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
2014 csat__math-B

30 maths questions

Q1 2 marks Matrices Matrix Algebra and Product Properties View
For two matrices $A = \left( \begin{array} { l l } 2 & 0 \\ 1 & 0 \end{array} \right) , B = \left( \begin{array} { r r } a & 0 \\ 2 & - 3 \end{array} \right)$, when the sum of all components of matrix $A + B$ is 6, what is the value of $a$? [2 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5
Q2 2 marks Vectors 3D & Lines Section Division and Coordinate Computation View
In coordinate space, for two points $\mathrm { A } ( a , 5,2 ) , \mathrm { B } ( - 2,0,7 )$, the point that divides segment AB internally in the ratio $3 : 2$ has coordinates $( 0 , b , 5 )$. What is the value of $a + b$? [2 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5
Q3 2 marks Addition & Double Angle Formulae Direct Double Angle Evaluation View
When $\tan \theta = \frac { \sqrt { 5 } } { 5 }$, what is the value of $\cos 2 \theta$? [2 points]
(1) $\frac { \sqrt { 2 } } { 3 }$
(2) $\frac { \sqrt { 3 } } { 3 }$
(3) $\frac { 2 } { 3 }$
(4) $\frac { \sqrt { 5 } } { 3 }$
(5) $\frac { \sqrt { 6 } } { 3 }$
Q4 3 marks Arithmetic Sequences and Series Find Specific Term from Given Conditions View
For an arithmetic sequence $\left\{ a _ { n } \right\}$ with first term 2, when $a _ { 9 } = 3 a _ { 3 }$, what is the value of $a _ { 5 }$? [3 points]
(1) 10
(2) 11
(3) 12
(4) 13
(5) 14
Q5 3 marks Independent Events View
For two events $A , B$, $$\mathrm { P } \left( A ^ { C } \cup B ^ { C } \right) = \frac { 4 } { 5 } , \quad \mathrm { P } \left( A \cap B ^ { C } \right) = \frac { 1 } { 4 }$$ what is the value of $\mathrm { P } \left( A ^ { C } \right)$? (Here, $A ^ { C }$ is the complement of $A$.) [3 points]
(1) $\frac { 1 } { 2 }$
(2) $\frac { 11 } { 20 }$
(3) $\frac { 3 } { 5 }$
(4) $\frac { 13 } { 20 }$
(5) $\frac { 7 } { 10 }$
Q6 3 marks Vectors 3D & Lines MCQ: Relationship Between Two Lines View
In coordinate space, when the line passing through two points $\mathrm { A } ( 5,5 , a ) , \mathrm { B } ( 0,0,3 )$ is perpendicular to the line $x = 4 - y = z - 1$, what is the value of $a$? [3 points]
(1) 3
(2) 5
(3) 7
(4) 9
(5) 11
Q7 3 marks Harmonic Form View
When the maximum value of the function $f ( x ) = 2 \cos ^ { 2 } x + k \sin 2 x - 1$ is $\sqrt { 10 }$, what is the value of the positive constant $k$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5
Q8 3 marks Circles Tangent Lines and Tangent Lengths View
In the coordinate plane, two lines $l _ { 1 } , l _ { 2 }$ tangent to the parabola $y ^ { 2 } = 8 x$ have slopes $m _ { 1 } , m _ { 2 }$ respectively. When $m _ { 1 } , m _ { 2 }$ are the two distinct roots of the equation $2 x ^ { 2 } - 3 x + 1 = 0$, what is the $x$-coordinate of the intersection point of $l _ { 1 }$ and $l _ { 2 }$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5
Q9 3 marks Combinations & Selection Counting Integer Solutions to Equations View
When selecting 5 numbers from the digits $1,2,3,4$ with repetition allowed, how many cases are there where the digit 4 appears at most once? [3 points]
(1) 45
(2) 42
(3) 39
(4) 36
(5) 33
Q10 3 marks Curve Sketching Number of Solutions / Roots via Curve Analysis View
As shown in the figure, the graphs of function $f ( x )$ defined on the closed interval $[ - 4,4 ]$ and function $g ( x ) = - \frac { 1 } { 2 } x + 1$ meet at three points, and the $x$-coordinates of these three points are $\alpha , \beta , 2$. The inequality $$\frac { g ( x ) } { f ( x ) } \leq 1$$ is satisfied. How many integers $x$ satisfy this inequality? (Here, $- 4 < \alpha < - 3,0 < \beta < 1$) [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5
Q11 3 marks Sequences and series, recurrence and convergence Auxiliary sequence transformation View
A sequence $\left\{ a _ { n } \right\}$ with all positive terms has $a _ { 1 } = 10$ and satisfies $$\left( a _ { n + 1 } \right) ^ { n } = 10 \left( a _ { n } \right) ^ { n + 1 } \quad ( n \geq 1 )$$ The following is the process of finding the general term $a _ { n }$.
Taking the common logarithm of both sides of the given equation, $$n \log a _ { n + 1 } = ( n + 1 ) \log a _ { n } + 1$$ Dividing both sides by $n ( n + 1 )$, $$\frac { \log a _ { n + 1 } } { n + 1 } = \frac { \log a _ { n } } { n } + ( \text { (가) } )$$ Let $b _ { n } = \frac { \log a _ { n } } { n }$. Then $b _ { 1 } = 1$ and $$b _ { n + 1 } = b _ { n } + \text { (가) }$$ Finding the general term of the sequence $\left\{ b _ { n } \right\}$, $$b _ { n } = \text { (나) }$$ Therefore, $$\log a _ { n } = n \times \text { (나) }$$ Thus $a _ { n } = 10 ^ { n \times ( \text { (나) } ) }$.
When the expressions for (가) and (나) are $f ( n )$ and $g ( n )$ respectively, what is the value of $\frac { g ( 10 ) } { f ( 4 ) }$? [3 points]
(1) 38
(2) 40
(3) 42
(4) 44
(5) 46
Q12 3 marks Composite & Inverse Functions Determine Domain or Range of a Composite Function View
For a quadratic function $f ( x )$ with leading coefficient 1 and the function $$g ( x ) = \left\{ \begin{array} { c c } \frac { 1 } { \ln ( x + 1 ) } & ( x \neq 0 ) \\ 8 & ( x = 0 ) \end{array} \right.$$ when the function $f ( x ) g ( x )$ is continuous on the interval $( - 1 , \infty )$, what is the value of $f ( 3 )$? [3 points]
(1) 6
(2) 9
(3) 12
(4) 15
(5) 18
Q13 3 marks Volumes of Revolution Volume of Revolution about a Horizontal Axis (Evaluate) View
As shown in the figure, there is a line $l : x - y - 1 = 0$ and a hyperbola $C : x ^ { 2 } - 2 y ^ { 2 } = 1$ with one focus at point $\mathrm { F } ( c , 0 )$ (where $c < 0$).
When the region enclosed by the line $l$ and the hyperbola $C$ is rotated about the $y$-axis, what is the volume of the solid of revolution? [3 points]
(1) $\frac { 5 } { 3 } \pi$
(2) $\frac { 3 } { 2 } \pi$
(3) $\frac { 4 } { 3 } \pi$
(4) $\frac { 7 } { 6 } \pi$
(5) $\pi$
Q14 4 marks Linear transformations View
As shown in the figure, there is a line $l : x - y - 1 = 0$ and a hyperbola $C : x ^ { 2 } - 2 y ^ { 2 } = 1$ with one focus at point $\mathrm { F } ( c , 0 )$ (where $c < 0$).
Under a rotation transformation by angle $\theta$ about the origin, the line $l$ is mapped to a line passing through the focus F of the hyperbola $C$. What is the value of $\sin 2 \theta$? [4 points]
(1) $- \frac { 2 } { 3 }$
(2) $- \frac { 5 } { 9 }$
(3) $- \frac { 4 } { 9 }$
(4) $- \frac { 1 } { 3 }$
(5) $- \frac { 2 } { 9 }$
Q15 4 marks Sequences and series, recurrence and convergence Summation of sequence terms View
In rectangle $\mathrm { A } _ { 1 } \mathrm {~B} _ { 1 } \mathrm { C } _ { 1 } \mathrm { D } _ { 1 }$, $\overline { \mathrm { A } _ { 1 } \mathrm {~B} _ { 1 } } = 1 , \overline { \mathrm {~A} _ { 1 } \mathrm { D } _ { 1 } } = 2$. As shown in the figure, let $\mathrm { M } _ { 1 } , \mathrm {~N} _ { 1 }$ be the midpoints of segments $\mathrm { A } _ { 1 } \mathrm { D } _ { 1 }$ and $\mathrm { B } _ { 1 } \mathrm { C } _ { 1 }$ respectively.
Draw a circular sector $\mathrm { N } _ { 1 } \mathrm { M } _ { 1 } \mathrm {~B} _ { 1 }$ with center $\mathrm { N } _ { 1 }$, radius $\overline { \mathrm { B } _ { 1 } \mathrm {~N} _ { 1 } }$, and central angle $\frac { \pi } { 2 }$, and draw a circular sector $\mathrm { D } _ { 1 } \mathrm { M } _ { 1 } \mathrm { C } _ { 1 }$ with center $\mathrm { D } _ { 1 }$, radius $\overline { \mathrm { C } _ { 1 } \mathrm { D } _ { 1 } }$, and central angle $\frac { \pi } { 2 }$.
The region enclosed by the arc $\mathrm { M } _ { 1 } \mathrm {~B} _ { 1 }$ and segment $\mathrm { M } _ { 1 } \mathrm {~B} _ { 1 }$ of sector $\mathrm { N } _ { 1 } \mathrm { M } _ { 1 } \mathrm {~B} _ { 1 }$ and the region enclosed by the arc $\mathrm { M } _ { 1 } \mathrm { C } _ { 1 }$ and segment $\mathrm { M } _ { 1 } \mathrm { C } _ { 1 }$ of sector $\mathrm { D } _ { 1 } \mathrm { M } _ { 1 } \mathrm { C } _ { 1 }$ are shaded to form a checkmark shape, and the resulting figure is called $R _ { 1 }$.
In figure $R _ { 1 }$, a rectangle $\mathrm { A } _ { 2 } \mathrm {~B} _ { 2 } \mathrm { C } _ { 2 } \mathrm { D } _ { 2 }$ is drawn with vertices at point $\mathrm { A } _ { 2 }$ on segment $\mathrm { M } _ { 1 } \mathrm {~B} _ { 1 }$, point $\mathrm { D } _ { 2 }$ on arc $\mathrm { M } _ { 1 } \mathrm { C } _ { 1 }$, and two points $\mathrm { B } _ { 2 } , \mathrm { C } _ { 2 }$ on side $\mathrm { B } _ { 1 } \mathrm { C } _ { 1 }$ such that $\overline { \mathrm { A } _ { 2 } \mathrm {~B} _ { 2 } } : \overline { \mathrm { A } _ { 2 } \mathrm { D } _ { 2 } } = 1 : 2$. A checkmark shape is shaded in rectangle $\mathrm { A } _ { 2 } \mathrm {~B} _ { 2 } \mathrm { C } _ { 2 } \mathrm { D } _ { 2 }$ using the same method as for figure $R _ { 1 }$, and the resulting figure is called $R _ { 2 }$.
Continuing this process, let $S _ { n }$ be the area of the shaded region in the $n$-th figure $R _ { n }$. What is the value of $\lim _ { n \rightarrow \infty } S _ { n }$? [4 points]
(1) $\frac { 25 } { 19 } \left( \frac { \pi } { 2 } - 1 \right)$
(2) $\frac { 5 } { 4 } \left( \frac { \pi } { 2 } - 1 \right)$
(3) $\frac { 25 } { 21 } \left( \frac { \pi } { 2 } - 1 \right)$
(4) $\frac { 25 } { 22 } \left( \frac { \pi } { 2 } - 1 \right)$
(5) $\frac { 25 } { 23 } \left( \frac { \pi } { 2 } - 1 \right)$
Q16 4 marks Continuous Probability Distributions and Random Variables PDF Graph Interpretation and Probability Computation View
A continuous probability density function is defined on the closed interval $[ 0 , a ]$ for a random variable $X$. When the random variable $X$ satisfies the following conditions, what is the value of the constant $k$? [4 points] (가) For all $x$ where $0 \leq x \leq a$, $\mathrm { P } ( 0 \leq X \leq x ) = k x ^ { 2 }$. (나) $\mathrm { E } ( X ) = 1$
(1) $\frac { 9 } { 16 }$
(2) $\frac { 4 } { 9 }$
(3) $\frac { 1 } { 4 }$
(4) $\frac { 1 } { 9 }$
(5) $\frac { 1 } { 16 }$
Q17 4 marks Matrices True/False or Multiple-Select Conceptual Reasoning View
Two $2 \times 2$ square matrices $A , B$ satisfy $$A B + A ^ { 2 } B = E , \quad ( A - E ) ^ { 2 } + B ^ { 2 } = O$$ Among the statements in the following, which are correct? (Here, $E$ is the identity matrix and $O$ is the zero matrix.) [4 points]
Statements ᄀ. The inverse matrix of $B$ exists. ㄴ. $A B = B A$ ㄷ. $\left( A ^ { 3 } - A \right) ^ { 2 } + E = O$
(1) ㄴ
(2) ㄷ
(3) ᄀ, ㄴ
(4) ᄀ, ㄷ
(5) ᄀ, ㄴ, ㄷ
Q18 4 marks Standard trigonometric equations Count zeros or intersection points involving trigonometric curves View
For a natural number $n$, let $a _ { n }$ be the $n$-th smallest $x$-coordinate among the intersection points of the line $y = n$ and the graph of the function $y = \tan x$ in the first quadrant.
What is the value of $\lim _ { n \rightarrow \infty } \frac { a _ { n } } { n }$? [4 points]
(1) $\frac { \pi } { 4 }$
(2) $\frac { \pi } { 2 }$
(3) $\frac { 3 } { 4 } \pi$
(4) $\pi$
(5) $\frac { 5 } { 4 } \pi$
Q19 4 marks Circles Sphere and 3D Circle Problems View
In coordinate space, a sphere $S$ with center coordinates all positive has center at $(x, y, z)$ where $x > 0, y > 0, z > 0$, is tangent to the $x$-axis and $y$-axis respectively, and intersects the $z$-axis at two distinct points. The area of the circle formed by the intersection of sphere $S$ and the $xy$-plane is $64 \pi$, and the distance between the two intersection points with the $z$-axis is 8. What is the radius of sphere $S$? [4 points]
(1) 11
(2) 12
(3) 13
(4) 14
(5) 15
Q20 4 marks Laws of Logarithms Characteristic and Mantissa of Common Logarithms View
For a real number $x > 1$, let $f ( x )$ and $g ( x )$ be the characteristic and mantissa of $\log x$ respectively. When the value of $3 f ( x ) + 5 g ( x )$ is a multiple of 10, the values of $x$ are listed in increasing order. Let the 2nd value be $a$ and the 6th value be $b$. What is the value of $\log a b$? [4 points]
(1) 8
(2) 10
(3) 12
(4) 14
(5) 16
Q21 4 marks Indefinite & Definite Integrals Finding a Function from an Integral Equation View
The graph of a continuous function $y = f ( x )$ is symmetric about the origin, and for all real numbers $x$, $$f ( x ) = \frac { \pi } { 2 } \int _ { 1 } ^ { x + 1 } f ( t ) d t$$ When $f ( 1 ) = 1$, what is the value of $$\pi ^ { 2 } \int _ { 0 } ^ { 1 } x f ( x + 1 ) d x$$ ? [4 points]
(1) $2 ( \pi - 2 )$
(2) $2 \pi - 3$
(3) $2 ( \pi - 1 )$
(4) $2 \pi - 1$
(5) $2 \pi$
Q22 3 marks Differentiating Transcendental Functions Evaluate derivative at a point or find tangent slope View
For the function $f ( x ) = 5 e ^ { 3 x - 3 }$, find the value of $f ^ { \prime } ( 1 )$. [3 points]
Q23 3 marks Conditional Probability Direct Conditional Probability Computation from Definitions View
Among 50 members of a marathon club who participated in a certain marathon, the number of members who completed the marathon and the number who withdrew are as follows. (Unit: persons)
CategoryMaleFemale
Completed279
Withdrew86

When one member is randomly selected from the participants and is found to be female, the probability that this member completed the marathon is $p$. Find the value of $100 p$. [3 points]
Q24 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 $\sqrt { 2 x ^ { 2 } - 6 x } = x ^ { 2 } - 3 x - 4$, and call it $k$. Find the value of $k ^ { 2 }$. [3 points]
Q25 3 marks Laws of Logarithms Logarithmic Formula Application (Modeling) View
In a cylindrical water pipe with cross-sectional radius $R ( R < 1 )$, water flows completely full. Let $v _ { c }$ be the speed of water at the center of the cross-section, and let $v$ be the speed of water at a point $x ( 0 < x \leq R )$ away from the wall toward the center. The following relationship holds: $$\frac { v _ { c } } { v } = 1 - k \log \frac { x } { R }$$ (Here, $k$ is a positive constant, and the unit of length is m and the unit of speed is m/s.) In this water pipe where $R < 1$, when the speed of water at a point $R ^ { \frac { 27 } { 23 } }$ away from the wall toward the center is $\frac { 1 } { 2 }$ of the speed at the center, the speed of water at a point $R ^ { a }$ away from the wall toward the center is $\frac { 1 } { 3 }$ of the speed at the center.
Find the value of $23 a$. [3 points]
Q26 4 marks Confidence intervals Determine minimum sample size for a desired interval width View
To determine the proportion of residents in a certain city who have experience using the central park, $n$ residents of the city were randomly sampled and surveyed. The result showed that 80\% had experience using the central park. Using this result, the 95\% confidence interval for the proportion of residents in the entire city who have experience using the central park is $[ a , b ]$. When $b - a = 0.098$, find the value of $n$. (Here, when $Z$ is a random variable following the standard normal distribution, calculate using $\mathrm { P } ( | Z | \leq 1.96 ) = 0.95$.) [4 points]
Q27 4 marks Circles Optimization on a Circle View
As shown in the figure, there is a point $\mathrm { A } ( 0 , a )$ on the $y$-axis and a point P moving on the ellipse $\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 9 } = 1$ with foci $\mathrm { F } , \mathrm { F } ^ { \prime }$. When the minimum value of $\overline { \mathrm { AP } } - \overline { \mathrm { FP } }$ is 1, find the value of $a ^ { 2 }$. [4 points]
Q28 4 marks Applied differentiation Limit evaluation involving derivatives or asymptotic analysis View
As shown in the figure, there is an isosceles triangle ABC with AB as one side of length 4, $\overline { \mathrm { AC } } = \overline { \mathrm { BC } }$, and $\angle \mathrm { ACB } = \theta$. On the extension of segment AB, a point D is taken such that $\overline { \mathrm { AC } } = \overline { \mathrm { AD } }$, and a point P is taken such that $\overline { \mathrm { AC } } = \overline { \mathrm { AP } }$ and $\angle \mathrm { PAB } = 2 \theta$. Let $S ( \theta )$ be the area of triangle BDP. Find the value of $\lim _ { \theta \rightarrow + 0 } ( \theta \times S ( \theta ) )$. (Here, $0 < \theta < \frac { \pi } { 6 }$) [4 points]
Q29 4 marks Vectors 3D & Lines Vector Algebra and Triple Product Computation View
In coordinate space, there are two points $\mathrm { P } , \mathrm { Q }$ moving on the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4$. Let $\mathrm { P } _ { 1 } , \mathrm { Q } _ { 1 }$ be the feet of the perpendiculars from points P and Q to the plane $y = 4$ respectively, and let $\mathrm { P } _ { 2 } , \mathrm { Q } _ { 2 }$ be the feet of the perpendiculars to the plane $y + \sqrt { 3 } z + 8 = 0$ respectively. Find the maximum value of $2 | \overrightarrow { \mathrm { PQ } } | ^ { 2 } - \left| \overrightarrow { \mathrm { P } _ { 1 } \mathrm { Q } _ { 1 } } \right| ^ { 2 } - \left| \overrightarrow { \mathrm { P } _ { 2 } \mathrm { Q } _ { 2 } } \right| ^ { 2 }$. [4 points]
Q30 4 marks Second order differential equations Verifying a particular solution satisfies a second-order ODE View
For a quadratic function $f ( x )$, the function $g ( x ) = f ( x ) e ^ { - x }$ satisfies the following conditions. (가) The points $( 1 , g ( 1 ) )$ and $( 4 , g ( 4 ) )$ are inflection points of the curve $y = g ( x )$. (나) The number of tangent lines drawn from the point $( 0 , k )$ to the curve $y = g ( x )$ is 3 when $k$ is in the range $- 1 < k < 0$. Find the value of $g ( - 2 ) \times g ( 4 )$. [4 points]