The graphs of the quadratic function $y = f ( x )$ and the cubic function $y = g ( x )$ are shown in the figure.
$f ( - 1 ) = f ( 3 ) = 0$, and the function $g ( x )$ has a local minimum value of $- 2$ at $x = 3$. What is the number of distinct real roots of the equation $\frac { g ( x ) + 2 } { f ( x ) } - \frac { 2 } { g ( x ) } = 1$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

When the graph of the function $y = f ( x )$ is as shown in the figure, which of the following statements are correct? [4 points]
ㄱ. $\lim _ { x \rightarrow +0 } f ( x ) = 1$ ㄴ. $\lim _ { x \rightarrow 1 } f ( x ) = f ( 1 )$ ㄷ. The function $( x - 1 ) f ( x )$ is continuous at $x = 1$.
(1) ㄱ
(2) ㄱ, ㄴ
(3) ㄱ, ㄷ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

A cubic function $f ( x )$ with leading coefficient 1 satisfies $f ( - x ) = - f ( x )$ for all real numbers $x$. When the equation $| f ( x ) | = 2$ has exactly 4 distinct real roots, what is the value of $f ( 3 )$? [4 points]
(1) 12
(2) 14
(3) 16
(4) 18
(5) 20

For the two functions $$f(x) = \begin{cases} -1 & (|x| \geq 1) \\ 1 & (|x| < 1) \end{cases}, \quad g(x) = \begin{cases} 1 & (|x| \geq 1) \\ -x & (|x| < 1) \end{cases}$$ which of the following statements are correct? Choose all that apply from $\langle$Remarks$\rangle$. [4 points]
Remarks ᄀ. $\lim_{x \rightarrow 1} f(x)g(x) = -1$ ㄴ. The function $g(x+1)$ is continuous at $x = 0$. ㄷ. The function $f(x)g(x+1)$ is continuous at $x = -1$.
(1) ᄀ
(2) ㄴ
(3) ᄀ, ㄴ
(4) ᄀ, ㄷ
(5) ᄀ, ㄴ, ㄷ

For the function
$$f ( x ) = \begin{cases} x + 1 & ( x \leq 0 ) \\ - \frac { 1 } { 2 } x + 7 & ( x > 0 ) \end{cases}$$
Find the sum of all real values of $a$ such that the function $f ( x ) f ( x - a )$ is continuous at $x = a$. [4 points]

On the coordinate plane, for a natural number $a > 1$, consider the two curves $y = 4 ^ { x } , y = a ^ { - x + 4 }$ and the line $y = 1$. Find the number of values of $a$ such that the number of points with integer coordinates inside or on the boundary of the region enclosed by these curves and line is between 20 and 40 (inclusive). [4 points]

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

For the function $f ( x ) = x ( x + 1 ) ( x - 4 )$, answer the following. When the line $y = 5 x + k$ and the graph of the function $y = f ( x )$ intersect at two distinct points, what is the value of the positive number $k$? [4 points]
(1) 5
(2) $\frac { 11 } { 2 }$
(3) 6
(4) $\frac { 13 } { 2 }$
(5) 7

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

Consider the function $$f ( x ) = \begin{cases} | 5 x ( x + 2 ) | & ( x < 0 ) \\ | 5 x ( x - 2 ) | & ( x \geq 0 ) \end{cases}$$ What is the number of distinct real roots of the irrational equation $\sqrt { 4 - f ( x ) } = 1 - x$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

As shown in the figure, in the coordinate plane, the circle $x ^ { 2 } + y ^ { 2 } = 1$ and the curve $y = \ln ( x + 1 )$ meet at point A in the first quadrant. For point $\mathrm { B } ( 1,0 )$, let H be the foot of the perpendicular from point P on arc AB to the $y$-axis, and let Q be the intersection of segment PH and the curve $y = \ln ( x + 1 )$. Let $\angle \mathrm { POB } = \theta$. If $S ( \theta )$ is the area of triangle OPQ and $L ( \theta )$ is the length of segment HQ, and $\lim _ { \theta \rightarrow + 0 } \frac { S ( \theta ) } { L ( \theta ) } = k$, find the value of $60 k$. (Here, $0 < \theta < \frac { \pi } { 6 }$ and O is the origin.) [4 points]

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

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

As shown in the figure, the graph of the function $f ( x )$ defined on the closed interval $[ 0,4 ]$ is formed by connecting the points $( 0,0 ) , ( 1,4 ) , ( 2,1 ) , ( 3,4 ) , ( 4,3 )$ in order with line segments. Find the number of sets $X = \{ a , b \}$ satisfying the following condition. (Here, $0 \leq a < b \leq 4$) [4 points]
A function $g ( x ) = f ( f ( x ) )$ from $X$ to $X$ exists and satisfies $g ( a ) = f ( a ) , g ( b ) = f ( b )$.
(1) 11
(2) 13
(3) 15
(4) 17
(5) 19

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

Find the maximum value of the real number $k$ such that the graphs of $y = \sqrt { x + 3 }$ and $y = \sqrt { 1 - x } + k$ intersect. [4 points]

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

For the function $y = \sqrt { 4 - 2 x } + 3$, what is the minimum value of the real number $k$ such that the graph of its inverse function and the line $y = - x + k$ intersect at two distinct points? [3 points]
(1) 1
(2) 3
(3) 5
(4) 7
(5) 9

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

Find the positive value of $k$ such that the curve $y = 4 x ^ { 3 } - 12 x + 7$ and the line $y = k$ intersect at exactly 2 points. [3 points]

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

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