Let $f$ be a function that is continuous on the closed interval $[ 2,4 ]$ with $f ( 2 ) = 10$ and $f ( 4 ) = 20$. Which of the following is guaranteed by the Intermediate Value Theorem?
(A) $f ( x ) = 13$ has at least one solution in the open interval $( 2,4 )$.
(B) $f ( 3 ) = 15$
(C) $f$ attains a maximum on the open interval $( 2,4 )$.
(D) $f ^ { \prime } ( x ) = 5$ has at least one solution in the open interval $( 2,4 )$.
(E) $f ^ { \prime } ( x ) > 0$ for all $x$ in the open interval $( 2,4 )$.

The Church of Saint Francis of Assisi, a modernist architectural work by Oscar Niemeyer, located at Pampulha Lake, in Belo Horizonte, has parabolic vaults. Figure 2 provides a front view of one of the vaults, with hypothetical measurements to simplify the calculations.
What is the measure of the height H, in meters, indicated in Figure 2?
(A) $\frac{16}{3}$
(B) $\frac{31}{5}$
(C) $\frac{25}{4}$
(D) $\frac{25}{3}$
(E) $\frac{75}{2}$

The figure on the right shows a circle with center at the origin O and radius 1, and the graph of a quadratic function $y = f ( x )$ passing through the point $( 0 , - 1 )$ on the coordinate plane. The equation $$\frac { 1 } { f ( x ) + 1 } - \frac { 1 } { f ( x ) - 1 } = \frac { 2 } { x ^ { 2 } }$$ has how many distinct real roots $x$? [3 points]
(1) 2
(2) 3
(3) 4
(4) 5
(5) 6

As shown in the figure, the graph of the cubic function $y = f ( x )$ is tangent to the $x$-axis at point $\mathrm { P } ( 2,0 )$ and meets the graph of the linear function $y = g ( x )$ only at point P. When $1 < f ( 0 ) < g ( 0 )$, what is the number of real roots of the equation $$f ( x ) + g ( x ) = \frac { 1 } { f ( x ) } + \frac { 1 } { g ( x ) }$$ ? [4 points]
(1) 7
(2) 6
(3) 5
(4) 4
(5) 3

A cubic function $f ( x )$ with leading coefficient 1 satisfies $f ( - x ) = - f ( x )$ for all real numbers $x$. When the equation $| f ( x ) | = 2$ has exactly 4 distinct real roots, what is the value of $f ( 3 )$? [4 points]
(1) 12
(2) 14
(3) 16
(4) 18
(5) 20

The graphs of the quadratic function $y = f ( x )$ and the cubic function $y = g ( x )$ are shown in the figure.
$f ( - 1 ) = f ( 3 ) = 0$, and the function $g ( x )$ has a local minimum value of $- 2$ at $x = 3$. What is the number of distinct real roots of the equation $\frac { g ( x ) + 2 } { f ( x ) } - \frac { 2 } { g ( x ) } = 1$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

As shown in the figure, the graphs of function $f ( x )$ defined on the closed interval $[ - 4,4 ]$ and function $g ( x ) = - \frac { 1 } { 2 } x + 1$ meet at three points, and the $x$-coordinates of these three points are $\alpha , \beta , 2$. The inequality $$\frac { g ( x ) } { f ( x ) } \leq 1$$ is satisfied. How many integers $x$ satisfy this inequality? (Here, $- 4 < \alpha < - 3,0 < \beta < 1$) [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

Consider the function $$f ( x ) = \begin{cases} | 5 x ( x + 2 ) | & ( x < 0 ) \\ | 5 x ( x - 2 ) | & ( x \geq 0 ) \end{cases}$$ What is the number of distinct real roots of the irrational equation $\sqrt { 4 - f ( x ) } = 1 - x$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

Find the number of integers $k$ such that the equation $2 x ^ { 3 } - 6 x ^ { 2 } + k = 0$ has exactly 2 distinct positive real roots. [3 points]

7. As shown in the figure, the graph of function $f ( x )$ is the broken line $A C B$. The solution set of the inequality $f ( x ) \geqslant \log _ { 2 } ( x + 1 )$ is [Figure]
A. $\{ x \mid - 1 < x \leqslant 0 \}$
B. $\{ x \mid - 1 \leqslant x \leqslant 1 \}$
C. $\{ x \mid - 1 < x \leqslant 1 \}$
D. $\{ x \mid - 1 < x \leqslant 2 \}$

13. The number of zeros of the function $f ( x ) = 2 \sin x \sin \left( x + \frac { \pi } { 2 } \right) - x ^ { 2 }$ is $\_\_\_\_$.

12. The number of zeros of the function $f(x) = 4\cos^2\frac{x}{2}\cos\left(\frac{\pi}{2} - x\right) - 2\sin x - |\ln(x+1)|$ is $\_\_\_\_$ .

14. If the function $f ( x ) = \left| 2 ^ { x } - 2 \right| - b$ has two zeros, then the range of the real number $b$ is $\_\_\_\_$

15. Given the function $\mathrm { f } ( \mathrm { x } ) = \left\{ \begin{array} { l l } x ^ { 3 } , & \mathrm { x } \leq \mathrm { a } , \\ \mathrm { x } ^ { 2 } , & \mathrm { x } > \mathrm { a } , \end{array} \right.$ if there exists a real number $b$ such that the function $\mathrm { g } ( \mathrm { x } ) = \mathrm { f } ( \mathrm { x } ) - \mathrm { b }$ has exactly two zeros, then the range of values for $a$ is $\_\_\_\_$.
III. Solution Questions: This section has 6 questions, for a total of 75 points. Show your work, proofs, or calculation steps in your answers.

13. Given functions $f ( x ) = | \ln x | , g ( x ) = \left\{ \begin{array} { c } 0,0 < x \leq 1 \\ \left| x ^ { 2 } - 4 \right| - 2 , x > 1 \end{array} \right.$, then the number of real roots of the equation $| f ( x ) + g ( x ) | = 1$ is $\_\_\_\_$.

8. Given the function $f ( x ) = \begin{cases} 2 - | x | , & x \leq 2 \\ ( x - 2 ) ^ { 2 } , & x > 2 \end{cases}$, and function $g ( x ) = 3 - f ( 2 - x )$, then the number of intersections of the graphs of $y = f(x)$ and $y = g(x)$ is
(A) 2
(B) 3
(C) 4
(D) 5

II. Fill-in-the-Blank Questions: This section has 6 questions, each worth 5 points, for a total of 30 points.

Given the function $F(x) = \left\{\begin{array}{l}2 - |x|, \quad x \leq 2 \\ (x - 2)^2, \quad x > 2\end{array}\right.$ and function $g(x) = b - f(2 - x)$, where $b \in \mathbb{R}$. If the function $y = f(x) - g(x)$ has exactly 4 zeros, then the range of $b$ is
(A) $\left(\frac{7}{4}, +\infty\right)$
(B) $\left(-\infty, \frac{7}{4}\right)$
(C) $\left(0, \frac{7}{4}\right)$
(D) $\left(\frac{7}{4}, 2\right)$

Given the function $f ( x ) = \left\{ \begin{array} { l l } \mathrm { e } ^ { x } , & x \leqslant 0 , \\ \ln x , & x > 0 , \end{array} \right.$ and $g ( x ) = f ( x ) + x + a$. If $g ( x )$ has 2 zeros, then the range of $a$ is
A. $( 0 , + \infty )$
B. $[ 0 , + \infty )$
C. $[ - 1 , + \infty )$
D. $[ 1 , + \infty )$

Let $f(x)$ be the function obtained by shifting $y = \cos\left(2x + \frac{\pi}{4}\right)$ to the left by $\frac{\pi}{6}$ units. The number of intersection points of $y = f(x)$ and $y = \frac{1}{2}x - \frac{1}{2}$ is
A. $1$
B. $2$
C. $3$
D. $4$

Let $f ( x ) = a ( x + 1 ) ^ { 2 } - 1 , g ( x ) = \cos x + 2 a x$. When $x \in ( - 1,1 )$, the curves $y = f ( x )$ and $y = g ( x )$ have exactly one intersection point. Then $a =$
A. $- 1$
B. $\frac { 1 } { 2 }$
C. 1
D. 2

Deduce that for all $x \in [0, 1[$
$$1 + \exp\left(\frac{x^{2}}{2}\right)(1 - x)^{x - 1} \leqslant \frac{4}{2 - x}$$

Deduce that for all $x \in [0, 1[$
$$1 + \exp\left(\frac{x^{2}}{2}\right)(1 - x)^{x - 1} \leqslant \frac{4}{2 - x}$$

Let $f(x) = xe^x$, and let $V$ and $W$ denote the inverses of $f|_{]-\infty,-1]}$ and $f|_{[-1,+\infty[}$ respectively. For a real parameter $m$, we consider the inequality with unknown $x \in \mathbb { R }$
$$x \mathrm { e } ^ { x } \leqslant m \tag{I.2}$$
Using the functions $V$ and $W$, determine, according to the values of $m$, the solutions of (I.2). Illustrate graphically the different cases.

Let $V$ and $W$ denote the inverses of $f|_{]-\infty,-1]}$ and $f|_{[-1,+\infty[}$ respectively, where $f(x) = xe^x$. For non-zero real parameters $a$ and $b$, we consider the equation with unknown $x \in \mathbb { R }$
$$\mathrm { e } ^ { a x } + b x = 0 \tag{I.3}$$
Determine, according to the values of $a$ and $b$, the number of solutions of (I.3). Explicitly express the possible solutions using the functions $V$ and $W$.

Find the number of intersection points of $y = \log x$ and $y = x^2$.