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
2009 csat__math-humanities

27 maths questions

Q1 2 marks Indices and Surds Evaluating Expressions Using Index Laws View
What is the value of $9 ^ { \frac { 3 } { 2 } } \times 27 ^ { - \frac { 2 } { 3 } }$? [2 points]
(1) $\frac { 1 } { 3 }$
(2) 1
(3) $\sqrt { 3 }$
(4) 3
(5) $3 \sqrt { 3 }$
Q2 2 marks Matrices Matrix Algebra and Product Properties View
For two matrices $A = \left( \begin{array} { l l } 2 & 1 \\ 1 & 1 \end{array} \right) , B = \left( \begin{array} { r r } - 1 & - 2 \\ 1 & 0 \end{array} \right)$, what is the sum of all components of the matrix $( A + B ) A$? [2 points]
(1) 9
(2) 10
(3) 11
(4) 12
(5) 13
Q3 2 marks Sequences and series, recurrence and convergence Multiple-choice on sequence properties View
What is the value of $\lim _ { n \rightarrow \infty } \frac { 2 } { \sqrt { n ^ { 2 } + 2 n } - \sqrt { n ^ { 2 } + 1 } }$? [2 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5
Q4 3 marks Laws of Logarithms Optimize a Logarithmic Expression View
What is the minimum value of the function $y = 3 + \log _ { 3 } \left( x ^ { 2 } - 4 x + 31 \right)$? [3 points]
(1) 4
(2) 5
(3) 6
(4) 7
(5) 8
Q5 3 marks Geometric Sequences and Series Finite Geometric Sum and Term Relationships View
Four numbers $1 , a , b , c$ form a geometric sequence with common ratio $r$ in this order, and satisfy $\log _ { 8 } c = \log _ { a } b$. What is the value of the common ratio $r$? (where $r > 1$) [3 points]
(1) 2
(2) $\frac { 5 } { 2 }$
(3) 3
(4) $\frac { 7 } { 2 }$
(5) 4
Q6 3 marks Laws of Logarithms Simplify or Evaluate a Logarithmic Expression View
When $a = \log _ { 2 } 10 , b = 2 \sqrt { 2 }$, what is the value of $a \log b$? [3 points]
(1) 1
(2) $\frac { 3 } { 2 }$
(3) 2
(4) $\frac { 5 } { 2 }$
(5) 3
Q7 3 marks Normal Distribution Sampling Distribution of the Mean View
A company manufactures women's general handball balls certified by the International Handball Federation. The weight of handball balls produced by this company follows a normal distribution with mean 350 g and standard deviation 16 g. The company determines that there is a problem in the production process if the average weight of 64 randomly selected handball balls is 346 g or less, or 355 g or more. Using the standard normal distribution table below, what is the probability that the company determines there is a problem in the production process? [3 points]
$z$$\mathrm { P } ( 0 \leqq Z \leqq z )$
2.000.4772
2.250.4878
2.500.4938
2.750.4970

(1) 0.0290
(2) 0.0258
(3) 0.0184
(4) 0.0152
(5) 0.0092
Q8 3 marks Exponential Functions Parameter Determination from Conditions View
Two exponential functions $f ( x ) = a ^ { b x - 1 } , g ( x ) = a ^ { 1 - b x }$ satisfy the following conditions.
(a) The graphs of $y = f ( x )$ and $y = g ( x )$ are symmetric with respect to the line $x = 2$.
(b) $f ( 4 ) + g ( 4 ) = \frac { 5 } { 2 }$
What is the value of the sum of the two constants $a + b$? (where $0 < a < 1$) [3 points]
(1) 1
(2) $\frac { 9 } { 8 }$
(3) $\frac { 5 } { 4 }$
(4) $\frac { 11 } { 8 }$
(5) $\frac { 3 } { 2 }$
Q9 4 marks Binomial Theorem (positive integer n) Find a Specific Coefficient in a Single Binomial Expansion View
In the expansion of $\left( x + \frac { 1 } { x ^ { 3 } } \right) ^ { 4 }$, what is the coefficient of $\frac { 1 } { x ^ { 4 } }$? [4 points]
(1) 4
(2) 6
(3) 8
(4) 10
(5) 12
Q10 3 marks Proof by induction Fill in missing steps of a given induction proof View
The sequence $\left\{ a _ { n } \right\}$ satisfies
$$\left\{ \begin{array} { l } a _ { 1 } = \frac { 1 } { 2 } \\ ( n + 1 ) ( n + 2 ) a _ { n + 1 } = n ^ { 2 } a _ { n } \quad ( n = 1,2,3 , \cdots ) \end{array} \right.$$
The following is a proof by mathematical induction that for all natural numbers $n$,
$$\sum _ { k = 1 } ^ { n } a _ { k } = \sum _ { k = 1 }^{n} \frac { 1 } { k ^ { 2 } } - \frac { n } { n + 1 }$$
holds. $\langle$Proof$\rangle$
(1) When $n = 1$, (left side) $= \frac { 1 } { 2 }$, (right side) $= 1 - \frac { 1 } { 2 } = \frac { 1 } { 2 }$, so (*) holds.
(2) Assume (*) holds when $n = m$: $$\sum _ { k = 1 } ^ { m } a _ { k } = \sum _ { k = 1 } ^ { m } \frac { 1 } { k ^ { 2 } } - \frac { m } { m + 1 }$$
Now show that (*) holds when $n = m + 1$.
$$\begin{aligned} & \sum _ { k = 1 } ^ { m + 1 } a _ { k } = \sum _ { k = 1 } ^ { m } \frac { 1 } { k ^ { 2 } } - \frac { m } { m + 1 } + a _ { m + 1 } \\ = & \sum _ { k = 1 } ^ { m } \frac { 1 } { k ^ { 2 } } - \frac { m } { m + 1 } + \square \text { (a) } a _ { m } \\ = & \sum _ { k = 1 } ^ { m } \frac { 1 } { k ^ { 2 } } - \frac { m } { m + 1 } \\ & \quad + \frac { m ^ { 2 } } { ( m + 1 ) ( m + 2 ) } \cdot \frac { ( m - 1 ) ^ { 2 } } { m ( m + 1 ) } \cdot \cdots \cdot \frac { 1 ^ { 2 } } { 2 \cdot 3 } a _ { 1 } \end{aligned}$$
Therefore, (*) also holds when $n = m + 1$. Thus, (*) holds for all natural numbers $n$.
Which expressions are correct for (a), (b), and (c) in the above proof? [3 points]
(1) $\dfrac{\text{(a)}}{m} \quad \dfrac{\text{(b)}}{(m+1)(m+2)} \quad \dfrac{\text{(c)}}{\frac{1}{(m+1)^2(m+2)}} \quad \dfrac{1}{(m+1)(m+2)^2}$
(2) $\dfrac{m}{(m+1)(m+2)} \quad \dfrac{m}{(m+1)^2(m+2)} \quad \dfrac{1}{(m+1)(m+2)}$
(3) $\dfrac{m^2}{(m+1)(m+2)} \quad \dfrac{1}{(m+1)^2(m+2)} \quad \dfrac{1}{(m+1)(m+2)^2}$
(4) $\dfrac{m^2}{(m+1)(m+2)} \quad \dfrac{1}{(m+1)^2(m+2)} \quad \dfrac{1}{(m+1)(m+2)}$
(5) $\dfrac{m^2}{(m+1)(m+2)} \quad \dfrac{m}{(m+1)^2(m+2)} \quad \dfrac{1}{(m+1)(m+2)^2}$
Q11 4 marks Laws of Logarithms Verify Truth of Logarithmic Statements View
For a constant $a$ with $0 < a < \frac { 1 } { 2 }$, let the point where the line $y = x$ meets the curve $y = \log _ { a } x$ be $( p , p )$, and let the point where the line $y = x$ meets the curve $y = \log _ { 2 a } x$ be $( q , q )$. Which of the following statements in $\langle$Remarks$\rangle$ are correct? [4 points]
$\langle$Remarks$\rangle$ ㄱ. If $p = \frac { 1 } { 2 }$, then $a = \frac { 1 } { 4 }$. ㄴ. $p < q$ ㄷ. $a ^ { p + q } = \frac { p q } { 2 ^ { q } }$
(1) ㄱ
(2) ㄱ, ㄴ
(3) ㄱ, ㄷ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ
Q12 4 marks Laws of Logarithms Verify Truth of Logarithmic Statements View
Let the set $U$ be
$$U = \left\{ \left. \left( \begin{array} { l l } a & b \\ c & d \end{array} \right) \right\rvert \, a , b , c , d \text { are positive numbers other than } 1 \right\}$$
Let the subset $S$ of $U$ be
$$S = \left\{ \left. \left( \begin{array} { l l } a & b \\ c & d \end{array} \right) \right\rvert \, \log _ { a } d = \log _ { b } c , \quad a \neq b , \quad b c \neq 1 \right\}$$
Which of the following statements in $\langle$Remarks$\rangle$ are correct? [4 points]
$\langle$Remarks$\rangle$ ㄱ. If $A = \left( \begin{array} { l l } 4 & 9 \\ 3 & 2 \end{array} \right)$, then $A \in S$. ㄴ. If $A \in U$ and $A$ has an inverse matrix, then $A \in S$. ㄷ. If $A \in S$, then $A$ has an inverse matrix.
(1) ㄱ
(2) ㄴ
(3) ㄱ, ㄷ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ
Q13 3 marks Stationary points and optimisation Geometric or applied optimisation problem View
For a natural number $n$, when two points $\mathrm { P } _ { n - 1 } , \mathrm { P } _ { n }$ are on the graph of the function $y = x ^ { 2 }$, the point $\mathrm { P } _ { n + 1 }$ is determined according to the following rule.
(a) The coordinates of the two points $\mathrm { P } _ { 0 } , \mathrm { P } _ { 1 }$ are $(0,0)$ and $(1,1)$, respectively.
(b) The point $\mathrm { P } _ { n + 1 }$ is the intersection of the line passing through point $\mathrm { P } _ { n }$ and perpendicular to the line $\mathrm { P } _ { n - 1 } \mathrm { P } _ { n }$ and the graph of the function $y = x ^ { 2 }$. (Here, $\mathrm { P } _ { n }$ and $\mathrm { P } _ { n + 1 }$ are distinct points.) Let $l _ { n } = \overline { \mathrm { P } _ { n - 1 } \mathrm { P } _ { n } }$. What is the value of $\lim _ { n \rightarrow \infty } \frac { l _ { n } } { n }$? [3 points]
(1) $2 \sqrt { 3 }$
(2) $2 \sqrt { 2 }$
(3) 2
(4) $\sqrt { 3 }$
(5) $\sqrt { 2 }$
Q14 4 marks Geometric Sequences and Series Fractal/Iterative Geometric Construction (Area, Length, or Perimeter Series) View
On the coordinate plane, there is a circle $C _ { 1 } : ( x - 4 ) ^ { 2 } + y ^ { 2 } = 1$. As shown in the figure, a tangent line $l$ with positive slope is drawn from the origin to the circle $C _ { 1 }$, and the point of tangency is called $\mathrm { P } _ { 1 }$.
A circle $C _ { 2 }$ has its center on the line $l$, passes through the point $\mathrm { P } _ { 1 }$, and is tangent to the $x$-axis. Let $\mathrm { P } _ { 2 }$ be the point of tangency between this circle and the $x$-axis.
A circle $C _ { 3 }$ has its center on the $x$-axis, passes through the point $\mathrm { P } _ { 2 }$, and is tangent to the line $l$. Let $\mathrm { P } _ { 3 }$ be the point of tangency between this circle and the line $l$.
A circle $C _ { 4 }$ has its center on the line $l$, passes through the point $\mathrm { P } _ { 3 }$, and is tangent to the $x$-axis. Let $\mathrm { P } _ { 4 }$ be the point of tangency between this circle and the $x$-axis. Continuing this process, let $S _ { n }$ be the area of circle $C _ { n }$. What is the value of $\sum _ { n = 1 } ^ { \infty } S _ { n }$? (Note: The radius of circle $C _ { n + 1 }$ is smaller than the radius of circle $C _ { n }$.) [4 points]
(1) $\frac { 3 } { 2 } \pi$
(2) $2 \pi$
(3) $\frac { 5 } { 2 } \pi$
(4) $3 \pi$
(5) $\frac { 7 } { 2 } \pi$
Q15 4 marks Permutations & Arrangements Distribution of Objects into Bins/Groups View
A certain volunteer service center operates the following 4 volunteer activity programs every day.
ProgramABCD
Volunteer Activity Hours1 hour2 hours3 hours4 hours

Chulsu wants to participate in one program each day for 5 days at this volunteer service center and create a volunteer activity plan so that the total volunteer activity hours is 8 hours. How many different volunteer activity plans can be created? [4 points]
(1) 47
(2) 44
(3) 41
(4) 38
(5) 35
Q16 4 marks Independent Events View
Bag A and Bag B each contain five marbles with the numbers $1,2,3,4,5$ written on them, one number per marble. Chulsu draws one marble from Bag A, and Younghee draws one marble from Bag B. They check the numbers on the two marbles and do not put them back. This process is repeated. What is the probability that the numbers on the two marbles drawn the first time are different, and the numbers on the two marbles drawn the second time are the same? [4 points]
(1) $\frac { 3 } { 20 }$
(2) $\frac { 1 } { 5 }$
(3) $\frac { 1 } { 4 }$
(4) $\frac { 3 } { 10 }$
(5) $\frac { 7 } { 20 }$
Q17 4 marks Probability Definitions Verifying Statements About Probability Properties View
In information theory, when an event $E$ occurs, the information content $I ( E )$ of event $E$ is defined as follows:
$$I ( E ) = - \log _ { 2 } \mathrm { P } ( E )$$
Which of the following statements in $\langle$Remarks$\rangle$ are correct? (Note: The probability $\mathrm { P } ( E )$ of event $E$ is positive, and the unit of information content is bits.) [4 points]
$\langle$Remarks$\rangle$ ㄱ. If event $E$ is rolling an odd number on a single die, then $I ( E ) = 1$. ㄴ. If two events $A$ and $B$ are independent and $\mathrm { P } ( A \cap B ) > 0$, then $I ( A \cap B ) = I ( A ) + I ( B )$. ㄷ. For two events $A$ and $B$ with $\mathrm { P } ( A ) > 0$ and $\mathrm { P } ( B ) > 0$, we have $2 I ( A \cup B ) \leqq I ( A ) + I ( B )$.
(1) ㄱ
(2) ㄱ, ㄴ
(3) ㄱ, ㄷ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ
Q18 3 marks Exponential Functions Parameter Determination from Conditions View
When the graph of the exponential function $y = 5 ^ { x - 1 }$ passes through the two points $( a , 5 ) , ( 3 , b )$, find the value of $a + b$. [3 points]
Q19 3 marks Laws of Logarithms Solve a Logarithmic Equation View
For two real numbers $a , b$ with $1 < a < b$,
$$\frac { 3 a } { \log _ { a } b } = \frac { b } { 2 \log _ { b } a } = \frac { 3 a + b } { 3 }$$
holds. Find the value of $10 \log _ { a } b$. [3 points]
Q20 3 marks Arithmetic Sequences and Series Find Specific Term from Given Conditions View
For an arithmetic sequence $\left\{ a _ { n } \right\}$ with common difference 2,
$$a _ { 1 } + a _ { 5 } + a _ { 9 } = 45$$
Find the value of $a _ { 1 } + a _ { 10 }$. [3 points]
Q21 3 marks Geometric Sequences and Series Sum of an Infinite Geometric Series (Direct Computation) View
For two infinite geometric sequences $\left\{ a _ { n } \right\} , \left\{ b _ { n } \right\}$ with the same common ratio, $a _ { 1 } - b _ { 1 } = 1$, $\sum _ { n = 1 } ^ { \infty } a _ { n } = 8$, and $\sum _ { n = 1 } ^ { \infty } b _ { n } = 6$. Find the value of $\sum _ { n = 1 } ^ { \infty } a _ { n } b _ { n }$. [3 points]
Q22 4 marks Complex Numbers Arithmetic Probability Involving Complex Number Conditions View
When rolling a die twice, let the outcomes be $m$ and $n$ in order. If the probability that $i ^ { m } \cdot ( - i ) ^ { n } = 1$ is $\frac { q } { p }$, find the value of $p + q$. (Here, $i = \sqrt { - 1 }$ and $p , q$ are coprime natural numbers.) [4 points]
Q23 4 marks Sequences and Series Recurrence Relations and Sequence Properties View
Let $a _ { n }$ be the sum of all natural numbers such that when divided by a natural number $n$ ($n \geqq 2$), the quotient and remainder are equal. For example, when divided by 4, the natural numbers with equal quotient and remainder are $5, 10, 15$, so $a _ { 4 } = 5 + 10 + 15 = 30$. Find the minimum value of the natural number $n$ satisfying $a _ { n } > 500$. [4 points]
Q24 4 marks Matrices Matrix Power Computation and Application View
A $2 \times 2$ square matrix $A$ has the sum of all components equal to 0 and satisfies
$$A ^ { 2 } + A ^ { 3 } = - 3 A - 3 E$$
Find the sum of all components of the matrix $A ^ { 4 } + A ^ { 5 }$. (Here, $E$ is the identity matrix.) [4 points]
Q25 4 marks Permutations & Arrangements Lattice Path / Grid Route Counting View
There is a walking path in a rectangular lawn. As shown in the figure, this walking path consists of 8 circles with equal radii that are externally tangent to each other.
Starting from point A and arriving at point B along the walking path by the shortest distance, find the number of possible routes. (Note: The points marked on the circles represent the points of tangency between the circles and the rectangle or between the circles.) [4 points]
Q26 3 marks Conditional Probability Direct Conditional Probability Computation from Definitions View
For two events $A$ and $B$, $\mathrm { P } ( A ) = \frac { 1 } { 2 } , \mathrm { P } \left( B ^ { C } \right) = \frac { 2 } { 3 }$, and $\mathrm { P } ( B \mid A ) = \frac { 1 } { 6 }$. What is the value of $\mathrm { P } \left( A ^ { C } \mid B \right)$? (Here, $A ^ { C }$ is the complement of $A$.) [3 points]
(1) $\frac { 1 } { 2 }$
(2) $\frac { 7 } { 12 }$
(3) $\frac { 2 } { 3 }$
(4) $\frac { 3 } { 4 }$
(5) $\frac { 5 } { 6 }$
Q27 3 marks Laws of Logarithms Characteristic and Mantissa of Common Logarithms View
For a natural number $n$, let $f ( n )$ be the mantissa of $\log n$. What is the number of elements in the set
$$A = \{ f ( n ) \mid 1 \leqq n \leqq 150 , n \text { is a natural number } \}$$
? [3 points]
(1) 131
(2) 133
(3) 135
(4) 137
(5) 139