If the domain of the function $\log _ { 5 } \left( 18 x - x ^ { 2 } - 77 \right)$ is $( \alpha , \beta )$ and the domain of the function $\log _ { ( x - 1 ) } \left( \frac { 2 x ^ { 2 } + 3 x - 2 } { x ^ { 2 } - 3 x - 4 } \right)$ is $( \gamma , \delta )$, then $\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }$ is equal to :
(1) 195
(2) 179
(3) 186
(4) 174

Let $\mathrm { f } ( x ) = \left\{ \begin{array} { l l } 3 x , & x < 0 \\ \min \{ 1 + x + [ x ] , x + 2 [ x ] \} , & 0 \leq x \leq 2 \\ 5 , & x > 2 , \end{array} \right.$ where [.] denotes greatest integer function. If $\alpha$ and $\beta$ are the number of points, where f is not continuous and is not differentiable, respectively, then $\alpha + \beta$ equals

If the set of all values of a, for which the equation $5 x ^ { 3 } - 15 x - a = 0$ has three distinct real roots, is the interval $( \alpha , \beta )$, then $\beta - 2 \alpha$ is equal to $\_\_\_\_$

Consider a quadratic function in $x$
$$y = ax^2 + bx + c$$
such that the graph of function (1) passes through the two points $(-1, -1)$ and $(2, 2)$.
(1) When we express $b$ and $c$ in terms of $a$, we have
$$b = \mathbf{A} - a, \quad c = \mathbf{BC}a.$$
(2) Suppose that one of the points of intersection of the graph of function (1) and the $x$-axis is within the interval $0 < x \leqq 1$. Then the range of values of $a$ is [see figure].
(3) When the value of $a$ varies within interval (2), the range of values of $a + bc$ is
$$\frac{\mathbf{GH}}{\square\mathbf{I}} \leqq a + bc \leqq \square.$$

Let $f(x) = 2x^3 - 3x + 1$. Select the correct statements about the graph of the function $y = f(x)$.
(1) The graph of $y = f(x)$ passes through the point $(1, 0)$
(2) The graph of $y = f(x)$ has only one intersection point with the $x$-axis
(3) The point $(1, 0)$ is a center of symmetry of the graph of $y = f(x)$
(4) The graph of $y = f(x)$ approximates a straight line $y = 3x - 3$ near the center of symmetry
(5) The graph of $y = 3x^3 - 6x^2 + 2x$ can be obtained from the graph of $y = f(x)$ by appropriate translation

Given a real-coefficient cubic polynomial function $f ( x ) = a x ^ { 3 } + b x ^ { 2 } + c x + 3$ . Let $g ( x ) = f ( - x ) - 3$ . It is known that the graph of $y = g ( x )$ has a center of symmetry at $( 1,0 )$ and $g ( - 1 ) < 0$ . Select the correct options.
(1) $g ( x ) = 0$ has three distinct integer roots
(2) $a < 0$
(3) The center of symmetry of the graph of $y = f ( x )$ is $( - 1 , - 3 )$
(4) $f ( 100 ) < 0$
(5) The graph of $y = f ( x )$ near the point $( - 1 , f ( - 1 ) )$ can be approximated by a line with slope $a$

A water pumping station found that its electricity consumption (unit: kilowatt-hours) is directly proportional to the cube of the pump motor speed (unit: rpm). Based on this, which of the following five graphs best describes the relationship between the electricity consumption $y$ (kilowatt-hours) and the pump motor speed $X$ (rpm) of this water pumping station?
(1) [Graph 1] (2) [Graph 2] (3) [Graph 3] (4) [Graph 4] (5) [Graph 5]

A person uses single-point perspective with a point on the horizon as the vanishing point to draw six vertical pillars $A , B , C , D , E , F$ on a coordinate plane. The coordinates of the top and base of each pillar are shown in the table below, with point $V ( 4,9 )$ representing the vanishing point, as shown in the figure. Since the base line and top line of pillars $A$ and $F$ in the figure are both parallel to the horizon, the actual heights of pillars $A$ and $F$ are equal. Based on the above, select the pillar with the maximum actual height.

Pillar	$A$	$B$	$C$	$D$	$E$	$F$
Top coordinate	$( 0,8 )$	$( 2,3 )$	$( 4,6 )$	$( 6,8 )$	$( 8,5 )$	$( 10,8 )$
Base coordinate	$( 0,6 )$	$( 2,0 )$	$( 4,3 )$	$( 6,5 )$	$( 8,1 )$	$( 10,6 )$

(1) $A$
(2) $B$
(3) $C$
(4) $D$
(5) $E$

Let $\Gamma$ be the graph of the function $y = x ^ { 3 } - x$ on the coordinate plane. Select the correct options.
(1) The center of symmetry of $\Gamma$ is the origin
(2) $\Gamma$ approximates the line $y = x$ near $x = 0$
(3) $\Gamma$ can coincide with the graph of the function $y = x ^ { 3 } + x + 3$ after appropriate translation
(4) $\Gamma$ and the graph of the function $y = x ^ { 3 } + x$ are symmetric about the $x$-axis
(5) $\Gamma$ and the graph of the function $y = - x ^ { 3 } + x$ are symmetric about the $y$-axis

Which of the following is the domain of the function $f$ whose graph is given above?
A) $[-3,0) \cup [4,7)$
B) $(-3,0) \cup (3,7]$
C) $[-3,2] \cup (3,7)$
D) $(-3,3) \cup (3,7]$
E) $[-3,2) \cup (4,7]$

The graph of the function $f: \mathbb{R}\setminus\{-1\} \rightarrow \mathbb{R}\setminus\{2\}$ is shown in the figure above.
Accordingly, $$\lim_{x \rightarrow -\infty} f(x) + \lim_{x \rightarrow 0} f(x)$$ What is the sum of these limits?
A) $-2$
B) $-1$
C) $0$
D) $1$
E) $3$

When 130 liters of milk in a dairy is used to make cheese, the graph of the linear relationship between the remaining milk and the amount of cheese produced is given.
According to this, when 10 kg of cheese is produced in this dairy, how many liters of milk remain?
A) 50
B) 60
C) 65
D) 75
E) 80

Below is the graph of the function f. If the function $( \mathbf { f } + \mathbf { g } )$ is continuous at the point $X = 1$, which of the following could be the graph of the function g?
A) [graph A]
B) [graph B]
C) [graph C]
D) [graph D]
E) [graph E]

$$f ( x ) = \frac { - k x ^ { 3 } + k ^ { 2 } x } { k ^ { 3 } x ^ { 2 } + x - ( k + 1 ) }$$
The function has a vertical asymptote at $x = 1$. Accordingly, what is the value of $f ( 2 )$?
A) - 5
B) - 4
C) - 3
D) - 2
E) - 1

The function
$$f ( x ) = \frac { a x } { | b x + 2 | }$$
defined on a subset of the set of positive real numbers has a vertical asymptote at $x = 2$ and a horizontal asymptote at $y = 4$.
Accordingly, what is the sum $a + b$?
A) 1
B) 2
C) 3
D) 4
E) 5

Below is the graph of the function $f$.
Accordingly, regarding the function f: I. The function f does not have an absolute maximum value on the interval $[ 0,4 ]$. II. There exists $a \in [ 0,4 ]$ such that $f ( a ) = 2$. III. $\lim _ { x \rightarrow 1 ^ { - } } ( f \circ f ) ( x ) = 1$.
Which of the following statements are true?
A) Only I
B) Only II
C) I and II
D) II and III
E) I, II and III

A function $f$ defined on the set of real numbers satisfies the inequalities $$1 \leq f ( x ) \leq 2$$ for every $x$.
Accordingly,\ I. $\lim _ { x \rightarrow 1 } \frac { 1 } { f ( x ) }$ exists.\ II. $\lim _ { x \rightarrow 1 } \frac { f ( x ) } { x }$ exists.\ III. $\lim _ { x \rightarrow 1 } ( | f ( x ) | - f ( x ) )$ exists.
Which of the following statements are always true?\ A) Only I\ B) Only II\ C) Only III\ D) I and II\ E) II and III

Let a be a real number, and $$f ( x ) = \ln ( 2 x + 8 )$$ The vertical asymptote of the function $$g ( x ) = \frac { \sin x } { x ^ { 2 } + a x }$$ is also a vertical asymptote of the function.\ Accordingly, what is a?\ A) 0\ B) 1\ C) 2\ D) 3\ E) 4

In the rectangular coordinate plane, the graphs of functions $f$, $g$, and $h$ are given in the figure.
Accordingly, for a real number $a$ satisfying the condition $0 < a < 2$
I. When $f(a) < g(a)$, then $g(a) < h(a)$ holds. II. When $g(a) < h(a)$, then $h(a) < f(a)$ holds. III. When $h(a) < f(a)$, then $f(a) < g(a)$ holds.
Which of the following statements are true?
A) Only I
B) Only II
C) Only III
D) I and II
E) I and III

Some digits in the 11-digit phone numbers of Ayla and Berk are given as follows. $$\begin{aligned} & \text{Ayla} \longrightarrow 05{*}{*}{*}{*}{*}7235 \\ & \text{Berk} \longrightarrow 05{*}{*}{*}{*}{*}9415 \end{aligned}$$ Let $A$ be the set of digits in Ayla's phone number and $B$ be the set of digits in Berk's phone number, where $$\begin{aligned} & s(A) = 9 \\ & s(B) = 6 \end{aligned}$$ It is known that $A \cap B = \{0, 1, 4, 5, 6\}$. What is the sum of the values of elements in the set $A \setminus B$?
A) 18
B) 20
C) 21
D) 26
E) 27

In the rectangular coordinate plane, the graph of a function $f$ defined on the closed interval $[-5,5]$ is given in the figure.
For distinct numbers $a, b, c$ and $d$ in the domain of this function
$$\begin{aligned} & f(a) = f(b) = 1 \\ & f(c) = f(d) = 3 \end{aligned}$$
the equalities are satisfied. Accordingly, regarding the ordering of $a, b, c$ and $d$ numbers I. $a < b < c < d$ II. $c < a < b < d$ III. $c < d < a < b$ Which of the following inequalities can be true?
A) Only I
B) Only II
C) I and II
D) II and III
E) I, II and III

Let $a, b, c$ and $d$ be real numbers such that
$$x + ay \leq b$$ $$x + cy \geq d$$
The solution set of the system of inequalities is shown in green on the coordinate plane below.
Accordingly, what are the signs of the numbers $a, b, c$ and $d$ in order?
A) $+, -, -, -$ B) $+, +, +, -$ C) $+, -, +, -$

For real numbers $\mathrm{a}$, $\mathrm{b}$ and $\mathrm{c}$ with $\mathrm{a} \cdot \mathrm{b} \cdot \mathrm{c} > 0$, a function $f$ is defined on the set of real numbers as
$$f(x) = ax^{2} + bx + c$$
Accordingly, the graph of function $f$ can be which of the graphs shown (I, II, III)?
A) Only I B) Only II C) I and III D) II and III E) I, II and III

In the rectangular coordinate plane, the graphs of functions $f$ and $g$ defined on the closed interval $[0,1]$ and the line $y = x$ are given below.
For real numbers $a, b$ and $c$ in the open interval $(0,1)$
$$\begin{aligned} & a < f(a) < g(a) \\ & g(b) < b < f(b) \\ & c < g(c) < f(c) \end{aligned}$$
If these inequalities are satisfied, which of the following orderings is correct?
A) $a < b < c$ B) $a < c < b$ C) $b < a < c$ D) $c < a < b$ E) $c < b < a$

In the rectangular coordinate plane, the graphs of linear functions $f$, $g$ and $h$ are shown in the figure.
Regarding these functions, the following equalities are given:
$$\begin{aligned} & f(x-5) = g(x) \\ & h(x) = -f(x) \end{aligned}$$
Which of the following orderings is correct for the values $f(0)$, $g(0)$ and $h(0)$?
A) $g(0) < f(0) < h(0)$ B) $f(0) < h(0) < g(0)$ C) $f(0) < g(0) < h(0)$ D) $g(0) < h(0) < f(0)$ E) $h(0) < g(0) < f(0)$