11. As shown in the figure, the rectangle's edge, is the midpoint, point moves along the edge, and moves with, denote, express the sum of distances from the moving point to two points as a function of, then the graph of is approximately
[Figure]
A.
[Figure]
B.
[Figure]
C.
[Figure]
D.

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 $\_\_\_\_$ .

Let $\mathrm { f } ^ { \prime } ( \mathrm { x } )$ be the derivative of the odd function $f ( x ) ( x \in \mathbf { R } )$. Given $\mathrm { f } ( - 1 ) = 0$, and when $\mathrm { x } > 0$, $x f ^ { \prime } ( x ) - f ( x ) < 0$. Then the range of $x$ for which $f ( x ) > 0$ holds is
(A) $( - \infty , - 1 ) \cup ( 0,1 )$
(B) $( - 1,0 ) \cup ( 1 , + \infty )$
(C) $( - \infty , - 1 ) \cup ( - 1,0 )$
(D) $( 0,1 ) \cup ( 1 , + \infty )$

12. Given the function $f ( x ) = \left\{ \begin{array} { l } x ^ { 2 } , x \leq 1 \\ x + \frac { 6 } { x } - 6 , x > 1 \end{array} \right.$ , then $f [ f ( - 2 ) ] =$ $\_\_\_\_$ , and the minimum value of $f ( x )$ is $\_\_\_\_$.

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

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 $\_\_\_\_$.

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.

15. Given functions $f ( x ) = 2 ^ { x } , g ( x ) = \hat { x } ^ { 2 } + a _ { 2 }$ (where $a \in R$). For unequal real numbers $x _ { 1 } , x _ { 2 }$, let $m = \frac { f \left( x _ { 1 } \right) - f \left( x _ { 2 } \right) } { x _ { 1 } - x _ { 2 } } , n = \frac { g \left( x _ { 1 } \right) - g \left( x _ { 2 } \right) } { x _ { 1 } - x _ { 2 } }$. Consider the following propositions:
(1) For any unequal real numbers $x _ { 1 } , x _ { 2 }$, we have $m > 0$; (2) For any $a$ and any unequal real numbers $x _ { 1 } , x _ { 2 }$, we have $n > 0$;
(3) For any $a$, there exist unequal real numbers $x _ { 1 } , x _ { 2 }$ such that $m = n$; (4) For any $a$, there exist unequal real numbers $x _ { 1 } , x _ { 2 }$ such that $m = - n$. The true propositions are \_\_\_\_ (write the numbers of all true propositions).
III. Solution Questions:

15. Given functions $f ( x ) = 2 ^ { x }$ and $g ( x ) = x ^ { 2 } + a x$ (where $a \in \mathbb{R}$). For unequal real numbers $x _ { 1 }$ and $x _ { 2 }$, let $m = \frac { f \left( x _ { 1 } \right) - f \left( x _ { 2 } \right) } { x _ { 1 } - x _ { 2 } }$ and $n = \frac { g \left( x _ { 1 } \right) - g \left( x _ { 2 } \right) } { x _ { 1 } - x _ { 2 } }$.
The following propositions are given:
(1) For any unequal real numbers $x _ { 1 }$ and $x _ { 2 }$, we have $m > 0$;
(2) For any $a$ and any unequal real numbers $x _ { 1 }$ and $x _ { 2 }$, we have $n > 0$;
(3) For any $a$, there exist unequal real numbers $x _ { 1 }$ and $x _ { 2 }$ such that $m = n$;
(4) For any $a$, there exist unequal real numbers $x _ { 1 }$ and $x _ { 2 }$ such that $m = - n$. The true propositions are $\_\_\_\_$ (write out the numbers of all true propositions).

III. Solution Questions

The graph of the function $y = 2 x ^ { 2 } - e ^ { | x | }$ on $[ - 2,2 ]$ is approximately
(A), (B), (C), (D) [as shown in the figures]

The range of the function $f(x) = \sqrt{x^2 - 2x - 3}$ is
A. $[-2, 2]$
B. $[-1, 1]$
C. $[0, 4]$
D. $[1, 3]$

The graph of the function $f ( x ) = \frac { \mathrm { e } ^ { x } - \mathrm { e } ^ { - x } } { x ^ { 2 } }$ is approximately (see options A, B, C, D in the figures).

The graph of function $f ( x ) = \frac { \mathrm { e } ^ { x } - \mathrm { e } ^ { - x } } { x ^ { 2 } }$ is approximately
A. [Graph A]
B. [Graph B]
C. [Graph C]
D. [Graph D]

The graph of the function $y = - x ^ { 4 } + x ^ { 2 } + 2$ is approximately (See figures A, B, C, D in the original paper.)

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. $[ 1 , + \infty )$

5. The graph of function $f ( x ) = \frac { \sin x + x } { \cos x + x ^ { 2 } }$ on $[ - \pi , \pi ]$ is approximately
A. [Figure]
B. [Figure]
C. [Figure]
D. [Figure]

The graph of the function $y = \frac { 2 x ^ { 3 } } { 2 ^ { x } + 2 ^ { - x } }$ on $[ - 6,6 ]$ is approximately
A. [graph A]
B. [graph B]
C. [graph C]
D. [graph D]

7. The graph of the function $y = \frac { 2 x ^ { 3 } } { 2 ^ { x } + 2 ^ { - x } }$ on $[ - 6,6 ]$ is approximately
A. [Figure]
B. [Figure]
C. [Figure]
D. [Figure]

There are many beautifully shaped and meaningful curves in mathematics. The curve $C : x ^ { 2 } + y ^ { 2 } = 1 + | x | y$ is one of them (as shown in the figure). Three conclusions are given: (1) The curve $C$ passes through exactly 6 lattice points (points with both integer coordinates); (2) The distance from any point on curve $C$ to the origin does not exceed $\sqrt { 2 }$; (3) The area of the ``heart-shaped'' region enclosed by curve $C$ is less than 3. The sequence numbers of all correct conclusions are (A) (1) (B) (2) (C) (1)(2) (D) (1)(2)(3)

Let the function $f ( x ) = \ln | 2 x + 1 | - \ln | 2 x - 1 |$ . Then $f ( x )$
A． is an even function and monotonically increasing on $\left( \frac { 1 } { 2 } , + \infty \right)$
B． is an odd function and monotonically decreasing on $\left( - \frac { 1 } { 2 } , \frac { 1 } { 2 } \right)$
C． is an even function and monotonically increasing on $\left( - \infty , - \frac { 1 } { 2 } \right)$
D． is an odd function and monotonically decreasing on $\left( - \infty , - \frac { 1 } { 2 } \right)$

Let $a \in \mathbb{R}$. If there exists a function $f ( x )$ with domain $\mathbb{R}$ that satisfies both ``for any $x _ { 0 } \in \mathbb{R}$, the value of $f \left( x _ { 0 } \right)$ is either $x _ { 0 } ^ { 2 }$ or $x _ { 0 }$'' and ``the equation $f ( x ) = a$ has no real solutions'', find the range of $a$ as $\_\_\_\_$

If there exists $a \in \mathbb{R}$ with $a \neq 0$ such that for all $x \in \mathbb{R}$, the inequality $f ( x + a ) < f ( x ) + f ( a )$ always holds, then function $f ( x )$ is said to have property $P$. Given: $q _ { 1 }$: $f ( x )$ is monotonically decreasing and $f ( x ) > 0$ always holds; $q _ { 2 }$: $f ( x )$ is monotonically increasing and there exists $x _ { 0 } < 0$ such that $f \left( x _ { 0 } \right) = 0$. Which is a sufficient condition for $f ( x )$ to have property $P$? ( )
A. Only $q _ { 1 }$
B. Only $q _ { 2 }$
C. Both $q _ { 1 }$ and $q _ { 2 }$
D. Neither $q _ { 1 }$ nor $q _ { 2 }$

4. Which of the following functions is an increasing function?
A. $f ( x ) = - x$
B. $f ( x ) = \left( \frac { 2 } { 3 } \right) ^ { x }$
C. $f ( x ) = x ^ { 2 }$
D. $f ( x ) = \sqrt [ 3 ] { x }$

The graph of the function $y = \left( 3 ^ { x } - 3 ^ { - x } \right) \cos x$ on the interval $\left[ - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right]$ is approximately: (see figures A, B, C, D)