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 $\mathrm{O}N_{0.2}P_{0.2}$.

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.

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

Given the function $f ( x ) = 2 \sin x + \sin 2 x$, the minimum value of $f ( x )$ is $\_\_\_\_$

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. $\pi$

If the triangle $P Q R$ varies, then the minimum value of
$$\cos ( P + Q ) + \cos ( Q + R ) + \cos ( R + P )$$
is
[A] $- \frac { 5 } { 3 }$
[B] $- \frac { 3 } { 2 }$
[C] $\frac { 3 } { 2 }$
[D] $\frac { 5 } { 3 }$

If $A > 0 , B > 0$ and $A + B = \frac { \pi } { 6 }$, then the minimum positive value of $( \tan A + \tan B )$ is :
(1) $\sqrt { 3 } - \sqrt { 2 }$
(2) $4 - 2 \sqrt { 3 }$
(3) $\frac { 2 } { \sqrt { 3 } }$
(4) $2 - \sqrt { 3 }$

Let the range of the function $f ( x ) = 6 + 16 \cos x \cdot \cos \left( \frac { \pi } { 3 } - x \right) \cdot \cos \left( \frac { \pi } { 3 } + x \right) \cdot \sin 3 x \cdot \cos 6 x , x \in \mathbf { R }$ be $[ \alpha , \beta ]$. Then the distance of the point $( \alpha , \beta )$ from the line $3 x + 4 y + 12 = 0$ is :
(1) 11
(2) 8
(3) 10
(4) 9

Consider vectors $\vec{a}$ and $\vec{b}$ on the coordinate plane satisfying $|\vec{a}| + |\vec{b}| = 9$ and $|\vec{a} - \vec{b}| = 7$. Let $|\vec{a}| = x$, where $1 < x < 8$, and let the angle between $\vec{a}$ and $\vec{b}$ be $\theta$. Using the triangle formed by vectors $\vec{a}$, $\vec{b}$, and $\vec{a} - \vec{b}$, we can express $\cos\theta$ in terms of $x$ as $\frac{c}{9x - x^2} + d$, where $c$ and $d$ are constants with $c > 0$. Let this expression be $f(x)$, with domain $\{x \mid 1 < x < 8\}$. Find $f(x)$ and its derivative. (Non-multiple choice question, 4 points)