The number of integers satisfying $|x - 3| \leq 2$ is:
(A) 3
(B) 4
(C) 5
(D) 6
(E) 7

6. The sign function is defined as $\operatorname{sgn} x = \begin{cases} 1, & x > 0, \\ 0, & x = 0, \\ -1, & x < 0. \end{cases}$ Let $f(x)$ be an increasing function on $\mathbf{R}$, and $g(x) = f(x) - f(ax)$ where $a > 1$. Then
A. $\operatorname{sgn}[g(x)] = \operatorname{sgn} x$
B. $\operatorname{sgn}[g(x)] = -\operatorname{sgn} x$
C. $\operatorname{sgn}[g(x)] = \operatorname{sgn}[f(x)]$
D. $\operatorname{sgn}[g(x)] = -\operatorname{sgn}[f(x)]$

7. For $x \in R$, define the sign function $\operatorname { sgn } x = \left\{ \begin{array} { c } 1 , x > 0 \\ 0 , x = 0 \\ - 1 , x < 0 \end{array} \right.$, then
A. $ \{ x | = x | \operatorname { sgn } x \mid \}$
B. $ \{ x | = \operatorname { sgn } | x \mid \}$
C. $ \{ x | = x | \operatorname { sgn } x \}$
D. $ \{ x \mid = x \operatorname { sgn } x \}$

12. Let $f ( x ) = \ln ( 1 + | x | ) - \frac { 1 } { 1 + x ^ { 2 } }$. The range of $x$ for which $f ( x ) > f ( 2 x - 1 )$ holds is
A. $\left( \frac { 1 } { 3 } , 1 \right)$
B. $\left( - \infty , \frac { 1 } { 3 } \right) \cup ( 1 , + \infty )$
C. $\left( - \frac { 1 } { 3 } , \frac { 1 } { 3 } \right)$
D. $\left( - \infty , - \frac { 1 } { 3 } \right) \cup \left( \frac { 1 } { 3 } , + \infty \right)$
II. Fill-in-the-Blank Questions: This section contains 4 questions, 5 points each, 20 points total

16. If the minimum value of the function $f ( x ) = | x + 1 | + 2 | x - a |$ is 5, then the real number $a = $ $\_\_\_\_$ . III. Solution Questions: This section contains 6 questions, for a total of 75 points. Show your work, proofs, or calculation steps. (17) (This question is worth 13 points: part (I) is worth 5 points, part (II) is worth 8 points) Eating zongzi on Dragon Boat Festival is a traditional custom in China. A plate contains 10 zongzi: 2 with red bean paste, 3 with meat, and 5 plain. The three types of zongzi look identical. Three zongzi are randomly selected. (I) Find the probability that one zongzi of each type is selected. (II) Let $X$ denote the number of red bean paste zongzi selected. Find the probability distribution and mathematical expectation of $X$. (18) (This question is worth 13 points: part (I) is worth 7 points, part (II) is worth 6 points) Given the function $f ( x ) = \sin \left( \frac { \pi } { 2 } - x \right) \sin x - \sqrt { 3 } \cos ^ { 2 } x$ (I) Find the minimum positive period and maximum value of $f ( x )$. (II) Discuss the monotonicity of $f ( x )$ on $\left[ \frac { \pi } { 6 } , \frac { 2 \pi } { 3 } \right]$. (19) (This question is worth 13 points: part (I) is worth 4 points, part (II) is worth 9 points) As shown in question (19), in the triangular pyramid $P - A B C$, $P C \perp$ plane $A B C$, $P C = 3$, $\angle A C B = \frac { \pi } { 2 }$. Points $D$ and $E$ are on segments $A B$ and $B C$ respectively, with $C D = D E = \sqrt { 2 }$, $C E = 2 E B = 2$. (I) Prove that $D E \perp$ plane $P C D$. (II) Find the cosine of the dihedral angle $A - P D - C$. (20) (This question is worth 12 points: part (I) is worth 7 points, part (II) is worth 5 points) Let the function $f ( x ) = \frac { 3 x ^ { 2 } + a x } { e ^ { x } } ( a \in R )$. (I) If $f ( x )$ has an extremum at $x = 0$, determine the value of $a$ and find the equation of the tangent line to the curve $y = f ( x )$ at the point $( 1 , f ( 1 ) )$. (II) If $f ( x )$ is decreasing on $[ 3 , + \infty )$, find the range of values for $a$. (21) (This question is worth 12 points: part (I) is worth 5 points, part (II) is worth 7 points)
[Figure]
Question (19) Figure
As shown in question (21), the ellipse $\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ {

Let $f(x) = a_0 + a_1 |x| + a_2 |x|^2 + a_3 |x|^3$, where $a_0, a_1, a_2, a_3$ are constants. Then
(A) $f(x)$ is differentiable at $x = 0$ whatever be $a_0, a_1, a_2, a_3$
(B) $f(x)$ is not differentiable at $x = 0$ whatever be $a_0, a_1, a_2, a_3$
(C) $f(x)$ is differentiable at $x = 0$ only if $a_1 = 0$
(D) $f(x)$ is differentiable at $x = 0$ only if $a_1 = 0, a_3 = 0$

Find the number of solutions of $|2x - [x]| = 4$, where $[x]$ denotes the greatest integer function.
(A) 2 (B) 3 (C) 4 (D) Infinitely many

For real numbers $x , y$ and $z$, show that $$| x | + | y | + | z | \leq | x + y - z | + | y + z - x | + | z + x - y |$$

For real numbers $x , y$ and $z$, show that $$| x | + | y | + | z | \leq | x + y - z | + | y + z - x | + | z + x - y |$$

Let $f ( x ) = a _ { 0 } + a _ { 1 } | x | + a _ { 2 } | x | ^ { 2 } + a _ { 3 } | x | ^ { 3 }$, where $a _ { 0 } , a _ { 1 } , a _ { 2 } , a _ { 3 }$ are constants. Then
(a) $f ( x )$ is differentiable at $x = 0$ whatever be $a _ { 0 } , a _ { 1 } , a _ { 2 } , a _ { 3 }$
(b) $f ( x )$ is not differentiable at $x = 0$ whatever be $a _ { 0 } , a _ { 1 } , a _ { 2 } , a _ { 3 }$
(c) $f ( x )$ is differentiable at $x = 0$ only if $a _ { 1 } = 0$
(d) $f ( x )$ is differentiable at $x = 0$ only if $a _ { 1 } = 0 , a _ { 3 } = 0$.

Let $f ( x ) = a _ { 0 } + a _ { 1 } | x | + a _ { 2 } | x | ^ { 2 } + a _ { 3 } | x | ^ { 3 }$, where $a _ { 0 } , a _ { 1 } , a _ { 2 } , a _ { 3 }$ are constants. Then
(a) $f ( x )$ is differentiable at $x = 0$ whatever be $a _ { 0 } , a _ { 1 } , a _ { 2 } , a _ { 3 }$
(b) $f ( x )$ is not differentiable at $x = 0$ whatever be $a _ { 0 } , a _ { 1 } , a _ { 2 } , a _ { 3 }$
(c) $f ( x )$ is differentiable at $x = 0$ only if $a _ { 1 } = 0$
(d) $f ( x )$ is differentiable at $x = 0$ only if $a _ { 1 } = 0 , a _ { 3 } = 0$.

Let $f(x) = a_0 + a_1 |x| + a_2 |x|^2 + a_3 |x|^3$, where $a_0, a_1, a_2, a_3$ are constants.
(A) $f(x)$ is differentiable at $x = 0$ whatever be $a_0, a_1, a_2, a_3$
(B) $f(x)$ is not differentiable at $x = 0$ whatever be $a_0, a_1, a_2, a_3$ Then
(C) $f(x)$ is differentiable at $x = 0$ only if $a_1 = 0$
(D) $f(x)$ is differentiable at $x = 0$ only if $a_1 = 0, a_3 = 0$

Let $f ( x ) = a _ { 0 } + a _ { 1 } | x | + a _ { 2 } | x | ^ { 2 } + a _ { 3 } | x | ^ { 3 }$, where $a _ { 0 } , a _ { 1 } , a _ { 2 } , a _ { 3 }$ are constants.
(A) $f ( x )$ is differentiable at $x = 0$ whatever be $a _ { 0 } , a _ { 1 } , a _ { 2 } , a _ { 3 }$
(B) $f ( x )$ is not differentiable at $x = 0$ whatever be $a _ { 0 } , a _ { 1 } , a _ { 2 } , a _ { 3 }$ Then
(C) $f ( x )$ is differentiable at $x = 0$ only if $a _ { 1 } = 0$
(D) $f ( x )$ is differentiable at $x = 0$ only if $a _ { 1 } = 0 , a _ { 3 } = 0$

Let $\alpha$ denote a real number. The range of values of $| \alpha - 4 |$ such that $| \alpha - 1 | + | \alpha + 3 | \leq 8$ is
(A) $( 0,7 )$
(B) $( 1,8 )$
(C) $[ 1,9 ]$
(D) $[ 2,5 ]$.

For a real number $x$, let $[ x ]$ denote the greatest integer less than or equal to $x$. Then the number of real solutions of $| 2 x - [ x ] | = 4$ is
(A) 4
(B) 3
(C) 2
(D) 1 .

What is the minimum value of the function $| x - 3 | + | x + 2 | + | x + 1 | + | x |$ for real $x$?
(A) 3
(B) 5
(C) 6
(D) 8

Let $f(x) = a_0 + a_1|x| + a_2|x|^2 + a_3|x|^3$, where $a_0, a_1, a_2, a_3$ are constants. Then $f'(x)$ exists at $x = 0$ if and only if
(A) $a_1 = 0$
(B) $a_1 = 0$ and $a_2 = 0$
(C) $a_1 = 0$ and $a_3 = 0$
(D) $a_2 = 0$ and $a_3 = 0$

The function $f ( x ) = 2 | x | + | x + 2 | - | | x + 2 | - 2 | x | |$ has a local minimum or a local maximum at $x =$
(A) $- 2$
(B) $\frac { - 2 } { 3 }$
(C) $2$
(D) $\frac { 2 } { 3 }$

The number of points at which the function $$f(x) = |2x+1| - 3|x+2| + |x^2 + x - 2|, \quad x \in \mathbb{R}$$ is NOT differentiable is ____.

Let $f ( x ) = 15 - | x - 10 | ; x \in R$. Then the set of all values of $x$, at which the function $g ( x ) = f ( f ( x ) )$ is not differentiable, is:
(1) $\{ 5,10,15 \}$
(2) $\{ 10 \}$
(3) $\{ 10,15 \}$
(4) $\{ 5,10,15,20 \}$

Let $S$ be the set of points where the function, $f(x) = |2 - |x - 3||$, $x \in R$, is not differentiable. Then $\sum _ { x \in S } f(f(x))$ is equal to

The number of real solutions of the equation, $x ^ { 2 } - | x | - 12 = 0$ is:
(1) 2
(2) 3
(3) 1
(4) 4

The number of elements in the set $\{ x \in R : ( | x | - 3 ) | x + 4 | = 6 \}$ is equal to
(1) 3
(2) 2
(3) 4
(4) 1

The number of points, at which the function $f ( x ) = | 2 x + 1 | - 3 | x + 2 | + \left| x ^ { 2 } + x - 2 \right| , x \in R$ is not differentiable, is

Let $[ t ]$ denote the greatest integer $\leq t$ and $\{ t \}$ denote the fractional part of $t$. Then integral value of $\alpha$ for which the left hand limit of the function $f ( x ) = [ 1 + x ] + \frac { \alpha ^ { 2 [ x ] + \{ x \} } + [ x ] - 1 } { 2 [ x ] + \{ x \} }$ at $x = 0$ is equal to $\alpha - \frac { 4 } { 3 }$ is $\_\_\_\_$