For a polynomial function $f ( x )$, define the function $g ( x )$ as $$g ( x ) = x ^ { 2 } f ( x )$$ If $f ( 2 ) = 1$ and $f ^ { \prime } ( 2 ) = 3$, what is the value of $g ^ { \prime } ( 2 )$? [3 points] (1) 12 (2) 14 (3) 16 (4) 18 (5) 20
The function $f ( x ) = 2 x ^ { 3 } - 9 x ^ { 2 } + a x + 5$ has a local maximum at $x = 1$ and a local minimum at $x = b$. What is the value of $a + b$? (Here, $a$ and $b$ are constants.) [3 points] (1) 12 (2) 14 (3) 16 (4) 18 (5) 20
An arithmetic sequence $\left\{ a _ { n } \right\}$ with all positive terms and equal first term and common difference satisfies $$\sum _ { k = 1 } ^ { 15 } \frac { 1 } { \sqrt { a _ { k } } + \sqrt { a _ { k + 1 } } } = 2$$ What is the value of $a _ { 4 }$? [3 points] (1) 6 (2) 7 (3) 8 (4) 9 (5) 10
What is the $x$-intercept of the tangent line drawn from the point $(0, 4)$ to the curve $y = x ^ { 3 } - x + 2$? [3 points] (1) $- \frac { 1 } { 2 }$ (2) $- 1$ (3) $- \frac { 3 } { 2 }$ (4) $- 2$ (5) $- \frac { 5 } { 2 }$
The function $$f ( x ) = a - \sqrt { 3 } \tan 2 x$$ has a maximum value of 7 and a minimum value of 3 on the closed interval $\left[ - \frac { \pi } { 6 } , b \right]$. What is the value of $a \times b$? (Here, $a$ and $b$ are constants.) [4 points] (1) $\frac { \pi } { 2 }$ (2) $\frac { 5 \pi } { 12 }$ (3) $\frac { \pi } { 3 }$ (4) $\frac { \pi } { 4 }$ (5) $\frac { \pi } { 6 }$
Let $A$ be the area enclosed by the two curves $y = x ^ { 3 } + x ^ { 2 }$, $y = - x ^ { 2 } + k$, and the $y$-axis, and let $B$ be the area enclosed by the two curves $y = x ^ { 3 } + x ^ { 2 }$, $y = - x ^ { 2 } + k$, and the line $x = 2$. When $A = B$, what is the value of the constant $k$? (Here, $4 < k < 5$) [4 points] (1) $\frac { 25 } { 6 }$ (2) $\frac { 13 } { 3 }$ (3) $\frac { 9 } { 2 }$ (4) $\frac { 14 } { 3 }$ (5) $\frac { 29 } { 6 }$
A function $f ( x )$ that is continuous on the set of all real numbers satisfies the following condition. When $n - 1 \leq x < n$, $| f ( x ) | = | 6 ( x - n + 1 ) ( x - n ) |$. (Here, $n$ is a natural number.) For the function $$g ( x ) = \int _ { 0 } ^ { x } f ( t ) d t - \int _ { x } ^ { 4 } f ( t ) d t$$ defined on the open interval $(0, 4)$, when $g ( x )$ has a minimum value of 0 at $x = 2$, what is the value of $\int _ { \frac { 1 } { 2 } } ^ { 4 } f ( x ) d x$? [4 points] (1) $- \frac { 3 } { 2 }$ (2) $- \frac { 1 } { 2 }$ (3) $\frac { 1 } { 2 }$ (4) $\frac { 3 } { 2 }$ (5) $\frac { 5 } { 2 }$
For a natural number $m$ ($m \geq 2$), let $f ( m )$ be the number of natural numbers $n \geq 2$ such that an integer exists among the $n$-th roots of $m ^ { 12 }$. What is the value of $\sum _ { m = 2 } ^ { 9 } f ( m )$? [4 points] (1) 37 (2) 42 (3) 47 (4) 52 (5) 57
For all sequences $\left\{ a _ { n } \right\}$ with all natural number terms satisfying the following conditions, let $M$ and $m$ be the maximum and minimum values of $a _ { 9 }$, respectively. What is the value of $M + m$? [4 points] (가) $a _ { 7 } = 40$ (나) For all natural numbers $n$, $$a _ { n + 2 } = \begin{cases} a _ { n + 1 } + a _ { n } & ( \text{when } a _ { n + 1 } \text{ is not a multiple of } 3 ) \\ \frac { 1 } { 3 } a _ { n + 1 } & ( \text{when } a _ { n + 1 } \text{ is a multiple of } 3 ) \end{cases}$$ (1) 216 (2) 218 (3) 220 (4) 222 (5) 224
For two sequences $\left\{ a _ { n } \right\}$ and $\left\{ b _ { n } \right\}$, $$\sum _ { k = 1 } ^ { 5 } \left( 3 a _ { k } + 5 \right) = 55 , \quad \sum _ { k = 1 } ^ { 5 } \left( a _ { k } + b _ { k } \right) = 32$$ What is the value of $\sum _ { k = 1 } ^ { 5 } b _ { k }$? [3 points]
The velocity $v ( t )$ and acceleration $a ( t )$ of a point P moving on a number line at time $t$ ($t \geq 0$) satisfy the following conditions. (가) When $0 \leq t \leq 2$, $v ( t ) = 2 t ^ { 3 } - 8 t$. (나) When $t \geq 2$, $a ( t ) = 6 t + 4$. Find the distance traveled by point P from time $t = 0$ to $t = 3$. [4 points]
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]
A cubic function $f ( x )$ with leading coefficient 1 and a function $g ( x )$ that is continuous on the set of all real numbers satisfy the following conditions. Find the value of $f ( 4 )$. [4 points] (가) For all real numbers $x$, $$f ( x ) = f ( 1 ) + ( x - 1 ) f ^ { \prime } ( g ( x ) )$$ (나) The minimum value of the function $g ( x )$ is $\frac { 5 } { 2 }$. (다) $f ( 0 ) = - 3$, $f ( g ( 1 ) ) = 6$
Among four-digit natural numbers that can be formed by selecting 4 numbers from the digits 1, 2, 3, 4, 5 with repetition allowed and arranging them in a line, how many are odd numbers greater than or equal to 4000? [3 points] (1) 125 (2) 150 (3) 175 (4) 200 (5) 225
A box contains 5 white masks and 9 black masks. When 3 masks are randomly drawn simultaneously from the box, what is the probability that at least one of the 3 masks is white? [3 points] (1) $\frac { 8 } { 13 }$ (2) $\frac { 17 } { 26 }$ (3) $\frac { 9 } { 13 }$ (4) $\frac { 19 } { 26 }$ (5) $\frac { 10 } { 13 }$
A bag contains 1 white ball marked with 1, 1 white ball marked with 2, 1 black ball marked with 1, and 3 black balls marked with 2. We perform a trial of simultaneously drawing 3 balls from the bag. Let $A$ be the event that among the 3 balls drawn, 1 is white and 2 are black, and let $B$ be the event that the product of the numbers on the 3 balls is 8. What is the value of $\mathrm { P } ( A \cup B )$? [3 points] (1) $\frac { 11 } { 20 }$ (2) $\frac { 3 } { 5 }$ (3) $\frac { 13 } { 20 }$ (4) $\frac { 7 } { 10 }$ (5) $\frac { 3 } { 4 }$
A company produces shampoo where the volume of 1 bottle follows a normal distribution $\mathrm { N } \left( m , \sigma ^ { 2 } \right)$. A sample of 16 bottles was randomly selected from the company's shampoo, and the 95\% confidence interval for $m$ using the sample mean is $746.1 \leq m \leq 755.9$. When a sample of $n$ bottles is randomly selected and a 99\% confidence interval for $m$ is constructed as $a \leq m \leq b$, what is the minimum natural number $n$ such that $b - a \leq 6$? (Here, the unit of volume is mL, and when $Z$ is a random variable following the standard normal distribution, $\mathrm { P } ( | Z | \leq 1.96 ) = 0.95$ and $\mathrm { P } ( | Z | \leq 2.58 ) = 0.99$.) [3 points] (1) 70 (2) 74 (3) 78 (4) 82 (5) 86
A continuous random variable $X$ has a range of $0 \leq X \leq a$, and the graph of the probability density function of $X$ is as shown in the figure. When $\mathrm { P } ( X \leq b ) - \mathrm { P } ( X \geq b ) = \frac { 1 } { 4 }$ and $\mathrm { P } ( X \leq \sqrt { 5 } ) = \frac { 1 } { 2 }$, what is the value of $a + b + c$? (Here, $a$, $b$, and $c$ are constants.) [4 points] (1) $\frac { 11 } { 2 }$ (2) 6 (3) $\frac { 13 } { 2 }$ (4) 7 (5) $\frac { 15 } { 2 }$
There are 6 cards with natural numbers 1 through 6 written on the front and 0 written on the back. These 6 cards are placed so that the natural number $k$ is visible in the $k$-th position for natural numbers $k$ not exceeding 6. Using these 6 cards and one die, we perform the following trial: Roll the die once. If the result is $k$, flip the card in the $k$-th position and place it back in its original position. After repeating this trial 3 times, given that the sum of all numbers visible on the 6 cards is even, what is the probability that the die shows 1 exactly once? The probability is $\frac { q } { p }$. Find the value of $p + q$. (Here, $p$ and $q$ are coprime natural numbers.) [4 points]
For the set $X = \{ x \mid x \text{ is a natural number not exceeding } 10 \}$, find the number of functions $f : X \rightarrow X$ satisfying the following conditions. [4 points] (가) For all natural numbers $x$ not exceeding 9, $f ( x ) \leq f ( x + 1 )$. (나) When $1 \leq x \leq 5$, $f ( x ) \leq x$, and when $6 \leq x \leq 10$, $f ( x ) \geq x$. (다) $f ( 6 ) = f ( 5 ) + 6$