18. A student uses the ``five-point method'' to draw the graph of the function $f ( \mathrm { x } ) = \mathrm { A } \sin ( \omega \mathrm { x } + \varphi ) \left( \omega > 0 , \varphi < \frac { \pi } { 2 } \right)$ during a certain period, creates a table and fills in partial data as follows:

$\omega \mathrm { x } + \varphi$	0	$\frac { \pi } { 2 }$	$\pi$	$\frac { 3 \pi } { 2 }$	$2 \pi$
x		$\frac { \pi } { 3 }$		$\frac { 5 \pi } { 6 }$
$\mathrm {~A} \sin ( \omega \mathrm { x } + \varphi )$	0	5		- 5	0

(I) Please complete the above data, fill in the corresponding positions on the answer sheet, and directly write the analytical expression of the function $f ( \mathrm { x } )$; (II) Shift all points on the graph of $y = f ( \mathrm { x } )$ to the left by $\frac { \pi } { 6 }$ units to obtain the graph of $y = g ( \mathrm { x } )$. Find the center of symmetry of the graph of $y = g ( \mathrm { x } )$ that is closest to the origin O.

19. The graph of function $f ( x )$ is obtained from the graph of function $g ( x ) = \cos x$ by the following transformations: first, stretch the vertical coordinates of all points on the graph of $g ( x )$ to 2 times their original length (horizontal coordinates unchanged), then shift the resulting graph to the right by $\frac { \pi } { 2 }$ units.
(1) Find the analytical expression of function $f ( x )$ and the equation of its axis of symmetry;
(2) Given that the equation $f ( x ) + g ( x ) = m$ has two distinct solutions $a$ and $b$ in $[ 0,2 \pi )$:
1) Find the range of the real number $m$;
2) Prove that $\cos ( a - b ) = \frac { 2 m ^ { 2 } } { 5 } - 1$.

The smallest positive period of the function $f(x) = \sin\left(2x + \frac{\pi}{3}\right)$ is
A. $4\pi$
B. $2\pi$
C. $\pi$
D. $\frac{\pi}{2}$

The partial graph of the function $y = \frac{\sin 2x}{1 - \cos x}$ is approximately (see options A, B, C, D in the figures provided).

14. The maximum value of the function $f ( x ) = \sin ^ { 2 } x + \sqrt { 3 } \cos x - \frac { 3 } { 4 } \left( x \in \left[ 0 , \frac { \pi } { 2 } \right] \right)$ is ______

5. The number of zeros of the function $f ( x ) = 2 \sin x - \sin 2 x$ on $[ 0,2 \pi ]$ is
A. 2
B. 3
C. 4
D. 5

6. The monotonically increasing interval of the function $f ( x ) = \cos \left( 3 x + \frac { \pi } { 2 } \right)$ is
A. $\left[ \frac { \pi } { 6 } + \frac { 2 k \pi } { 3 } , \frac { \pi } { 2 } + \frac { 2 k \pi } { 3 } \right] ( k \in \mathbb{Z} )$
[Figure]
Front View
[Figure]
Top View
B. $\left[ \frac { \pi } { 6 } + \frac { k \pi } { 3 } , \frac { \pi } { 2 } + \frac { k \pi } { 3 } \right] ( k \in \mathbb{Z} )$
C. $\left[ \frac { \pi } { 6 } + \frac { k \pi } { 3 } , \frac { \pi } { 6 } + \frac { k \pi } { 3 } \right] ( k \in \mathbb{Z} )$
D. $\left[ - \frac { \pi } { 6 } + \frac { 2 k \pi } { 3 } , \frac { \pi } { 6 } + \frac { 2 k \pi } { 3 } \right] ( k \in \mathbb{Z} )$

8. If $x _ { 1 } = \frac { \pi } { 4 } , x _ { 2 } = \frac { 3 \pi } { 4 }$ are two adjacent extreme points of the function $f ( x ) = \sin \omega x ( \omega > 0 )$, then $\omega =$
A. 2
B. $\frac { 3 } { 2 }$
C. 1
D. $\frac { 1 } { 2 }$

Among the following functions, which one has period $\frac { \pi } { 2 }$ and is monotonically increasing on the interval $\left( \frac { \pi } { 4 } , \frac { \pi } { 2 } \right)$?
A．$f ( x ) = | \cos 2 x |$
B．$f ( x ) = | \sin 2 x |$
C．$f ( x ) = \cos | x |$
D．$f ( x ) = \sin | x |$

The minimum positive period of the function $f ( x ) = \sin ^ { 2 } 2 x$ is $\_\_\_\_$.

9. Among the following functions, which one has period $\frac { \pi } { 2 }$ and is monotonically increasing on the interval $\left( \frac { \pi } { 4 } , \frac { \pi } { 2 } \right)$?
A. $f ( x ) = | \cos 2 x |$
B. $f ( x ) = | \sin 2 x |$
C. $f ( x ) = \cos | x |$
D. $f ( x ) = \sin | x |$

11. Regarding the function $f ( x ) = \sin | x | + | \sin x |$, there are four conclusions:
(1) $f ( x )$ is an even function
(2) $f ( x )$ is monotonically increasing on the interval $\left( \frac { \pi } { 2 } , \pi \right)$
(3) $f ( x )$ has 4 zeros on $[ - \pi , \pi ]$
(4) The maximum value of $f ( x )$ is 2
The numbers of all correct conclusions are
A. (1)(2)(4)
B. (2)(4)
C. (1)(4)
D. (1)(3)

11. Regarding the function $f ( x ) = \sin | x | + | \sin x |$ , there are four conclusions:
(1) $f ( x )$ is an even function
(2) $f ( x )$ is monotonically increasing on the interval $\left( \frac { \pi } { 2 } , \pi \right)$
(3) $f ( x )$ has 4 zeros on $[ - \pi , \pi ]$
(4) The maximum value of $f ( x )$ is 2
The numbers of all correct conclusions are
A. (1)(2)(4)
B. (2)(4)
C. (1)(4)
D. (1)(3)

12. Let $f ( x ) = \sin \left( \omega x + \frac { \pi } { 5 } \right) ( \omega > 0 )$ . Given that $f ( x )$ has exactly 5 zeros on $[ 0,2 \pi ]$ , consider the following 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)
II. Fill-in-the-Blank Questions: This section has 4 questions, each worth 5 points, for a total of 20 points.

The function $f ( x ) = \cos \left( \omega x + \frac { \pi } { 6 } \right)$ has a graph on $[ - \pi , \pi ]$ as shown. The minimum positive period of $f ( x )$ is
A. $\frac { 10 \pi } { 9 }$
B. $\frac { 7 \pi } { 6 }$
C. $\frac { 4 \pi } { 3 }$
D. $\frac { 3 \pi } { 2 }$

Let the function $f ( x ) = \cos \left( \omega x + \frac { \pi } { 6 } \right)$ on $[ - \pi , \pi ]$ have a graph as shown. Then the smallest positive period of $f ( x )$ is
A. $\frac { 10 \pi } { 9 }$
B. $\frac { 7 \pi } { 6 }$
C. $\frac { 4 \pi } { 3 }$
D. $\frac { 3 \pi } { 2 }$

Given the function $f ( x ) = \sin x + \frac { 1 } { \sin x }$, then
A. the minimum value of $f ( x )$ is 2
B. the graph of $f ( x )$ is symmetric about the $y$-axis
C. the graph of $f ( x )$ is symmetric about the line $x = \pi$
D. the graph of $f ( x )$ is symmetric about the line $x = \frac { \pi } { 2 }$

Regarding the function $f ( x ) = \sin x + \frac { 1 } { \sin x }$ , there are four propositions:
(1) The graph of $f ( x )$ is symmetric about the $y$-axis.
(2) The graph of $f ( x )$ is symmetric about the origin.
(3) The graph of $f ( x )$ is symmetric about the line $x = \frac { \pi } { 2 }$ .
(4) The minimum value of $f ( x )$ is 2 .
The sequence numbers of all true propositions are $\_\_\_\_$ .

4. A

Solution: Based on the graph and properties of translation, the function $f ( x )$ is monotonically increasing on $\left( - \frac { \pi } { 3 } , \frac { 2 \pi } { 3 } \right)$ and monotonically decreasing on $\left( - \frac { 2 \pi } { 3 } , \frac { 5 \pi } { 3 } \right)$.

15. The function $f ( x ) = 2 \cos ( \omega x + \varphi )$ has a partial graph shown in the figure. Then $f \left( \frac { \pi } { 2 } \right) =$ $\_\_\_\_$ . [Figure]

16. Given that the function $f(x) = 2\cos(\omega x + \varphi)$ has a partial graph as shown in the figure, then the minimum positive integer $x$ satisfying the condition $\left(f(x) - f\left(-\frac{7\pi}{4}\right)\right)\left(f(x) - f\left(\frac{4\pi}{3}\right)\right) > 0$ is $\_\_\_\_$.

III. Solution Questions (70 points total. Show all work, proofs, and calculations. Questions 17-21 are required for all students. Questions 22-23 are optional; students should answer according to requirements.)

6. Let the function $f ( x ) = \sin \left( \omega x + \frac { \pi } { 4 } \right) + b$ ( $\omega > 0$ ) have minimum positive period $T$. If $\frac { 2 \pi } { 3 } < T < \pi$ and the graph of $y = f ( x )$ is symmetric about the point $\left( \frac { 3 \pi } { 2 } , 2 \right)$, then $f \left( \frac { \pi } { 2 } \right) =$
A. $1$
B. $\frac { 3 } { 2 }$
C. $\frac { 5 } { 2 }$
D. $3$

Given $f ( x ) = \sin \omega x , f \left( x _ { 1 } \right) = - 1 , f \left( x _ { 2 } \right) = 1 , \left| x _ { 1 } - x _ { 2 } \right| _ { \text {min} } = \frac { \pi } { 2 }$, then $\omega =$ \_\_\_\_

When $x \in [ 0,2 \pi ]$ , the number of intersection points of the curves $y = \sin x$ and $y = 2 \sin \left( 3 x - \frac { \pi } { 6 } \right)$ is
A. $3$
B. $4$
C. $6$
D. $8$

For functions $f ( x ) = \sin 2 x$ and $g ( x ) = \sin \left( 2 x - \frac { \pi } { 4 } \right)$, the correct statements are
A. $f ( x )$ and $g ( x )$ have the same zeros
B. $f ( x )$ and $g ( x )$ have the same maximum value
C. $f ( x )$ and $g ( x )$ have the same minimum positive period
D. The graphs of $f ( x )$ and $g ( x )$ have the same axes of symmetry