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
2016 csat__math-B

30 maths questions

Q1 2 marks Matrices Matrix Algebra and Product Properties View
For two matrices $A = \left( \begin{array} { l l } a & 3 \\ 0 & 1 \end{array} \right) , B = \left( \begin{array} { r r } 4 & 1 \\ - 1 & 0 \end{array} \right)$, when the sum of all entries of matrix $A + B$ is 9, 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 three points $\mathrm { A } ( a , 0,5 ) , \mathrm { B } ( 1 , b , - 3 ) , \mathrm { C } ( 1,1,1 )$ that are vertices of a triangle, when the centroid of the triangle has coordinates $( 2,2,1 )$, what is the value of $a + b$? [2 points]
(1) 6
(2) 7
(3) 8
(4) 9
(5) 10
Q3 2 marks Small angle approximation View
What is the value of $\lim _ { x \rightarrow 0 } \frac { \ln ( 1 + 5 x ) } { \sin 3 x }$? [2 points]
(1) 1
(2) $\frac { 4 } { 3 }$
(3) $\frac { 5 } { 3 }$
(4) 2
(5) $\frac { 7 } { 3 }$
Q4 3 marks Integration by Substitution Substitution to Evaluate a Definite Integral (Numerical Answer) View
What is the value of $\int _ { 0 } ^ { e } \frac { 5 } { x + e } d x$? [3 points]
(1) $\ln 2$
(2) $2 \ln 2$
(3) $3 \ln 2$
(4) $4 \ln 2$
(5) $5 \ln 2$
Q5 3 marks Independent Events View
Two events $A$ and $B$ are independent, and $$\mathrm { P } \left( A ^ { C } \right) = \frac { 1 } { 4 } , \quad \mathrm { P } ( A \cap B ) = \frac { 1 } { 2 }$$ What is the value of $\mathrm { P } \left( B \mid A ^ { C } \right)$? (Here, $A ^ { C }$ is the complement of $A$.) [3 points]
(1) $\frac { 5 } { 12 }$
(2) $\frac { 1 } { 2 }$
(3) $\frac { 7 } { 12 }$
(4) $\frac { 2 } { 3 }$
(5) $\frac { 3 } { 4 }$
Q6 3 marks Linear transformations View
When point $\mathrm { P } ( 2 , - 1 )$ is mapped to point Q by the linear transformation represented by the matrix $\left( \begin{array} { r r } 1 & 2 \\ - 2 & 1 \end{array} \right)$, what is the slope of line PQ? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5
Q7 3 marks Tangents, normals and gradients Find tangent line with a specified slope or from an external point View
For the curve $y = 3 e ^ { x - 1 }$, when the tangent line at point A passes through the origin O, what is the length of segment OA? [3 points]
(1) $\sqrt { 6 }$
(2) $\sqrt { 7 }$
(3) $2 \sqrt { 2 }$
(4) 3
(5) $\sqrt { 10 }$
Q8 3 marks Permutations & Arrangements Probability via Permutation Counting View
When a coin is tossed 5 times, what is the probability that the product of the number of heads and the number of tails is 6? [3 points]
(1) $\frac { 5 } { 8 }$
(2) $\frac { 9 } { 16 }$
(3) $\frac { 1 } { 2 }$
(4) $\frac { 7 } { 16 }$
(5) $\frac { 3 } { 8 }$
Q9 3 marks Circles Area and Geometric Measurement Involving Circles View
For the parabola $y ^ { 2 } = 4 x$, let $l$ be the tangent line at point $\mathrm { A } ( 4,4 )$. Let B be the intersection of line $l$ and the directrix of the parabola, C be the intersection of line $l$ and the $x$-axis, and D be the intersection of the directrix and the $x$-axis. What is the area of triangle BCD? [3 points]
(1) $\frac { 7 } { 4 }$
(2) 2
(3) $\frac { 9 } { 4 }$
(4) $\frac { 5 } { 2 }$
(5) $\frac { 11 } { 4 }$
Q10 3 marks Exponential Equations & Modelling Evaluate Expression Given Exponential/Logarithmic Conditions View
For a certain financial product, when an initial asset $W _ { 0 }$ is invested, the expected asset $W$ after $t$ years is given as follows. $$\begin{aligned} & W = \frac { W _ { 0 } } { 2 } 10 ^ { a t } \left( 1 + 10 ^ { a t } \right) \\ & \text { (where } W _ { 0 } > 0 , t \geq 0 \text {, and } a \text { is a constant.) } \end{aligned}$$ When an initial asset $w _ { 0 }$ is invested in this financial product, the expected asset after 15 years is 3 times the initial asset. When an initial asset $w _ { 0 }$ is invested in this financial product, the expected asset after 30 years is $k$ times the initial asset. What is the value of the real number $k$? (where $w _ { 0 } > 0$) [3 points]
(1) 9
(2) 10
(3) 11
(4) 12
(5) 13
Q11 3 marks Volumes of Revolution Volume of Revolution about a Horizontal Axis (Evaluate) View
Consider the function $$f ( x ) = \begin{cases} | 5 x ( x + 2 ) | & ( x < 0 ) \\ | 5 x ( x - 2 ) | & ( x \geq 0 ) \end{cases}$$ On the closed interval $[ 0,1 ]$, what is the volume of the solid of revolution generated by rotating the region enclosed by the graph of $y = f ( x )$, the $x$-axis, and the line $x = 1$ about the $x$-axis? [3 points]
(1) $\frac { 65 } { 6 } \pi$
(2) $\frac { 35 } { 3 } \pi$
(3) $\frac { 25 } { 2 } \pi$
(4) $\frac { 40 } { 3 } \pi$
(5) $\frac { 85 } { 6 } \pi$
Q12 3 marks Curve Sketching Number of Solutions / Roots via Curve Analysis View
Consider the function $$f ( x ) = \begin{cases} | 5 x ( x + 2 ) | & ( x < 0 ) \\ | 5 x ( x - 2 ) | & ( x \geq 0 ) \end{cases}$$ What is the number of distinct real roots of the irrational equation $\sqrt { 4 - f ( x ) } = 1 - x$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5
Q13 3 marks Sequences and series, recurrence and convergence Summation of sequence terms View
As shown in the figure, for a square ABCD with side length 5, let the five division points of diagonal BD be $\mathrm { P } _ { 1 } , \mathrm { P } _ { 2 } , \mathrm { P } _ { 3 } , \mathrm { P } _ { 4 }$ in order from point B. Draw squares with diagonals $\mathrm { BP } _ { 1 } , \mathrm { P } _ { 2 } \mathrm { P } _ { 3 } , \mathrm { P } _ { 4 } \mathrm { D }$ and circles with diameters $\mathrm { P } _ { 1 } \mathrm { P } _ { 2 } , \mathrm { P } _ { 3 } \mathrm { P } _ { 4 }$, then color the figure-eight-shaped region to obtain figure $R _ { 1 }$. In figure $R _ { 1 }$, let $\mathrm { Q } _ { 1 }$ be the vertex of the square with diagonal $\mathrm { P } _ { 2 } \mathrm { P } _ { 3 }$ closest to point A, and $\mathrm { Q } _ { 2 }$ be the vertex closest to point C. Draw squares with diagonals $\mathrm { AQ } _ { 1 }$ and $\mathrm { CQ } _ { 2 }$, and in these 2 new squares, draw figure-eight-shaped figures using the same method as for $R _ { 1 }$ and color them to obtain figure $R _ { 2 }$. In figure $R _ { 2 }$, in the squares with diagonals $\mathrm { AQ } _ { 1 }$ and $\mathrm { CQ } _ { 2 }$, draw figure-eight-shaped figures using the same method as obtaining $R _ { 2 }$ from $R _ { 1 }$ and color them to obtain figure $R _ { 3 }$. Continuing this process, let $S _ { n }$ be the area of the colored region in the $n$-th figure $R _ { n }$. What is the value of $\lim _ { n \rightarrow \infty } S _ { n }$? [3 points]
(1) $\frac { 24 } { 17 } ( \pi + 3 )$
(2) $\frac { 25 } { 17 } ( \pi + 3 )$
(3) $\frac { 26 } { 17 } ( \pi + 3 )$
(4) $\frac { 24 } { 17 } ( 2 \pi + 1 )$
(5) $\frac { 25 } { 17 } ( 2 \pi + 1 )$
Q14 4 marks Combinations & Selection Selection with Arithmetic or Divisibility Conditions View
For three integers $a , b , c$ satisfying $$1 \leq | a | \leq | b | \leq | c | \leq 5$$ what is the number of all ordered pairs $( a , b , c )$? [4 points]
(1) 360
(2) 320
(3) 280
(4) 240
(5) 200
Q15 4 marks Stationary points and optimisation Geometric or applied optimisation problem View
In the coordinate plane, point A has coordinates $( 1,0 )$, and for $\theta$ with $0 < \theta < \frac { \pi } { 2 }$, point B has coordinates $( \cos \theta , \sin \theta )$. For point C in the first quadrant such that quadrilateral OACB is a parallelogram, let $f ( \theta )$ be the area of quadrilateral OACB and $g ( \theta )$ be the square of the length of segment OC. What is the maximum value of $f ( \theta ) + g ( \theta )$? (Here, O is the origin.) [4 points]
(1) $2 + \sqrt { 5 }$
(2) $2 + \sqrt { 6 }$
(3) $2 + \sqrt { 7 }$
(4) $2 + 2 \sqrt { 2 }$
(5) 5
Q16 4 marks Matrices True/False or Multiple-Select Conceptual Reasoning View
Two square matrices $A$ and $B$ satisfy $$A + B = ( B A ) ^ { 2 } , \quad A B A = B + E$$ Among the following statements, which are correct? (Here, $E$ is the identity matrix.) [4 points]
Statements ㄱ. $A = B ^ { 2 }$ ㄴ. $B ^ { - 1 } = A ^ { 2 } + E$ ㄷ. $A ^ { 5 } - B ^ { 5 } = A + B$
(1) ㄱ
(2) ㄴ
(3) ㄱ, ㄷ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ
Q17 4 marks Sequences and series, recurrence and convergence Auxiliary sequence transformation View
A sequence $\left\{ a _ { n } \right\}$ with all positive terms has $a _ { 1 } = a _ { 2 } = 1$, and with $S _ { n } = \sum _ { k = 1 } ^ { n } a _ { k }$, it satisfies $$a _ { n + 1 } = \frac { S _ { n } ^ { 2 } } { S _ { n - 1 } } + ( 2 n - 1 ) S _ { n } \quad ( n \geq 2 )$$ The following is the process of finding the general term $a _ { n }$. $$\begin{gathered} \text{Since } a _ { n + 1 } = S _ { n + 1 } - S _ { n } \text{, from the given equation we have} \\ S _ { n + 1 } = \frac { S _ { n } ^ { 2 } } { S _ { n - 1 } } + 2 n S _ { n } \quad ( n \geq 2 ) \end{gathered}$$ Dividing both sides by $S _ { n }$, we get $$\frac { S _ { n + 1 } } { S _ { n } } = \frac { S _ { n } } { S _ { n - 1 } } + 2 n$$ Let $b _ { n } = \frac { S _ { n + 1 } } { S _ { n } }$. Then $b _ { 1 } = 2$ and $$b _ { n } = b _ { n - 1 } + 2 n \quad ( n \geq 2 )$$ The general term of sequence $\left\{ b _ { n } \right\}$ is $$b _ { n } = \text { (가) } \times ( n + 1 ) \quad ( n \geq 1 )$$ Therefore, $$S _ { n } = ( \text{가} ) \times \{ ( n - 1 ) ! \} ^ { 2 } \quad ( n \geq 1 )$$ Thus $a _ { 1 } = 1$, and for $n \geq 2$, $$\begin{aligned} a _ { n } & = S _ { n } - S _ { n - 1 } \\ & = \text { (나) } \times \{ ( n - 2 ) ! \} ^ { 2 } \end{aligned}$$ Let $f ( n )$ and $g ( n )$ be the expressions that fit (가) and (나), respectively. What is the value of $f ( 10 ) + g ( 6 )$? [4 points]
(1) 110
(2) 125
(3) 140
(4) 155
(5) 170
Q18 4 marks Normal Distribution Sampling Distribution of the Mean View
From a population following a normal distribution $\mathrm { N } \left( 50,8 ^ { 2 } \right)$, a sample of size 16 is randomly extracted to obtain the sample mean $\bar { X }$. From a population following a normal distribution $\mathrm { N } \left( 75 , \sigma ^ { 2 } \right)$, a sample of size 25 is randomly extracted to obtain the sample mean $\bar { Y }$. When $\mathrm { P } ( \bar { X } \leq 53 ) + \mathrm { P } ( \bar { Y } \leq 69 ) = 1$, what is the value of $\mathrm { P } ( \bar { Y } \geq 71 )$ using the standard normal distribution table below?
$z$$\mathrm { P } ( 0 \leq Z \leq z )$
1.00.3413
1.20.3849
1.40.4192
1.60.4452

[4 points]
(1) 0.8413
(2) 0.8644
(3) 0.8849
(4) 0.9192
(5) 0.9452
Q19 4 marks Vectors 3D & Lines Dihedral Angle Computation View
In coordinate space, there are a point $\mathrm { A } ( 2,2,1 )$ and a plane $\alpha : x + 2 y + 2 z - 14 = 0$. When point P on plane $\alpha$ satisfies $\overline { \mathrm { AP } } \leq 3$, what is the area of the projection of the figure traced by point P onto the $xy$-plane? [4 points]
(1) $\frac { 14 } { 3 } \pi$
(2) $\frac { 13 } { 3 } \pi$
(3) $4 \pi$
(4) $\frac { 11 } { 3 } \pi$
(5) $\frac { 10 } { 3 } \pi$
Q20 4 marks Laws of Logarithms Characteristic and Mantissa of Common Logarithms View
For a positive number $x$, let $f ( x )$ be the characteristic (integer part) of $\log x$.
How many natural numbers $n$ not exceeding 100 satisfy $$f ( n + 10 ) = f ( n ) + 1$$ ? [4 points]
(1) 11
(2) 13
(3) 15
(4) 17
(5) 19
Q21 4 marks Parametric differentiation View
For a real number $0 < t < 41$, the curve $y = x ^ { 3 } + 2 x ^ { 2 } - 15 x + 5$ and the line $y = t$ intersect at three points. Let the point with the largest $x$-coordinate be $( f ( t ) , t )$ and the point with the smallest $x$-coordinate be $( g ( t ) , t )$. Let $h ( t ) = t \times \{ f ( t ) - g ( t ) \}$. What is the value of $h ^ { \prime } ( 5 )$? [4 points]
(1) $\frac { 79 } { 12 }$
(2) $\frac { 85 } { 12 }$
(3) $\frac { 91 } { 12 }$
(4) $\frac { 97 } { 12 }$
(5) $\frac { 103 } { 12 }$
Q22 3 marks Arithmetic Sequences and Series Find Common Difference from Given Conditions View
For an arithmetic sequence $\left\{ a _ { n } \right\}$ with first term 2, $$2 \left( a _ { 2 } + a _ { 3 } \right) = a _ { 9 }$$ Find the common difference of the sequence $\left\{ a _ { n } \right\}$. [3 points]
Q23 3 marks Differentiating Transcendental Functions Evaluate derivative at a point or find tangent slope View
For the function $f ( x ) = 4 \sin 7 x$, find the value of $f ^ { \prime } ( 2 \pi )$. [3 points]
Q24 3 marks Continuous Probability Distributions and Random Variables PDF Graph Interpretation and Probability Computation View
For a continuous random variable $X$ that takes all real values in the closed interval $[ 0,1 ]$, the probability density function is $$f ( x ) = k x \left( 1 - x ^ { 3 } \right) \quad ( 0 \leq x \leq 1 )$$ Find the value of $24 k$. (Here, $k$ is a constant.) [3 points]
Q25 3 marks Geometric Sequences and Series Determine the Limit of a Sequence via Geometric Series View
For a geometric sequence $\left\{ a _ { n } \right\}$ with first term 1 and common ratio $r$ ($r > 1$), let $S _ { n } = \sum _ { k = 1 } ^ { n } a _ { k }$. When $\lim _ { n \rightarrow \infty } \frac { a _ { n } } { S _ { n } } = \frac { 3 } { 4 }$, find the value of $r$. [3 points]
Q26 4 marks Conic sections Focal Distance and Point-on-Conic Metric Computation View
As shown in the figure, there is an ellipse $\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$ with foci at $\mathrm { F } ( c , 0 )$ and $\mathrm { F } ^ { \prime } ( - c , 0 )$. For point P on the ellipse in the second quadrant, let Q be the midpoint of segment $\mathrm { PF } ^ { \prime }$, and let R be the point that divides segment PF internally in the ratio $1 : 3$. When $\angle \mathrm { PQR } = \frac { \pi } { 2 }$, $\overline { \mathrm { QR } } = \sqrt { 5 }$, and $\overline { \mathrm { RF } } = 9$, find the value of $a ^ { 2 } + b ^ { 2 }$. (Here, $a$, $b$, and $c$ are positive numbers.) [4 points]
Q27 4 marks Vectors: Lines & Planes Distance Computation (Point-to-Plane or Line-to-Line) View
In coordinate space, there are two mutually perpendicular planes $\alpha$ and $\beta$. For two points $\mathrm { A }$ and $\mathrm { B }$ on plane $\alpha$, $\overline { \mathrm { AB } } = 3 \sqrt { 5 }$, and line AB is parallel to plane $\beta$. The distance between point A and plane $\beta$ is 2, and the distance between a point P on plane $\beta$ and plane $\alpha$ is 4. Find the area of triangle PAB. [4 points]
Q28 4 marks Differentiating Transcendental Functions Limit involving transcendental functions View
As shown in the figure, in the coordinate plane, the circle $x ^ { 2 } + y ^ { 2 } = 1$ and the curve $y = \ln ( x + 1 )$ meet at point A in the first quadrant. For point $\mathrm { B } ( 1,0 )$, let H be the foot of the perpendicular from point P on arc AB to the $y$-axis, and let Q be the intersection of segment PH and the curve $y = \ln ( x + 1 )$. Let $\angle \mathrm { POB } = \theta$. If $S ( \theta )$ is the area of triangle OPQ and $L ( \theta )$ is the length of segment HQ, and $\lim _ { \theta \rightarrow + 0 } \frac { S ( \theta ) } { L ( \theta ) } = k$, find the value of $60 k$. (Here, $0 < \theta < \frac { \pi } { 6 }$ and O is the origin.) [4 points]
Q29 4 marks Vectors 3D & Lines Vector Algebra and Triple Product Computation View
For two points $\mathrm { A } ( 2 , \sqrt { 2 } , \sqrt { 3 } )$ and $\mathrm { B } ( 1 , - \sqrt { 2 } , 2 \sqrt { 3 } )$ in coordinate space, point P satisfies the following conditions. (가) $| \overrightarrow { \mathrm { AP } } | = 1$ (나) The angle between $\overrightarrow { \mathrm { AP } }$ and $\overrightarrow { \mathrm { AB } }$ is $\frac { \pi } { 6 }$.
For point Q on a sphere centered at the origin with radius 1, the maximum value of $\overrightarrow { \mathrm { AP } } \cdot \overrightarrow { \mathrm { AQ } }$ is $a + b \sqrt { 33 }$. Find the value of $16 \left( a ^ { 2 } + b ^ { 2 } \right)$. (Here, $a$ and $b$ are rational numbers.) [4 points]
Q30 4 marks Differential equations Integral Equations Reducible to DEs View
A function $f ( x )$ that is continuous on the entire set of real numbers satisfies the following conditions. (가) For $x \leq b$, $f ( x ) = a ( x - b ) ^ { 2 } + c$. (Here, $a$, $b$, and $c$ are constants.) (나) For all real numbers $x$, $f ( x ) = \int _ { 0 } ^ { x } \sqrt { 4 - 2 f ( t ) } \, dt$. When $\int _ { 0 } ^ { 6 } f ( x ) \, dx = \frac { q } { p }$, find the value of $p + q$. (Here, $p$ and $q$ are coprime natural numbers.) [4 points]