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
2005 csat__math-humanities

27 maths questions

Q1 2 marks Indices and Surds Evaluating Expressions Using Index Laws View
What is the value of $3 ^ { \frac { 2 } { 3 } } \times 9 ^ { \frac { 3 } { 2 } } \div 27 ^ { \frac { 8 } { 9 } }$? [2 points]
(1) 1
(2) $\sqrt { 3 }$
(3) 3
(4) $3 \sqrt { 3 }$
(5) 9
Q2 2 marks Matrices Linear System and Inverse Existence View
For two matrices $A = \left( \begin{array} { l l } 1 & 2 \\ 2 & 5 \end{array} \right) , B = \left( \begin{array} { l l } 2 & - 3 \\ 1 & - 2 \end{array} \right)$, what is the sum of all components of matrix $X$ that satisfies $A X = B$? [2 points]
(1) $- 2$
(2) $- 1$
(3) 0
(4) 1
(5) 2
Q3 2 marks Arithmetic Sequences and Series Find Specific Term from Given Conditions View
For an arithmetic sequence $\left\{ a _ { n } \right\}$ $$a _ { 1 } + a _ { 2 } = 10 , \quad a _ { 3 } + a _ { 4 } + a _ { 5 } = 45$$ When this holds, what is the value of $a _ { 10 }$? [2 points]
(1) 47
(2) 45
(3) 43
(4) 41
(5) 39
Q4 3 marks Sequences and series, recurrence and convergence Multiple-choice on sequence properties View
What is the value of $\lim _ { n \rightarrow \infty } \left( \sqrt { n ^ { 2 } + 6 n + 4 } - n \right)$? [3 points]
(1) $\frac { 1 } { 3 }$
(2) $\frac { 1 } { 2 }$
(3) 1
(4) 2
(5) 3
Q5 3 marks Laws of Logarithms Verify Truth of Logarithmic Statements View
From the following , select all correct statements. [3 points]
ㄱ. $2 ^ { \log _ { 2 } 1 + \log _ { 2 } 2 + \log _ { 2 } 3 + \cdots + \log _ { 2 } 10 } = 10$ ㄴ. $\log _ { 2 } \left( 2 ^ { 1 } \times 2 ^ { 2 } \times 2 ^ { 3 } \times \cdots \times 2 ^ { 10 } \right) ^ { 2 } = 55 ^ { 2 }$ ㄷ. $\left( \log _ { 2 } 2 ^ { 1 } \right) \left( \log _ { 2 } 2 ^ { 2 } \right) \left( \log _ { 2 } 2 ^ { 3 } \right) \cdots \left( \log _ { 2 } 2 ^ { 10 } \right) = 55$
(1) ㄱ
(2) ㄴ
(3) ㄷ
(4) ㄱ, ㄷ
(5) ㄱ, ㄴ, ㄷ
Q6 3 marks Matrices Matrix Algebra and Product Properties View
For square matrices $A$ and $B$ of order 2, select all statements that are always true from . (Here, $E$ is the identity matrix and $O$ is the zero matrix.) [3 points]
ㄱ. $( A + B ) ^ { 2 } = A ^ { 2 } + 2 A B + B ^ { 2 }$ ㄴ. If $A ^ { 2 } + A - 2 E = O$, then $A$ has an inverse matrix. ㄷ. If $A \neq O$ and $A ^ { 2 } = A$, then $A = E$.
(1) ㄱ
(2) ㄴ
(3) ㄷ
(4) ㄱ, ㄴ
(5) ㄴ, ㄷ
Q7 3 marks Sequences and series, recurrence and convergence Convergence proof and limit determination View
For a sequence $\left\{ a _ { n } \right\}$ where the sum of the first $n$ terms $S _ { n } = 2 n + \frac { 1 } { 2 ^ { n } }$, what is the value of $\lim _ { n \rightarrow \infty } a _ { n }$? [3 points]
(1) 2
(2) 1
(3) $\frac { 1 } { 2 }$
(4) $\frac { 1 } { 4 }$
(5) 0
Q8 3 marks Matrices Matrix Algebra and Product Properties View
The following table shows the manufacturing cost per unit, selling price, and sales volume for two products A and B produced by a company last year.
CategoryProduct AProduct B
Manufacturing Cost$a _ { 11 }$$a _ { 12 }$
Selling Price$a _ { 21 }$$a _ { 22 }$

Sales VolumeFirst HalfSecond Half
A$b _ { 11 }$$b _ { 12 }$
B$b _ { 21 }$$b _ { 22 }$

Represent the above tables as matrices $A = \left( \begin{array} { l l } a _ { 11 } & a _ { 12 } \\ a _ { 21 } & a _ { 22 } \end{array} \right)$ and $B = \left( \begin{array} { l l } b _ { 11 } & b _ { 12 } \\ b _ { 21 } & b _ { 22 } \end{array} \right)$ respectively, and let the product of these two matrices be $A B = \left( \begin{array} { l l } a & b \\ c & d \end{array} \right)$. When the profit per unit is defined as the selling price minus the manufacturing cost, select all correct statements from . [3 points]
ㄱ. $a + b$ is the total manufacturing cost of products sold in the first half of last year. ㄴ. $c + d$ is the total selling amount of products sold throughout last year. ㄷ. $d - b$ is the total profit from products sold in the second half of last year.
(1) ㄱ
(2) ㄴ
(3) ㄱ, ㄷ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ
Q9 3 marks Probability Definitions Finite Equally-Likely Probability Computation View
There are four people of different heights. When they stand in a line, what is the probability that the third person from the front is shorter than both of their neighbors? [3 points]
(1) $\frac { 1 } { 3 }$
(2) $\frac { 1 } { 2 }$
(3) $\frac { 3 } { 5 }$
(4) $\frac { 2 } { 3 }$
(5) $\frac { 3 } { 4 }$
Q10 4 marks Exponential Functions True/False or Multiple-Statement Verification View
For the function $f ( x ) = \frac { 4 ^ { x } } { 4 ^ { x } + 2 }$, select all correct statements from . [4 points]
ㄱ. $f \left( \frac { 1 } { 2 } \right) = \frac { 1 } { 2 }$ ㄴ. $f ( x ) + f ( 1 - x ) = 1$ ㄷ. $\sum _ { k = 1 } ^ { 100 } f \left( \frac { k } { 101 } \right) = 50$
(1) ㄱ
(2) ㄴ
(3) ㄷ
(4) ㄱ, ㄴ
(5) ㄱ, ㄴ, ㄷ
Q11 4 marks Number Theory Combinatorial Number Theory and Counting View
As shown in the figure below, for a natural number $n$, $n$ terms $$\left[ \frac { n } { 1 } \right] \left[ \frac { n } { 2 } \right] \left[ \frac { n } { 3 } \right] , \cdots , \left[ \frac { n } { n } \right]$$ are arranged in the $n$-th row from column 1 to column $n$ in order. (Here, $[ x ]$ is the greatest integer not exceeding $x$.)
Select all correct statements from . [4 points]
ㄱ. In row $n$, the number of terms with value 1 is $\left[ \frac { n + 1 } { 2 } \right]$. ㄴ. In row 100, the number of terms with value 3 is 8. ㄷ. In column 3, the number of terms with value 5 is 5.
(1) ㄱ
(2) ㄴ
(3) ㄷ
(4) ㄱ, ㄴ
(5) ㄱ, ㄴ, ㄷ
Q12 3 marks Proof by induction Fill in missing steps of a given induction proof View
The following is a proof by mathematical induction that the inequality $$\sum _ { i = 1 } ^ { 2 n + 1 } \frac { 1 } { n + i } = \frac { 1 } { n + 1 } + \frac { 1 } { n + 2 } + \cdots + \frac { 1 } { 3 n + 1 } > 1$$ holds for all natural numbers $n$.
For a natural number $n$, let $a _ { n } = \frac { 1 } { n + 1 } + \frac { 1 } { n + 2 } + \cdots + \frac { 1 } { 3 n + 1 }$. We need to show that $a _ { n } > 1$.
(1) When $n = 1$, $a _ { 1 } = \frac { 1 } { 2 } + \frac { 1 } { 3 } + \frac { 1 } { 4 } > 1$.
(2) Assume that when $n = k$, $a _ { k } > 1$. $$\begin{aligned} & \text{When } n = k + 1 \\ & \begin{aligned} a _ { k + 1 } & = \frac { 1 } { k + 2 } + \frac { 1 } { k + 3 } + \cdots + \frac { 1 } { 3 k + 4 } \\ & = a _ { k } + \left( \frac { 1 } { 3 k + 2 } + \frac { 1 } { 3 k + 3 } + \frac { 1 } { 3 k + 4 } \right) \end{aligned} \end{aligned}$$ On the other hand, $( 3 k + 2 ) ( 3 k + 4 )$ $\square$ (b) $( 3 k + 3 ) ^ { 2 }$, so $$\frac { 1 } { 3 k + 2 } + \frac { 1 } { 3 k + 4 } > \text{(c)}$$ Since $a _ { k } > 1$, $$a _ { k + 1 } > a _ { k } + \left( \frac { 1 } { 3 k + 3 } + \text{(c)} \right) \text{(a)} > 1$$ Therefore, by (1) and (2), $a _ { n } > 1$ for all natural numbers $n$.
What are the correct values for (a), (b), and (c) in the above proof? [3 points]
(a)(b)(c)
(1) $\frac { 1 } { k + 1 }$$>$$\frac { 2 } { 3 k + 3 }$
(2) $\frac { 1 } { k + 1 }$$<$$\frac { 2 } { 3 k + 3 }$
(3) $\frac { 1 } { k + 1 }$$<$$\frac { 4 } { 3 k + 3 }$
(4) $\frac { 2 } { k + 1 }$$>$$\frac { 4 } { 3 k + 3 }$
(5) $\frac { 2 } { k + 1 }$$<$$\frac { 1 } { k + 1 }$
Q13 3 marks Confidence intervals Conceptual reasoning about confidence level and sample size effects View
The following explains the relationship between confidence interval, confidence level, and sample size.
There is a population following a normal distribution $N \left( m , \sigma ^ { 2 } \right)$. When a sample of size $n$ is randomly extracted from this population, the sample mean follows a normal distribution $\square$ (a). Using the distribution of this sample mean, let the confidence interval for the population mean $m$ with confidence level $\alpha$ be $a \leqq m \leqq b$. When the sample size is fixed at $n$ and the confidence level is set higher than $\alpha$, let the confidence interval be $c \leqq m \leqq d$. Then $d - c$ is $\square$ (b) than $b - a$. On the other hand, when the confidence level is fixed at $\alpha$ and the sample size is $2 n$, let the confidence interval be $e \leqq m \leqq f$. Then $f - e$ is $\square$ (c) times $b - a$.
What are the correct values for (a), (b), and (c) in the above process? [3 points]
(a)(b)
(1)$N \left( m , \sigma ^ { 2 } \right)$larger
(2)$N \left( m , \sigma ^ { 2 } \right)$smaller
(3)$N \left( m , \frac { \sigma ^ { 2 } } { n } \right)$smaller
(4)$N \left( m , \frac { \sigma ^ { 2 } } { n } \right)$larger
(5)$N \left( m , \frac { \sigma ^ { 2 } } { n } \right)$smaller

(c)
(1)$\frac{1}{2}$
(2)$\frac{1}{2}$
(3)$\frac{1}{\sqrt{2}}$
(4)$\sqrt{2}$
(5)$\frac{1}{\sqrt{2}}$
Q14 4 marks Combinations & Selection Counting Arrangements with Run or Pattern Constraints View
Among 12-character strings made using all eight $a$'s and four $b$'s, how many strings satisfy all of the following conditions? [4 points]
(a) $b$ cannot appear consecutively.
(b) If the first character is $b$, then the last character is $a$.
(1) 70
(2) 105
(3) 140
(4) 175
(5) 210
Q15 4 marks Laws of Logarithms Logarithmic Formula Application (Modeling) View
When sound passes through a building wall, a certain proportion is transmitted into the interior while the rest is reflected or absorbed. The ratio of sound transmitted into the interior is called the transmission rate. When the acoustic output of a speaker is $W$ (watts), the intensity $P$ (decibels) of sound transmitted into the interior at a distance of $r$ (m) from the speaker in a building with transmission rate $\alpha$ is as follows. $$\begin{aligned} & P = 10 \log \frac { \alpha W } { I _ { 0 } } - 20 \log r - 11 \\ & \text{(where } I _ { 0 } = 10 ^ { - 12 } \text{ (watts/m}^2\text{) and } r > 1 \text{.)} \end{aligned}$$ A speaker is emitting sound with an acoustic output of 100 (watts). When the intensity of sound transmitted into the interior of a building with transmission rate $\frac { 1 } { 100 }$ is 59 (decibels) or less, what is the minimum distance between the speaker and the building? (Assume that sound spreads uniformly in space and that factors other than transmission rate are not considered.) [4 points]
(1) $10 ^ { 2 } \mathrm{~m}$
(2) $10 ^ { \frac { 17 } { 8 } } \mathrm{~m}$
(3) $10 ^ { \frac { 13 } { 6 } } \mathrm{~m}$
(4) $10 ^ { \frac { 9 } { 4 } } \mathrm{~m}$
(5) $10 ^ { \frac { 5 } { 2 } } \mathrm{~m}$
Q16 3 marks Approximating Binomial to Normal Distribution View
The following table shows the results of a survey on customer preferences for hiking boots by manufacturer at a certain department store.
ManufacturerABCDTotal
Preference (\%)20282527100

When 192 customers each purchase one pair of hiking boots, what is the probability that 42 or more customers will choose product C, using the standard normal distribution table on the right? [3 points]
$z$$\mathrm { P } ( 0 \leqq Z \leqq z )$
0.50.1915
1.00.3413
1.50.4332
2.00.4772

(1) 0.6915
(2) 0.7745
(3) 0.8256
(4) 0.8332
(5) 0.8413
Q17 4 marks Geometric Sequences and Series Applied Geometric Model with Contextual Interpretation View
A society where the proportion of the population aged 65 and over in the total population is 20\% or more is called a 'super-aged society'. In 2000, a certain country had a total population of 10 million and a population aged 65 and over of 500,000. Assuming that the total population increases by 0.3\% each year compared to the previous year and the population aged 65 and over increases by 4\% each year compared to the previous year, when is the first time a 'super-aged society' predicted to occur? (Given: $\log 1.003 = 0.0013$, $\log 1.04 = 0.0170$, $\log 2 = 0.3010$) [4 points]
(1) 2048--2050
(2) 2038--2040
(3) 2028--2030
(4) 2018--2020
(5) 2008--2010
Q18 Matrices Matrix Algebra and Product Properties View
For the quadratic equation $x ^ { 2 } - 4 x - 1 = 0$ with roots $\alpha$ and $\beta$, find the sum of all components of the product of two matrices $\left( \begin{array} { l l } \alpha & \beta \\ 0 & \alpha \end{array} \right) \left( \begin{array} { l l } \beta & \alpha \\ 0 & \beta \end{array} \right)$.
Q19 3 marks Laws of Logarithms Solve a Logarithmic Inequality View
Solve the system of inequalities $$\left\{ \begin{array} { l } \log _ { 3 } | x - 3 | < 4 \\ \log _ { 2 } x + \log _ { 2 } ( x - 2 ) \geqq 3 \end{array} \right.$$ and find the number of integers $x$ that satisfy it. [3 points]
Q20 3 marks Discrete Probability Distributions Probability Distribution Table Completion and Expectation Calculation View
The probability distribution table of random variable $X$ is shown below. Find the variance of random variable $Y = 10 X + 5$. [3 points]
$X$0123Total
$\mathrm { P } ( X )$$\frac { 2 } { 10 }$$\frac { 3 } { 10 }$$\frac { 3 } { 10 }$$\frac { 2 } { 10 }$1
Q21 3 marks Geometric Sequences and Series Finite Geometric Sum and Term Relationships View
For a geometric sequence $\left\{ a _ { n } \right\}$ with common ratio $r$ and $a _ { 2 } = 1$, let $\omega = a _ { 1 } a _ { 2 } a _ { 3 } \cdots a _ { 10 }$ be the product of the first 10 terms. Find the value of $\log _ { r } \omega$. (Here, $r > 0$ and $r \neq 1$.) [3 points]
Q22 4 marks Matrices Matrix Entry and Coefficient Identities View
Natural numbers are arranged at regular intervals on the sides and vertices of squares with side lengths $1, 3, 5, \cdots, 2 n - 1, \cdots$ as shown in the figure below. In each square, 1 is placed directly above the lower left vertex.
Let the $2 \times 2$ matrices with the natural numbers at the four vertices of each square as components be $A _ { 1 } , A _ { 2 } , A _ { 3 } , \cdots , A _ { n } , \cdots$ in order. For example, $A _ { 1 } = \left( \begin{array} { l l } 1 & 2 \\ 4 & 3 \end{array} \right) , A _ { 2 } = \left( \begin{array} { c c } 3 & 6 \\ 12 & 9 \end{array} \right)$. Find the sum of all components of matrix $A _ { 15 }$. [4 points]
Q26 3 marks Geometric Sequences and Series True/False or Multiple-Statement Verification View
For an infinite geometric sequence $\left\{ a _ { n } \right\}$, choose all correct statements from \textless Remarks\textgreater. [3 points]
\textless Remarks\textgreater ㄱ. If the infinite geometric series $\sum _ { n = 1 } ^ { \infty } a _ { n }$ converges, then $\sum _ { n = 1 } ^ { \infty } a _ { 2 n }$ also converges. ㄴ. If the infinite geometric series $\sum _ { n = 1 } ^ { \infty } a _ { n }$ diverges, then $\sum _ { n = 1 } ^ { \infty } a _ { 2 n }$ also diverges. ㄷ. If the infinite geometric series $\sum _ { n = 1 } ^ { \infty } a _ { n }$ converges, then $\sum _ { n = 1 } ^ { \infty } \left( a _ { n } + \frac { 1 } { 2 } \right)$ also converges.
(1) ㄱ
(2) ㄴ
(3) ㄱ, ㄷ
(4) ㄱ, ㄴ
(5) ㄴ, ㄷ
Q27 4 marks Laws of Logarithms Characteristic and Mantissa of Common Logarithms View
Let $a$ be the largest integer among numbers whose common logarithm characteristic is 2, and let $b$ be the smallest number among numbers whose common logarithm characteristic is $-2$. What is the value of $ab$? [4 points]
(1) 0.9
(2) 0.99
(3) 1
(4) 9.99
(5) 10
Q28 4 marks Chain Rule Geometric Limit with Parametric Chain Rule View
For two points $\mathrm { P } ( n , f ( n ) )$ and $\mathrm { Q } ( n + 1 , f ( n + 1 ) )$ on the graph of the quadratic function $f ( x ) = 3 x ^ { 2 }$, let $a _ { n }$ be the distance between them. Find the value of $\lim _ { n \rightarrow \infty } \frac { a _ { n } } { n }$. (Here, $n$ is a natural number.) [4 points]
(1) 9
(2) 8
(3) 7
(4) 6
(5) 5
Q29 4 marks Probability Definitions Finite Equally-Likely Probability Computation View
When two dice are rolled simultaneously, what is the probability that the number on one die is a multiple of the number on the other die? [4 points]
(1) $\frac { 7 } { 18 }$
(2) $\frac { 1 } { 2 }$
(3) $\frac { 11 } { 18 }$
(4) $\frac { 13 } { 18 }$
(5) $\frac { 5 } { 6 }$
Q30 4 marks Combinations & Selection Counting Arrangements with Run or Pattern Constraints View
When arranging $1, 2, 2, 4, 5, 5$ in a line to form a six-digit natural number, find the number of natural numbers greater than 300000. [4 points]