For two matrices $A = \left( \begin{array} { l l } 2 & 0 \\ 1 & 1 \end{array} \right) , B = \left( \begin{array} { l l } 1 & 1 \\ 0 & 1 \end{array} \right)$, what is the sum of all components of matrix $A + 2B$? [2 points]
Q5
3 marksDifferentiation from First PrinciplesView
For the function $f ( x ) = 2 x ^ { 2 } + a x$, when $\lim _ { h \rightarrow 0 } \frac { f ( 1 + h ) - f ( 1 ) } { h } = 6$, what is the value of the constant $a$? [3 points] (1) $-4$ (2) $-2$ (3) $0$ (4) $2$ (5) $4$
For an arithmetic sequence $\left\{ a _ { n } \right\}$ with first term 6 and common difference $d$, let $S _ { n }$ denote the sum of the first $n$ terms. When $$\frac { a _ { 8 } - a _ { 6 } } { S _ { 8 } - S _ { 6 } } = 2$$ holds, what is the value of $d$? [3 points] (1) $-1$ (2) $-2$ (3) $-3$ (4) $-4$ (5) $-5$
What is the area of the region enclosed by the curve $y = x ^ { 2 } - 4 x + 3$ and the line $y = 3$? [3 points] (1) 10 (2) $\frac { 31 } { 3 }$ (3) $\frac { 32 } { 3 }$ (4) 11 (5) $\frac { 34 } { 3 }$
A pharmaceutical company produces medicine bottles with capacity following a normal distribution with mean $m$ and standard deviation 10. When a random sample of 25 bottles is taken from the company's production, the probability that the sample mean capacity is at least 2000 is 0.9772. Using the standard normal distribution table below, what is the value of $m$? (Here, the unit of capacity is mL.) [3 points]
For a natural number $n$, $f ( n )$ is defined as follows: $$f ( n ) = \begin{cases} \log _ { 3 } n & ( n \text{ is odd} ) \\ \log _ { 2 } n & ( n \text{ is even} ) \end{cases}$$ For the sequence $\left\{ a _ { n } \right\}$ where $a _ { n } = f \left( 6 ^ { n } \right) - f \left( 3 ^ { n } \right)$, what is the value of $\sum _ { n = 1 } ^ { 15 } a _ { n }$? [3 points] (1) $120 \left( \log _ { 2 } 3 - 1 \right)$ (2) $105 \log _ { 3 } 2$ (3) $105 \log _ { 2 } 3$ (4) $120 \log _ { 2 } 3$ (5) $120 \left( \log _ { 3 } 2 + 1 \right)$
For a natural number $n$, $f ( n )$ is defined as follows: $$f ( n ) = \begin{cases} \log _ { 3 } n & ( n \text{ is odd} ) \\ \log _ { 2 } n & ( n \text{ is even} ) \end{cases}$$ For two natural numbers $m , n$ with $m, n \leq 20$, how many ordered pairs $( m , n )$ satisfy $f ( m n ) = f ( m ) + f ( n )$? (1) 220 (2) 230 (3) 240 (4) 250 (5) 260
Bag A contains 2 white balls and 3 black balls, and Bag B contains 1 white ball and 3 black balls. One ball is randomly drawn from Bag A. If it is white, 2 white balls are put into Bag B; if it is black, 2 black balls are put into Bag B. Then one ball is randomly drawn from Bag B. What is the probability that the drawn ball is white? [4 points] (1) $\frac { 1 } { 6 }$ (2) $\frac { 1 } { 5 }$ (3) $\frac { 7 } { 30 }$ (4) $\frac { 4 } { 15 }$ (5) $\frac { 3 } { 10 }$
A sequence $\left\{ a _ { n } \right\}$ with all positive terms has $a _ { 1 } = 10$ and satisfies $$\left( a _ { n + 1 } \right) ^ { n } = 10 \left( a _ { n } \right) ^ { n + 1 } \quad ( n \geq 1 )$$ The following is the process of finding the general term $a _ { n }$. Taking the common logarithm of both sides of the given equation: $$n \log a _ { n + 1 } = ( n + 1 ) \log a _ { n } + 1$$ Dividing both sides by $n ( n + 1 )$: $$\frac { \log a _ { n + 1 } } { n + 1 } = \frac { \log a _ { n } } { n } + ( \text{(가)} )$$ Let $b _ { n } = \frac { \log a _ { n } } { n }$. Then $b _ { 1 } = 1$ and $$b _ { n + 1 } = b _ { n } + \text{(가)}$$ Finding the general term of the sequence $\left\{ b _ { n } \right\}$: $$b _ { n } = \text{(나)}$$ Therefore, $$\log a _ { n } = n \times \text{(나)}$$ Thus $a _ { n } = 10 ^ { n \times \text{(나)} }$. Let $f ( n )$ and $g ( n )$ be the expressions that fit in (가) and (나), respectively. What is the value of $\frac { g ( 10 ) } { f ( 4 ) }$? [4 points] (1) 38 (2) 40 (3) 42 (4) 44 (5) 46
There are 8 white ping-pong balls and 7 orange ping-pong balls to be distributed entirely among 3 students. In how many ways can the balls be distributed so that each student receives at least one white ball and at least one orange ball? [4 points] (1) 295 (2) 300 (3) 305 (4) 310 (5) 315
Two square matrices $A , B$ satisfy $$A B + A ^ { 2 } B = E , \quad ( A - E ) ^ { 2 } + B ^ { 2 } = O$$ Which of the following statements in the given options are correct? (Here, $E$ is the identity matrix and $O$ is the zero matrix.) [4 points] Options $\text{ᄀ}$. The inverse matrix of $B$ exists. $\text{ㄴ}$. $A B = B A$ $\text{ㄷ}$. $\left( A ^ { 3 } - A \right) ^ { 2 } + E = O$ (1) ㄴ (2) ㄷ (3) ᄀ, ㄴ (4) ᄀ, ㄷ (5) ᄀ, ㄴ, ㄷ
For a positive real number $x$, let $f ( x )$ and $g ( x )$ be the characteristic and mantissa of $\log x$, respectively. For a natural number $n$, let $a _ { n }$ be the product of all values of $x$ satisfying $f ( x ) - ( n + 1 ) g ( x ) = n$. What is the value of $\lim _ { n \rightarrow \infty } \frac { \log a _ { n } } { n ^ { 2 } }$? [4 points] (1) 1 (2) $\frac { 3 } { 2 }$ (3) 2 (4) $\frac { 5 } { 2 }$ (5) 3
On the coordinate plane, for a cubic function $f ( x ) = x ^ { 3 } + a x ^ { 2 } + b x$ and a real number $t$, let P be the point where the tangent line to the curve $y = f ( x )$ at the point $( t , f ( t ) )$ intersects the $y$-axis. Let $g ( t )$ be the distance from the origin to point P. The function $f ( x )$ and the function $g ( t )$ satisfy the following conditions. (가) $f ( 1 ) = 2$ (나) The function $g ( t )$ is differentiable on the entire set of real numbers. What is the value of $f ( 3 )$? (Here, $a , b$ are constants.) [4 points] (1) 21 (2) 24 (3) 27 (4) 30 (5) 33
A sequence $\left\{ a _ { n } \right\}$ satisfies the following conditions. (가) $a _ { 1 } = a _ { 2 } + 3$ (나) $a _ { n + 1 } = - 2 a _ { n } ( n \geq 1 )$ Find the value of $a _ { 9 }$. [3 points]
For the system of linear equations in $x , y$: $$\left( \begin{array} { l l }
5 & a \\
a & 3
\end{array} \right) \binom { x } { y } = \binom { x + 5 y } { 6 x + y }$$ Find the sum of all real values of $a$ such that the system has a solution other than $x = 0 , y = 0$. [4 points]
The function $f ( x ) = 2 x ^ { 3 } - 12 x ^ { 2 } + a x - 4$ has a local maximum value $M$ at $x = 1$. Find the value of $a + M$. (Here, $a$ is a constant.) [3 points]
There are 5 drawers, each labeled with a natural number from 1 to 5. Two drawers are randomly assigned to Younghee. Let $X$ be the random variable representing the smaller of the two natural numbers on the assigned drawers. Find the value of $\mathrm { E } ( 10 X )$. [4 points]
For the function $$f ( x ) = \begin{cases} x + 1 & ( x \leq 0 ) \\ - \frac { 1 } { 2 } x + 7 & ( x > 0 ) \end{cases}$$ Find the sum of all real values of $a$ such that the function $f ( x ) f ( x - a )$ is continuous at $x = a$. [4 points]
The function $f ( x ) = 3 x ^ { 2 } - a x$ satisfies $$\lim _ { n \rightarrow \infty } \frac { 1 } { n } \sum _ { k = 1 } ^ { n } f \left( \frac { 3 k } { n } \right) = f ( 1 )$$ Find the value of the constant $a$. [4 points]
On the coordinate plane, for a natural number $a > 1$, consider the two curves $y = 4 ^ { x } , y = a ^ { - x + 4 }$ and the line $y = 1$. Find the number of values of $a$ such that the number of points with integer coordinates inside or on the boundary of the region enclosed by these curves and line is between 20 and 40 (inclusive). [4 points]