The graph of the function $y = f ( x )$ is shown in the figure. What is the value of $\lim _ { x \rightarrow 0 - } f ( x ) + \lim _ { x \rightarrow 1 + } f ( x )$? [3 points]
(1) - 1
(2) - 2
(3) - 3
(4) - 4
(5) - 5

For two functions $$\begin{aligned} & f ( x ) = \left\{ \begin{array} { c c } x ^ { 2 } - 4 x + 6 & ( x < 2 ) \\ 1 & ( x \geq 2 ) \end{array} , \right. \\ & g ( x ) = a x + 1 \end{aligned}$$ When the function $\frac { g ( x ) } { f ( x ) }$ is continuous on the entire set of real numbers, what is the value of the constant $a$? [4 points]
(1) $- \frac { 5 } { 4 }$
(2) $- 1$
(3) $- \frac { 3 } { 4 }$
(4) $- \frac { 1 } { 2 }$
(5) $- \frac { 1 } { 4 }$

On the coordinate plane, the function $$f ( x ) = \begin{cases} - x + 10 & ( x < 10 ) \\ ( x - 10 ) ^ { 2 } & ( x \geq 10 ) \end{cases}$$ and for a natural number $n$, there is a circle $O _ { n }$ centered at $( n , f ( n ) )$ with radius 3. Let $A _ { n }$ be the number of all points with integer coordinates that are inside circle $O _ { n }$ and below the graph of the function $y = f ( x )$, and let $B _ { n }$ be the number of all points with integer coordinates that are inside circle $O _ { n }$ and above the graph of the function $y = f ( x )$. What is the value of $\sum _ { n = 1 } ^ { 20 } \left( A _ { n } - B _ { n } \right)$? [4 points]
(1) 19
(2) 21
(3) 23
(4) 25
(5) 27

The graph of the function $y = f ( x )$ is shown in the figure. Find the value of $\lim _ { x \rightarrow 0 - } f ( x ) + \lim _ { x \rightarrow 1 + } f ( x )$. [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

In the coordinate plane, find the number of points with both natural number coordinates that are contained in the interior of the region enclosed by the curve $y = \frac { 1 } { 2 x - 8 } + 3$ and the $x$-axis and $y$-axis. [3 points]
(1) 3
(2) 4
(3) 5
(4) 6
(5) 7

The graph of the function $y = f ( x )$ is shown in the figure. [Figure] What is the value of $\lim _ { x \rightarrow - 1 - } f ( x ) - \lim _ { x \rightarrow 1 + } f ( x )$? [3 points]
(1) - 2
(2) - 1
(3) 0
(4) 1
(5) 2

As shown in the figure, let $\mathrm { A }$ and $\mathrm { B }$ be the $x$-intercept and $y$-intercept, respectively, of the graph of the function $y = \frac { k } { x - 1 } + 3$ where $0 < k < 3$. [Figure] Let $\mathrm { P }$ be the point (other than $\mathrm { B }$) where the line passing through the intersection of the two asymptotes of this graph and point $\mathrm { B }$ meets the graph, and let $\mathrm { Q }$ be the foot of the perpendicular from point $\mathrm { P }$ to the $x$-axis. Which of the following statements are correct? [4 points] ㄱ. When $k = 1$, the coordinates of point $\mathrm { P }$ are $( 2,4 )$. ㄴ. For real numbers $0 < k < 3$, the sum of the slope of line AB and the slope of line AP is 0. ㄷ. When the area of quadrilateral PBAQ is a natural number, the slope of line BP is between 0 and 1.
(1) ㄱ
(2) ㄱ, ㄴ
(3) ㄱ, ㄷ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

The graph of the function $y = f ( x )$ is shown in the figure. What is the value of $\lim _ { x \rightarrow 0 + } f ( x ) - \lim _ { x \rightarrow 1 - } f ( x )$? [3 points]
(1) $-2$
(2) $-1$
(3) 0
(4) 1
(5) 2

What is the value of $\lim _ { x \rightarrow 2 } \frac { x ^ { 2 } + 2 x - 8 } { x - 2 }$? [2 points]
(1) 2
(2) 4
(3) 6
(4) 8
(5) 10

For a constant $a$, define the function $f ( x )$ as $$f ( x ) = \lim _ { n \rightarrow \infty } \frac { ( a - 2 ) x ^ { 2 n + 1 } + 2 x } { 3 x ^ { 2 n } + 1 }$$ What is the sum of all values of $a$ such that $( f \circ f ) ( 1 ) = \frac { 5 } { 4 }$? [4 points]
(1) $\frac { 11 } { 2 }$
(2) $\frac { 13 } { 2 }$
(3) $\frac { 15 } { 2 }$
(4) $\frac { 17 } { 2 }$
(5) $\frac { 19 } { 2 }$

The graph of the function $y = f ( x )$ is shown in the figure.
What is the value of $\lim _ { x \rightarrow - 1 - } f ( x ) + \lim _ { x \rightarrow 2 } f ( x )$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

A function $f ( x )$ continuous on the entire set of real numbers satisfies $$\{ f ( x ) \} ^ { 3 } - \{ f ( x ) \} ^ { 2 } - x ^ { 2 } f ( x ) + x ^ { 2 } = 0$$ for all real numbers $x$. When the maximum value of $f ( x )$ is 1 and the minimum value is 0, what is the value of $f \left( - \frac { 4 } { 3 } \right) + f ( 0 ) + f \left( \frac { 1 } { 2 } \right)$? [4 points]
(1) $\frac { 1 } { 2 }$
(2) 1
(3) $\frac { 3 } { 2 }$
(4) 2
(5) $\frac { 5 } { 2 }$

What is the value of $\lim _ { x \rightarrow \infty } \frac { \sqrt { x ^ { 2 } - 2 } + 3 x } { x + 5 }$? [2 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]

Consider the function $$f(x) = \begin{cases} 3x - a & (x < 2) \\ x^2 + a & (x \geq 2) \end{cases}$$ If $f$ is continuous on the set of all real numbers, find the value of the constant $a$. [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

For two natural numbers $a$ and $b$, the function $f(x)$ is defined as $$f(x) = \begin{cases} 2x^3 - 6x + 1 & (x \leq 2) \\ a(x-2)(x-b) + 9 & (x > 2) \end{cases}$$ For a real number $t$, let $g(t)$ denote the number of intersection points of the graph of $y = f(x)$ and the line $y = t$. $$g(k) + \lim_{t \rightarrow k-} g(t) + \lim_{t \rightarrow k+} g(t) = 9$$ If the number of real numbers $k$ satisfying this condition is 1, find the maximum value of $a + b$ for the ordered pair $(a, b)$ of two natural numbers. [4 points]
(1) 51
(2) 52
(3) 53
(4) 54
(5) 55

The function $$f ( x ) = \begin{cases} 3 x - 2 & ( x < 1 ) \\ x ^ { 2 } - 3 x + a & ( x \geq 1 ) \end{cases}$$ is continuous on the set of all real numbers. What is the value of the constant $a$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

Let $f ( x )$ be a cubic function with positive leading coefficient, and for a real number $t$, let the function $$g ( x ) = \left\{ \begin{array} { r r } - f ( x ) & ( x < t ) \\ f ( x ) & ( x \geq t ) \end{array} \right.$$ be continuous on the set of all real numbers and satisfy the following conditions. (가) For all real numbers $a$, the value of $\lim _ { x \rightarrow a + } \frac { g ( x ) } { x ( x - 2 ) }$ exists. (나) The set of natural numbers $m$ such that $\lim _ { x \rightarrow m + } \frac { g ( x ) } { x ( x - 2 ) }$ is negative is $\left\{ g ( - 1 ) , - \frac { 7 } { 2 } g ( 1 ) \right\}$. Find the value of $g ( - 5 )$. (Given that $g ( - 1 ) \neq - \frac { 7 } { 2 } g ( 1 )$) [4 points]

4. Among the following functions, which one is both an even function and has a zero point?
(A) $y = \ln x$
(B) $y = x ^ { 2 } + 1$
(C) $y = \sin x$
(D) $y = \cos x$

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. $\{ x \mid - 1 < x \leqslant 2 \}$

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) $\left(\frac{7}{4}, 2\right)$

10. The graph of the function $f ( x ) = a x ^ { 3 } + b x ^ { 2 } + c x + d$ is shown in the figure. Then the correct conclusion is [Figure]
(A) $a > 0 , b < 0 , c > 0 , d > 0$
(B) $a > 0 , b < 0 , c < 0 , d > 0$
(C) $a < 0 , b < 0 , c < 0 , d > 0$
(D) $a > 0 , b > 0 , c > 0 , d < 0$

II. Fill in the Blank Questions

(11) $\lg \frac { 5 } { 2 } + 2 \lg 2 - \left( \frac { 1 } { 2 } \right) ^ { - 1 } =$ $\_\_\_\_$. (12) In $\triangle A B C$, $A B = \sqrt { 6 } , \angle A = 75 ^ { \circ } , \angle B = 45 ^ { \circ }$. Then $A C =$ $\_\_\_\_$. (13) In the sequence $\left\{ a _ { n } \right\}$, $a _ { 1 } = 1 , a _ { n } = a _ { n - 1 } + \frac { 1 } { 2 } ( n \geq 2 )$. Then the sum of the first 9 terms of the sequence $\left\{ a _ { n } \right\}$ equals $\_\_\_\_$. (14) In the rectangular coordinate system $x O y$, if the line $y = 2 a$ and the graph of the function $y = | x - a | - 1$ have only one intersection point, then the value of $a$ is $\_\_\_\_$. (15) $\triangle A B C$ is an equilateral triangle with side length 2. Given that vectors $\vec { a } , \vec { b }$ satisfy $\overrightarrow { A B } = 2 \vec { a } , \overrightarrow { A C } = 2 \vec { a } + \vec { b }$, then the correct conclusions among the following are $\_\_\_\_$. (Write out the serial numbers of all correct conclusions)
(1) $\vec { a }$ is a unit vector; (2) $\vec { b }$ is a unit vector; (3) $\vec { a } \perp \vec { b }$; (4) $\vec { b } \parallel \overrightarrow { B C }$; (5) $( 4 \vec { a } + \vec { b } ) \perp \overrightarrow { B C }$.

III. Solution Questions

As shown in the figure, rectangle $ABCD$ has sides $\mathrm { AB } = 2 , \mathrm { BC } = 1$, and $O$ is the midpoint of $AB$. Point $P$ moves along edges $\mathrm { BC } , \mathrm { CD }$, and $DA$, with $\angle \mathrm { BOP } = \mathrm { x }$. The sum of distances from moving point $P$ to points $A$ and $B$ is expressed as a function of $x$, denoted $f ( x )$. The graph of $f ( x )$ is approximately
(A), (B), (C), or (D) [as shown in figures]

10. Given the function $f ( x ) = \left\{ \begin{array} { l } x + \frac { 2 } { x } - 1 , x \geq 1 \\ \lg \left( x ^ { 2 } + 1 \right) , x < 1 \end{array} \right.$ , then $f ( f ( - 3 ) ) =$ $\_\_\_\_$ , and the minimum value of $f ( x )$ is $\_\_\_\_$ .