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-science

26 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 Composite & Inverse Functions Recover a Function from a Composition or Functional Equation View
When two constants $a , b$ satisfy $\lim _ { x \rightarrow 2 } \frac { x ^ { 2 } - ( a + 2 ) x + 2 a } { x ^ { 2 } - b } = 3$, what is the value of $a + b$? [2 points]
(1) $- 6$
(2) $- 4$
(3) $- 2$
(4) 0
(5) 2
Q4 3 marks Vectors Introduction & 2D True/False or Multiple-Statement Verification View
On the coordinate plane, there are two arbitrary distinct vectors $\overrightarrow { \mathrm { OP } } , \overrightarrow { \mathrm { OQ } }$ with initial point at the origin O. When the endpoints $\mathrm { P } , \mathrm { Q }$ of the two vectors are translated 3 units in the $x$-direction and 1 unit in the $y$-direction to points $\mathrm { P } ^ { \prime } , \mathrm { Q } ^ { \prime }$ respectively, which of the following statements in are always true? [3 points]

ㄱ. $\left| \overrightarrow { \mathrm { OP } } - \overrightarrow { \mathrm { OP } ^ { \prime } } \right| = \sqrt { 10 }$ ㄴ. $| \overrightarrow { \mathrm { OP } } - \overrightarrow { \mathrm { OQ } } | = \left| \overrightarrow { \mathrm { OP } ^ { \prime } } - \overrightarrow { \mathrm { OQ } ^ { \prime } } \right|$ ㄷ. $\overrightarrow { \mathrm { OP } } \cdot \overrightarrow { \mathrm { OQ } } = \overrightarrow { \mathrm { OP } ^ { \prime } } \cdot \overrightarrow { \mathrm { OQ } ^ { \prime } }$
(1) ㄱ
(2) ㄷ
(3) ㄱ, ㄴ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ
Q5 3 marks Conic sections Triangle or Quadrilateral Area and Perimeter with Foci View
Let $\mathrm { F } , \mathrm { F } ^ { \prime }$ be the two foci of the hyperbola $\frac { x ^ { 2 } } { 5 } - \frac { y ^ { 2 } } { 4 } = 1$, and let Q be the point symmetric to a point P on the hyperbola (not a vertex) with respect to the origin. When the area of quadrilateral $\mathrm { F } ^ { \prime } \mathrm { QFP }$ is 24, and the coordinates of point P are $( a , b )$, what is the value of $| a | + | b |$? [3 points]
(1) 9
(2) 10
(3) 11
(4) 12
(5) 13
Q6 3 marks Composite & Inverse Functions Existence or Properties of Functions and Inverses (Proof-Based) View
For a function $y = f ( x )$ defined on all real numbers, let $N ( f )$ denote the smallest natural number $k$ such that the function $y = x ^ { k } f ( x )$ is continuous at $x = 0$. For example,
$$f ( x ) = \left\{ \begin{array} { l l } \frac { 1 } { x } & ( x \neq 0 ) \\ 0 & ( x = 0 ) \end{array} \text { then } N ( f ) = 2 \right. \text { . }$$
For the following functions $g _ { i } ( i = 1,2,3 )$, let $N \left( g _ { i } \right) = a _ { i }$. Which correctly represents the order of $a _ { i }$? [3 points]
$$\begin{aligned} & g _ { 1 } ( x ) = \begin{cases} \frac { | x | } { x } & ( x \neq 0 ) \\ 0 & ( x = 0 ) \end{cases} \\ & g _ { 2 } ( x ) = \begin{cases} - x ^ { 2 } + 1 & ( x \neq 0 ) \\ 0 & ( x = 0 ) \end{cases} \\ & g _ { 3 } ( x ) = \begin{cases} \frac { 1 } { x ^ { 2 } } & ( x \neq 0 ) \\ 0 & ( x = 0 ) \end{cases} \end{aligned}$$
(1) $a _ { 1 } = a _ { 2 } < a _ { 3 }$
(2) $a _ { 1 } < a _ { 2 } = a _ { 3 }$
(3) $a _ { 1 } = a _ { 2 } = a _ { 3 }$
(4) $a _ { 2 } = a _ { 3 } < a _ { 1 }$
(5) $a _ { 3 } < a _ { 1 } = a _ { 2 }$
Q7 3 marks Conic sections Triangle or Quadrilateral Area and Perimeter with Foci View
The figure on the right shows 6 ellipses, each with a side of a regular hexagon ABCDEF with side length 10 as the major axis, and with equal minor axis lengths. As shown in the figure, the sum of the areas of 6 triangles formed by a vertex of the regular hexagon and the foci of the two adjacent ellipses is $6 \sqrt { 3 }$. What is the length of the minor axis of the ellipse? [3 points]
(1) $4 \sqrt { 2 }$
(2) 6
(3) $4 \sqrt { 3 }$
(4) 8
(5) $6 \sqrt { 2 }$
Q8 3 marks Inequalities Integer Solutions of an Inequality View
For two natural numbers $a , b ( a < b )$, the fractional inequality
$$\frac { x } { x - a } + \frac { x } { x - b } \leqq 0$$
is satisfied by exactly 2 integers $x$. What is the number of ordered pairs $( a , b )$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5
Q9 3 marks Stationary points and optimisation Analyze function behavior from graph or table of derivative View
The function $y = f ( x )$ is continuous on all real numbers, and for all $x$ with $| x | \neq 1$, the derivative $f ^ { \prime } ( x )$ is
$$f ^ { \prime } ( x ) = \left\{ \begin{array} { c c } x ^ { 2 } & ( | x | < 1 ) \\ - 1 & ( | x | > 1 ) \end{array} \right.$$
Which of the following statements in are true? [3 points]

ㄱ. The function $y = f ( x )$ has an extremum at $x = - 1$. ㄴ. For all real numbers $x$, $f ( x ) = f ( - x )$. ㄷ. If $f ( 0 ) = 0$, then $f ( 1 ) > 0$.
(1) ㄱ
(2) ㄴ
(3) ㄷ
(4) ㄱ, ㄷ
(5) ㄱ, ㄴ, ㄷ
Q10 4 marks Circles Sphere and 3D Circle Problems View
In coordinate space, the $xy$-plane, $yz$-plane, and $zx$-plane divide the space into 8 regions. How many of these 8 regions does the sphere
$$( x + 2 ) ^ { 2 } + ( y - 3 ) ^ { 2 } + ( z - 4 ) ^ { 2 } = 24$$
pass through? [4 points]
(1) 8
(2) 7
(3) 6
(4) 5
(5) 4
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 statements in are true? [3 points]

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
On the coordinate plane, for two points $\mathrm { A } ( 1 , \sqrt { 3 } ) , \mathrm { B } ( 1 , - \sqrt { 3 } )$, what is the total length of the figure represented by points $\mathrm { P } ( x , y )$ satisfying the following two conditions? [4 points]
(가) $x ^ { 2 } + y ^ { 2 } = 4$
(나) For any point $( 1 , a )$ on the line segment AB, the matrix $\left( \begin{array} { l l } x & y \\ 1 & a \end{array} \right)$ 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 statements in are true? [4 points]

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
The weight of products manufactured at a certain factory follows a normal distribution $\mathrm { N } \left( 11,2 ^ { 2 } \right)$. Two people A and B each independently randomly extracted a sample of size 4. Using the standard normal distribution table on the right, what is the probability that the sample means of 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 $\mathrm { OA } _ { 1 }$ connecting the origin O and the point $\mathrm { A } _ { 1 } ( 0,8 )$, and with 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 the line 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 the line 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 the arc $\mathrm { A } _ { n } \mathrm { B } _ { n }$ of the 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 is a proof 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 the equation holds when $n = m$:
$$\begin{aligned} & \sum _ { k = 1 } ^ { m } ( 5 k - 3 ) \left( \frac { 1 } { k } + \frac { 1 } { k + 1 } + \frac { 1 } { k + 2 } + \cdots + \frac { 1 } { m } \right) \\ = & \frac { m ( 5 m + 3 ) } { 4 } \end{aligned}$$
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 should be filled in 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 cubes of equal size. When 4 of these glass boxes are replaced with black glass boxes of the same size, and the view from above looks like (가) and the side view 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 Chain Rule Limit Involving Derivative Definition of Composed Functions View
For the function $f ( x ) = x ^ { 4 } + 4 x ^ { 2 } + 1$, find the value of $$\lim _ { h \rightarrow 0 } \frac { f ( 1 + 2 h ) - f ( 1 ) } { h }$$. [3 points]
Q19 3 marks Volumes of Revolution Volume of Revolution with Parameter Determination View
When a solid of revolution is created by rotating the figure enclosed by the curve $y = a \left( 1 - x ^ { 2 } \right)$ and the $x$-axis around the $y$-axis, and the volume of the solid of revolution is $16 \pi$, find the positive value of $a$. [3 points]
Q20 3 marks Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View
The graph of the function $f ( x ) = x ^ { 3 }$ is translated $a$ units in the $x$-direction and $b$ units in the $y$-direction to obtain the graph of the function $y = g ( x )$.
$$g ( 0 ) = 0 \text { and } \int _ { a } ^ { 3 a } g ( x ) dx - \int _ { 0 } ^ { 2 a } f ( x ) dx = 32$$
Find the value of $a ^ { 4 }$. [3 points]
Q21 4 marks Vectors 3D & Lines Volume of a 3D Solid View
Two spheres $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 81 , x ^ { 2 } + ( y - 5 ) ^ { 2 } + z ^ { 2 } = 56$ are denoted by $S _ { 1 } , S _ { 2 }$ respectively. Let P be a point on the circle formed by the intersection of the two spheres $S _ { 1 } , S _ { 2 }$, and let $\mathrm { P } ^ { \prime }$ be the orthogonal projection of point P onto the $xy$-plane. Let Q and R be the points where the sphere $S _ { 1 }$ intersects the $y$-axis. Find the maximum volume of the tetrahedron $\mathrm { PQP } ^ { \prime } \mathrm { R }$. [4 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, color region A in the figure on the right; if the number is 2, color region B. Continue throwing this box until both regions are colored. Find the probability that the process is completed on the 3rd throw, expressed as $\frac { q } { p }$. Find the value of $p + q$. (Here, $p , q$ are coprime natural numbers.) [4 points]
Q24 4 marks Vectors: Lines & Planes Sphere-Plane Intersection and Projection of Circles View
Let C be the circle formed by the intersection of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4$ and the plane $z = - 1$. When a plane $\alpha$ containing the $x$-axis intersects the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4$ to form a circle that meets C at exactly one point, and a normal vector to plane $\alpha$ is $\vec { n } = ( a , 3 , b )$, find the value of $a ^ { 2 } + b ^ { 2 }$. [4 points]
Q27 3 marks Harmonic Form View
There is a line $l$ passing through the origin O with slope $\tan \theta$. Let $\mathrm { A } ^ { \prime } , \mathrm { B } ^ { \prime }$ be the feet of the perpendiculars from two points $\mathrm { A } ( 0,2 ) , \mathrm { B } ( 2 \sqrt { 3 } , 0 )$ to line $l$, respectively. What is the value of $\theta$ that maximizes the sum of the distances from the origin O to point $\mathrm { A } ^ { \prime }$ and to point $\mathrm { B } ^ { \prime }$, $\overline { \mathrm { OA } ^ { \prime } } + \overline { \mathrm { OB } ^ { \prime } }$? (Given that $0 < \theta < \frac { \pi } { 2 }$.) [3 points]
(1) $\frac { \pi } { 12 }$
(2) $\frac { \pi } { 6 }$
(3) $\frac { \pi } { 4 }$
(4) $\frac { \pi } { 3 }$
(5) $\frac { 5 } { 12 } \pi$
Q29 4 marks Connected Rates of Change Geometric Related Rates with Distance or Angle View
The distance between point O and point E is 40 m. As shown in the figure on the right, person A departs from point O and runs along the half-line OS perpendicular to segment OE at a constant speed of 3 m/s, and person B departs from point E 10 seconds after person A starts and runs along the half-line EN perpendicular to segment OE at a constant speed of 4 m/s. The angle formed by the intersection of the segment connecting the positions of persons A and B with segment OE is $\theta$ (in radians). What is the rate of change of $\theta$ at the moment 20 seconds after person A departs? [4 points]
(1) $\frac { 21 } { 290 }$ radians/second
(2) $\frac { 13 } { 290 }$ radians/second
(3) $\frac { 7 } { 290 }$ radians/second
(4) $\frac { 3 } { 290 }$ radians/second
(5) $\frac { 1 } { 290 }$ radians/second
Q30 4 marks Curve Sketching Range and Image Set Determination View
For a positive number $a$, on the closed interval $[ - a , a ]$, the function
$$f ( x ) = \frac { x - 5 } { ( x - 5 ) ^ { 2 } + 36 }$$
has maximum value $M$ and minimum value $m$. Find the minimum value of $a$ such that $M + m = 0$. [4 points]