For the function $$f ( x ) = \begin{cases} - x & ( x \leq 0 ) \\ x - 1 & ( 0 < x \leq 2 ) \\ 2 x - 3 & ( x > 2 ) \end{cases}$$ and a non-constant polynomial $p ( x )$, which of the following statements are correct? [4 points]
ㄱ. If the function $p ( x ) f ( x )$ is continuous on the entire set of real numbers, then $p ( 0 ) = 0$. ㄴ. If the function $p ( x ) f ( x )$ is differentiable on the entire set of real numbers, then $p ( 2 ) = 0$. ㄷ. If the function $p ( x ) \{ f ( x ) \} ^ { 2 }$ is differentiable on the entire set of real numbers, then $p ( x )$ is divisible by $x ^ { 2 } ( x - 2 ) ^ { 2 }$.
(1) ㄱ
(2) ㄱ, ㄴ
(3) ㄱ, ㄷ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

For a positive real number $t$, let $f ( t )$ be the value of the real number $a$ such that the curve $y = t ^ { 3 } \ln ( x - t )$ meets the curve $y = 2 e ^ { x - a }$ at exactly one point. Find the value of $\left\{ f ^ { \prime } \left( \frac { 1 } { 3 } \right) \right\} ^ { 2 }$. [4 points]

Two polynomial functions $f ( x )$ and $g ( x )$ satisfy $$\lim _ { x \rightarrow 0 } \frac { f ( x ) + g ( x ) } { x } = 3 , \quad \lim _ { x \rightarrow 0 } \frac { f ( x ) + 3 } { x g ( x ) } = 2$$ For the function $h ( x ) = f ( x ) g ( x )$, what is the value of $h ^ { \prime } ( 0 )$? [4 points]
(1) 27
(2) 30
(3) 33
(4) 36
(5) 39

Given that the function $f(x)=a\mathrm{e}^x-\ln x$ is monotonically increasing on the interval $(1,2)$, the minimum value of $a$ is
A. $\mathrm{e}^2$
B. $\mathrm{e}$
C. $\mathrm{e}^{-1}$
D. $\mathrm{e}^{-2}$

(17 points) Given function $f ( x ) = \ln \frac { x } { 2 - x } + a x + b ( x - 1 ) ^ { 3 }$ .
(1) If $b = 0$ and $f ^ { \prime } ( x ) \geqslant 0$ , find the minimum value of $a$ ;
(2) Prove that the curve $y = f ( x )$ is centrally symmetric;
(3) If $f ( x ) > - 2$ if and only if $1 < x < 2$ , find the range of $b$ .

If $y = \left[ x + \sqrt { x ^ { 2 } - 1 } \right] ^ { 15 } + \left[ x - \sqrt { x ^ { 2 } - 1 } \right] ^ { 15 }$, then $\left( x ^ { 2 } - 1 \right) \frac { d ^ { 2 } y } { d x ^ { 2 } } + x \frac { d y } { d x }$ is equal to
(1) $224 y ^ { 2 }$
(2) $125 y$
(3) $225 y$
(4) $225 y ^ { 2 }$

The function $f ( x ) = \left\{ \begin{array} { l l } \frac { \pi } { 4 } + \tan ^ { - 1 } x , & | x | \leq 1 \\ \frac { 1 } { 2 } ( | x | - 1 ) , & | x | > 1 \end{array} \right.$ is:
(1) continuous on $R - \{ 1 \}$ and differentiable on $R - \{ - 1,1 \}$.
(2) both continuous and differentiable on $R - \{ 1 \}$
(3) continuous on $R - \{ - 1 \}$ and differentiable on $R - \{ - 1,1 \}$
(4) both continuous and differentiable on $R - \{ - 1 \}$

$$f ( x ) = \begin{cases} 1 , & x \leq 1 \\ x ^ { 2 } + a x + b , & 1 < x < 3 \\ 5 , & x \geq 3 \end{cases}$$
If the function is continuous on the set of real numbers, what is the difference $a - b$?
A) $-4$
B) $-1$
C) 2
D) 3
E) 5