O valor de $\sin(30^\circ) + \cos(60^\circ) + \tan(45^\circ)$ é
(A) 1 (B) $\dfrac{3}{2}$ (C) 2 (D) $\dfrac{5}{2}$ (E) 3

QUESTION 153
The value of $\sin 30^\circ + \cos 60^\circ$ is
(A) $\frac{1}{2}$
(B) $\frac{\sqrt{2}}{2}$
(C) 1
(D) $\frac{\sqrt{3}}{2}$
(E) $\sqrt{2}$

QUESTION 166
The value of $\tan 45^\circ + \cos 0^\circ$ is
(A) 1
(B) 2
(C) $\sqrt{2}$
(D) $1 + \sqrt{2}$
(E) $2\sqrt{2}$

Rays of sunlight are hitting the surface of a lake forming an angle $x$ with its surface, as shown in the figure.
Under certain conditions, one can assume that the light intensity of these rays, on the lake surface, is given approximately by $I(x) = K \cdot \sin(x)$, where $k$ is a constant, and assuming that $x$ is between $0^{\circ}$ and $90^{\circ}$.
When $x = 30^{\circ}$, the light intensity is reduced to what percentage of its maximum value?
(A) $33\%$
(B) $50\%$
(C) $57\%$
(D) $70\%$
(E) $86\%$

In a right triangle, $\sin(\theta) = \dfrac{3}{5}$. What is the value of $\cos(\theta)$?
(A) $\dfrac{1}{5}$
(B) $\dfrac{2}{5}$
(C) $\dfrac{3}{5}$
(D) $\dfrac{4}{5}$
(E) $\dfrac{5}{4}$

The value of $\cos(60^\circ) + \sin(30^\circ)$ is:
(A) 0
(B) $\dfrac{1}{2}$
(C) 1
(D) $\dfrac{3}{2}$
(E) 2

Find the value of the following sum of 100 terms. (Possible hint: also consider the same sum with $\sin^{2}$ instead of $\cos^{2}$.)
$$\cos^{2}\left(\frac{\pi}{101}\right) + \cos^{2}\left(\frac{2\pi}{101}\right) + \cos^{2}\left(\frac{3\pi}{101}\right) + \cdots + \cos^{2}\left(\frac{99\pi}{101}\right) + \cos^{2}\left(\frac{100\pi}{101}\right)$$

Statements
(13) As $x \rightarrow - \infty$ the function $\cos \left( e ^ { x } \right)$ tends to a finite limit. (14) As $x \rightarrow \infty$ the function $\cos \left( e ^ { x } \right)$ changes sign infinitely many times. (15) As $x \rightarrow \infty$, the function $\sin ( \ln ( x ) )$ tends to a finite limit. (16) $\sin ( \ln ( x ) )$ changes sign only finitely many times as $x$ goes towards 0 from 1.

For $\theta$ satisfying $\frac { \pi } { 2 } < \theta < \pi$ and $\sin \theta = \frac { \sqrt { 21 } } { 7 }$, what is the value of $\tan \theta$? [2 points]
(1) $- \frac { \sqrt { 3 } } { 2 }$
(2) $- \frac { \sqrt { 3 } } { 4 }$
(3) 0
(4) $\frac { \sqrt { 3 } } { 4 }$
(5) $\frac { \sqrt { 3 } } { 2 }$

What is the maximum value of the function $f ( x ) = 4 \cos x + 3$? [3 points]
(1) 6
(2) 7
(3) 8
(4) 9
(5) 10

For $\theta$ satisfying $\frac{3}{2}\pi < \theta < 2\pi$ and $\sin(-\theta) = \frac{1}{3}$, find the value of $\tan\theta$. [3 points]
(1) $-\frac{\sqrt{2}}{2}$
(2) $-\frac{\sqrt{2}}{4}$
(3) $-\frac{1}{4}$
(4) $\frac{1}{4}$
(5) $\frac{\sqrt{2}}{4}$

The function $f(x) = a\cos bx + 3$ is defined on the closed interval $[0, 2\pi]$ and has a maximum value of 13 at $x = \frac{\pi}{3}$. For the ordered pair $(a, b)$ of two natural numbers $a$ and $b$ satisfying this condition, what is the minimum value of $a + b$? [4 points]
(1) 12
(2) 14
(3) 16
(4) 18
(5) 20

Calculate the exact value of: $\cos \left( \frac { \pi } { 11 } \right)$

3. As shown in the figure, the water depth change curve of a certain port from 6 to 18 o'clock approximately satisfies the function $y = 3 \sin \left( \frac { \pi } { 6 } x + \varphi \right) + k$. Based on this function, the maximum water depth (in meters) during this period is
A. 5
B. 6
C. 8
D. 10 [Figure]

3. Execute the program flowchart shown in the figure, the output value of $S$ is
A. $- \frac { \sqrt { 3 } } { 2 }$ B. $\frac { \sqrt { 3 } } { 2 }$ C. $- \frac { 1 } { 2 }$ D. $\frac { 1 } { 2 }$ [Figure]

4. Among the following functions, the one with minimum positive period $\pi$ and whose graph is symmetric about the origin is
A. $\mathrm { y } = \cos \left( 2 x + \frac { \pi } { 2 } \right)$
B. $y = \sin \left( 2 x + \frac { \pi } { 2 } \right)$
C. $y = \sin 2 x + \cos 2 x$
D. $y = \sin x + \cos x$

10. Let $x \in \mathbf{R}$ and $[x]$ denote the greatest integer not exceeding $x$. If there exists a real number $t$ such that $[t] = 1$, $[t^2] = 2$, $\ldots$, $[t^n] = n$ all hold simultaneously, then the maximum value of the positive integer $n$ is
A. 3
B. 4
C. 5
D. 6
II. Fill-in-the-Blank Questions: This section has 6 questions. Candidates must answer 5 of them, each worth 5 points, for a total of 25 points. Write your answers in the corresponding positions on the answer sheet. Answers in wrong positions, illegible writing, or ambiguous answers will receive no credit.

(A) Compulsory Questions (Questions 11-14)

11. The function $f ( x ) = \sin ^ { 2 } x + \sin x \cos x + 1$ has minimum positive period $\_\_\_\_$ and minimum value $\_\_\_\_$.

11. The minimum positive period of the function $f ( x ) = \sin ^ { 2 } x + \sin x \cos x + 1$ is $\_\_\_\_$ , and the decreasing interval is $\_\_\_\_$ .

14. Given the function $f ( x ) = \sin \omega x + \cos \omega x ( \omega > 0 ) , x \in \mathbb{R}$. If the function $f ( x )$ is monotonically increasing on the interval $( - \omega , \omega )$, and the graph of $f ( x )$ is symmetric about the line $x = \omega$, then the value of $\omega$ is $\_\_\_\_$.

III. Solution Questions: This section has 6 questions, for a total of 80 points.

Given the function $f ( x ) = \sin x - 2 \sqrt { 3 } \sin ^ { 2 } \frac { x } { 2 }$\n(I) Find the minimum positive period of $f ( x )$;\n(II) Find the minimum value of $f ( x )$ on the interval $\left[ 0 , \frac { 2 \pi } { 3 } \right]$.

15. (This question is worth 13 points) Given the function $f ( x ) = \sqrt { 2 } \sin \frac { x } { 2 } \cos \frac { x } { 2 } - \sqrt { 2 } \sin ^ { 2 } \frac { x } { 2 }$. (I) Find the minimum positive period of $f ( x )$; (II) Find the minimum value of $f ( x )$ on the interval $[ - \pi , 0 ]$.

Given the function $f(x) = \sin^2 x - \sin^2\left(x - \frac{\pi}{6}\right)$, $x \in \mathbb{R}$.
(I) Find the minimum positive period of $f(x)$;
(II) Find the maximum and minimum values of $f(x)$ on the interval $\left[-\frac{\pi}{3}, \frac{\pi}{4}\right]$.

16. (14 points) In $\triangle A B C$ , the sides opposite to angles $\mathrm { A } , \mathrm { B }$ , C are $a , b , c$ respectively. Given that $\tan \left( \frac { \pi } { 4 } + \mathrm { A } \right) = 2$ .
(1) Find the value of $\frac { \sin 2 A } { \sin 2 A + \cos ^ { 2 } A }$ ;
(2) If $\mathrm { B } = \frac { \pi } { 4 } , a = 3$ , find the area of $\triangle A B C$ .

Given the function $\mathrm { f } ( \mathrm { x } ) = \frac { 1 } { 2 } \sin 2 \mathrm { x } - \sqrt { 3 } \cos ^ { 2 } x$ .
(I) Find the minimum positive period and minimum value of $\mathrm { f } ( \mathrm { x } )$;
(II) The graph of function $\mathrm { f } ( \mathrm { x } )$ is transformed by stretching each point's horizontal coordinate to twice the original length while keeping the vertical coordinate unchanged, resulting in the graph of function $\mathrm { g } ( \mathrm { x } )$. When $\mathrm { x } \in \left[ \frac { \pi } { 2 } , \pi \right]$, find the range of $\mathrm { g } ( \mathrm { x } )$.