When three numbers $\frac { 9 } { 4 } , a , 4$ form a geometric sequence in this order, what is the value of the positive number $a$? [3 points] (1) $\frac { 8 } { 3 }$ (2) 3 (3) $\frac { 10 } { 3 }$ (4) $\frac { 11 } { 3 }$ (5) 4
For two conditions on the real number $x$: $$\begin{aligned}
& p : | x - 1 | \leq 3 , \\
& q : | x | \leq a
\end{aligned}$$ What is the minimum value of the natural number $a$ such that $p$ is a sufficient condition for $q$? [3 points] (1) 1 (2) 2 (3) 3 (4) 4 (5) 5
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, when the graph of the function $y = \frac { 3 } { x - 5 } + k$ is symmetric with respect to the line $y = x$, what is the value of the constant $k$? [3 points] (1) 1 (2) 2 (3) 3 (4) 4 (5) 5
The velocity $v ( t )$ at time $t ( t \geq 0 )$ of a point P moving on a number line is $$v ( t ) = - 2 t + 4$$ What is the distance traveled by point P from $t = 0$ to $t = 4$? [3 points] (1) 8 (2) 9 (3) 10 (4) 11 (5) 12
The total number of students at a certain school is 360, and each student chose either experiential learning A or experiential learning B. Among the students at this school, those who chose experiential learning A are 90 male students and 70 female students. When one student is randomly selected from the students at this school who chose experiential learning B, the probability that this student is male is $\frac { 2 } { 5 }$. What is the number of female students at this school? [3 points] (1) 180 (2) 185 (3) 190 (4) 195 (5) 200
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) $- \frac { 1 } { 4 }$
The weight of pomegranates produced at a certain farm follows a normal distribution with mean $m$ and standard deviation 40. A sample of size 64 was taken from the pomegranates produced at this farm, and the sample mean of the pomegranate weights was $\bar { x }$. Using this result, the 99\% confidence interval for the mean $m$ of the pomegranate weights produced at this farm is $\bar { x } - c \leq m \leq \bar { x } + c$. What is the value of $c$? (Here, the unit of weight is g, and when $Z$ is a random variable following the standard normal distribution, $\mathrm { P } ( 0 \leq Z \leq 2.58 ) = 0.495$.) [4 points] (1) 25.8 (2) 21.5 (3) 17.2 (4) 12.9 (5) 8.6
A quadratic function $f ( x )$ with leading coefficient 1 satisfies $$\lim _ { x \rightarrow a } \frac { f ( x ) - ( x - a ) } { f ( x ) + ( x - a ) } = \frac { 3 } { 5 }$$ When the two roots of the equation $f ( x ) = 0$ are $\alpha$ and $\beta$, what is the value of $| \alpha - \beta |$? (Here, $a$ is a constant.) [4 points] (1) 1 (2) 2 (3) 3 (4) 4 (5) 5
A jump is defined as moving from a point $( x , y )$ on the coordinate plane to one of the three points $( x + 1 , y )$, $( x , y + 1 )$, or $( x + 1 , y + 1 )$. Let $X$ be the random variable representing the number of jumps that occur when one case is randomly selected from all cases of moving from point $( 0,0 )$ to point $( 4,3 )$ by repeating jumps. The following is the process of finding the mean $\mathrm { E } ( X )$ of the random variable $X$. (Here, each case is selected with equal probability.) Let $N$ be the total number of cases of moving from point $( 0,0 )$ to point $( 4,3 )$ by repeating jumps. If the smallest value that the random variable $X$ can take is $k$, then $k =$ (가), and the largest value is $k + 3$. $$\begin{aligned}
& \mathrm { P } ( X = k ) = \frac { 1 } { N } \times \frac { 4 ! } { 3 ! } = \frac { 4 } { N } \\
& \mathrm { P } ( X = k + 1 ) = \frac { 1 } { N } \times \frac { 5 ! } { 2 ! 2 ! } = \frac { 30 } { N } \\
& \mathrm { P } ( X = k + 2 ) = \frac { 1 } { N } \times \text { (나) } \\
& \mathrm { P } ( X = k + 3 ) = \frac { 1 } { N } \times \frac { 7 ! } { 3 ! 4 ! } = \frac { 35 } { N }
\end{aligned}$$ and $$\sum _ { i = k } ^ { k + 3 } \mathrm { P } ( X = i ) = 1$$ so $N =$ (다). Therefore, the mean $\mathrm { E } ( X )$ of the random variable $X$ is as follows. $$\mathrm { E } ( X ) = \sum _ { i = k } ^ { k + 3 } \{ i \times \mathrm { P } ( X = i ) \} = \frac { 257 } { 43 }$$ When the numbers that fit (가), (나), and (다) are $a$, $b$, and $c$, respectively, what is the value of $a + b + c$? [4 points] (1) 190 (2) 193 (3) 196 (4) 199 (5) 202
A cubic function $f ( x )$ with positive leading coefficient satisfies the following conditions. (가) The function $f ( x )$ has a local maximum at $x = 0$ and a local minimum at $x = k$. (Here, $k$ is a constant.) (나) For all real numbers $t$ greater than 1, $\int _ { 0 } ^ { t } \left| f ^ { \prime } ( x ) \right| d x = f ( t ) + f ( 0 )$ Which of the following statements in the given options are correct? [4 points] Options ᄀ. $\int _ { 0 } ^ { k } f ^ { \prime } ( x ) d x < 0$ ㄴ. $0 < k \leq 1$ ㄷ. The local minimum value of the function $f ( x )$ is 0. (1) ᄀ (2) ㄷ (3) ᄀ, ㄴ (4) ㄴ, ㄷ (5) ᄀ, ㄴ, ㄷ
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 universal set is $U = \{ x \mid x$ is a natural number not exceeding 9 $\}$, and two subsets of $U$ are $$A = \{ 3,6,7 \} , B = \{ a - 4,8,9 \}$$ If $$A \cap B ^ { C } = \{ 6,7 \}$$ find the value of the natural number $a$. [3 points]
For the curve $y = x ^ { 3 } - a x + b$, the slope of the line perpendicular to the tangent line at the point $( 1,1 )$ is $- \frac { 1 } { 2 }$. For the two constants $a$ and $b$, find the value of $a + b$. [4 points]
Find the number of all ordered pairs $( a , b , c )$ of non-negative integers satisfying the following conditions. [4 points] (가) $a + b + c = 7$ (나) $2 ^ { a } \times 4 ^ { b }$ is a multiple of 8.
For a natural number $n$, let $\mathrm { P } _ { n }$ be the point where the line $x = 4 ^ { n }$ meets the curve $y = \sqrt { x }$. Let $L _ { n }$ be the length of the segment $\mathrm { P } _ { n } \mathrm { P } _ { n + 1 }$. Find the value of $\lim _ { n \rightarrow \infty } \left( \frac { L _ { n + 1 } } { L _ { n } } \right) ^ { 2 }$. [4 points]
The random variable $X$ follows a normal distribution with mean $m$ and standard deviation 5, and the probability density function $f ( x )$ of the random variable $X$ satisfies the following conditions. (가) $f ( 10 ) > f ( 20 )$ (나) $f ( 4 ) < f ( 22 )$ When $m$ is a natural number, $\mathrm { P } ( 17 \leq X \leq 18 ) = a$. Find the value of $1000a$ using the standard normal distribution table below. [4 points]
For a real number $k$, let $g ( x )$ be the inverse function of $f ( x ) = x ^ { 3 } - 3 x ^ { 2 } + 6 x + k$. For the equation $4 f ^ { \prime } ( x ) + 12 x - 18 = \left( f ^ { \prime } \circ g \right) ( x )$ to have a real root in the closed interval $[ 0,1 ]$, let $m$ be the minimum value of $k$ and $M$ be the maximum value of $k$. Find the value of $m ^ { 2 } + M ^ { 2 }$. [4 points]