The sum of all components of the inverse matrix $A ^ { - 1 }$ of the matrix $A = \left( \begin{array} { r r } 1 & - 2 \\ 0 & 1 \end{array} \right)$ is? [2 points] (1) 5 (2) 4 (3) 3 (4) 2 (5) 1
What is the value of $\lim _ { x \rightarrow 0 } \frac { e ^ { x } - 1 } { 5 x }$? [2 points] (1) 5 (2) $e$ (3) 1 (4) $\frac { 1 } { e }$ (5) $\frac { 1 } { 5 }$
A random variable $X$ follows a binomial distribution $\mathrm { B } ( 200 , p )$ and the mean of $X$ is 40. What is the variance of $X$? [2 points] (1) 32 (2) 33 (3) 34 (4) 35 (5) 36
For two sets $$A = \left\{ x \left\lvert \, \frac { ( x - 2 ) ^ { 2 } } { x - 4 } \leq 0 \right. \right\} , \quad B = \left\{ x \mid x ^ { 2 } - 8 x + a \leq 0 \right\}$$ When $A \cup B = \{ x \mid x \leq 5 \}$, what is the value of the constant $a$? [3 points] (1) 7 (2) 10 (3) 12 (4) 15 (5) 16
When arranging 5 white flags and 5 blue flags in a line, how many ways are there to place white flags at both ends? (Note: flags of the same color are indistinguishable from each other.) [3 points] (1) 56 (2) 63 (3) 70 (4) 77 (5) 84
On the coordinate plane, three points $\mathrm { A } ( 3,0 ) , \mathrm { B } ( 3,3 ) , \mathrm { C } ( 0,3 )$ are mapped to $\mathrm { A } ^ { \prime } , \mathrm { B } ^ { \prime } , \mathrm { C } ^ { \prime }$ respectively by the linear transformation represented by the matrix $\left( \begin{array} { l l } k & 0 \\ 0 & k \end{array} \right) ( k > 1 )$. When the area of the common part of the interior of triangle ABC and the interior of triangle $\mathrm { A } ^ { \prime } \mathrm { B } ^ { \prime } \mathrm { C } ^ { \prime }$ is $\frac { 1 } { 2 }$, what is the value of $k$? [3 points] (1) $\frac { 3 } { 2 }$ (2) $\frac { 5 } { 3 }$ (3) $\frac { 7 } { 4 }$ (4) $\frac { 9 } { 5 }$ (5) $\frac { 11 } { 6 }$
The female silkworm moth secretes pheromone to attract males. When $t$ seconds have passed since the female silkworm moth secreted pheromone, the concentration $y$ of pheromone measured at a distance $x$ from the secretion site satisfies the following equation. $$\log y = A - \frac { 1 } { 2 } \log t - \frac { K x ^ { 2 } } { t } \text { (where } A \text { and } K \text { are positive constants.) }$$ When 1 second has passed since the female silkworm moth secreted pheromone, the pheromone concentration measured at a distance 2 from the secretion site is $a$, and when 4 seconds have passed, the pheromone concentration measured at a distance $d$ from the secretion site is $\frac { a } { 2 }$. What is the value of $d$? [3 points] (1) 7 (2) 6 (3) 5 (4) 4 (5) 3
The calcium content in one bottle of beverage produced by a certain company follows a normal distribution with population mean $m$ and population standard deviation $\sigma$. When 16 bottles of beverage produced by this company were randomly sampled and the calcium content was measured, the sample mean was 12.34. When the 95\% confidence interval for the population mean $m$ of the calcium content in one bottle of beverage produced by this company is $11.36 \leq m \leq a$, what is the value of $a + \sigma$? (Note: When $Z$ follows the standard normal distribution, $\mathrm { P } ( 0 \leq Z \leq 1.96 ) = 0.4750$, and the unit of calcium content is mg.) [3 points] (1) 14.32 (2) 14.82 (3) 15.32 (4) 15.82 (5) 16.32
On the coordinate plane, a rotation transformation $f$ centered at the origin maps the point $( 1,0 )$ to a point $\left( \frac { \sqrt { 3 } } { 2 } , a \right)$ in the first quadrant. When the sum of all components of the matrix representing the rotation transformation $f$ is $b$, what is the value of $a ^ { 2 } + b ^ { 2 }$? [3 points] (1) $\frac { 31 } { 12 }$ (2) $\frac { 11 } { 4 }$ (3) $\frac { 35 } { 12 }$ (4) $\frac { 37 } { 12 }$ (5) $\frac { 13 } { 4 }$
For a rhombus ABCD with side length 10, an ellipse with diagonal BD as the major axis and diagonal AC as the minor axis has a distance between the two foci of $10 \sqrt { 2 }$. What is the area of rhombus ABCD? [3 points] (1) $55 \sqrt { 3 }$ (2) $65 \sqrt { 2 }$ (3) $50 \sqrt { 3 }$ (4) $45 \sqrt { 3 }$ (5) $45 \sqrt { 2 }$
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
Box A contains 3 red balls and 5 black balls, and box B is empty. When 2 balls are randomly drawn from box A, if a red ball appears, perform [Execution 1], and if no red ball appears, perform [Execution 2]. What is the probability that the number of red balls in box B is 1? [3 points] [Execution 1] Put the drawn balls into box B. [Execution 2] Put the drawn balls into box B, and then randomly draw 2 more balls from box A and put them into box B. (1) $\frac { 1 } { 2 }$ (2) $\frac { 7 } { 12 }$ (3) $\frac { 2 } { 3 }$ (4) $\frac { 3 } { 4 }$ (5) $\frac { 5 } { 6 }$
There is a circle with radius 1. As shown in the figure, a rectangle with a ratio of horizontal length to vertical length of $3 : 1$ is inscribed in this circle, and the common part of the interior of the circle and the exterior of the rectangle is colored to obtain a figure $R _ { 1 }$. In figure $R _ { 1 }$, 2 circles are drawn tangent to three sides of the rectangle. A rectangle is drawn in each of the newly drawn circles using the same method as obtaining figure $R _ { 1 }$, and the colored figure obtained is $R _ { 2 }$. In figure $R _ { 2 }$, 4 circles are drawn tangent to three sides of the newly drawn rectangles. A rectangle is drawn in each of the newly drawn circles using the same method as obtaining figure $R _ { 1 }$, and the colored figure obtained is $R _ { 3 }$. Continuing this process, let $S _ { n }$ be the area of the colored part in the $n$-th obtained figure $R _ { n }$. What is the value of $\lim _ { n \rightarrow \infty } S _ { n }$? [4 points] (1) $\frac { 5 } { 4 } \pi - \frac { 5 } { 3 }$ (2) $\frac { 5 } { 4 } \pi - \frac { 3 } { 2 }$ (3) $\frac { 4 } { 3 } \pi - \frac { 8 } { 5 }$ (4) $\frac { 5 } { 4 } \pi - 1$ (5) $\frac { 4 } { 3 } \pi - \frac { 16 } { 15 }$
In the figure, let $a$ be the area of region $A$ enclosed by the two curves $y = e ^ { x } , y = x e ^ { x }$ and the $y$-axis, and let $b$ be the area of region $B$ enclosed by the two curves $y = e ^ { x } , y = x e ^ { x }$ and the line $x = 2$. What is the value of $b - a$? [4 points] (1) $\frac { 3 } { 2 }$ (2) $e - 1$ (3) 2 (4) $\frac { 5 } { 2 }$ (5) $e$
For a sequence $\left\{ a _ { n } \right\}$ with first term 1, let $S _ { n } = \sum _ { k = 1 } ^ { n } a _ { k }$. The following relation holds: $$n S _ { n + 1 } = ( n + 2 ) S _ { n } + ( n + 1 ) ^ { 3 } \quad ( n \geq 1 )$$ The following is part of the process of finding the general term of the sequence $\left\{ a _ { n } \right\}$. Since $S _ { n + 1 } = S _ { n } + a _ { n + 1 }$ for natural numbers $n$, $$n a _ { n + 1 } = 2 S _ { n } + ( n + 1 ) ^ { 3 } \quad \cdots (\text{ㄱ})$$ For natural numbers $n \geq 2$, $$( n - 1 ) a _ { n } = 2 S _ { n - 1 } + n ^ { 3 } \quad \cdots (\text{ㄴ})$$ By subtracting (ㄴ) from (ㄱ), we obtain $$n a _ { n + 1 } = ( n + 1 ) a _ { n } + \quad \text { (가) }$$ Dividing both sides by $n ( n + 1 )$, $$\frac { a _ { n + 1 } } { n + 1 } = \frac { a _ { n } } { n } + \frac { ( \text{가} ) } { n ( n + 1 ) }$$ If $b _ { n } = \frac { a _ { n } } { n }$, then $$b _ { n + 1 } = b _ { n } + \text{ (나) } \quad ( n \geq 2 )$$ so $$b _ { n } = b _ { 2 } + \text{ (다) } \quad ( n \geq 3 )$$ When the expressions that go in (가), (나), and (다) are $f ( n ) , g ( n ) , h ( n )$ respectively, what is the value of $\frac { f ( 3 ) } { g ( 3 ) h ( 6 ) }$? [4 points] (1) 30 (2) 36 (3) 42 (4) 48 (5) 54
For the function $f ( x ) = 2 x \cos x$ with domain $\{ x \mid 0 \leq x \leq \pi \}$, which of the following are correct? Choose all that apply from . [4 points] Remarks ㄱ. If $f ^ { \prime } ( a ) = 0$, then $\tan a = \frac { 1 } { a }$. ㄴ. There exists $a$ in the interval $\left( \frac { \pi } { 4 } , \frac { \pi } { 3 } \right)$ where the function $f ( x )$ has a local maximum value at $x = a$. ㄷ. On the interval $\left[ 0 , \frac { \pi } { 2 } \right]$, the number of distinct real roots of the equation $f ( x ) = 1$ is 2. (1) ㄱ (2) ㄷ (3) ㄱ, ㄴ (4) ㄴ, ㄷ (5) ㄱ, ㄴ, ㄷ
For a real number $m$, let $f ( m )$ be the number of intersection points of the line passing through the point $( 0,2 )$ with slope $m$ and the curve $y = x ^ { 3 } - 3 x ^ { 2 } + 1$. What is the maximum value of the real number $a$ such that the function $f ( m )$ is continuous on the interval $( - \infty , a )$? [4 points] (1) $- 3$ (2) $- \frac { 3 } { 4 }$ (3) $\frac { 3 } { 2 }$ (4) $\frac { 15 } { 4 }$ (5) 6
On the coordinate plane, let $\theta _ { 1 }$ be the acute angle that the line $y = m x ( 0 < m < \sqrt { 3 } )$ makes with the $x$-axis, and let $\theta _ { 2 }$ be the acute angle that the line $y = m x$ makes with the line $y = \sqrt { 3 } x$. What is the value of $m$ that maximizes $3 \sin \theta _ { 1 } + 4 \sin \theta _ { 2 }$? [4 points] (1) $\frac { \sqrt { 3 } } { 6 }$ (2) $\frac { \sqrt { 3 } } { 7 }$ (3) $\frac { \sqrt { 3 } } { 8 }$ (4) $\frac { \sqrt { 3 } } { 9 }$ (5) $\frac { \sqrt { 3 } } { 10 }$
In coordinate space, triangle ABC satisfies the following conditions. (가) The area of triangle ABC is 6. (나) The area of the orthogonal projection of triangle ABC onto the $yz$-plane is 3. What is the maximum area of the orthogonal projection of triangle ABC onto the plane $x - 2 y + 2 z = 1$? [4 points] (1) $2 \sqrt { 6 } + 1$ (2) $2 \sqrt { 2 } + 3$ (3) $3 \sqrt { 5 } - 1$ (4) $2 \sqrt { 5 } + 1$ (5) $3 \sqrt { 6 } - 2$
In coordinate space, there is a point $\mathrm { A } ( 9,0,5 )$, and on the $xy$-plane there is an ellipse $\frac { x ^ { 2 } } { 9 } + y ^ { 2 } = 1$. For a point P on the ellipse, find the maximum value of $\overline { \mathrm { AP } }$. [3 points]
Let $d$ be the distance between the focus of the parabola $y ^ { 2 } = n x$ and the tangent line to the parabola at the point $( n , n )$. Find the minimum natural number $n$ satisfying $d ^ { 2 } \geq 40$. [4 points]
Three numbers $a , a + b , 2 a - b$ form an arithmetic sequence in this order, and three numbers $1 , a - 1, 3 b + 1$ form a geometric sequence with positive common ratio in this order. Find the value of $a ^ { 2 } + b ^ { 2 }$. [3 points]
As shown in the figure, let Q be the foot of the perpendicular from point P on a circle with center O and diameter AB of length 2 to the line segment AB, let R be the foot of the perpendicular from point Q to the line segment OP, and let S be the foot of the perpendicular from point O to the line segment AP. When $\angle \mathrm { PAQ } = \theta \left( 0 < \theta < \frac { \pi } { 4 } \right)$, let $f ( \theta )$ be the area of triangle AOS and $g ( \theta )$ be the area of triangle PRQ. When $\lim _ { \theta \rightarrow +0 } \frac { \theta ^ { 2 } f ( \theta ) } { g ( \theta ) } = \frac { q } { p }$, find the value of $p ^ { 2 } + q ^ { 2 }$. (Here, $p$ and $q$ are coprime natural numbers.) [4 points]
For the function $f ( x ) = 3 ( x - 1 ) ^ { 2 } + 5$, define the function $F ( x )$ as $F ( x ) = \int _ { 0 } ^ { x } f ( t ) \, dt$. A differentiable function $g ( x )$ satisfies the following for all real numbers $x$: $$F ( g ( x ) ) = \frac { 1 } { 2 } F ( x )$$ When $g ^ { \prime } ( 2 ) = p$, find the value of $30 p$. [4 points]
As shown in the figure, a cylinder with base radius 7 and a cone with base radius 5 and height 12 are placed on a plane $\alpha$, and the circumference of the base of the cone is inscribed in the circumference of the base of the cylinder. Let O be the center of the base of the cylinder that meets plane $\alpha$, and let A be the apex of the cone. A sphere $S$ with center B and radius 4 satisfies the following conditions. (가) The sphere $S$ is tangent to both the cylinder and the cone. (나) When $\mathrm { A } ^ { \prime }$ and $\mathrm { B } ^ { \prime }$ are the orthogonal projections of points $\mathrm { A }$ and $\mathrm { B }$ onto plane $\alpha$ respectively, $\angle \mathrm { A } ^ { \prime } \mathrm { OB } ^ { \prime } = 180 ^ { \circ }$. When the acute angle between line AB and plane $\alpha$ is $\theta$, $\tan \theta = p$. Find the value of $100 p$. (Note: The center of the base of the cone and point $\mathrm { A } ^ { \prime }$ coincide.) [4 points]
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 $t \geq 1$, $\overline { \mathrm { PQ } } \leq 10$.