Find the number of real solutions to the equation $x = 99 \sin ( \pi x )$.

For a natural number $n$, let $a _ { n }$ be the $n$-th smallest $x$-coordinate among the intersection points of the line $y = n$ and the graph of the function $y = \tan x$ in the first quadrant.
What is the value of $\lim _ { n \rightarrow \infty } \frac { a _ { n } } { n }$? [4 points]
(1) $\frac { \pi } { 4 }$
(2) $\frac { \pi } { 2 }$
(3) $\frac { 3 } { 4 } \pi$
(4) $\pi$
(5) $\frac { 5 } { 4 } \pi$

The number of zeros of the function $f ( x ) = \cos \left( 3 x + \frac { \pi } { 6 } \right)$ on $[ 0, \pi ]$ is $\_\_\_\_$.

Let the function $f ( x ) = \sin \left( \omega x + \frac { \pi } { 5 } \right) ( \omega > 0 )$. It is known that $f ( x )$ has exactly 5 zeros on $[ 0,2 \pi ]$. The following are four conclusions:
(1) $f ( x )$ has exactly 3 local maximum points on $( 0,2 \pi )$
(2) $f ( x )$ has exactly 2 local minimum points on $( 0,2 \pi )$
(3) $f ( x )$ is monotonically increasing on $\left( 0 , \frac { \pi } { 10 } \right)$
(4) The range of $\omega$ is $\left[ \frac { 12 } { 5 } , \frac { 29 } { 10 } \right)$
The numbers of all correct conclusions are
A. (1)(4)
B. (2)(3)
C. (1)(2)(3)
D. (1)(3)(4)

Let the function $f : \mathbb { R } \rightarrow \mathbb { R }$ be defined by
$$f ( x ) = \frac { \sin x } { e ^ { \pi x } } \frac { \left( x ^ { 2023 } + 2024 x + 2025 \right) } { \left( x ^ { 2 } - x + 3 \right) } + \frac { 2 } { e ^ { \pi x } } \frac { \left( x ^ { 2023 } + 2024 x + 2025 \right) } { \left( x ^ { 2 } - x + 3 \right) }$$
Then the number of solutions of $f ( x ) = 0$ in $\mathbb { R }$ is $\_\_\_\_$ .

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]

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

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

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 $0 < x < \frac { 3 } { 2 } \pi$ ?