Exercise 1 (5 points)
The plane is equipped with an orthogonal coordinate system $(\mathrm{O}, \mathrm{I}, \mathrm{J})$.
We denote by $\Gamma$ the representative curve of the function $g$ defined on the interval $]0; 1]$ by $g(x) = \ln x$. Let $a$ be a real number in the interval $]0; 1]$. We denote by $M_a$ the point on the curve $\Gamma$ with abscissa $a$ and $d_a$ the tangent line to the curve $\Gamma$ at the point $M_a$. This line $d_a$ intersects the $x$-axis at point $N_a$ and the $y$-axis at point $P_a$. We are interested in the area of triangle $\mathrm{O}N_aP_a$ as the real number $a$ varies in the interval $]0; 1]$.
In this question, we study the particular case where $a = 0.2$.
a. Determine graphically an estimate of the area of triangle $\mathrm{O}N_{0.2}P_{0.2}$ in square units.
b. Determine an equation of the tangent line $d_{0.2}$.
c. Calculate the exact value of the area of triangle $\mathrm{O}N_{0.2}P_{0.2}$.

Using the steps below, find the value of $x ^ { 2012 } + x ^ { - 2012 }$, where $x + x ^ { - 1 } = \frac { \sqrt { 5 } + 1 } { 2 }$. a) For any real $r$, show that $\left| r + r ^ { - 1 } \right| \geq 2$. What does this tell you about the given $x$? b) Show that $\cos \left( \frac { \pi } { 5 } \right) = \frac { \sqrt { 5 } + 1 } { 4 }$, e.g. compare $\sin \left( \frac { 2 \pi } { 5 } \right)$ and $\sin \left( \frac { 3 \pi } { 5 } \right)$. c) Combine conclusions of parts a and b to express $x$ and therefore the desired quantity in a suitable form.

Answer the following questions
(a) Evaluate $$\lim_{x \rightarrow 0^{+}} \left( x^{x^{x}} - x^{x} \right)$$ (b) Let $A = \frac{2\pi}{9}$, i.e., $A = 40$ degrees. Calculate the following $$1 + \cos A + \cos 2A + \cos 4A + \cos 5A + \cos 7A + \cos 8A$$ (c) Find the number of solutions to $e^{x} = \frac{x}{2017} + 1$.

(Calculus) When $\sin \alpha = \frac { 3 } { 4 }$, what is the value of $\cos 2 \alpha$? [3 points]
(1) $- \frac { 1 } { 32 }$
(2) $- \frac { 1 } { 16 }$
(3) $- \frac { 1 } { 8 }$
(4) $- \frac { 1 } { 4 }$
(5) $- \frac { 1 } { 2 }$

[Calculus] When $\tan \theta = - \sqrt { 2 }$, what is the value of $\sin \theta \tan 2 \theta$? (where $\frac { \pi } { 2 } < \theta < \pi$ ) [3 points]
(1) $\frac { 2 \sqrt { 3 } } { 3 }$
(2) $\sqrt { 3 }$
(3) $\frac { 4 \sqrt { 3 } } { 3 }$
(4) $\frac { 5 \sqrt { 3 } } { 3 }$
(5) $2 \sqrt { 3 }$

As shown in the figure, there is an equilateral triangle ABC and a circle O with diameter AC on a plane. Point D on segment BC is determined such that $\angle \mathrm { DAB } = \frac { \pi } { 15 }$. When point X moves on circle O, let P be the point where the dot product $\overrightarrow { \mathrm { AD } } \cdot \overrightarrow { \mathrm { CX } }$ of the two vectors $\overrightarrow { \mathrm { AD } } , \overrightarrow { \mathrm { CX } }$ is minimized. If $\angle \mathrm { ACP } = \frac { q } { p } \pi$, find the value of $p + q$. (Note: $p$ and $q$ are coprime natural numbers.) [4 points]

On the coordinate plane, let $\theta _ { 1 }$ be the acute angle that the line $y = m x ( 0 < m < \sqrt { 3 } )$ makes with the $x$-axis, and let $\theta _ { 2 }$ be the acute angle that the line $y = m x$ makes with the line $y = \sqrt { 3 } x$. What is the value of $m$ that maximizes $3 \sin \theta _ { 1 } + 4 \sin \theta _ { 2 }$? [4 points]
(1) $\frac { \sqrt { 3 } } { 6 }$
(2) $\frac { \sqrt { 3 } } { 7 }$
(3) $\frac { \sqrt { 3 } } { 8 }$
(4) $\frac { \sqrt { 3 } } { 9 }$
(5) $\frac { \sqrt { 3 } } { 10 }$

When $\sin \theta = \frac { 1 } { 3 }$, what is the value of $\sin 2 \theta$? (Given that $0 < \theta < \frac { \pi } { 2 }$.) [2 points]
(1) $\frac { 7 \sqrt { 2 } } { 18 }$
(2) $\frac { 4 \sqrt { 2 } } { 9 }$
(3) $\frac { \sqrt { 2 } } { 2 }$
(4) $\frac { 5 \sqrt { 2 } } { 9 }$
(5) $\frac { 11 \sqrt { 2 } } { 18 }$

When $\tan \theta = \frac { \sqrt { 5 } } { 5 }$, what is the value of $\cos 2 \theta$? [2 points]
(1) $\frac { \sqrt { 2 } } { 3 }$
(2) $\frac { \sqrt { 3 } } { 3 }$
(3) $\frac { 2 } { 3 }$
(4) $\frac { \sqrt { 5 } } { 3 }$
(5) $\frac { \sqrt { 6 } } { 3 }$

As shown in the figure, in triangle ABC with $\overline { \mathrm { AB } } = 5 , \overline { \mathrm { AC } } = 2 \sqrt { 5 }$, let D be the foot of the perpendicular from vertex A to segment BC.
For point E that divides segment AD internally in the ratio $3 : 1$, we have $\overline { \mathrm { EC } } = \sqrt { 5 }$. If $\angle \mathrm { ABD } = \alpha , \angle \mathrm { DCE } = \beta$, what is the value of $\cos ( \alpha - \beta )$? [4 points]
(1) $\frac { \sqrt { 5 } } { 5 }$
(2) $\frac { \sqrt { 5 } } { 4 }$
(3) $\frac { 3 \sqrt { 5 } } { 10 }$
(4) $\frac { 7 \sqrt { 5 } } { 20 }$
(5) $\frac { 2 \sqrt { 5 } } { 5 }$

In an isosceles triangle ABC with $\overline { \mathrm { AB } } = \overline { \mathrm { AC } }$, let $\angle \mathrm { A } = \alpha , \angle \mathrm { B } = \beta$. If $\tan ( \alpha + \beta ) = - \frac { 3 } { 2 }$, what is the value of $\tan \alpha$? [3 points]
(1) $\frac { 5 } { 2 }$
(2) $\frac { 12 } { 5 }$
(3) $\frac { 23 } { 10 }$
(4) $\frac { 11 } { 5 }$
(5) $\frac { 21 } { 10 }$

If $\tan a = \frac { 1 } { 3 } , \tan ( a + b ) = \frac { 1 } { 2 }$, then $\tan b =$
(A) $\frac { 1 } { 7 }$
(B) $\frac { 1 } { 6 }$
(C) $\frac { 5 } { 7 }$
(D) $\frac { 5 } { 6 }$

8. Given $\tan \alpha = - 2 , \tan ( \alpha + \beta ) = \frac { 1 } { 7 }$, then the value of $\tan \beta$ is $\_\_\_\_$ .

9. If $\tan \alpha = 2 \tan \frac { \pi } { 5 }$, then $\frac { \cos \left( \alpha - \frac { 3 \pi } { 10 } \right) } { \sin \left( \alpha - \frac { \pi } { 5 } \right) } =$
A. $1$
B. $2$
C. $3$
D. $4$

12. $\sin 15 ^ { \circ } + \sin 75 ^ { \circ } = \_\_\_\_$.

13. Given $\sin \alpha + 2 \cos \alpha = 0$, the value of $2 \sin \alpha \cos \alpha - \cos ^ { 2 } \alpha$ is \_\_\_\_.

15. Given $w > 0$, the two closest intersection points of the graphs of $y = 2 \sin w x$ and $y = 2 \cos w x$ have a distance of $2 \sqrt { 3 }$. Then $w =$ $\_\_\_\_$.
III. Solution Questions: This section has 6 questions, for a total of 75 points. Solutions should include written explanations, proofs, or calculation steps.

19. (This question is worth 12 points)

Let $A , B , C$ be the interior angles of $\triangle A B C$. $\tan A , \tan B$ are the two real roots of the equation $x ^ { 2 } + \sqrt { 3 } p x - p + 1 = 0 ( p \in R )$. (1) Find the size of $C$; (2) If $A B = 3 , A C = \sqrt { 6 }$, find the value of $p$.

19. As shown in the figure, $A$, $B$, $C$, $D$ are the four interior angles of quadrilateral $ABCD$.
(1) Prove: $\tan \frac { A } { 2 } = \frac { 1 - \cos A } { \sin A }$;
(2) If $A + C = 180 ^ { \circ }$, $AB = 6$, $BC = 3$, $CD = 4$, $AD = 5$, find the value of $\tan \frac { A } { 2 } + \tan \frac { B } { 2 } + \tan \frac { C } { 2 } + \tan \frac { D } { 2 }$. [Figure]

If $\sin \alpha = \frac { 1 } { 3 }$, then $\cos 2 \alpha =$
A. $\frac { 8 } { 9 }$
B. $\frac { 7 } { 9 }$
C. $- \frac { 7 } { 9 }$
D. $- \frac { 8 } { 9 }$

Given the function $f ( x ) = 2 \cos ^ { 2 } x - \sin ^ { 2 } x + 2$, then
A. The minimum positive period of $f ( x )$ is $\pi$, and the maximum value is 3
B. The minimum positive period of $f ( x )$ is $\pi$, and the maximum value is 4
C. The minimum positive period of $f ( x )$ is $2 \pi$, and the maximum value is 3
D. The minimum positive period of $f ( x )$ is $2 \pi$, and the maximum value is 4

If $f ( x ) = \cos x - \sin x$ is decreasing on $[ 0 , a ]$, then the maximum value of $a$ is
A. $\frac { \pi } { 4 }$
B. $\frac { \pi } { 2 }$
C. $\frac { 3 \pi } { 4 }$
D. $\pi$

If $f ( x ) = \cos x - \sin x$ is an even function on $[ - a , a ]$, then the maximum value of $a$ is
A. $\frac { \pi } { 4 }$
B. $\frac { \pi } { 2 }$
C. $\frac { 3 \pi } { 4 }$
D. $\pi$

The vertex of angle $\alpha$ is at the origin, its initial side coincides with the positive $x$-axis, and two points on its terminal side are $A ( 1 , a )$ and $B ( 2 , b )$. If $\cos 2 \alpha = \frac { 2 } { 3 }$, then $| a - b | =$
A. $\frac { 1 } { 5 }$
B. $\frac { \sqrt { 5 } } { 5 }$
C. $\frac { 2 \sqrt { 5 } } { 5 }$
D. (incomplete)

Given $\tan \left( \alpha - \frac { 5 \pi } { 4 } \right) = \frac { 1 } { 5 }$, then $\tan \alpha = $ \_\_\_\_ .