For two matrices $A = \left( \begin{array} { l l } 0 & 0 \\ 1 & 1 \end{array} \right) , B = \left( \begin{array} { l l } 1 & 0 \\ 1 & 1 \end{array} \right)$, what is the sum of all components of the matrix $2 A + B$? [2 points] (1) 10 (2) 9 (3) 8 (4) 7 (5) 6
In coordinate space, for two points $\mathrm { A } ( a , 1,3 ) , \mathrm { B } ( a + 6,4,12 )$, the point that divides the line segment AB internally in the ratio $1 : 2$ has coordinates $( 5,2 , b )$. What is the value of $a + b$? [2 points] (1) 7 (2) 8 (3) 9 (4) 10 (5) 11
As shown in the figure, there is a road network connected in a diamond shape. Starting from point A and traveling the shortest distance to point B without passing through point C or point D, how many ways are there? [3 points] (1) 26 (2) 24 (3) 22 (4) 20 (5) 18
The temperature of a fire room changes over time. Let the initial temperature of a certain fire room be $T _ { 0 } \left( { } ^ { \circ } \mathrm { C } \right)$, and the temperature $t$ minutes after the fire starts be $T \left( { } ^ { \circ } \mathrm { C } \right)$. The following equation holds. $T = T _ { 0 } + k \log ( 8 t + 1 ) \quad ($ where $k$ is a constant. $)$ In this fire room with an initial temperature of $20 ^ { \circ } \mathrm { C }$, the temperature was $365 ^ { \circ } \mathrm { C }$ at $\frac { 9 } { 8 }$ minutes after the fire started, and the temperature was $710 ^ { \circ } \mathrm { C }$ at $a$ minutes after the fire started. What is the value of $a$? [3 points] (1) $\frac { 99 } { 8 }$ (2) $\frac { 109 } { 8 }$ (3) $\frac { 119 } { 8 }$ (4) $\frac { 129 } { 8 }$ (5) $\frac { 139 } { 8 }$
The tangent line at the point $( b , 1 )$ on the hyperbola $x ^ { 2 } - 4 y ^ { 2 } = a$ is perpendicular to one asymptote of the hyperbola. What is the value of $a + b$? (Given that $a , b$ are positive numbers.) [3 points] (1) 68 (2) 77 (3) 86 (4) 95 (5) 104
At a certain school, $60 \%$ of all students commute by bus, and the remaining $40 \%$ walk to school. Of the students who commute by bus, $\frac { 1 } { 20 }$ were late, and of the students who walk, $\frac { 1 } { 15 }$ were late. When one student is randomly selected from all students at this school and is found to be late, what is the probability that this student commuted by bus? [3 points] (1) $\frac { 3 } { 7 }$ (2) $\frac { 9 } { 20 }$ (3) $\frac { 9 } { 19 }$ (4) $\frac { 1 } { 2 }$ (5) $\frac { 9 } { 17 }$
In the coordinate plane, let $f$ be the rotation transformation that rotates by $\frac { \pi } { 3 }$ about the origin, and let $g$ be the reflection transformation about the line $y = x$. When the line $x + 2 y + 5 = 0$ is mapped to the line $a x + b y + 5 = 0$ by the composite transformation $g ^ { - 1 } \circ f \circ g$, what is the value of $a + 2 b$? (Given that $a , b$ are constants.) [3 points] (1) $\frac { 1 } { 2 }$ (2) 1 (3) $\frac { 3 } { 2 }$ (4) 2 (5) $\frac { 5 } { 2 }$
Starting from point A, one travels to point B which is 6 km away, and then returns to point A along the same route. For the first 1 km, one walks at a constant speed, and for the remaining 5 km, one travels at twice the initial walking speed. On the return trip, one travels at a speed 2 km/h faster than the initial walking speed. When the total time for the round trip is at most 2 hours 30 minutes, what is the minimum value of the initial walking speed? (Given that the unit of speed is km/h.) [3 points] (1) $\frac { 12 } { 5 }$ (2) $\frac { 13 } { 5 }$ (3) $\frac { 14 } { 5 }$ (4) 3 (5) $\frac { 16 } { 5 }$
There is a bag containing 4 white balls and 3 black balls. Two balls are drawn simultaneously from the bag. If the two balls are of different colors, one coin is flipped 3 times. If the two balls are of the same color, one coin is flipped 2 times. What is the probability that the coin shows heads exactly 2 times in this trial? [3 points] (1) $\frac { 9 } { 28 }$ (2) $\frac { 19 } { 56 }$ (3) $\frac { 5 } { 14 }$ (4) $\frac { 3 } { 8 }$ (5) $\frac { 11 } { 28 }$
A continuous function $f ( x )$ satisfies $$f ( x ) = e ^ { x ^ { 2 } } + \int _ { 0 } ^ { 1 } t f ( t ) d t$$ What is the value of $\int _ { 0 } ^ { 1 } x f ( x ) d x$? [3 points] (1) $e - 2$ (2) $\frac { e - 1 } { 2 }$ (3) $\frac { e } { 2 }$ (4) $e - 1$ (5) $\frac { e + 1 } { 2 }$
A random variable $X$ follows a normal distribution $\mathrm { N } \left( m , \sigma ^ { 2 } \right)$ and satisfies the following conditions. (a) $\mathrm { P } ( X \geq 64 ) = \mathrm { P } ( X \leq 56 )$ (b) $\mathrm { E } \left( X ^ { 2 } \right) = 3616$ What is the value of $\mathrm { P } ( X \leq 68 )$ obtained using the table on the right? [3 points] (1) 0.9104 (2) 0.9332 (3) 0.9544 (4) 0.9772 (5) 0.9938
The graph of a function $y = f ( x )$ defined on all real numbers is as shown in the figure, and a cubic function $g ( x )$ has leading coefficient 1 and $g ( 0 ) = 3$. When the composite function $( g \circ f ) ( x )$ is continuous on all real numbers, what is the value of $g ( 3 )$? [4 points] (1) 31 (2) 30 (3) 29 (4) 28 (5) 27
Two $2 \times 2$ square matrices $A , B$ satisfy $$2 A ^ { 2 } + A B = E , \quad A B + B A = 2 A + E$$ Which of the following statements are correct? Choose all that apply from . (Given that $E$ is the identity matrix.) [4 points]
ㄱ. $A ^ { - 1 } = 2 A + B$ ㄴ. $B = 2 A + 2 E$ ㄷ. $( B - E ) ^ { 2 } = O$ (where $O$ is the zero matrix.) (1) ㄴ (2) ㄷ (3) ㄱ, ㄴ (4) ㄱ, ㄷ (5) ㄱ, ㄴ, ㄷ
The sequence $\left\{ a _ { n } \right\}$ has $a _ { 1 } = 4$ and satisfies $$a _ { n + 1 } = n \cdot 2 ^ { n } + \sum _ { k = 1 } ^ { n } \frac { a _ { k } } { k } \quad ( n \geq 1 )$$ The following is the process of finding the general term $a _ { n }$. From the given equation, $$a _ { n } = ( n - 1 ) \cdot 2 ^ { n - 1 } + \sum _ { k = 1 } ^ { n - 1 } \frac { a _ { k } } { k } \quad ( n \geq 2 )$$ Therefore, for natural numbers $n \geq 2$, $$a _ { n + 1 } - a _ { n } = \text { (a) } + \frac { a _ { n } } { n }$$ so $$a _ { n + 1 } = \frac { ( n + 1 ) a _ { n } } { n } + \text { (a) }$$ If $b _ { n } = \frac { a _ { n } } { n }$, then $$b _ { n + 1 } = b _ { n } + \frac { ( \text { a } ) } { n + 1 } ( n \geq 2 )$$ and since $b _ { 2 } = 3$, $$b _ { n } = \text { (b) } \quad ( n \geq 2 )$$ Therefore, $$a _ { n } = \left\{ \begin{array} { c c }
4 & ( n = 1 ) \\
n \times ( \boxed { ( \text{b} ) } ) & ( n \geq 2 )
\end{array} \right.$$ Let $f ( n )$ and $g ( n )$ be the expressions that fit (a) and (b), respectively. What is the value of $f ( 4 ) + g ( 7 )$? [4 points] (1) 90 (2) 95 (3) 100 (4) 105 (5) 110
For a natural number $n$, a line passing through the focus F of the parabola $y ^ { 2 } = \frac { x } { n }$ intersects the parabola at two points P and Q, respectively. If $\overline { \mathrm { PF } } = 1$ and $\overline { \mathrm { FQ } } = a _ { n }$, what is the value of $\sum _ { n = 1 } ^ { 10 } \frac { 1 } { a _ { n } }$? [4 points] (1) 210 (2) 205 (3) 200 (4) 195 (5) 190
A cubic function $f ( x )$ satisfies $f ( 0 ) > 0$. Define the function $g ( x )$ as $$g ( x ) = \left| \int _ { 0 } ^ { x } f ( t ) d t \right|$$ The graph of the function $y = g ( x )$ is as shown in the figure. Which of the following statements are correct? Choose all that apply. [4 points]
ㄱ. The equation $f ( x ) = 0$ has three distinct real roots. ㄴ. $f ^ { \prime } ( 0 ) < 0$ ㄷ. The number of natural numbers $m$ satisfying $\int _ { m } ^ { m + 2 } f ( x ) d x > 0$ is 3. (1) ㄴ (2) ㄷ (3) ㄱ, ㄴ (4) ㄱ, ㄷ (5) ㄱ, ㄴ, ㄷ
In coordinate space, one face ABC of a regular tetrahedron ABCD lies on the plane $2 x - y + z = 4$, and the vertex D lies on the plane $x + y + z = 3$. When the centroid of triangle ABC has coordinates $( 1,1,3 )$, what is the length of one edge of the regular tetrahedron ABCD? [4 points] (1) $2 \sqrt { 2 }$ (2) 3 (3) $2 \sqrt { 3 }$ (4) 4 (5) $3 \sqrt { 2 }$
For the function $f ( x ) = k x ^ { 2 } e ^ { - x } ( k > 0 )$ and a real number $t$, let $g ( t )$ be the smaller of the distance from the point $( t , f ( t ) )$ on the curve $y = f ( x )$ to the $x$-axis and the distance to the $y$-axis. What is the maximum value of $k$ such that the function $g ( t )$ is not differentiable at exactly one point? [4 points] (1) $\frac { 1 } { e }$ (2) $\frac { 1 } { \sqrt { e } }$ (3) $\frac { e } { 2 }$ (4) $\sqrt { e }$ (5) $e$
Find the maximum value $a$ of the function $f ( x ) = 2 \cos \left( x - \frac { \pi } { 3 } \right) + 2 \sqrt { 3 } \sin x$. Find the value of $a ^ { 2 }$. [3 points]
Let $A$ be the matrix representing the linear transformation $f : ( x , y ) \rightarrow ( 2 x - y , x - 2 y )$. When the point $( 5 , - 1 )$ is mapped to the point $( a , b )$ by the linear transformation represented by the matrix $A ^ { 4 }$, find the value of $a + b$. [3 points]
In an equilateral triangle ABC with side length 2, let H be the foot of the perpendicular from vertex A to side BC. When point P moves on line segment AH, find the maximum value of $| \overrightarrow { \mathrm { PA } } \cdot \overrightarrow { \mathrm { PB } } |$, which is $\frac { q } { p }$. Find the value of $p + q$. (Given that $p$ and $q$ are coprime natural numbers.)
For a normal distribution with known standard deviation $\sigma$, a sample of size $n$ is randomly extracted from the population. The 95\% confidence interval for the population mean obtained from this sample is [100.4, 139.6]. Using the same sample, find the number of natural numbers contained in the 99\% confidence interval for the population mean. (Given that when $Z$ is a random variable following the standard normal distribution, $\mathrm { P } ( 0 \leq Z \leq 1.96 ) = 0.475$ and $\mathrm { P } ( 0 \leq Z \leq 2.58 ) = 0.495$.) [3 points]
For a natural number $n$, the point $\mathrm { P } _ { n }$ on the coordinate plane is determined according to the following rules. (a) The coordinates of the three points $\mathrm { P } _ { 1 } , \mathrm { P } _ { 2 } , \mathrm { P } _ { 3 }$ are $( - 1,0 ) , ( 1,0 )$, and $( - 1,2 )$, respectively. (b) The midpoint of line segment $\mathrm { P } _ { n } \mathrm { P } _ { n + 1 }$ and the midpoint of line segment $\mathrm { P } _ { n + 2 } \mathrm { P } _ { n + 3 }$ are the same. For example, the coordinates of point $\mathrm { P } _ { 4 }$ are $( 1 , - 2 )$. When the coordinates of point $\mathrm { P } _ { 25 }$ are $( a , b )$, find the value of $a + b$. [4 points]
As shown in the figure, there is a rectangular piece of paper ABCD with $\overline { \mathrm { AB } } = 9$ and $\overline { \mathrm { AD } } = 3$. Using the line connecting point E on segment AB and point F on segment DC as the fold line, the paper is folded so that the orthogonal projection of point B onto the plane AEFD is point D. When $\overline { \mathrm { AE } } = 3$, the angle between the two planes AEFD and EFCB is $\theta$. Find the value of $60 \cos \theta$. (Given that $0 < \theta < \frac { \pi } { 2 }$ and the thickness of the paper is negligible.) [4 points]
In triangle ABC, $\overline { \mathrm { AB } } = 1$, $\angle \mathrm { A } = \theta$, and $\angle \mathrm { B } = 2 \theta$. Point D on side AB is chosen so that $\angle \mathrm { ACD } = 2 \angle \mathrm { BCD }$. When $\lim _ { \theta \rightarrow + 0 } \frac { \overline { \mathrm { CD } } } { \theta } = a$, find the value of $27 a ^ { 2 }$. (Given that $0 < \theta < \frac { \pi } { 4 }$.) [4 points]
In the coordinate plane, for a natural number $n$, let $a _ { n }$ be the number of points in the region $$\left\{ ( x , y ) \mid 2 ^ { x } - n \leq y \leq \log _ { 2 } ( x + n ) \right\}$$ that satisfy the following conditions. (a) The $x$-coordinate and $y$-coordinate are equal. (b) The $x$-coordinate and $y$-coordinate are both integers. For example, $a _ { 1 } = 2$ and $a _ { 2 } = 4$. Find the value of $\sum _ { n = 1 } ^ { 30 } a _ { n }$. [4 points]