For two matrices $A = \left( \begin{array} { l l } 2 & 1 \\ 5 & 0 \end{array} \right) , B = \left( \begin{array} { l l } 1 & 0 \\ 1 & 1 \end{array} \right)$, what is the sum of all components of matrix $A - B$? [2 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 $\lim _ { x \rightarrow - 1 - 0 } f ( x ) + \lim _ { x \rightarrow + 0 } f ( x )$? [3 points] (1) 1 (2) 2 (3) 3 (4) 4 (5) 5
Let $\bar { X }$ be the sample mean obtained by randomly extracting a sample of size $n$ from a population with population standard deviation 14. When $\sigma ( \bar { X } ) = 2$, what is the value of $n$? [3 points] (1) 9 (2) 16 (3) 25 (4) 36 (5) 49
For a sequence $\left\{ a _ { n } \right\}$, the curve $y = x ^ { 2 } - ( n + 1 ) x + a _ { n }$ intersects the $x$-axis, and the curve $y = x ^ { 2 } - n x + a _ { n }$ does not intersect the $x$-axis. What is the value of $\lim _ { n \rightarrow \infty } \frac { a _ { n } } { n ^ { 2 } }$? [3 points] (1) $\frac { 1 } { 20 }$ (2) $\frac { 1 } { 10 }$ (3) $\frac { 3 } { 20 }$ (4) $\frac { 1 } { 5 }$ (5) $\frac { 1 } { 4 }$
For the logarithmic inequality in $x$ $$\log _ { 5 } ( x - 1 ) \leq \log _ { 5 } \left( \frac { 1 } { 2 } x + k \right)$$ when the number of all integers $x$ satisfying this inequality is 3, what is the value of the natural number $k$? [3 points] (1) 1 (2) 2 (3) 3 (4) 4 (5) 5
At a rice collection event, the weight of rice donated by each student follows a normal distribution with mean 1.5 kg and standard deviation 0.2 kg. When one student is randomly selected from those who participated in the event, what is the probability that the weight of rice donated by this student is at least 1.3 kg and at most 1.8 kg, using the standard normal distribution table below? [3 points]
For a natural number $n$, let P be the point with coordinates $( 0,2 n + 1 )$, and let Q be the point on the graph of the function $f ( x ) = n x ^ { 2 }$ with $y$-coordinate 1 in the first quadrant. When $n = 1$, what is the area of the region enclosed by the line segment PQ, the curve $y = f ( x )$, and the $y$-axis? [3 points] (1) $\frac { 3 } { 2 }$ (2) $\frac { 19 } { 12 }$ (3) $\frac { 5 } { 3 }$ (4) $\frac { 7 } { 4 }$ (5) $\frac { 11 } { 6 }$
For a natural number $n$, let P be the point with coordinates $( 0,2 n + 1 )$, and let Q be the point on the graph of the function $f ( x ) = n x ^ { 2 }$ with $y$-coordinate 1 in the first quadrant. For the point $\mathrm { R } ( 0,1 )$, let $S _ { n }$ be the area of triangle PRQ and $l _ { n }$ be the length of line segment PQ. What is the value of $\lim _ { n \rightarrow \infty } \frac { S _ { n } ^ { 2 } } { l _ { n } }$? [4 points] (1) $\frac { 3 } { 2 }$ (2) $\frac { 5 } { 4 }$ (3) 1 (4) $\frac { 3 } { 4 }$ (5) $\frac { 1 } { 2 }$
For a certain financial product, the expected asset $W$ after $t$ years of investing an initial asset $W _ { 0 }$ is given as follows: $$W = \frac { W _ { 0 } } { 2 } 10 ^ { a t } \left( 1 + 10 ^ { a t } \right)$$ (where $W _ { 0 } > 0 , t \geq 0$, and $a$ is a constant.) When an initial asset $w _ { 0 }$ is invested in this financial product, the expected asset after 15 years is 3 times the initial asset. When an initial asset $w _ { 0 }$ is invested in this financial product, the expected asset after 30 years is $k$ times the initial asset. What is the value of the real number $k$? (where $w _ { 0 } > 0$) [4 points] (1) 9 (2) 10 (3) 11 (4) 12 (5) 13
How many ordered pairs $( a , b , c , d , e )$ of non-negative integers satisfy the following conditions? [4 points] (가) Among $a , b , c , d , e$, the number of 0's is 2. (나) $a + b + c + d + e = 10$ (1) 240 (2) 280 (3) 320 (4) 360 (5) 400
A sequence $\left\{ a _ { n } \right\}$ with all positive terms has $a _ { 1 } = a _ { 2 } = 1$, and when $S _ { n } = \sum _ { k = 1 } ^ { n } a _ { k }$, $$a _ { n + 1 } = \frac { S _ { n } ^ { 2 } } { S _ { n - 1 } } + ( 2 n - 1 ) S _ { n } \quad ( n \geq 2 )$$ The following is the process of finding the general term $a _ { n }$. Since $a _ { n + 1 } = S _ { n + 1 } - S _ { n }$, from the given equation we have $$S _ { n + 1 } = \frac { S _ { n } ^ { 2 } } { S _ { n - 1 } } + 2 n S _ { n } \quad ( n \geq 2 )$$ Dividing both sides by $S _ { n }$, $$\frac { S _ { n + 1 } } { S _ { n } } = \frac { S _ { n } } { S _ { n - 1 } } + 2 n$$ Let $b _ { n } = \frac { S _ { n + 1 } } { S _ { n } }$. Then $b _ { 1 } = 2$ and $$b _ { n } = b _ { n - 1 } + 2 n \quad ( n \geq 2 )$$ Finding the general term of the sequence $\left\{ b _ { n } \right\}$, $$b _ { n } = \text { (가) } \times ( n + 1 ) \quad ( n \geq 1 )$$ Therefore, $$S _ { n } = ( \text{가} ) \times \{ ( n - 1 ) ! \} ^ { 2 } \quad ( n \geq 1 )$$ Thus $a _ { 1 } = 1$, and for $n \geq 2$, $$\begin{aligned} a _ { n } & = S _ { n } - S _ { n - 1 } \\ & = \text { (나) } \times \{ ( n - 2 ) ! \} ^ { 2 } \end{aligned}$$ When the expressions for (가) and (나) are $f ( n )$ and $g ( n )$ respectively, what is the value of $f ( 10 ) + g ( 6 )$? [4 points] (1) 110 (2) 125 (3) 140 (4) 155 (5) 170
Two polynomial functions $f ( x ) , g ( x )$ satisfy for all real numbers $x$ $$f ( - x ) = - f ( x ) , \quad g ( - x ) = g ( x )$$ For the function $h ( x ) = f ( x ) g ( x )$, $$\int _ { - 3 } ^ { 3 } ( x + 5 ) h ^ { \prime } ( x ) d x = 10$$ What is the value of $h ( 3 )$? [4 points] (1) 1 (2) 2 (3) 3 (4) 4 (5) 5
For all cubic functions $f ( x )$ satisfying $f ( 0 ) = 0$ and the following conditions, let $M$ be the maximum value and $m$ be the minimum value of $\frac { f ^ { \prime } ( 0 ) } { f ( 0 ) }$. What is the value of $M m$? [4 points] (가) The function $| f ( x ) |$ is not differentiable only at $x = - 1$. (나) The equation $f ( x ) = 0$ has at least one real root in the closed interval $[ 3,5 ]$. (1) $\frac { 1 } { 15 }$ (2) $\frac { 1 } { 10 }$ (3) $\frac { 2 } { 15 }$ (4) $\frac { 1 } { 6 }$ (5) $\frac { 1 } { 5 }$
For an arithmetic sequence $\left\{ a _ { n } \right\}$, when $a _ { 8 } - a _ { 4 } = 28$, find the common difference of the sequence $\left\{ a _ { n } \right\}$. [3 points]
For the system of linear equations in $x , y$ $$\left( \begin{array} { c c } 1 & a - 2 \\ 2 & - 1 \end{array} \right) \binom { x } { y } = 3 \binom { x } { y }$$ Find the value of the constant $a$ such that the system has a solution other than $x = 0 , y = 0$. [3 points]
A company has a total of 60 employees, and each employee belongs to one of two departments, A or B. Department A has 20 employees and Department B has 40 employees. 50\% of the employees in Department A are women. 60\% of the women employees in the company belong to Department B. When one employee is randomly selected from the 60 employees and is found to belong to Department B, the probability that this employee is a woman is $p$. Find the value of $80p$. [4 points]
Two functions $$f ( x ) = \left\{ \begin{array} { l l } x + 3 & ( x \leq a ) \\ x ^ { 2 } - x & ( x > a ) \end{array} , \quad g ( x ) = x - ( 2 a + 7 ) \right.$$ Find the product of all real values of $a$ such that the function $f ( x ) g ( x )$ is continuous on the entire set of real numbers. [4 points]
Two polynomial functions $f ( x ) , g ( x )$ satisfy the following conditions. (가) $g ( x ) = x ^ { 3 } f ( x ) - 7$ (나) $\lim _ { x \rightarrow 2 } \frac { f ( x ) - g ( x ) } { x - 2 } = 2$ When the equation of the tangent line to the curve $y = g ( x )$ at the point $( 2 , g ( 2 ) )$ is $y = a x + b$, find the value of $a ^ { 2 } + b ^ { 2 }$. (where $a , b$ are constants.) [4 points]
A quadratic function $f ( x )$ satisfies $f ( 0 ) = 0$ and the following conditions. (가) $\int _ { 0 } ^ { 2 } | f ( x ) | d x = - \int _ { 0 } ^ { 2 } f ( x ) d x = 4$ (나) $\int _ { 2 } ^ { 3 } | f ( x ) | d x = \int _ { 2 } ^ { 3 } f ( x ) d x$ Find the value of $f ( 5 )$. [4 points]
For a real number $x \geq \frac { 1 } { 100 }$, let $f ( x )$ be the mantissa of $\log x$. Let $R$ be the region representing the ordered pairs $( a , b )$ of two real numbers satisfying the following conditions on the coordinate plane. (가) $a < 0$ and $b > 10$. (나) The graph of the function $y = 9 f ( x )$ and the line $y = a x + b$ meet at exactly one point. For a point $( a , b )$ in region $R$, the minimum value of $( a + 20 ) ^ { 2 } + b ^ { 2 }$ is $100 \times \frac { q } { p }$. Find the value of $p + q$. (where $p$ and $q$ are coprime natural numbers.) [4 points]