For positive numbers $a , b$ and natural numbers $m , n$ satisfying the inequality $a ^ { m } < a ^ { n } < b ^ { n } < b ^ { m }$, which of the following is correct? [3 points]
(1) $a < 1 < b , m > n$
(2) $a < 1 < b , m < n$
(3) $a < b < 1 , m < n$
(4) $1 < a < b , m > n$
(5) $1 < a < b , m < n$

When the two roots of the equation $4 ^ { x } - 7 \cdot 2 ^ { x } + 12 = 0$ are $\alpha , \beta$, find the value of $2 ^ { 2 \alpha } + 2 ^ { 2 \beta }$. [3 points]

For two exponential functions $f ( x ) = 4 ^ { x }$, $g ( x ) = \left( \frac { 1 } { 2 } \right) ^ { x }$ with domain $\{ x \mid - 1 \leqq x \leqq 3 \}$, let $M$ be the maximum value of $f ( x )$ and $m$ be the minimum value of $g ( x )$. What is the value of $M m$? [3 points]
(1) 8
(2) 6
(3) 4
(4) 2
(5) 1

Even when the surroundings suddenly become dark, the human eye perceives the change gradually. After the light intensity suddenly changes from 1000 to 10 and $t$ seconds have elapsed, the light intensity $I ( t )$ perceived by a person is $$I ( t ) = 10 + 990 \times a ^ { - 5 t } ( \text { where } a \text { is a constant with } a > 1 )$$ After the light intensity suddenly changes from 1000 to 10, let $s$ seconds elapse until the person perceives the light intensity as 21. What is the value of $s$? (Note: The unit of light intensity is Td (troland).) [3 points]
(1) $\frac { 1 + 2 \log 3 } { 5 \log a }$
(2) $\frac { 1 + 3 \log 3 } { 5 \log a }$
(3) $\frac { 2 + \log 3 } { 5 \log a }$
(4) $\frac { 2 + 2 \log 3 } { 5 \log a }$
(5) $\frac { 2 + 3 \log 3 } { 5 \log a }$

When the graph of the function $f ( x ) = 2 ^ { x }$ is translated by $m$ in the $x$-direction and by $n$ in the $y$-direction, the graph of the function $y = g ( x )$ is obtained. By this translation, point $\mathrm { A } ( 1 , f ( 1 ) )$ moves to point $\mathrm { A } ^ { \prime } ( 3 , g ( 3 ) )$. When the graph of the function $y = g ( x )$ passes through the point $( 0,1 )$, what is the value of $m + n$? [3 points]
(1) $\frac { 11 } { 4 }$
(2) 3
(3) $\frac { 13 } { 4 }$
(4) $\frac { 7 } { 2 }$
(5) $\frac { 15 } { 4 }$

Two exponential functions $f ( x ) = a ^ { b x - 1 } , g ( x ) = a ^ { 1 - b x }$ satisfy the following conditions.
(a) The graphs of $y = f ( x )$ and $y = g ( x )$ are symmetric with respect to the line $x = 2$.
(b) $f ( 4 ) + g ( 4 ) = \frac { 5 } { 2 }$
What is the value of the sum of the two constants $a + b$? (where $0 < a < 1$) [3 points]
(1) 1
(2) $\frac { 9 } { 8 }$
(3) $\frac { 5 } { 4 }$
(4) $\frac { 11 } { 8 }$
(5) $\frac { 3 } { 2 }$

When the graph of the exponential function $y = 5 ^ { x - 1 }$ passes through the two points $( a , 5 ) , ( 3 , b )$, find the value of $a + b$. [3 points]

On the coordinate plane, the graph of the exponential function $y = a ^ { x }$ is reflected about the $y$-axis, then translated 3 units in the $x$-direction and 2 units in the $y$-direction. The resulting graph passes through the point $( 1,4 )$. What is the value of the positive number $a$? [3 points]
(1) $\sqrt { 2 }$
(2) 2
(3) $2 \sqrt { 2 }$
(4) 4
(5) $4 \sqrt { 2 }$

On the coordinate plane, the two points where the two curves $y = \left| \log _ { 2 } x \right|$ and $y = \left( \frac { 1 } { 2 } \right) ^ { x }$ meet are $\mathrm { P } \left( x _ { 1 } , y _ { 1 } \right) , \mathrm { Q } \left( x _ { 2 } , y _ { 2 } \right) \left( x _ { 1 } < x _ { 2 } \right)$, and the point where the two curves $y = \left| \log _ { 2 } x \right|$ and $y = 2 ^ { x }$ meet is $\mathrm { R } \left( x _ { 3 } , y _ { 3 } \right)$. Which of the following statements in are correct? [4 points]
ㄱ. $\frac { 1 } { 2 } < x _ { 1 } < 1$ ㄴ. $x _ { 2 } y _ { 2 } - x _ { 3 } y _ { 3 } = 0$ ㄷ. $x _ { 2 } \left( x _ { 1 } - 1 \right) > y _ { 1 } \left( y _ { 2 } - 1 \right)$
(1) ㄱ
(2) ㄷ
(3) ㄱ, ㄴ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

In the coordinate plane, let the two points where the curves $y = \left| \log _ { 2 } x \right|$ and $y = \left( \frac { 1 } { 2 } \right) ^ { x }$ meet be $\mathrm { P } \left( x _ { 1 } , y _ { 1 } \right) , \mathrm { Q } \left( x _ { 2 } , y _ { 2 } \right) \left( x _ { 1 } < x _ { 2 } \right)$, and let the point where the curves $y = \left| \log _ { 2 } x \right|$ and $y = 2 ^ { x }$ meet be $\mathrm { R } \left( x _ { 3 } , y _ { 3 } \right)$. Which of the following are correct? Choose all that apply from $\langle$Remarks$\rangle$. [4 points]
$\langle$Remarks$\rangle$ ㄱ. $\frac { 1 } { 2 } < x _ { 1 } < 1$ ㄴ. $x _ { 2 } y _ { 2 } - x _ { 3 } y _ { 3 } = 0$ ㄷ. $x _ { 2 } \left( x _ { 1 } - 1 \right) > y _ { 1 } \left( y _ { 2 } - 1 \right)$
(1) ㄱ
(2) ㄷ
(3) ㄱ, ㄴ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

For natural numbers $a , b$, let P and Q be the points where the curve $y = a ^ { x + 1 }$ and the curve $y = b ^ { x }$ meet the line $x = t ( t \geq 1 )$, respectively. Find the number of all ordered pairs $( a , b )$ of $a , b$ satisfying the following condition. For example, $a = 4 , b = 5$ satisfies the following condition. [4 points]
(A) $2 \leq a \leq 10, 2 \leq b \leq 10$
(B) For some real number $t \geq 1$, $\overline { \mathrm { PQ } } \leq 10$.

For natural numbers $a , b$, let P and Q be the points where the curve $y = a ^ { x + 1 }$ and the curve $y = b ^ { x }$ meet the line $x = t ( t \geq 1 )$ respectively. Find the number of all ordered pairs $( a , b )$ satisfying the following conditions. For example, $a = 4 , b = 5$ satisfies the following conditions. [4 points] (가) $2 \leq a \leq 10, 2 \leq b \leq 10$ (나) For some real number $t \geq 1$, $\overline { \mathrm { PQ } } \leq 10$.

For a constant $a > 3$, two curves $y = a ^ { x - 1 }$ and $y = 3 ^ { x }$ meet at point P. Let the $x$-coordinate of point P be $k$.
What is the value of $\lim _ { n \rightarrow \infty } \frac { \left( \frac { a } { 3 } \right) ^ { n + k } } { \left( \frac { a } { 3 } \right) ^ { n + 1 } + 1 }$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

Find the sum of all natural numbers $x$ satisfying the inequality $\left( \frac { 1 } { 2 } \right) ^ { x - 5 } \geq 4$. [3 points]

What is the maximum value of the function $f ( x ) = 1 + \left( \frac { 1 } { 3 } \right) ^ { x - 1 }$ on the closed interval $[ 1,3 ]$? [3 points]
(1) $\frac { 5 } { 3 }$
(2) 2
(3) $\frac { 7 } { 3 }$
(4) $\frac { 8 } { 3 }$
(5) 3

Let A be the point where the graph of the exponential function $y = a ^ { x } ( a > 1 )$ meets the line $y = \sqrt { 3 }$. For the point $\mathrm { B } ( 4,0 )$, if the line OA and the line AB are perpendicular to each other, what is the product of all values of $a$? (Here, O is the origin.) [4 points]
(1) $3 ^ { \frac { 1 } { 3 } }$
(2) $3 ^ { \frac { 2 } { 3 } }$
(3) 3
(4) $3 ^ { \frac { 4 } { 3 } }$
(5) $3 ^ { \frac { 5 } { 3 } }$

How many natural numbers $x$ satisfy the inequality $\left( \frac { 1 } { 9 } \right) ^ { x } < 3 ^ { 21 - 4 x }$? [3 points]
(1) 6
(2) 7
(3) 8
(4) 9
(5) 10

How many natural numbers $x$ satisfy the inequality $\left( \frac { 1 } { 9 } \right) ^ { x } < 3 ^ { 21 - 4 x }$? [3 points]
(1) 6
(2) 7
(3) 8
(4) 9
(5) 10

For a natural number $n$, define the function $f ( x )$ as $$f ( x ) = \begin{cases} \left| 3 ^ { x + 2 } - n \right| & ( x < 0 ) \\ \left| \log _ { 2 } ( x + 4 ) - n \right| & ( x \geq 0 ) \end{cases}$$ Let $g ( t )$ be the number of distinct real roots of the equation $f ( x ) = t$ for a real number $t$. Find the sum of all natural numbers $n$ such that the maximum value of the function $g ( t )$ is 4. [4 points]

For a constant $a$ ($a > 1$), let A be a point in the first quadrant on the curve $y = a ^ { x } - 2$. Let B be the point where the line passing through A and parallel to the $y$-axis meets the $x$-axis, and let C be the point where this line meets the asymptote of the curve $y = a ^ { x } - 2$. If $\overline { \mathrm { AB } } = \overline { \mathrm { BC } }$ and the area of triangle AOC is 8, what is the value of $a \times \overline { \mathrm { OB } }$? (Here, O is the origin.) [4 points]
(1) $2 ^ { \frac { 13 } { 6 } }$
(2) $2 ^ { \frac { 7 } { 3 } }$
(3) $2 ^ { \frac { 5 } { 2 } }$
(4) $2 ^ { \frac { 8 } { 3 } }$
(5) $2 ^ { \frac { 17 } { 6 } }$

7. The solution set of the inequality $2 ^ { x ^ { 2 } - x } < 4$ is $\_\_\_\_$ .

7. Given the function $f ( x ) = 2 ^ { | x - m | } - 1$ defined on $\mathbb{R}$ (where $m$ is a real number), let $a = f \left( \log _ { 0.5 } 3 \right)$, $b = f \left( \log _ { 2 } 5 \right)$, $c = f ( 2m )$. Then the size relationship of $a, b, c$ is
(A) $a < b < c$
(B) $c < a < b$
(C) $a < c < b$
(D)

Given the function $\mathrm{f}(x) = 2^{|x-1|} - 1$ defined on $\mathbb{R}$ (where m is a real number) is an even function, let $\mathrm{a} = \mathrm{f}(\log_{0.5}3)$, $b = f(\log_2 5)$, $c = f(2m)$. Then the size relationship of $a, b, c$ is
(A) $a < b < c$
(B) $a < c < b$
(C) $c < a < b$
(D) $c < b < a$

9. Let $f ( x ) = \ln x , 0 < a < b$. If $p = f ( \sqrt { a b } ) , q = f \left( \frac { a + b } { 2 } \right)$, $r = \frac { 1 } { 2 } ( f ( a ) + f ( b ) )$, then the correct relation is
A. $q = r < p$
B. $q = r > p$
C. $p = r < q$
D. $p = r > q$

13. Given that the graph of function $f ( x ) = a x ^ { 3 } - 2 x$ passes through the point $( - 1,4 )$, then $a = $ $\_\_\_\_$ .