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
2013 csat__math-science

30 maths questions

Q1 2 marks Matrices Matrix Algebra and Product Properties View
For two matrices $A = \left( \begin{array} { l l } 0 & 0 \\ 1 & 1 \end{array} \right) , B = \left( \begin{array} { l l } 1 & 0 \\ 1 & 1 \end{array} \right)$, what is the sum of all components of the matrix $2 A + B$? [2 points]
(1) 10
(2) 9
(3) 8
(4) 7
(5) 6
Q2 2 marks Vectors 3D & Lines Section Division and Coordinate Computation View
In coordinate space, for two points $\mathrm { A } ( a , 1,3 ) , \mathrm { B } ( a + 6,4,12 )$, the point that divides the line segment AB internally in the ratio $1 : 2$ has coordinates $( 5,2 , b )$. What is the value of $a + b$? [2 points]
(1) 7
(2) 8
(3) 9
(4) 10
(5) 11
Q3 2 marks Addition & Double Angle Formulae Direct Double Angle Evaluation View
When $\sin \theta = \frac { 1 } { 3 }$, what is the value of $\sin 2 \theta$? (Given that $0 < \theta < \frac { \pi } { 2 }$.) [2 points]
(1) $\frac { 7 \sqrt { 2 } } { 18 }$
(2) $\frac { 4 \sqrt { 2 } } { 9 }$
(3) $\frac { \sqrt { 2 } } { 2 }$
(4) $\frac { 5 \sqrt { 2 } } { 9 }$
(5) $\frac { 11 \sqrt { 2 } } { 18 }$
Q4 3 marks Inequalities Solve Polynomial/Rational Inequality for Solution Set View
The product of all real roots of the irrational equation $x ^ { 2 } - 2 x + 2 \sqrt { x ^ { 2 } - 2 x } = 8$ is? [3 points]
(1) - 5
(2) - 4
(3) - 3
(4) - 2
(5) - 1
Q5 3 marks Combinations & Selection Lattice Path Counting View
As shown in the figure, there is a road network connected in a diamond shape. Starting from point A and traveling the shortest distance to point B without passing through point C or point D, how many ways are there? [3 points]
(1) 26
(2) 24
(3) 22
(4) 20
(5) 18
Q6 3 marks Laws of Logarithms Logarithmic Formula Application (Modeling) View
The temperature of a fire room changes over time. Let the initial temperature of a certain fire room be $T _ { 0 } \left( { } ^ { \circ } \mathrm { C } \right)$, and the temperature $t$ minutes after the fire starts be $T \left( { } ^ { \circ } \mathrm { C } \right)$. The following equation holds. $T = T _ { 0 } + k \log ( 8 t + 1 ) \quad ($ where $k$ is a constant. $)$ In this fire room with an initial temperature of $20 ^ { \circ } \mathrm { C }$, the temperature was $365 ^ { \circ } \mathrm { C }$ at $\frac { 9 } { 8 }$ minutes after the fire started, and the temperature was $710 ^ { \circ } \mathrm { C }$ at $a$ minutes after the fire started. What is the value of $a$? [3 points]
(1) $\frac { 99 } { 8 }$
(2) $\frac { 109 } { 8 }$
(3) $\frac { 119 } { 8 }$
(4) $\frac { 129 } { 8 }$
(5) $\frac { 139 } { 8 }$
Q7 3 marks Conic sections Tangent and Normal Line Problems View
The tangent line at the point $( b , 1 )$ on the hyperbola $x ^ { 2 } - 4 y ^ { 2 } = a$ is perpendicular to one asymptote of the hyperbola. What is the value of $a + b$? (Given that $a , b$ are positive numbers.) [3 points]
(1) 68
(2) 77
(3) 86
(4) 95
(5) 104
Q8 3 marks Conditional Probability Bayes' Theorem with Production/Source Identification View
At a certain school, $60 \%$ of all students commute by bus, and the remaining $40 \%$ walk to school. Of the students who commute by bus, $\frac { 1 } { 20 }$ were late, and of the students who walk, $\frac { 1 } { 15 }$ were late. When one student is randomly selected from all students at this school and is found to be late, what is the probability that this student commuted by bus? [3 points]
(1) $\frac { 3 } { 7 }$
(2) $\frac { 9 } { 20 }$
(3) $\frac { 9 } { 19 }$
(4) $\frac { 1 } { 2 }$
(5) $\frac { 9 } { 17 }$
Q9 3 marks Linear transformations View
In the coordinate plane, let $f$ be the rotation transformation that rotates by $\frac { \pi } { 3 }$ about the origin, and let $g$ be the reflection transformation about the line $y = x$. When the line $x + 2 y + 5 = 0$ is mapped to the line $a x + b y + 5 = 0$ by the composite transformation $g ^ { - 1 } \circ f \circ g$, what is the value of $a + 2 b$? (Given that $a , b$ are constants.) [3 points]
(1) $\frac { 1 } { 2 }$
(2) 1
(3) $\frac { 3 } { 2 }$
(4) 2
(5) $\frac { 5 } { 2 }$
Q10 3 marks Inequalities Inequality Word Problem (Applied/Contextual) View
Starting from point A, one travels to point B which is 6 km away, and then returns to point A along the same route. For the first 1 km, one walks at a constant speed, and for the remaining 5 km, one travels at twice the initial walking speed. On the return trip, one travels at a speed 2 km/h faster than the initial walking speed. When the total time for the round trip is at most 2 hours 30 minutes, what is the minimum value of the initial walking speed? (Given that the unit of speed is km/h.) [3 points]
(1) $\frac { 12 } { 5 }$
(2) $\frac { 13 } { 5 }$
(3) $\frac { 14 } { 5 }$
(4) 3
(5) $\frac { 16 } { 5 }$
Q11 3 marks Tree Diagrams Multi-Stage Sequential Process View
There is a bag containing 4 white balls and 3 black balls.
Two balls are drawn simultaneously from the bag. If the two balls are of different colors, one coin is flipped 3 times. If the two balls are of the same color, one coin is flipped 2 times. What is the probability that the coin shows heads exactly 2 times in this trial? [3 points]
(1) $\frac { 9 } { 28 }$
(2) $\frac { 19 } { 56 }$
(3) $\frac { 5 } { 14 }$
(4) $\frac { 3 } { 8 }$
(5) $\frac { 11 } { 28 }$
Q12 3 marks Standard Integrals and Reverse Chain Rule Integral Equation to Determine a Function Value View
A continuous function $f ( x )$ satisfies
$$f ( x ) = e ^ { x ^ { 2 } } + \int _ { 0 } ^ { 1 } t f ( t ) d t$$
What is the value of $\int _ { 0 } ^ { 1 } x f ( x ) d x$? [3 points]
(1) $e - 2$
(2) $\frac { e - 1 } { 2 }$
(3) $\frac { e } { 2 }$
(4) $e - 1$
(5) $\frac { e + 1 } { 2 }$
Q13 3 marks Normal Distribution Algebraic Relationship Between Normal Parameters and Probability View
A random variable $X$ follows a normal distribution $\mathrm { N } \left( m , \sigma ^ { 2 } \right)$ and satisfies the following conditions.
(a) $\mathrm { P } ( X \geq 64 ) = \mathrm { P } ( X \leq 56 )$
(b) $\mathrm { E } \left( X ^ { 2 } \right) = 3616$ What is the value of $\mathrm { P } ( X \leq 68 )$ obtained using the table on the right? [3 points]
(1) 0.9104
(2) 0.9332
(3) 0.9544
(4) 0.9772
(5) 0.9938
$x$$\mathrm { P } ( m \leq X \leq x )$
$m + 1.5 \sigma$0.4332
$m + 2 \sigma$0.4772
$m + 2.5 \sigma$0.4938
Q14 4 marks Geometric Sequences and Series Fractal/Iterative Geometric Construction (Area, Length, or Perimeter Series) View
As shown in the figure, there is a circle O with diameter AB of length 2. A line passing through the center of circle O and perpendicular to line segment AB intersects the circle at two points, one of which is C.
A circle centered at C passing through points A and B is drawn. The region that is outside this circle and inside circle O is colored to form a triangular shape, creating figure $R _ { 1 }$. The semicircle of circle O that does not include the colored part is divided into 2 quarter circles, and circles inscribed in each quarter circle are drawn. Inside these 2 circles, two triangular shapes are created using the same method as for figure $R _ { 1 }$ and colored, creating figure $R _ { 2 }$. The semicircles of the 2 newly created circles in figure $R _ { 2 }$ that do not include the colored parts are each divided into 2 quarter circles, and circles inscribed in each of the 4 quarter circles are drawn. Inside these 4 circles, 4 shapes are created using the same method as for figure $R _ { 1 }$ and colored, creating figure $R _ { 3 }$. Continuing this process, let $S _ { n }$ be the area of the colored part in figure $R _ { n }$ obtained at the $n$-th step. What is the value of $\lim _ { n \rightarrow \infty } S _ { n }$? [4 points]
(1) $\frac { 5 + 2 \sqrt { 2 } } { 7 }$
(2) $\frac { 5 + 3 \sqrt { 2 } } { 7 }$
(3) $\frac { 5 + 4 \sqrt { 2 } } { 7 }$
(4) $\frac { 5 + 5 \sqrt { 2 } } { 7 }$
(5) $\frac { 5 + 6 \sqrt { 2 } } { 7 }$
Q15 4 marks Curve Sketching Finding Parameters for Continuity View
The graph of a function $y = f ( x )$ defined on all real numbers is as shown in the figure, and a cubic function $g ( x )$ has leading coefficient 1 and $g ( 0 ) = 3$. When the composite function $( g \circ f ) ( x )$ is continuous on all real numbers, what is the value of $g ( 3 )$? [4 points]
(1) 31
(2) 30
(3) 29
(4) 28
(5) 27
Q16 4 marks Matrices Matrix Algebra and Product Properties View
Two $2 \times 2$ square matrices $A , B$ satisfy
$$2 A ^ { 2 } + A B = E , \quad A B + B A = 2 A + E$$
Which of the following statements are correct? Choose all that apply from . (Given that $E$ is the identity matrix.) [4 points]
ㄱ. $A ^ { - 1 } = 2 A + B$ ㄴ. $B = 2 A + 2 E$ ㄷ. $( B - E ) ^ { 2 } = O$ (where $O$ is the zero matrix.)
(1) ㄴ
(2) ㄷ
(3) ㄱ, ㄴ
(4) ㄱ, ㄷ
(5) ㄱ, ㄴ, ㄷ
Q17 4 marks Sequences and Series Recurrence Relations and Sequence Properties View
The sequence $\left\{ a _ { n } \right\}$ has $a _ { 1 } = 4$ and satisfies
$$a _ { n + 1 } = n \cdot 2 ^ { n } + \sum _ { k = 1 } ^ { n } \frac { a _ { k } } { k } \quad ( n \geq 1 )$$
The following is the process of finding the general term $a _ { n }$.
From the given equation,
$$a _ { n } = ( n - 1 ) \cdot 2 ^ { n - 1 } + \sum _ { k = 1 } ^ { n - 1 } \frac { a _ { k } } { k } \quad ( n \geq 2 )$$
Therefore, for natural numbers $n \geq 2$,
$$a _ { n + 1 } - a _ { n } = \text { (a) } + \frac { a _ { n } } { n }$$
so
$$a _ { n + 1 } = \frac { ( n + 1 ) a _ { n } } { n } + \text { (a) }$$
If $b _ { n } = \frac { a _ { n } } { n }$, then
$$b _ { n + 1 } = b _ { n } + \frac { ( \text { a } ) } { n + 1 } ( n \geq 2 )$$
and since $b _ { 2 } = 3$,
$$b _ { n } = \text { (b) } \quad ( n \geq 2 )$$
Therefore,
$$a _ { n } = \left\{ \begin{array} { c c } 4 & ( n = 1 ) \\ n \times ( \boxed { ( \text{b} ) } ) & ( n \geq 2 ) \end{array} \right.$$
Let $f ( n )$ and $g ( n )$ be the expressions that fit (a) and (b), respectively. What is the value of $f ( 4 ) + g ( 7 )$? [4 points]
(1) 90
(2) 95
(3) 100
(4) 105
(5) 110
Q18 4 marks Conic sections Focal Chord and Parabola Segment Relations View
For a natural number $n$, a line passing through the focus F of the parabola $y ^ { 2 } = \frac { x } { n }$ intersects the parabola at two points P and Q, respectively. If $\overline { \mathrm { PF } } = 1$ and $\overline { \mathrm { FQ } } = a _ { n }$, what is the value of $\sum _ { n = 1 } ^ { 10 } \frac { 1 } { a _ { n } }$? [4 points]
(1) 210
(2) 205
(3) 200
(4) 195
(5) 190
Q19 4 marks Stationary points and optimisation Analyze function behavior from graph or table of derivative View
A cubic function $f ( x )$ satisfies $f ( 0 ) > 0$. Define the function $g ( x )$ as
$$g ( x ) = \left| \int _ { 0 } ^ { x } f ( t ) d t \right|$$
The graph of the function $y = g ( x )$ is as shown in the figure. Which of the following statements are correct? Choose all that apply. [4 points]
ㄱ. The equation $f ( x ) = 0$ has three distinct real roots. ㄴ. $f ^ { \prime } ( 0 ) < 0$ ㄷ. The number of natural numbers $m$ satisfying $\int _ { m } ^ { m + 2 } f ( x ) d x > 0$ is 3.
(1) ㄴ
(2) ㄷ
(3) ㄱ, ㄴ
(4) ㄱ, ㄷ
(5) ㄱ, ㄴ, ㄷ
Q20 4 marks Vectors 3D & Lines Volume of a 3D Solid View
In coordinate space, one face ABC of a regular tetrahedron ABCD lies on the plane $2 x - y + z = 4$, and the vertex D lies on the plane $x + y + z = 3$. When the centroid of triangle ABC has coordinates $( 1,1,3 )$, what is the length of one edge of the regular tetrahedron ABCD? [4 points]
(1) $2 \sqrt { 2 }$
(2) 3
(3) $2 \sqrt { 3 }$
(4) 4
(5) $3 \sqrt { 2 }$
Q21 4 marks Stationary points and optimisation Composite or piecewise function extremum analysis View
For the function $f ( x ) = k x ^ { 2 } e ^ { - x } ( k > 0 )$ and a real number $t$, let $g ( t )$ be the smaller of the distance from the point $( t , f ( t ) )$ on the curve $y = f ( x )$ to the $x$-axis and the distance to the $y$-axis. What is the maximum value of $k$ such that the function $g ( t )$ is not differentiable at exactly one point? [4 points]
(1) $\frac { 1 } { e }$
(2) $\frac { 1 } { \sqrt { e } }$
(3) $\frac { e } { 2 }$
(4) $\sqrt { e }$
(5) $e$
Q22 3 marks Differentiating Transcendental Functions Evaluate derivative at a point or find tangent slope View
For the function $f ( x ) = x \ln x + 13 x$, find the value of $f ^ { \prime } ( 1 )$. [3 points]
Q23 3 marks Harmonic Form View
Find the maximum value $a$ of the function $f ( x ) = 2 \cos \left( x - \frac { \pi } { 3 } \right) + 2 \sqrt { 3 } \sin x$. Find the value of $a ^ { 2 }$. [3 points]
Q24 3 marks Linear transformations View
Let $A$ be the matrix representing the linear transformation $f : ( x , y ) \rightarrow ( 2 x - y , x - 2 y )$. When the point $( 5 , - 1 )$ is mapped to the point $( a , b )$ by the linear transformation represented by the matrix $A ^ { 4 }$, find the value of $a + b$. [3 points]
Q25 Vectors Introduction & 2D Optimization of a Vector Expression View
In an equilateral triangle ABC with side length 2, let H be the foot of the perpendicular from vertex A to side BC. When point P moves on line segment AH, find the maximum value of $| \overrightarrow { \mathrm { PA } } \cdot \overrightarrow { \mathrm { PB } } |$, which is $\frac { q } { p }$. Find the value of $p + q$. (Given that $p$ and $q$ are coprime natural numbers.)
Q26 3 marks Confidence intervals Count integers or determine length of a confidence interval View
For a normal distribution with known standard deviation $\sigma$, a sample of size $n$ is randomly extracted from the population. The 95\% confidence interval for the population mean obtained from this sample is [100.4, 139.6]. Using the same sample, find the number of natural numbers contained in the 99\% confidence interval for the population mean. (Given that when $Z$ is a random variable following the standard normal distribution, $\mathrm { P } ( 0 \leq Z \leq 1.96 ) = 0.475$ and $\mathrm { P } ( 0 \leq Z \leq 2.58 ) = 0.495$.) [3 points]
Q27 4 marks Arithmetic Sequences and Series Sequence Defined by Recurrence with AP Connection View
For a natural number $n$, the point $\mathrm { P } _ { n }$ on the coordinate plane is determined according to the following rules.
(a) The coordinates of the three points $\mathrm { P } _ { 1 } , \mathrm { P } _ { 2 } , \mathrm { P } _ { 3 }$ are $( - 1,0 ) , ( 1,0 )$, and $( - 1,2 )$, respectively.
(b) The midpoint of line segment $\mathrm { P } _ { n } \mathrm { P } _ { n + 1 }$ and the midpoint of line segment $\mathrm { P } _ { n + 2 } \mathrm { P } _ { n + 3 }$ are the same. For example, the coordinates of point $\mathrm { P } _ { 4 }$ are $( 1 , - 2 )$. When the coordinates of point $\mathrm { P } _ { 25 }$ are $( a , b )$, find the value of $a + b$. [4 points]
Q28 4 marks Variable acceleration (vectors) View
As shown in the figure, there is a rectangular piece of paper ABCD with $\overline { \mathrm { AB } } = 9$ and $\overline { \mathrm { AD } } = 3$. Using the line connecting point E on segment AB and point F on segment DC as the fold line, the paper is folded so that the orthogonal projection of point B onto the plane AEFD is point D. When $\overline { \mathrm { AE } } = 3$, the angle between the two planes AEFD and EFCB is $\theta$. Find the value of $60 \cos \theta$. (Given that $0 < \theta < \frac { \pi } { 2 }$ and the thickness of the paper is negligible.) [4 points]
Q29 4 marks SUVAT in 2D & Gravity View
In triangle ABC, $\overline { \mathrm { AB } } = 1$, $\angle \mathrm { A } = \theta$, and $\angle \mathrm { B } = 2 \theta$. Point D on side AB is chosen so that $\angle \mathrm { ACD } = 2 \angle \mathrm { BCD }$. When $\lim _ { \theta \rightarrow + 0 } \frac { \overline { \mathrm { CD } } } { \theta } = a$, find the value of $27 a ^ { 2 }$. (Given that $0 < \theta < \frac { \pi } { 4 }$.) [4 points]
Q30 4 marks Laws of Logarithms Logarithmic Function Graph Intersection or Geometric Analysis View
In the coordinate plane, for a natural number $n$, let $a _ { n }$ be the number of points in the region
$$\left\{ ( x , y ) \mid 2 ^ { x } - n \leq y \leq \log _ { 2 } ( x + n ) \right\}$$
that satisfy the following conditions.
(a) The $x$-coordinate and $y$-coordinate are equal.
(b) The $x$-coordinate and $y$-coordinate are both integers. For example, $a _ { 1 } = 2$ and $a _ { 2 } = 4$. Find the value of $\sum _ { n = 1 } ^ { 30 } a _ { n }$. [4 points]