csat-suneung

Papers (39)
2026
csat__math 30
2025
csat__math 43
2024
csat__math 44
2023
csat__math 30
2022
csat__math 41
2021
csat__math-humanities 30 csat__math-science 29
2020
csat__math-humanities 30 csat__math-science 30
2019
csat__math-humanities 30 csat__math-science 30
2018
csat__math-humanities 30 csat__math-science 30
2017
csat__math-humanities 30 csat__math-science 30
2016
csat__math-A 30 csat__math-B 30
2015
csat__math-A 30 csat__math-B 30
2014
csat__math-A 30 csat__math-B 30
2013
csat__math-humanities 30 csat__math-science 30
2012
csat__math-humanities 29 csat__math-science 30
2011
csat__math-humanities 30 csat__math-science 36
2010
csat__math-humanities 27 csat__math-science 33
2009
csat__math-humanities 27 csat__math-science 23
2008
csat__math-humanities 29 csat__math-science 14
2007
csat__math-humanities 30 csat__math-science 17
2006
csat__math-humanities 30 csat__math-science 26
2005
csat__math-humanities 27 csat__math-science 32
2006 csat__math-humanities

30 maths questions

Q1 2 marks Indices and Surds Evaluating Expressions Using Index Laws View
What is the value of $5 ^ { \frac { 2 } { 3 } } \times 25 ^ { - \frac { 5 } { 6 } }$? [2 points]
(1) $\frac { 1 } { 25 }$
(2) $\frac { 1 } { 5 }$
(3) 1
(4) 5
(5) 25
Q2 2 marks Matrices Matrix Algebra and Product Properties View
For two matrices $A = \left( \begin{array} { l l } 1 & 1 \\ 1 & 0 \end{array} \right) , B = \left( \begin{array} { l l } 1 & 2 \\ 3 & 4 \end{array} \right)$, what is the matrix $X$ that satisfies $2 A + X = A B$? [2 points]
(1) $\left( \begin{array} { r r } 1 & 5 \\ 3 & - 1 \end{array} \right)$
(2) $\left( \begin{array} { r r } 2 & 4 \\ - 1 & 2 \end{array} \right)$
(3) $\left( \begin{array} { l l } 2 & 5 \\ 7 & 0 \end{array} \right)$
(4) $\left( \begin{array} { l l } 2 & 7 \\ 4 & 5 \end{array} \right)$
(5) $\left( \begin{array} { l l } 4 & 6 \\ 1 & 2 \end{array} \right)$
Q3 2 marks Arithmetic Sequences and Series Find Specific Term from Given Conditions View
For an arithmetic sequence $\left\{ a _ { n } \right\}$, $$a _ { 5 } = 4 a _ { 3 } , \quad a _ { 2 } + a _ { 4 } = 4$$ When these conditions hold, what is the value of $a _ { 6 }$? [2 points]
(1) 5
(2) 8
(3) 11
(4) 13
(5) 16
Q4 3 marks Probability Definitions Probability Using Set/Event Algebra View
For two events $A$ and $B$ in the sample space $S$, if they are mutually exclusive events, $A \cup B = S$, and $\mathrm { P } ( A ) = 2 \mathrm { P } ( B )$, what is the value of $\mathrm { P } ( A )$? [3 points]
(1) $\frac { 2 } { 3 }$
(2) $\frac { 1 } { 2 }$
(3) $\frac { 2 } { 5 }$
(4) $\frac { 1 } { 3 }$
(5) $\frac { 1 } { 4 }$
Q5 3 marks Binomial Distribution Compute Expectation, Variance, or Standard Deviation View
When a random variable $X$ follows a binomial distribution $\mathrm { B } \left( 100 , \frac { 1 } { 5 } \right)$, what is the standard deviation of the random variable $3 X - 4$? [3 points]
(1) 12
(2) 15
(3) 18
(4) 21
(5) 24
Q6 3 marks Matrices Matrix Algebra and Product Properties View
For all non-zero $2 \times 2$ square matrices $A , B$ satisfying the following three conditions, which matrix is always equal to $B ^ { 3 } + 2 B A ^ { 3 }$? (Here, $E$ is the identity matrix.) [3 points] (가) $A B = B A$ (나) $( E - B ) ^ { 2 } = E - B$ (다) $A B = - B$
(1) $2 A$
(2) $- A$
(3) $E$
(4) $2 B$
(5) $- B$
Q7 3 marks Sequences and series, recurrence and convergence Multiple-choice on sequence properties View
When a sequence $\left\{ a _ { n } \right\}$ satisfies $n < a _ { n } < n + 1$ for all natural numbers $n$, what is the value of $\lim _ { n \rightarrow \infty } \frac { n ^ { 2 } } { a _ { 1 } + a _ { 2 } + \cdots + a _ { n } }$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5
Q8 3 marks Continuous Probability Distributions and Random Variables PDF Graph Interpretation and Probability Computation View
A continuous random variable $X$ has a range of $0 \leqq X \leqq 3$, and the graph of its probability density function is as follows. When $\mathrm { P } ( m \leqq X \leqq 2 ) = \mathrm { P } ( 2 \leqq X \leqq 3 )$, what is the value of $m$? (Here, $0 < m < 2$.) [3 points]
(1) $\frac { \sqrt { 2 } } { 2 }$
(2) $\frac { \sqrt { 3 } } { 2 }$
(3) 1
(4) $\sqrt { 2 }$
(5) $\sqrt { 3 }$
Q9 3 marks Exponential Functions Ordering and Comparing Exponential Values View
For positive numbers $a , b$ and natural numbers $m , n$ satisfying the inequality $a ^ { m } < a ^ { n } < b ^ { n } < b ^ { m }$, which of the following is correct? [3 points]
(1) $a < 1 < b , m > n$
(2) $a < 1 < b , m < n$
(3) $a < b < 1 , m < n$
(4) $1 < a < b , m > n$
(5) $1 < a < b , m < n$
Q10 4 marks Curve Sketching Lattice Points and Counting via Graph Geometry View
The figure on the right shows 6 semicircles with center at $( 1,1 )$ and radii of lengths $\frac { 1 } { 3 } , \frac { 2 } { 3 } , 1 , \frac { 4 } { 3 } , \frac { 5 } { 3 } , 2$ respectively. Three functions $$\begin{aligned} & y = \log _ { \frac { 1 } { 4 } } x \\ & y = \left( \frac { 2 } { 3 } \right) ^ { x } \\ & y = 3 ^ { x } \end{aligned}$$ Let $a , b , c$ be the number of intersection points where the graphs of these functions meet the semicircles, respectively. Which of the following correctly represents the relationship between $a , b , c$? (Here, $x \geqq 1$ and the semicircles include the endpoints of the diameter.) [4 points]
(1) $a < b < c$
(2) $a < c < b$
(3) $b < c < a$
(4) $c < a < b$
(5) $c < b < a$
Q11 3 marks Laws of Logarithms Characteristic and Mantissa of Common Logarithms View
For a positive number $a$, let the characteristic and mantissa of $\log a$ be $f ( a )$ and $g ( a )$ respectively. Which of the following in are correct? [3 points] 〈Remarks〉 ㄱ. $f ( 2006 ) = 3$ ㄴ. $g ( 2 ) + g ( 6 ) = g ( 12 ) + 1$ ㄷ. If $f ( a b ) = f ( a ) + f ( b )$, then $g ( a b ) = g ( a ) + g ( b )$.
(1) ㄱ
(2) ㄱ, ㄴ
(3) ㄱ, ㄷ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ
Q12 4 marks Circles Circle-Line Intersection and Point Conditions View
For two points $\mathrm { A } ( 1 , \sqrt { 3 } ) , \mathrm { B } ( 1 , - \sqrt { 3 } )$ on the coordinate plane, what is the total length of the figure represented by point $\mathrm { P } ( x , y )$ satisfying the following two conditions? [4 points] (가) $x ^ { 2 } + y ^ { 2 } = 4$ (나) For any point $( 1 , a )$ on segment AB, the matrix $\left( \begin{array} { c c } x & y \\ 1 & a \end{array} \right)$ has an inverse matrix.
(1) $\frac { 1 } { 3 } \pi$
(2) $\frac { 1 } { 2 } \pi$
(3) $\pi$
(4) $\frac { 4 } { 3 } \pi$
(5) $\frac { 3 } { 2 } \pi$
Q13 4 marks Arithmetic Sequences and Series Properties of AP Terms under Transformation View
Two sequences $\left\{ a _ { n } \right\} , \left\{ b _ { n } \right\}$ are given by $$\begin{aligned} & a _ { n } = \frac { 1 } { 2 ^ { n - 1 } } \cos \frac { ( n - 1 ) \pi } { 2 } \\ & b _ { n } = \frac { 1 + ( - 1 ) ^ { n - 1 } } { 2 ^ { n } } \end{aligned}$$ Which of the following in are correct? [4 points] 〈Remarks〉 ㄱ. For all natural numbers $k$, $a _ { 3 k } < 0$. ㄴ. For all natural numbers $k$, $a _ { 4 k - 1 } + b _ { 4 k - 1 } = 0$. ㄷ. $\sum _ { n = 1 } ^ { \infty } a _ { n } = \frac { 3 } { 5 } \sum _ { n = 1 } ^ { \infty } b _ { n }$
(1) ㄱ
(2) ㄴ
(3) ㄷ
(4) ㄱ, ㄴ
(5) ㄴ, ㄷ
Q14 3 marks Normal Distribution Sampling Distribution of the Mean View
Suppose the weight of products produced at a certain factory follows a normal distribution $\mathrm { N } \left( 11,2 ^ { 2 } \right)$. Two people, $A$ and $B$, each independently extracted a sample of size 4. Using the standard normal distribution table on the right, what is the probability that the sample means extracted by both $A$ and $B$ are between 10 and 14 inclusive? [3 points]
$z$$\mathrm { P } ( 0 \leqq Z \leqq z )$
10.3413
20.4772
30.4987

(1) 0.8123
(2) 0.7056
(3) 0.6587
(4) 0.5228
(5) 0.2944
Q15 4 marks Radians, Arc Length and Sector Area View
As shown in the figure, a sector $\mathrm { OA } _ { 1 } \mathrm {~B} _ { 1 }$ with radius equal to the line segment $\mathrm { OA } _ { 1 } ( 0,8 )$ connecting the origin $O$ and point $\mathrm { A } _ { 1 } ( 0,8 )$ and central angle $\theta$ is drawn. The foot of the perpendicular from point $\mathrm { B } _ { 1 }$ to the $x$-axis is $\mathrm { A } _ { 2 }$, and a sector $\mathrm { OA } _ { 2 } \mathrm {~B} _ { 2 }$ with radius equal to segment $\mathrm { OA } _ { 2 }$ and central angle $\theta$ is drawn. The foot of the perpendicular from point $\mathrm { B } _ { 2 }$ to the $y$-axis is $\mathrm { A } _ { 3 }$, and a sector $\mathrm { OA } _ { 3 } \mathrm {~B} _ { 3 }$ with radius equal to segment $\mathrm { OA } _ { 3 }$ and central angle $\theta$ is drawn. Continuing this process of alternately dropping perpendiculars to the $x$-axis and $y$-axis in the clockwise direction, let $l _ { n }$ be the length of arc $\mathrm { A } _ { n } \mathrm {~B} _ { n }$ of sector $\mathrm { OA } _ { n } \mathrm {~B} _ { n }$. When $\sum _ { n = 1 } ^ { \infty } l _ { n } = 12 \theta$, what is the value of $\sin \theta$? (Here, $0 < \theta < \frac { \pi } { 2 }$.) [4 points]
(1) $\frac { 1 } { 7 }$
(2) $\frac { 1 } { 6 }$
(3) $\frac { 1 } { 5 }$
(4) $\frac { 1 } { 4 }$
(5) $\frac { 1 } { 3 }$
Q16 4 marks Proof by induction Fill in missing steps of a given induction proof View
The following proves by mathematical induction that for all natural numbers $n$, $$\sum _ { k = 1 } ^ { n } ( 5 k - 3 ) \left( \frac { 1 } { k } + \frac { 1 } { k + 1 } + \frac { 1 } { k + 2 } + \cdots + \frac { 1 } { n } \right) = \frac { n ( 5 n + 3 ) } { 4 }$$ holds.
(1) When $n = 1$, (left side) $= 2$, (right side) $= 2$, so the given equation holds.
(2) Assume that when $n = m$, the equation holds: $$\begin{aligned} & \sum _ { k = 1 } ^ { m } ( 5 k - 3 ) \left( \frac { 1 } { k } + \frac { 1 } { k + 1 } + \frac { 1 } { k + 2 } + \cdots + \frac { 1 } { m } \right) \\ = & \frac { m ( 5 m + 3 ) } { 4 } \end{aligned}$$ Now we show that it holds when $n = m + 1$. $$\begin{aligned} & \sum _ { k = 1 } ^ { m + 1 } ( 5 k - 3 ) \left( \frac { 1 } { k } + \frac { 1 } { k + 1 } + \cdots + \frac { 1 } { m + 1 } \right) \\ = & \sum _ { k = 1 } ^ { m } ( 5 k - 3 ) \left( \frac { 1 } { k } + \frac { 1 } { k + 1 } + \cdots + \frac { 1 } { m + 1 } \right) + \frac { \text { (가) } } { m + 1 } \\ = & \sum _ { k = 1 } ^ { m } ( 5 k - 3 ) \left( \frac { 1 } { k } + \frac { 1 } { k + 1 } + \cdots + \frac { 1 } { \sqrt { \text { (나) } } } \right) \\ \quad & \quad + \frac { 1 } { m + 1 } \sum _ { k = 1 } ^ { m } ( 5 k - 3 ) + \frac { \text { (가) } } { m + 1 } \\ = & \frac { m ( 5 m + 3 ) } { 4 } + \frac { 1 } { m + 1 } \sum _ { k = 1 } ^ { m + 1 } \text { (다) } \\ = & \frac { ( m + 1 ) ( 5 m + 8 ) } { 4 } \end{aligned}$$ Therefore, the equation also holds when $n = m + 1$. Thus, the given equation holds for all natural numbers $n$. What are the correct values for (가), (나), and (다) in the above proof? [4 points]
(가)(나)(다)
(1)$5 m - 3$$m$$5 k + 2$
(2)$5 m - 3$$m + 1$$5 k + 2$
(3)$5 m + 2$$m$$5 k - 3$
(4)$5 m + 2$$m$$5 k + 2$
(5)$5 m + 2$$m + 1$$5 k - 3$
Q17 4 marks Permutations & Arrangements Distribution of Objects into Bins/Groups View
As shown in the figure, a rectangular solid is made from 12 transparent glass boxes in the shape of identical cubes. If 4 of these glass boxes are replaced with glass boxes of the same size but black in color such that the rectangular solid viewed from above looks like (가) and viewed from the side looks like (나), how many ways can this be done? [4 points]
(1) 54
(2) 48
(3) 42
(4) 36
(5) 30
Q18 3 marks Sequences and series, recurrence and convergence Convergence proof and limit determination View
Find the value of $\lim _ { n \rightarrow \infty } \frac { 5 \cdot 3 ^ { n + 1 } - 2 ^ { n + 1 } } { 3 ^ { n } + 2 ^ { n } }$. [3 points]
Q19 3 marks Arithmetic Sequences and Series Sequence Defined by Recurrence with AP Connection View
In the sequence $\left\{ a _ { n } \right\}$, $a _ { 1 } = 1 , a _ { 2 } = 4 , a _ { 3 } = 10$, and the sequence $\left\{ a _ { n + 1 } - a _ { n } \right\}$ is a geometric sequence. Find the value of $a _ { 5 }$. [3 points]
Q20 3 marks Laws of Logarithms Solve a Logarithmic Equation View
For two positive numbers $a , b$, $$\left\{ \begin{array} { l } ab = 27 \\ \log _ { 3 } \frac { b } { a } = 5 \end{array} \right.$$ When these conditions hold, find the value of $4 \log _ { 3 } a + 9 \log _ { 3 } b$. [3 points]
Q21 3 marks Exponential Functions Exponential Equation Solving View
When the two roots of the equation $4 ^ { x } - 7 \cdot 2 ^ { x } + 12 = 0$ are $\alpha , \beta$, find the value of $2 ^ { 2 \alpha } + 2 ^ { 2 \beta }$. [3 points]
Q22 3 marks Discrete Probability Distributions Probability Distribution Table Completion and Expectation Calculation View
The following is a probability distribution table for the random variable $X$.
$X$$k$$2 k$$4 k$Total
$\mathrm { P } ( X = x )$$\frac { 4 } { 7 }$$a$$b$1

If $\frac { 4 } { 7 } , a , b$ form a geometric sequence in this order and the mean of $X$ is 24, find the value of $k$. [3 points]
Q23 4 marks Discrete Probability Distributions Probability Computation for Compound or Multi-Stage Random Experiments View
There is a regular tetrahedron-shaped box with the numbers $1,1,1,2$ written one on each face. When this box is thrown, if the number on the bottom face is 1, region A in the figure on the right is colored, and if the number is 2, region B is colored. When the box is thrown repeatedly until both regions are colored, find the probability that the process is completed on the 3rd throw. If this probability is $\frac { q } { p }$, find the value of $p + q$. (Here, $p , q$ are coprime natural numbers.) [4 points]
Q24 4 marks Laws of Logarithms Optimize a Logarithmic Expression View
For the function with domain $\{ x \mid 1 \leqq x \leqq 81 \}$, $$y = \left( \log _ { 3 } x \right) \left( \log _ { \frac { 1 } { 3 } } x \right) + 2 \log _ { 3 } x + 10$$ Let $M$ be the maximum value and $m$ be the minimum value. Find the value of $M + m$. [4 points]
Q25 4 marks Laws of Logarithms Logarithmic Formula Application (Modeling) View
To remove bacteria living in a water tank, a chemical is to be administered. Let $C _ { 0 }$ be the initial number of bacteria per 1 mL of water in the tank, and let $C$ be the number of bacteria per 1 mL at time $t$ hours after the chemical is administered. The following relationship holds: $$\log \frac { C } { C _ { 0 } } = - k t \quad ( k \text { is a positive constant } )$$ The initial number of bacteria per 1 mL of water is $8 \times 10 ^ { 5 }$, and at time 3 hours after the chemical is administered, the number of bacteria per 1 mL becomes $2 \times 10 ^ { 5 }$. After $a$ hours from administering the chemical, the number of bacteria per 1 mL first becomes $8 \times 10 ^ { 3 }$ or less. Find the value of $a$. (Here, calculate using $\log 2 = 0.3$.) [4 points]
Q26 3 marks Conditional Probability Direct Conditional Probability Computation from Definitions View
A certain class consists of 18 male students and 16 female students. All students in this class take a class in either Chinese or Japanese, but not both. Among the male students, 12 take Chinese class, and among the female students, 7 take Japanese class. When a student selected from this class is taking Chinese class, what is the probability that this student is female? [3 points]
(1) $\frac { 1 } { 7 }$
(2) $\frac { 2 } { 7 }$
(3) $\frac { 3 } { 7 }$
(4) $\frac { 4 } { 7 }$
(5) $\frac { 5 } { 7 }$
Q27 4 marks Matrices Matrix Power Computation and Application View
For two matrices $A = \left( \begin{array} { l l } 1 & 1 \\ 0 & 1 \end{array} \right) , B = \left( \begin{array} { l l } 1 & 0 \\ 1 & 1 \end{array} \right)$, define sets $S , T$ as $$\begin{aligned} & S = \left\{ \binom { x } { y } \left\lvert \, \binom { x } { y } = A ^ { n } \binom { 1 } { 1 } \right. , n \text { is a natural number} \right\} \\ & T = \left\{ \binom { x } { y } \left\lvert \, \binom { x } { y } = B ^ { n } \binom { 1 } { 1 } \right. , n \text { is a natural number} \right\} \end{aligned}$$ Which of the following in are correct? [4 points] 〈Remarks〉 ㄱ. If $\binom { a } { b } \in S$, then $\binom { b } { a } \in T$. ㄴ. If $\binom { a } { b } \in S , \binom { c } { d } \in S$, then $\binom { a + c } { b + d } \in S$. ㄷ. If $\binom { a } { b } \in S , \binom { p } { q } \in T$, then the matrix $\left( \begin{array} { l l } a & p \\ b & q \end{array} \right)$ has an inverse matrix.
(1) ㄱ
(2) ㄱ, ㄴ
(3) ㄱ, ㄷ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ
Q28 4 marks Combinations & Selection Selection with Arithmetic or Divisibility Conditions View
When selecting two different odd numbers from the odd numbers from 1 to 30, how many cases are there where the sum of the two numbers is a multiple of 3? [4 points]
(1) 43
(2) 41
(3) 39
(4) 37
(5) 35
Q29 4 marks Sequences and Series Recurrence Relations and Sequence Properties View
For a natural number $p \geqq 2$, a sequence $\left\{ a _ { n } \right\}$ satisfies the following three conditions. Which of the following in are correct? [4 points] Conditions (가) $a _ { 1 } = 0$ (나) $a _ { k + 1 } = a _ { k } + 1 \quad ( 1 \leqq k \leqq p - 1 )$ (다) $a _ { k + p } = a _ { k } \quad ( k = 1,2,3 , \cdots )$ 〈Remarks〉 ㄱ. $a _ { 2 k } = 2 a _ { k }$ ㄴ. $a _ { 1 } + a _ { 2 } + \cdots + a _ { p } = \frac { p ( p - 1 ) } { 2 }$ ㄷ. $a _ { p } + a _ { 2 p } + \cdots + a _ { k p } = k ( p - 1 )$
(1) ㄱ
(2) ㄴ
(3) ㄷ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ
Q30 4 marks Binomial Theorem (positive integer n) Determine Parameters from Conditions on Coefficients or Terms View
In the expansion of the polynomial $2 ( x + a ) ^ { n }$, the coefficient of $x ^ { n - 1 }$ and the coefficient of $x ^ { n - 1 }$ in the expansion of the polynomial $( x - 1 ) ( x + a ) ^ { n }$ are equal. Find the maximum value of $a n$ for all ordered pairs $( a , n )$ satisfying this condition. (Here, $a$ is a natural number and $n$ is a natural number with $n \geqq 2$.) [4 points]