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-science

32 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 Inequalities Integer Solutions of an Inequality View
System of inequalities $$\left\{ \begin{array} { l } \frac { x + 2 } { x ^ { 2 } - 4 x + 3 } \geqq 0 \\ \frac { 9 } { x - 8 } \leqq - 1 \end{array} \right.$$ What is the number of integers $x$ that satisfy the system? [2 points]
(1) 10
(2) 9
(3) 8
(4) 7
(5) 6
Q5 3 marks Laws of Logarithms Verify Truth of Logarithmic Statements View
Which of the following in $\langle$Remarks$\rangle$ are correct? [3 points]
$\langle$Remarks$\rangle$ 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 Vectors 3D & Lines Line-Plane Intersection View
Let $\alpha$ be the plane passing through point $\mathrm { A } ( 1,2,3 )$ and perpendicular to the line $l : x - 1 = \frac { y - 2 } { - 2 } = \frac { z - 3 } { 3 }$. When the intersection point of plane $\alpha$ and line $m : x - 2 = y = \frac { z - 6 } { 5 }$ is B, what is the length of segment AB? [3 points]
(1) $\sqrt { 19 }$
(2) $\sqrt { 17 }$
(3) $\sqrt { 15 }$
(4) $\sqrt { 13 }$
(5) $\sqrt { 11 }$
Q7 3 marks Vectors 3D & Lines Dihedral Angle Computation View
As shown in the figure on the right, in a cube ABCD-EFGH with edge length 3, there are three points $\mathrm { P } , \mathrm { Q } , \mathrm { R }$ on the three edges AD, BC, FG such that $\overline { \mathrm { DP } } = \overline { \mathrm { BQ } } = \overline { \mathrm { GR } } = 1$. The angle between plane PQR and plane CGHD is $\theta$. What is the value of $\cos \theta$? (where $0 < \theta < \frac { \pi } { 2 }$) [3 points]
(1) $\frac { \sqrt { 10 } } { 5 }$
(2) $\frac { \sqrt { 10 } } { 10 }$
(3) $\frac { \sqrt { 11 } } { 11 }$
(4) $\frac { 2 \sqrt { 11 } } { 11 }$
(5) $\frac { 3 \sqrt { 11 } } { 11 }$
Q8 4 marks Indefinite & Definite Integrals Antiderivative Verification and Construction View
The following shows the graph of a continuous function $y = f ( x )$ and two distinct points $\mathrm { P } ( a , f ( a ) ) , \mathrm { Q } ( b , f ( b ) )$ on this graph.
When the function $F ( x )$ satisfies $F ^ { \prime } ( x ) = f ( x )$, which of the following statements in $\langle$Remarks$\rangle$ are always correct? [4 points]
$\langle$Remarks$\rangle$ ㄱ. The function $F ( x )$ is increasing on the interval $[ a , b ]$. ㄴ. $\frac { F ( b ) - F ( a ) } { b - a }$ is equal to the slope of line PQ. ㄷ. $\int _ { a } ^ { b } \{ f ( x ) - f ( b ) \} d x \leqq \frac { ( b - a ) \{ f ( a ) - f ( b ) \} } { 2 }$
(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 are arranged in a line, what is the probability that the third person from the front is shorter than the two people adjacent to him? [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 Indefinite & Definite Integrals Integral Equation with Symmetry or Substitution View
The following is the graph of a continuous function $y = f ( x )$.
When the inverse function $g ( x )$ of function $f ( x )$ exists and is continuous on the interval $[ 0,1 ]$, the limit value $$\lim _ { n \rightarrow \infty } \sum _ { k = 1 } ^ { n } \left\{ g \left( \frac { k } { n } \right) - g \left( \frac { k - 1 } { n } \right) \right\} \frac { k } { n }$$ has the same value as which of the following? [4 points]
(1) $\int _ { 0 } ^ { 1 } g ( x ) d x$
(2) $\int _ { 0 } ^ { 1 } x g ( x ) d x$
(3) $\int _ { 0 } ^ { 1 } f ( x ) d x$
(4) $\int _ { 0 } ^ { 1 } x f ( x ) d x$
(5) $\int _ { 0 } ^ { 1 } \{ f ( x ) - g ( x ) \} d x$
Q11 4 marks Number Theory Combinatorial Number Theory and Counting View
As shown in the figure below, for a natural number $n$, the $n$ 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 $x$.)
Which of the following in $\langle$Remarks$\rangle$ are correct? [4 points]
$\langle$Remarks$\rangle$ ㄱ. In the $n$-th row, the number of terms with value 1 is $\left[ \frac { n + 1 } { 2 } \right]$. ㄴ. In the 100th row, the number of terms with value 3 is 8. ㄷ. In the 3rd column, 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 $n$.
$\langle$Proof$\rangle$ For a natural number $n$, let $a _ { n } = \frac { 1 } { n + 1 } + \frac { 1 } { n + 2 } + \cdots + \frac { 1 } { 3 n + 1 }$. It suffices 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 $a _ { k } > 1$ when $n = k$. $$\begin{aligned} & \text{When } n = k + 1, \\ & \begin{aligned} a _ { k + 1 } & = \frac { 1 } { k + 2 } + \frac { 1 } { k + 3 } + \cdots + \frac { 1 } { 3 k + 4 } \\ & = a _ { k } + \left( \frac { 1 } { 3 k + 2 } + \frac { 1 } { 3 k + 3 } + \frac { 1 } { 3 k + 4 } \right) \end{aligned} \end{aligned}$$ □
On the other hand, $( 3 k + 2 ) ( 3 k + 4 )$ (나) $( 3 k + 3 ) ^ { 2 }$, so $$\frac { 1 } { 3 k + 2 } + \frac { 1 } { 3 k + 4 } > \text{ (다) }$$ Since $a _ { k } > 1$, $$a _ { k + 1 } > a _ { k } + \left( \frac { 1 } { 3 k + 3 } + \text{ (다) } \right) \text{ (가) } > 1$$ Therefore, by (1) and (2), $a _ { n } > 1$ for all natural numbers $n$.
What are the correct expressions for (가), (나), and (다) in the above proof? [3 points]
(가)(나)(다)
(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 { 2 } { 3 k + 4 }$
(4) $\frac { 2 } { k + 1 }$$>$$\frac { 2 } { 3 k + 4 }$
(5) $\frac { 2 } { k + 1 }$$<$$\frac { 2 } { 3 k + 3 }$
Q13 3 marks Stationary points and optimisation Count or characterize roots using extremum values View
For $a > 1$, consider the function $f ( x ) = 2 x ^ { 3 } - 3 ( a + 1 ) x ^ { 2 } + 6 a x - 4 a + 2$. Let $b$ be one real root of the equation $f ( x ) = 0$. The following is a process for comparing the magnitudes of two numbers $a$ and $b$. $f ^ { \prime } ( x ) =$ (가) and since $a > 1$, $f ( x )$ has a (나) at $x = 1$. Since $f ( 1 ) < 0$ and $f ( b ) = 0$, $a$ (다) $b$.
What are the correct expressions for (가), (나), and (다) in the above process? [3 points]
(가)(나)(다)
(1) $6 ( x - 1 ) ( x - a )$local minimum$>$
(2) $6 ( x - 1 ) ( x - a )$local minimum$<$
(3) $6 ( x - 1 ) ( x - a )$local maximum$>$
(4) $6 ( x - a ) ( x - 1 )$local maximum$<$
(5) $6 ( x - a ) ( x - 1 )$local maximum$>$
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] (가) $b$ cannot appear consecutively. (나) 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 Vectors 3D & Lines MCQ: Distance or Length Optimization on a Line View
In coordinate space, there are two points $\mathrm { A } ( 3,1,1 ) , \mathrm { B } ( 1 , - 3 , - 1 )$. For a point P on the plane $x - y + z = 0$, what is the minimum value of $| \overrightarrow { \mathrm { PA } } + \overrightarrow { \mathrm { PB } } |$? [4 points]
(1) $\frac { 4 \sqrt { 3 } } { 3 }$
(2) $\frac { 5 \sqrt { 3 } } { 3 }$
(3) $2 \sqrt { 3 }$
(4) $\frac { 7 \sqrt { 3 } } { 3 }$
(5) $\frac { 8 \sqrt { 3 } } { 3 }$
Q16 3 marks Approximating Binomial to Normal Distribution View
The following is a table showing customer preference by manufacturer for hiking boots sold 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 company C's product, 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 is 20\% or more of the total population is called a 'super-aged society'.
In the year 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 3 marks Chain Rule Limit Involving Derivative Definition of Composed Functions View
Two real numbers $a$ and $b$ satisfy $\lim _ { x \rightarrow 2 } \frac { \sqrt { x ^ { 2 } + a } - b } { x - 2 } = \frac { 2 } { 5 }$. Find the value of $a + b$. [3 points]
Q19 3 marks Laws of Logarithms Solve a Logarithmic Inequality View
System of inequalities $$\left\{ \begin{array} { l } \log _ { 3 } | x - 3 | < 4 \\ \log _ { 2 } x + \log _ { 2 } ( x - 2 ) \geqq 3 \end{array} \right.$$ Find the number of integers $x$ that satisfy the system. [3 points]
Q20 3 marks Solving quadratics and applications Solving an equation via substitution to reduce to quadratic form View
Find the product of all real roots of the irrational equation $x ^ { 2 } + 7 x + 10 + \sqrt { x ^ { 2 } + 7 x + 12 } = 0$. [3 points]
Q21 Vectors 3D & Lines Distance from a Point to a Line (Show/Compute) View
A sphere with center $\mathrm { C } ( 0,1,1 )$ and radius $2 \sqrt { 2 }$ intersects the line $\frac { x } { 2 } = y = - z$ at two points A and B. Let $S$ be the area of triangle CAB. Find the value of $S ^ { 2 }$.
Q22 4 marks Conic sections Focal Distance and Point-on-Conic Metric Computation View
For the ellipse $\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 20 } = 1$, let F and $\mathrm { F } ^ { \prime }$ be the two foci, and let A be the vertex closest to focus F. For a point P on this ellipse such that $\angle \mathrm { PFF } ^ { \prime } = \frac { \pi } { 3 }$, find the value of $\overline { \mathrm { PA } } ^ { 2 }$. [4 points]
Q23 4 marks Laws of Logarithms Simplify or Evaluate a Logarithmic Expression View
For a real number $a$ ($a > 1$), let $b = \sum _ { n = 1 } ^ { \infty } \left( \frac { 1 } { a } \right) ^ { n }$ be represented as in [Figure 1], and for a real number $c$, let $d = 16 ^ { c }$ be represented as in [Figure 2].
For the real numbers $x$, $y$, $z$ in the figure below, find the value of $\frac { x z } { y }$. [4 points]
Q24 4 marks Stationary points and optimisation Count or characterize roots using extremum values View
The cubic equation in $x$, $\frac { 1 } { 3 } x ^ { 3 } - x = k$, has three distinct real roots $\alpha$, $\beta$, $\gamma$. For a real number $k$, let $m$ be the minimum value of $| \alpha | + | \beta | + | \gamma |$. Find the value of $m ^ { 2 }$. [4 points]
Q25 4 marks Geometric Sequences and Series Fractal/Iterative Geometric Construction (Area, Length, or Perimeter Series) View
As shown in the figure below, from a square with side length 1, a square with side length $\frac { 1 } { 2 }$ is cut out, and the remaining shape is called $A _ { 1 }$. From a square with side length $\frac { 1 } { 4 }$, a square with side length $\frac { 1 } { 8 }$ is cut out, and two resulting shapes are attached to the upper two sides of $A _ { 1 }$ to form a figure called $A _ { 2 }$. From a square with side length $\frac { 1 } { 16 }$, a square with side length $\frac { 1 } { 32 }$ is cut out, and four resulting shapes are attached to the upper four sides of $A _ { 2 }$ to form a figure called $A _ { 3 }$.
Continuing this process, let $A _ { n }$ be the $n$-th figure obtained and $S _ { n }$ be its area. If $\lim _ { n \rightarrow \infty } S _ { n } = \frac { q } { p }$, find the value of $p + q$. (Here, $p$ and $q$ are coprime natural numbers.) [4 points]
Q26 (Probability and Statistics) 3 marks Measures of Location and Spread View
The following is a stem-and-leaf plot showing the number of push-ups performed by 10 high school students in 3 minutes.
(Unit: times)
Stem\multicolumn{3}{|c|}{Leaf}
159
2378
3266
415

Let $m$ be the mean, $n$ be the median, and $f$ be the mode of the number of push-ups. Which of the following is correct? [3 points]
(1) $m < n < f$
(2) $m < f < n$
(3) $f < m < n$
(4) $n < m < f$
(5) $n < f < m$
Q27 (Probability and Statistics) 3 marks Discrete Probability Distributions Multiple Choice: Direct Probability or Distribution Calculation View
A discrete random variable $X$ can take values $0,1,2,3,4,5,6,7$ and its probability mass function is $$\mathrm { P } ( X = x ) = \left\{ \begin{array} { l l } c , & x = 0,1,2 \\ 2 c , & x = 3,4,5 \\ 5 c ^ { 2 } , & x = 6,7 \end{array} \quad ( \text { where } c \text { is a positive number } ) \right.$$ Let $A$ be the event that the random variable $X$ is at least 6, and let $B$ be the event that the random variable $X$ is at least 3. What is the value of $\mathrm { P } ( A \mid B )$? [3 points]
(1) $\frac { 1 } { 5 }$
(2) $\frac { 1 } { 6 }$
(3) $\frac { 1 } { 7 }$
(4) $\frac { 1 } { 8 }$
(5) $\frac { 1 } { 9 }$
Q27 (Discrete Mathematics) 3 marks Permutations & Arrangements Distribution of Objects into Bins/Groups View
How many ordered pairs $( A , B )$ of disjoint subsets of the set $\{ 1,2,3,4,5,6 \}$ are there? [3 points]
(1) 729
(2) 720
(3) 243
(4) 64
(5) 36
Q28 (Probability and Statistics) 3 marks Tree Diagrams Multi-Stage Sequential Process View
A bag contains 5 red balls, 4 yellow balls, 2 blue balls, and 9 white balls. A ball is drawn from the bag, its color is noted, and then it is returned. This procedure is repeated 3 times. What is the probability of drawing one red ball, one yellow ball, and one blue ball, regardless of the order? [3 points]
(1) $\frac { 1 } { 200 }$
(2) $\frac { 3 } { 100 }$
(3) $\frac { 7 } { 100 }$
(4) $\frac { 11 } { 100 }$
(5) $\frac { 11 } { 20 }$
Q29 (Probability and Statistics) 4 marks Normal Distribution Convergence in Distribution / Central Limit Theorem Application View
A music club is preparing for its regular concert this year. Based on past experience, the attendance rate among invited guests is 0.5. When 100 people are randomly selected from the invited guests,
$z$$\mathrm { P } ( 0 \leqq Z \leqq z )$
1.00.3413
1.20.3849
1.40.4192
1.60.4452

what is the probability that the attendance rate is at least 0.43 and at most 0.56, using the standard normal distribution table on the right? [4 points]
(1) 0.8041
(2) 0.7698
(3) 0.7605
(4) 0.7262
(5) 0.6826
Q30 (Calculus and Differentiation) 4 marks Areas by integration View
Find the area of the region enclosed by the curve $y = 3 \sqrt { x - 9 }$, the tangent line to this curve at the point $( 18,9 )$, and the $x$-axis. [4 points]
Q30 (Probability and Statistics) 4 marks Continuous Probability Distributions and Random Variables Combinatorial Probability and Limiting Probability View
The following is a probability distribution table of a certain population.
$X$123Total
$\mathrm { P } ( X )$0.50.30.21

When a sample of size 2 is drawn with replacement from this population, the probability distribution table of the sample mean $\bar { X }$ is as follows.
$\bar { X }$11.522.53
Frequency1$a$$b$21
$\mathrm { P } ( \bar { X } )$0.25$c$$d$0.120.04

Find the value of $100 ( b + c )$. [4 points]
Q30 (Discrete Mathematics) 4 marks Sequences and Series Evaluation of a Finite or Infinite Sum View
For a natural number $k$, when $n = 5 ^ { k }$, $f ( n )$ satisfies $$f ( 5 n ) = f ( n ) + 3 , \quad f ( 5 ) = 4$$ Find the value of $\sum _ { k = 1 } ^ { 10 } f \left( 5 ^ { k } \right)$. [4 points]