For two matrices $A = \left( \begin{array} { l l } 1 & 1 \\ 0 & 2 \end{array} \right) , B = \left( \begin{array} { l l } 1 & 1 \\ 3 & 0 \end{array} \right)$, what is the sum of all components of the matrix $A + B$? [2 points] (1) 5 (2) 6 (3) 7 (4) 8 (5) 9
What is the maximum value of the function $f ( x ) = \sin x + \sqrt { 7 } \cos x - \sqrt { 2 }$? [2 points] (1) $\sqrt { 2 }$ (2) $\sqrt { 3 }$ (3) 2 (4) $\sqrt { 5 }$ (5) $\sqrt { 6 }$
In coordinate space, for two points $\mathrm { A } ( 2 , a , - 2 ) , \mathrm { B } ( 5 , - 3 , b )$, when the point that divides segment AB internally in the ratio $2 : 1$ lies on the $x$-axis, what is the value of $a + b$? [3 points] (1) 6 (2) 7 (3) 8 (4) 9 (5) 10
Let the matrices representing two linear transformations $f , g$ be $\left( \begin{array} { l l } 2 & 1 \\ 4 & 2 \end{array} \right) , \left( \begin{array} { r r } 2 & 0 \\ 1 & - 1 \end{array} \right)$ respectively. When the point $( 1,2 )$ is mapped to the point $( a , 6 )$ by the composite transformation $f \circ g$, what is the value of $a$? [3 points] (1) 1 (2) 2 (3) 3 (4) 4 (5) 5
For two events $A , B$, $A ^ { C }$ and $B$ are mutually exclusive events, and $$\mathrm { P } ( A ) = 2 \mathrm { P } ( B ) = \frac { 3 } { 5 }$$ When this condition is satisfied, what is the value of $\mathrm { P } \left( A \cap B ^ { C } \right)$? (Here, $A ^ { C }$ is the complement of $A$.) [3 points] (1) $\frac { 3 } { 20 }$ (2) $\frac { 1 } { 5 }$ (3) $\frac { 1 } { 4 }$ (4) $\frac { 3 } { 10 }$ (5) $\frac { 7 } { 20 }$
For the function $f ( x ) = \frac { 1 } { x }$, what is the value of $\lim _ { n \rightarrow \infty } \sum _ { k = 1 } ^ { n } f \left( 1 + \frac { 2 k } { n } \right) \frac { 2 } { n }$? [3 points] (1) $\ln 6$ (2) $\ln 5$ (3) $2 \ln 2$ (4) $\ln 3$ (5) $\ln 2$
As shown in the figure, a line passes through the focus F of the parabola $y ^ { 2 } = 12 x$ and meets the parabola at two points $\mathrm { A } , \mathrm { B }$. Let C and D be the feet of the perpendiculars from A and B to the directrix $l$ respectively. When $\overline { \mathrm { AC } } = 4$, what is the length of segment BD? [3 points] (1) 12 (2) $\frac { 25 } { 2 }$ (3) 13 (4) $\frac { 27 } { 2 }$ (5) 14
A snack factory produces snacks where the weight of one package follows a normal distribution with mean 75 g and standard deviation 2 g. Using the standard normal distribution table below, what is the probability that the weight of a randomly selected package of snacks from this factory is at least 76 g and at most 78 g? [3 points]
Let $l$ be the line passing through two distinct points $\mathrm { A } , \mathrm { B }$ on plane $\alpha$, and let H be the foot of the perpendicular from point P (not on plane $\alpha$) to plane $\alpha$. When $\overline { \mathrm { AB } } = \overline { \mathrm { PA } } = \overline { \mathrm { PB } } = 6 , \overline { \mathrm { PH } } = 4$, what is the distance between point H and line $l$? [3 points] (1) $\sqrt { 11 }$ (2) $2 \sqrt { 3 }$ (3) $\sqrt { 13 }$ (4) $\sqrt { 14 }$ (5) $\sqrt { 15 }$
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
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 $k$. Let A be the point where the tangent line to the curve $y = 3 ^ { x }$ at point P meets the $x$-axis, and let B be the point where the tangent line to the curve $y = a ^ { x - 1 }$ at point P meets the $x$-axis. For point $\mathrm { H } ( k , 0 )$, when $\overline { \mathrm { AH } } = 2 \overline { \mathrm { BH } }$, what is the value of $a$? [4 points] (1) 6 (2) 7 (3) 8 (4) 9 (5) 10
A survey of 320 students at a school regarding membership in the mathematics club found that 60\% of male students and 50\% of female students joined the mathematics club. Let $p _ { 1 }$ be the probability that a randomly selected student from those who joined the mathematics club is male, and let $p _ { 2 }$ be the probability that a randomly selected student from those who joined the mathematics club is female. When $p _ { 1 } = 2 p _ { 2 }$, what is the number of male students at this school? [4 points] (1) 170 (2) 180 (3) 190 (4) 200 (5) 210
Two square matrices $A , B$ satisfy $$A ^ { 2 } - A B = 3 E , \quad A ^ { 2 } B - B ^ { 2 } A = A + B$$ From the statements below, select all correct ones. (Here, $E$ is the identity matrix.) [4 points] Statements ᄀ. The inverse matrix of $A$ exists. ㄴ. $A B = B A$ ㄷ. $( A + 2 B ) ^ { 2 } = 24 E$ (1) ᄀ (2) ㄷ (3) ᄀ, ㄴ (4) ㄴ, ㄷ (5) ᄀ, ㄴ, ㄷ
The sequence $\left\{ a _ { n } \right\}$ satisfies $a _ { 1 } = 1$, and with $S _ { n } = \sum _ { k = 1 } ^ { n } a _ { k }$, $$a _ { n + 1 } = ( n + 1 ) S _ { n } + n ! \quad ( n \geq 1 )$$ The following is the process of finding the general term $a _ { n }$. For a natural number $n$, since $a _ { n + 1 } = S _ { n + 1 } - S _ { n }$, by the given equation, $$S _ { n + 1 } = ( n + 2 ) S _ { n } + n ! \quad ( n \geq 1 )$$ Dividing both sides by $( n + 2 ) !$, $$\frac { S _ { n + 1 } } { ( n + 2 ) ! } = \frac { S _ { n } } { ( n + 1 ) ! } + \frac { 1 } { ( n + 1 ) ( n + 2 ) }$$ Let $b _ { n } = \frac { S _ { n } } { ( n + 1 ) ! }$. Then $b _ { 1 } = \frac { 1 } { 2 }$ and $$b _ { n + 1 } = b _ { n } + \frac { 1 } { ( n + 1 ) ( n + 2 ) }$$ Finding the general term of the sequence $\left\{ b _ { n } \right\}$, $$b _ { n } = \frac { ( \text{(가)} ) } { n + 1 }$$ Therefore, $$S _ { n } = \text{(가)} \times n!$$ Thus, $$a _ { n } = \text{(나)} \times ( n - 1 ) ! \quad ( n \geq 1 )$$ When the expressions that fit (가) and (나) are $f ( n )$ and $g ( n )$ respectively, what is the value of $f ( 7 ) + g ( 6 )$? [4 points] (1) 44 (2) 41 (3) 38 (4) 35 (5) 32
A bag contains 1 ball with the number 1, 2 balls with the number 2, and 5 balls with the number 3. One ball is randomly drawn from the bag, the number on the ball is confirmed, and then it is returned. This trial is repeated 2 times. Let $\bar { X }$ be the average of the numbers on the drawn balls. What is the value of $\mathrm { P } ( \bar { X } = 2 )$? [4 points] (1) $\frac { 5 } { 32 }$ (2) $\frac { 11 } { 64 }$ (3) $\frac { 3 } { 16 }$ (4) $\frac { 13 } { 64 }$ (5) $\frac { 7 } { 32 }$
In coordinate space, a line $l : \frac { x } { 2 } = 6 - y = z - 6$ and plane $\alpha$ meet perpendicularly at point $\mathrm { P } ( 2,5,7 )$. For a point $\mathrm { A } ( a , b , c )$ on line $l$ and a point Q on plane $\alpha$, when $\overrightarrow { \mathrm { AP } } \cdot \overrightarrow { \mathrm { AQ } } = 6$, what is the value of $a + b + c$? (Here, $a > 0$) [4 points] (1) 15 (2) 16 (3) 17 (4) 18 (5) 19
As shown in the figure, there is an isosceles triangle ABC with $\angle \mathrm { CAB } = \angle \mathrm { BCA } = \theta$ that is externally tangent to a circle of radius 1. On the extension of segment AB, a point D (not equal to A) is chosen such that $\angle \mathrm { DCB } = \theta$. Let the area of triangle BDC be $S ( \theta )$. What is the value of $\lim _ { \theta \rightarrow + 0 } \{ \theta \times S ( \theta ) \}$? (Here, $0 < \theta < \frac { \pi } { 4 }$) [4 points] (1) $\frac { 2 } { 3 }$ (2) $\frac { 8 } { 9 }$ (3) $\frac { 10 } { 9 }$ (4) $\frac { 4 } { 3 }$ (5) $\frac { 14 } { 9 }$
For a natural number $n$, let $a _ { n }$ be the smallest natural number $m$ satisfying the following conditions. What is the value of $\sum _ { n = 1 } ^ { 10 } a _ { n }$? [4 points] (가) The coordinates of point A are $\left( 2 ^ { n } , 0 \right)$. (나) Let D be the point on the line passing through two points $\mathrm { B } ( 1,0 )$ and $\mathrm { C } \left( 2 ^ { m } , m \right)$ whose $x$-coordinate is $2 ^ { n }$. The area of triangle ABD is less than or equal to $\frac { m } { 2 }$. (1) 109 (2) 111 (3) 113 (4) 115 (5) 117
For the irrational equation $x ^ { 2 } - 6 x - \sqrt { x ^ { 2 } - 6 x - 1 } = 3$, let $k$ be the product of all real roots. Find the value of $k ^ { 2 }$. [3 points]
Find the number of all ordered pairs $( a , b , c )$ of natural numbers satisfying the following conditions. [4 points] (가) $a \times b \times c$ is odd. (나) $a \leq b \leq c \leq 20$
When compressing digital images, let $P$ be the peak signal-to-noise ratio, which is an index indicating the degree of difference between the original and compressed images, and let $E$ be the mean squared error between the original and compressed images. The following relationship holds: $$P = 20 \log 255 - 10 \log E \quad ( E > 0 )$$ When two original images $A$ and $B$ are compressed, let the peak signal-to-noise ratios be $P _ { A }$ and $P _ { B }$ respectively, and the mean squared errors be $E _ { A } \left( E _ { A } > 0 \right)$ and $E _ { B } \left( E _ { B } > 0 \right)$ respectively. When $E _ { B } = 100 E _ { A }$, find the value of $P _ { A } - P _ { B }$. [3 points]
For the ellipse $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 4 } = 1$, let F be the focus with positive $x$-coordinate and $\mathrm { F } ^ { \prime }$ be the focus with negative $x$-coordinate. A point P on this ellipse is chosen in the first quadrant such that $\angle \mathrm { FPF } ^ { \prime } = \frac { \pi } { 2 }$, and a point Q with positive $y$-coordinate is chosen on the extension of segment FP such that $\overline { \mathrm { FQ } } = 6$. Find the area of triangle $\mathrm { QF } ^ { \prime } \mathrm { F}$. [4 points]
For a positive number $a$, the function $f ( x ) = \int _ { 0 } ^ { x } ( a - t ) e ^ { t } \, d t$ has a maximum value of 32. Find the area enclosed by the curve $y = 3 e ^ { x }$ and the two lines $x = a$ and $y = 3$. [4 points]
In coordinate space, there is a sphere $S : x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 50$ and a point $\mathrm { P } ( 0,5,5 )$. For all circles $C$ satisfying the following conditions, find the maximum area of the orthogonal projection of $C$ onto the $xy$-plane, expressed as $\frac { q } { p } \pi$. Find the value of $p + q$. (Here, $p$ and $q$ are coprime natural numbers.) [4 points] (가) Circle $C$ is formed by the intersection of a plane passing through point P and the sphere $S$. (나) The radius of circle $C$ is 1.
For the function $f ( x ) = e ^ { x + 1 } - 1$ and a natural number $n$, let the function $g ( x )$ be defined as $$g ( x ) = 100 | f ( x ) | - \sum _ { k = 1 } ^ { n } \left| f \left( x ^ { k } \right) \right|$$ Find the sum of all natural numbers $n$ such that $g ( x )$ is differentiable on the entire set of real numbers. [4 points]