\theta_3$. (Non-multiple choice, 4 points)
Note: The tilt angle $\theta^{\circ}$ is the angle between the tower body and a vertical dashed line ($0 \leq \theta < 90$), and the offset distance is the distance from the tower top to the vertical dashed line.

Suppose there are two iron towers with equal tower heights. Their tilt angles $\alpha^{\circ}, \beta^{\circ}$ satisfy $\sin\alpha^{\circ} = \frac{1}{5}$ and $\sin\beta^{\circ} = \frac{7}{25}$ respectively. It is known that the offset distances of the two towers differ by 20 meters. Find the difference in the distance from the tower tops to the ground. (Non-multiple choice, 6 points)
Note: The tilt angle $\theta^{\circ}$ is the angle between the tower body and a vertical dashed line ($0 \leq \theta < 90$), the offset distance is the distance from the tower top to the vertical dashed line, and the distance from the tower top to the ground is the vertical height.

On the coordinate plane, the graph of the function $y = \sin x$ is symmetric about $x = \frac{\pi}{2}$, as shown in the figure. Find the value of $\theta$ in the range $0 < \theta \leq \pi$ that satisfies $\sin \theta = \sin\left(\theta + \frac{\pi}{5}\right)$.
(1) $\frac{\pi}{5}$
(2) $\frac{2\pi}{5}$
(3) $\frac{3\pi}{5}$
(4) $\frac{4\pi}{5}$
(5) $\pi$

Let $0 \leq \theta \leq 2 \pi$. All $\theta$ satisfying $\sin 2 \theta > \sin \theta$ and $\cos 2 \theta > \cos \theta$ can be expressed as $a \pi < \theta < b \pi$, where $a$ and $b$ are real numbers. What is the value of $b - a$?
(1) $\frac { 1 } { 3 }$
(2) $\frac { 1 } { 2 }$
(3) $\frac { 2 } { 3 }$
(4) $\frac { 3 } { 4 }$
(5) 1

Continuing from question 19, where $f ( x ) = a \sin ( b x )$ with $f(0) = f(12) = 0$ and $f(2) = 4$, a person wants to sunbathe when the UVI value is between $4 \sqrt { 2 }$ and $4 \sqrt { 3 }$ (inclusive). The time during which he can sunbathe is set as $t$ hours after sunrise. Find the maximum possible range of $t$.

18. The angle $x$ is measured in radians and is such that $0 \leq x \leq \pi$.
The total length of any intervals for which $- 1 \leq \tan x \leq 1$ and $\sin 2 x \geq 0.5$ is
A $\frac { \pi } { 12 }$
B $\frac { \pi } { 6 }$
C $\frac { \pi } { 4 }$
D $\frac { \pi } { 3 }$
E $\frac { 5 \pi } { 12 }$ F $\frac { \pi } { 2 }$ G $\quad \frac { 5 \pi } { 6 }$

What is the value, in radians, of the largest angle $x$ in the range $0 \leq x \leq 2 \pi$ that satisfies the equation $8 \sin ^ { 2 } x + 4 \cos ^ { 2 } x = 7$ ?

How many solutions does the equation $x \tan x = 1$ have in the interval $- 2 \pi \leq x \leq 2 \pi$ ?
A 0 B 1 C 2 D 3 E 4 F 5 G 6

The following question appeared in an examination:
Given that $\tan x = \sqrt { 3 }$, find the possible values of $\sin 2 x$.
A student gave the following answer:
$$\begin{aligned} & \tan x = \sqrt { 3 } \text { so } x = 60 ^ { \circ } \text { and } 2 x = 120 ^ { \circ } \\ & \text { therefore } \sin 2 x = \frac { \sqrt { 3 } } { 2 } \end{aligned}$$
Which one of the following statements is correct?
A $\frac { \sqrt { 3 } } { 2 }$ is the only possible value, and this is fully supported by the reasoning given in the student's answer.
B $\frac { \sqrt { 3 } } { 2 }$ is the only possible value, but the reasoning given should consider other possible values of $x$ for which $\tan x = \sqrt { 3 }$.
C $\frac { \sqrt { 3 } } { 2 }$ is the only possible value, but the reasoning given should consider other possible values of $x$ for which $\sin 2 x = \frac { \sqrt { 3 } } { 2 }$.
D $\frac { \sqrt { 3 } } { 2 }$ is not the only possible value because the reasoning given should have considered other possible values of $x$ for which $\tan x = \sqrt { 3 }$.
E $\frac { \sqrt { 3 } } { 2 }$ is not the only possible value because the reasoning given should have considered other possible values of $x$ for which $\sin 2 x = \frac { \sqrt { 3 } } { 2 }$.

The non-zero real number $c$ is such that the equation $\cos x = c$ has two solutions for $0 < x < \frac { 3 } { 2 } \pi$.
How many solutions of the equation $\cos ^ { 2 } 2 x = c ^ { 2 }$ are there in the range $0 < x < \frac { 3 } { 2 } \pi$ ?

Find the number of solutions of the equation
$$x \sin 2 x = \cos 2 x$$
with $0 \leq x \leq 2 \pi$.
A 0
B 1
C 2
D 3
E 4

Find the fraction of the interval $0 \leq \theta \leq \pi$ for which the inequality
$$\left(\sin(2\theta) - \frac{1}{2}\right)(\sin\theta - \cos\theta) \geq 0$$
is satisfied.