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
2012 csat__math-science

30 maths questions

Q1 2 marks Matrices Linear System and Inverse Existence View
The sum of all components of the inverse matrix $A ^ { - 1 }$ of the matrix $A = \left( \begin{array} { r r } 1 & - 2 \\ 0 & 1 \end{array} \right)$ is? [2 points]
(1) 5
(2) 4
(3) 3
(4) 2
(5) 1
Q2 2 marks Differentiating Transcendental Functions Limit involving transcendental functions View
What is the value of $\lim _ { x \rightarrow 0 } \frac { e ^ { x } - 1 } { 5 x }$? [2 points]
(1) 5
(2) $e$
(3) 1
(4) $\frac { 1 } { e }$
(5) $\frac { 1 } { 5 }$
Q3 2 marks Binomial Distribution Find Parameters from Moment Conditions View
A random variable $X$ follows a binomial distribution $\mathrm { B } ( 200 , p )$ and the mean of $X$ is 40. What is the variance of $X$? [2 points]
(1) 32
(2) 33
(3) 34
(4) 35
(5) 36
Q4 3 marks Inequalities Set Operations Using Inequality-Defined Sets View
For two sets
$$A = \left\{ x \left\lvert \, \frac { ( x - 2 ) ^ { 2 } } { x - 4 } \leq 0 \right. \right\} , \quad B = \left\{ x \mid x ^ { 2 } - 8 x + a \leq 0 \right\}$$
When $A \cup B = \{ x \mid x \leq 5 \}$, what is the value of the constant $a$? [3 points]
(1) 7
(2) 10
(3) 12
(4) 15
(5) 16
Q5 3 marks Combinations & Selection Counting Arrangements with Run or Pattern Constraints View
When arranging 5 white flags and 5 blue flags in a line, how many ways are there to place white flags at both ends? (Note: flags of the same color are indistinguishable from each other.) [3 points]
(1) 56
(2) 63
(3) 70
(4) 77
(5) 84
Q6 3 marks Linear transformations View
On the coordinate plane, three points $\mathrm { A } ( 3,0 ) , \mathrm { B } ( 3,3 ) , \mathrm { C } ( 0,3 )$ are mapped to $\mathrm { A } ^ { \prime } , \mathrm { B } ^ { \prime } , \mathrm { C } ^ { \prime }$ respectively by the linear transformation represented by the matrix $\left( \begin{array} { l l } k & 0 \\ 0 & k \end{array} \right) ( k > 1 )$. When the area of the common part of the interior of triangle ABC and the interior of triangle $\mathrm { A } ^ { \prime } \mathrm { B } ^ { \prime } \mathrm { C } ^ { \prime }$ is $\frac { 1 } { 2 }$, what is the value of $k$? [3 points]
(1) $\frac { 3 } { 2 }$
(2) $\frac { 5 } { 3 }$
(3) $\frac { 7 } { 4 }$
(4) $\frac { 9 } { 5 }$
(5) $\frac { 11 } { 6 }$
Q7 3 marks Laws of Logarithms Logarithmic Formula Application (Modeling) View
The female silkworm moth secretes pheromone to attract males.
When $t$ seconds have passed since the female silkworm moth secreted pheromone, the concentration $y$ of pheromone measured at a distance $x$ from the secretion site satisfies the following equation.
$$\log y = A - \frac { 1 } { 2 } \log t - \frac { K x ^ { 2 } } { t } \text { (where } A \text { and } K \text { are positive constants.) }$$
When 1 second has passed since the female silkworm moth secreted pheromone, the pheromone concentration measured at a distance 2 from the secretion site is $a$, and when 4 seconds have passed, the pheromone concentration measured at a distance $d$ from the secretion site is $\frac { a } { 2 }$. What is the value of $d$? [3 points]
(1) 7
(2) 6
(3) 5
(4) 4
(5) 3
Q8 3 marks Vectors Introduction & 2D Magnitude of Vector Expression View
In triangle ABC,
$$\overline { \mathrm { AB } } = 2 , \quad \angle \mathrm {~B} = 90 ^ { \circ } , \quad \angle \mathrm { C } = 30 ^ { \circ }$$
When point P satisfies $\overrightarrow { \mathrm { PB } } + \overrightarrow { \mathrm { PC } } = \overrightarrow { 0 }$, what is the value of $| \overrightarrow { \mathrm { PA } } | ^ { 2 }$? [3 points]
(1) 5
(2) 6
(3) 7
(4) 8
(5) 9
Q9 3 marks Confidence intervals Algebraic problem using two confidence intervals View
The calcium content in one bottle of beverage produced by a certain company follows a normal distribution with population mean $m$ and population standard deviation $\sigma$. When 16 bottles of beverage produced by this company were randomly sampled and the calcium content was measured, the sample mean was 12.34. When the 95\% confidence interval for the population mean $m$ of the calcium content in one bottle of beverage produced by this company is $11.36 \leq m \leq a$, what is the value of $a + \sigma$? (Note: When $Z$ follows the standard normal distribution, $\mathrm { P } ( 0 \leq Z \leq 1.96 ) = 0.4750$, and the unit of calcium content is mg.) [3 points]
(1) 14.32
(2) 14.82
(3) 15.32
(4) 15.82
(5) 16.32
Q10 3 marks Linear transformations View
On the coordinate plane, a rotation transformation $f$ centered at the origin maps the point $( 1,0 )$ to a point $\left( \frac { \sqrt { 3 } } { 2 } , a \right)$ in the first quadrant. When the sum of all components of the matrix representing the rotation transformation $f$ is $b$, what is the value of $a ^ { 2 } + b ^ { 2 }$? [3 points]
(1) $\frac { 31 } { 12 }$
(2) $\frac { 11 } { 4 }$
(3) $\frac { 35 } { 12 }$
(4) $\frac { 37 } { 12 }$
(5) $\frac { 13 } { 4 }$
Q11 3 marks Conic sections Triangle or Quadrilateral Area and Perimeter with Foci View
For a rhombus ABCD with side length 10, an ellipse with diagonal BD as the major axis and diagonal AC as the minor axis has a distance between the two foci of $10 \sqrt { 2 }$. What is the area of rhombus ABCD? [3 points]
(1) $55 \sqrt { 3 }$
(2) $65 \sqrt { 2 }$
(3) $50 \sqrt { 3 }$
(4) $45 \sqrt { 3 }$
(5) $45 \sqrt { 2 }$
Q12 3 marks Curve Sketching Number of Solutions / Roots via Curve Analysis View
The graphs of the quadratic function $y = f ( x )$ and the cubic function $y = g ( x )$ are shown in the figure.
$f ( - 1 ) = f ( 3 ) = 0$, and the function $g ( x )$ has a local minimum value of $- 2$ at $x = 3$. What is the number of distinct real roots of the equation $\frac { g ( x ) + 2 } { f ( x ) } - \frac { 2 } { g ( x ) } = 1$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5
Q13 3 marks Probability Definitions Conditional Probability and Bayes' Theorem View
Box A contains 3 red balls and 5 black balls, and box B is empty. When 2 balls are randomly drawn from box A, if a red ball appears, perform [Execution 1], and if no red ball appears, perform [Execution 2]. What is the probability that the number of red balls in box B is 1? [3 points] [Execution 1] Put the drawn balls into box B. [Execution 2] Put the drawn balls into box B, and then randomly draw 2 more balls from box A and put them into box B.
(1) $\frac { 1 } { 2 }$
(2) $\frac { 7 } { 12 }$
(3) $\frac { 2 } { 3 }$
(4) $\frac { 3 } { 4 }$
(5) $\frac { 5 } { 6 }$
Q14 4 marks Geometric Sequences and Series Fractal/Iterative Geometric Construction (Area, Length, or Perimeter Series) View
There is a circle with radius 1. As shown in the figure, a rectangle with a ratio of horizontal length to vertical length of $3 : 1$ is inscribed in this circle, and the common part of the interior of the circle and the exterior of the rectangle is colored to obtain a figure $R _ { 1 }$. In figure $R _ { 1 }$, 2 circles are drawn tangent to three sides of the rectangle. A rectangle is drawn in each of the newly drawn circles using the same method as obtaining figure $R _ { 1 }$, and the colored figure obtained is $R _ { 2 }$. In figure $R _ { 2 }$, 4 circles are drawn tangent to three sides of the newly drawn rectangles. A rectangle is drawn in each of the newly drawn circles using the same method as obtaining figure $R _ { 1 }$, and the colored figure obtained is $R _ { 3 }$. Continuing this process, let $S _ { n }$ be the area of the colored part in the $n$-th obtained figure $R _ { n }$. What is the value of $\lim _ { n \rightarrow \infty } S _ { n }$? [4 points]
(1) $\frac { 5 } { 4 } \pi - \frac { 5 } { 3 }$
(2) $\frac { 5 } { 4 } \pi - \frac { 3 } { 2 }$
(3) $\frac { 4 } { 3 } \pi - \frac { 8 } { 5 }$
(4) $\frac { 5 } { 4 } \pi - 1$
(5) $\frac { 4 } { 3 } \pi - \frac { 16 } { 15 }$
Q15 4 marks Matrices True/False or Multiple-Select Conceptual Reasoning View
Two square matrices $A , B$ satisfy
$$A ^ { 2 } + B = 3 E , \quad A ^ { 4 } + B ^ { 2 } = 7 E$$
Which of the following are correct? Choose all that apply from . (Here, $E$ is the identity matrix.) [4 points]
Remarks ㄱ. $A B = B A$ ㄴ. $B ^ { - 1 } = A ^ { 2 }$ ㄷ. $A ^ { 6 } + B ^ { 3 } = 18 E$
(1) ㄱ
(2) ㄴ
(3) ㄱ, ㄴ
(4) ㄱ, ㄷ
(5) ㄱ, ㄴ, ㄷ
Q16 4 marks Areas by integration View
In the figure, let $a$ be the area of region $A$ enclosed by the two curves $y = e ^ { x } , y = x e ^ { x }$ and the $y$-axis, and let $b$ be the area of region $B$ enclosed by the two curves $y = e ^ { x } , y = x e ^ { x }$ and the line $x = 2$. What is the value of $b - a$? [4 points]
(1) $\frac { 3 } { 2 }$
(2) $e - 1$
(3) 2
(4) $\frac { 5 } { 2 }$
(5) $e$
Q17 4 marks Sequences and series, recurrence and convergence Auxiliary sequence transformation View
For a sequence $\left\{ a _ { n } \right\}$ with first term 1, let $S _ { n } = \sum _ { k = 1 } ^ { n } a _ { k }$. The following relation holds:
$$n S _ { n + 1 } = ( n + 2 ) S _ { n } + ( n + 1 ) ^ { 3 } \quad ( n \geq 1 )$$
The following is part of the process of finding the general term of the sequence $\left\{ a _ { n } \right\}$.
Since $S _ { n + 1 } = S _ { n } + a _ { n + 1 }$ for natural numbers $n$,
$$n a _ { n + 1 } = 2 S _ { n } + ( n + 1 ) ^ { 3 } \quad \cdots (\text{ㄱ})$$
For natural numbers $n \geq 2$,
$$( n - 1 ) a _ { n } = 2 S _ { n - 1 } + n ^ { 3 } \quad \cdots (\text{ㄴ})$$
By subtracting (ㄴ) from (ㄱ), we obtain
$$n a _ { n + 1 } = ( n + 1 ) a _ { n } + \quad \text { (가) }$$
Dividing both sides by $n ( n + 1 )$,
$$\frac { a _ { n + 1 } } { n + 1 } = \frac { a _ { n } } { n } + \frac { ( \text{가} ) } { n ( n + 1 ) }$$
If $b _ { n } = \frac { a _ { n } } { n }$, then
$$b _ { n + 1 } = b _ { n } + \text{ (나) } \quad ( n \geq 2 )$$
so
$$b _ { n } = b _ { 2 } + \text{ (다) } \quad ( n \geq 3 )$$
When the expressions that go in (가), (나), and (다) are $f ( n ) , g ( n ) , h ( n )$ respectively, what is the value of $\frac { f ( 3 ) } { g ( 3 ) h ( 6 ) }$? [4 points]
(1) 30
(2) 36
(3) 42
(4) 48
(5) 54
Q18 4 marks Differentiating Transcendental Functions Monotonicity or convexity of transcendental functions View
For the function $f ( x ) = 2 x \cos x$ with domain $\{ x \mid 0 \leq x \leq \pi \}$, which of the following are correct? Choose all that apply from . [4 points]
Remarks ㄱ. If $f ^ { \prime } ( a ) = 0$, then $\tan a = \frac { 1 } { a }$. ㄴ. There exists $a$ in the interval $\left( \frac { \pi } { 4 } , \frac { \pi } { 3 } \right)$ where the function $f ( x )$ has a local maximum value at $x = a$. ㄷ. On the interval $\left[ 0 , \frac { \pi } { 2 } \right]$, the number of distinct real roots of the equation $f ( x ) = 1$ is 2.
(1) ㄱ
(2) ㄷ
(3) ㄱ, ㄴ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ
Q19 4 marks Curve Sketching Continuity and Discontinuity Analysis of Piecewise Functions View
For a real number $m$, let $f ( m )$ be the number of intersection points of the line passing through the point $( 0,2 )$ with slope $m$ and the curve $y = x ^ { 3 } - 3 x ^ { 2 } + 1$. What is the maximum value of the real number $a$ such that the function $f ( m )$ is continuous on the interval $( - \infty , a )$? [4 points]
(1) $- 3$
(2) $- \frac { 3 } { 4 }$
(3) $\frac { 3 } { 2 }$
(4) $\frac { 15 } { 4 }$
(5) 6
Q20 4 marks Addition & Double Angle Formulae Multi-Step Composite Problem Using Identities View
On the coordinate plane, let $\theta _ { 1 }$ be the acute angle that the line $y = m x ( 0 < m < \sqrt { 3 } )$ makes with the $x$-axis, and let $\theta _ { 2 }$ be the acute angle that the line $y = m x$ makes with the line $y = \sqrt { 3 } x$. What is the value of $m$ that maximizes $3 \sin \theta _ { 1 } + 4 \sin \theta _ { 2 }$? [4 points]
(1) $\frac { \sqrt { 3 } } { 6 }$
(2) $\frac { \sqrt { 3 } } { 7 }$
(3) $\frac { \sqrt { 3 } } { 8 }$
(4) $\frac { \sqrt { 3 } } { 9 }$
(5) $\frac { \sqrt { 3 } } { 10 }$
Q21 4 marks Vectors 3D & Lines MCQ: Cross-Section or Surface Area of a Solid View
In coordinate space, triangle ABC satisfies the following conditions. (가) The area of triangle ABC is 6. (나) The area of the orthogonal projection of triangle ABC onto the $yz$-plane is 3.
What is the maximum area of the orthogonal projection of triangle ABC onto the plane $x - 2 y + 2 z = 1$? [4 points]
(1) $2 \sqrt { 6 } + 1$
(2) $2 \sqrt { 2 } + 3$
(3) $3 \sqrt { 5 } - 1$
(4) $2 \sqrt { 5 } + 1$
(5) $3 \sqrt { 6 } - 2$
Q22 3 marks Combinations & Selection Basic Combination Computation View
For a natural number $r$, when ${}_{3}\mathrm{H}_{r} = {}_{7}\mathrm{C}_{2}$, find the value of ${}_{5}\mathrm{H}_{r}$. [3 points]
Q23 3 marks Standard trigonometric equations Evaluate trigonometric expression given a constraint View
For $x$ satisfying the equation $3 \cos 2 x + 17 \cos x = 0$, find the value of $\tan ^ { 2 } x$. [3 points]
Q24 3 marks Conic sections Optimization on Conics View
In coordinate space, there is a point $\mathrm { A } ( 9,0,5 )$, and on the $xy$-plane there is an ellipse $\frac { x ^ { 2 } } { 9 } + y ^ { 2 } = 1$. For a point P on the ellipse, find the maximum value of $\overline { \mathrm { AP } }$. [3 points]
Q25 4 marks Conic sections Tangent and Normal Line Problems View
Let $d$ be the distance between the focus of the parabola $y ^ { 2 } = n x$ and the tangent line to the parabola at the point $( n , n )$. Find the minimum natural number $n$ satisfying $d ^ { 2 } \geq 40$. [4 points]
Q26 3 marks Arithmetic Sequences and Series Arithmetic-Geometric Hybrid Problem View
Three numbers $a , a + b , 2 a - b$ form an arithmetic sequence in this order, and three numbers $1 , a - 1, 3 b + 1$ form a geometric sequence with positive common ratio in this order. Find the value of $a ^ { 2 } + b ^ { 2 }$. [3 points]
Q27 4 marks Stationary points and optimisation Geometric or applied optimisation problem View
As shown in the figure, let Q be the foot of the perpendicular from point P on a circle with center O and diameter AB of length 2 to the line segment AB, let R be the foot of the perpendicular from point Q to the line segment OP, and let S be the foot of the perpendicular from point O to the line segment AP. When $\angle \mathrm { PAQ } = \theta \left( 0 < \theta < \frac { \pi } { 4 } \right)$, let $f ( \theta )$ be the area of triangle AOS and $g ( \theta )$ be the area of triangle PRQ. When $\lim _ { \theta \rightarrow +0 } \frac { \theta ^ { 2 } f ( \theta ) } { g ( \theta ) } = \frac { q } { p }$, find the value of $p ^ { 2 } + q ^ { 2 }$. (Here, $p$ and $q$ are coprime natural numbers.) [4 points]
Q28 4 marks Indefinite & Definite Integrals Finding a Function from an Integral Equation View
For the function $f ( x ) = 3 ( x - 1 ) ^ { 2 } + 5$, define the function $F ( x )$ as $F ( x ) = \int _ { 0 } ^ { x } f ( t ) \, dt$. A differentiable function $g ( x )$ satisfies the following for all real numbers $x$:
$$F ( g ( x ) ) = \frac { 1 } { 2 } F ( x )$$
When $g ^ { \prime } ( 2 ) = p$, find the value of $30 p$. [4 points]
Q29 4 marks SUVAT in 2D & Gravity View
As shown in the figure, a cylinder with base radius 7 and a cone with base radius 5 and height 12 are placed on a plane $\alpha$, and the circumference of the base of the cone is inscribed in the circumference of the base of the cylinder. Let O be the center of the base of the cylinder that meets plane $\alpha$, and let A be the apex of the cone. A sphere $S$ with center B and radius 4 satisfies the following conditions. (가) The sphere $S$ is tangent to both the cylinder and the cone. (나) When $\mathrm { A } ^ { \prime }$ and $\mathrm { B } ^ { \prime }$ are the orthogonal projections of points $\mathrm { A }$ and $\mathrm { B }$ onto plane $\alpha$ respectively, $\angle \mathrm { A } ^ { \prime } \mathrm { OB } ^ { \prime } = 180 ^ { \circ }$.
When the acute angle between line AB and plane $\alpha$ is $\theta$, $\tan \theta = p$. Find the value of $100 p$. (Note: The center of the base of the cone and point $\mathrm { A } ^ { \prime }$ coincide.) [4 points]
Q30 4 marks Exponential Functions Intersection and Distance between Curves View
For natural numbers $a , b$, let P and Q be the points where the curve $y = a ^ { x + 1 }$ and the curve $y = b ^ { x }$ meet the line $x = t ( t \geq 1 )$ respectively. Find the number of all ordered pairs $( a , b )$ satisfying the following conditions. For example, $a = 4 , b = 5$ satisfies the following conditions. [4 points] (가) $2 \leq a \leq 10, 2 \leq b \leq 10$ (나) For some real number $t \geq 1$, $\overline { \mathrm { PQ } } \leq 10$.