Let $f$ be the function defined by $f ( x ) = \sqrt { | x - 2 | }$ for all $x$. Which of the following statements is true?
(A) $f$ is continuous but not differentiable at $x = 2$.
(B) $f$ is differentiable at $x = 2$.
(C) $f$ is not continuous at $x = 2$.
(D) $\lim _ { x \rightarrow 2 } f ( x ) \neq 0$
(E) $x = 2$ is a vertical asymptote of the graph of $f$.

10. If the function $f ( x ) = a | x - b | + 2$ is increasing on $[ 0 , + \infty )$, then the range of real numbers $a$ and $b$ is $\_\_\_\_$.

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 \}$

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 ^ {

Given the function $f ( x ) = \left| x - a ^ { 2 } \right| + | x - 2 a + 1 |$ .
(1) When $a = 2$, find the solution set of the inequality $f ( x ) \geqslant 4$;
(2) If $f ( x ) \geqslant 4$ for all $x$, find the range of values of $a$.

Let $f(x) = |x|\sin x + |x - \pi|\cos x$ for $x \in \mathbb{R}$. Then
(a) $f$ is differentiable at $x = 0$ and $x = \pi$
(b) $f$ is not differentiable at $x = 0$ and $x = \pi$
(c) $f$ is differentiable at $x = 0$ but not differentiable at $x = \pi$
(d) $f$ is not differentiable at $x = 0$ but differentiable at $x = \pi$

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 the function on the real line $\mathbb { R }$ given by $f ( x ) = | x | + | x + 1 | + e ^ { x }$, which of the following is true ?
(A) It is differentiable everywhere.
(B) It is differentiable everywhere except at $x = 0$ and $x = - 1$.
(C) It is differentiable everywhere except at $x = 1 / 2$.
(D) It is differentiable everywhere except at $x = - 1 / 2$.

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

17. The function $f ( x ) = ( x 2 - 1 ) | x 2 - 3 x + 2 |$ is NOT differentiable at:
(A) - 1
(B) 0
(C) 1
(D) 2

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 $[ \mathrm { t } ]$ denote the greatest integer $\leq \mathrm { t }$. Then the equation in $\mathrm { x } , [ \mathrm { x } ] ^ { 2 } + 2 [ \mathrm { x } + 2 ] - 7 = 0$ has :
(1) exactly two solutions
(2) exactly four integral solutions
(3) no integral solution
(4) infinitely many solutions

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

Let $[ x ]$ denote the greatest integer $\leq x$, where $x \in R$. If the domain of the real valued function $f ( x ) = \sqrt { \frac { | [ x ] | - 2 } { | [ x ] | - 3 } }$ is $( - \infty , a ) \cup [ b , c ) \cup [ 4 , \infty ) , a < b < c$, then the value of $a + b + c$ is:
(1) 8
(2) 1
(3) $- 2$
(4) $- 3$

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

The number of points, where the function $f : R \rightarrow R , f ( x ) = | x - 1 | \cos | x - 2 | \sin | x - 1 | + ( x - 3 ) \left| x ^ { 2 } - 5 x + 4 \right|$, is NOT differentiable, is
(1) 1
(2) 2
(3) 3
(4) 4

Let $f ( x ) = 2 + | x | - | x - 1 | + | x + 1 | , x \in R$. Consider $( S 1 ) : f ^ { \prime } \left( - \frac { 3 } { 2 } \right) + f ^ { \prime } \left( - \frac { 1 } { 2 } \right) + f ^ { \prime } \left( \frac { 1 } { 2 } \right) + f ^ { \prime } \left( \frac { 3 } { 2 } \right) = 2$ $( S 2 ) : \int _ { - 2 } ^ { 2 } f ( x ) d x = 12$ Then,
(1) both ( $S 1$ ) and ( $S 2$ ) are correct
(2) both $( S 1 )$ and $( S 2 )$ are wrong
(3) only ( $S 1$ ) is correct
(4) only ( $S 2$ ) is correct

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 $\_\_\_\_$

The number of real roots of the equation $x | x | - 5 | x + 2 | + 6 = 0$, is
(1) 5
(2) 4
(3) 6
(4) 3

The number of distinct real roots of the equation $| x + 1 | | x + 3 | - 4 | x + 2 | + 5 = 0$, is

The sum of the squares of all the roots of the equation $x ^ { 2 } + | 2 x - 3 | - 4 = 0$, is
(1) $3 ( 3 - \sqrt { 2 } )$
(2) $6 ( 3 - \sqrt { 2 } )$
(3) $6 ( 2 - \sqrt { 2 } )$
(4) $3 ( 2 - \sqrt { 2 } )$