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
2015 csat__math-B

30 maths questions

Q1 2 marks Matrices Matrix Algebra and Product Properties View
For two matrices $A = \left( \begin{array} { l l } 1 & 1 \\ 0 & 2 \end{array} \right) , B = \left( \begin{array} { l l } 1 & 1 \\ 3 & 0 \end{array} \right)$, what is the sum of all components of the matrix $A + B$? [2 points]
(1) 5
(2) 6
(3) 7
(4) 8
(5) 9
Q2 2 marks Harmonic Form View
What is the maximum value of the function $f ( x ) = \sin x + \sqrt { 7 } \cos x - \sqrt { 2 }$? [2 points]
(1) $\sqrt { 2 }$
(2) $\sqrt { 3 }$
(3) 2
(4) $\sqrt { 5 }$
(5) $\sqrt { 6 }$
Q3 2 marks Laws of Logarithms Simplify or Evaluate a Logarithmic Expression View
What is the value of $\lim _ { x \rightarrow 0 } \frac { \ln ( 1 + x ) } { 3 x }$? [2 points]
(1) 1
(2) $\frac { 1 } { 2 }$
(3) $\frac { 1 } { 3 }$
(4) $\frac { 1 } { 4 }$
(5) $\frac { 1 } { 5 }$
Q4 3 marks Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View
What is the value of $\int _ { 0 } ^ { 1 } 3 \sqrt { x } \, d x$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5
Q5 3 marks Vectors 3D & Lines Section Division and Coordinate Computation View
In coordinate space, for two points $\mathrm { A } ( 2 , a , - 2 ) , \mathrm { B } ( 5 , - 3 , b )$, when the point that divides segment AB internally in the ratio $2 : 1$ lies on the $x$-axis, what is the value of $a + b$? [3 points]
(1) 6
(2) 7
(3) 8
(4) 9
(5) 10
Q6 3 marks Geometric Sequences and Series Sum of an Infinite Geometric Series (Direct Computation) View
For a geometric sequence $\left\{ a _ { n } \right\}$ with $a _ { 1 } = 3 , a _ { 2 } = 1$, what is the value of $\sum _ { n = 1 } ^ { \infty } \left( a _ { n } \right) ^ { 2 }$? [3 points]
(1) $\frac { 89 } { 8 }$
(2) $\frac { 87 } { 8 }$
(3) $\frac { 85 } { 8 }$
(4) $\frac { 83 } { 8 }$
(5) $\frac { 81 } { 8 }$
Q7 3 marks Linear transformations View
Let the matrices representing two linear transformations $f , g$ be $\left( \begin{array} { l l } 2 & 1 \\ 4 & 2 \end{array} \right) , \left( \begin{array} { r r } 2 & 0 \\ 1 & - 1 \end{array} \right)$ respectively. When the point $( 1,2 )$ is mapped to the point $( a , 6 )$ by the composite transformation $f \circ g$, what is the value of $a$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5
Q8 3 marks Probability Definitions Probability Using Set/Event Algebra View
For two events $A , B$, $A ^ { C }$ and $B$ are mutually exclusive events, and $$\mathrm { P } ( A ) = 2 \mathrm { P } ( B ) = \frac { 3 } { 5 }$$ When this condition is satisfied, what is the value of $\mathrm { P } \left( A \cap B ^ { C } \right)$? (Here, $A ^ { C }$ is the complement of $A$.) [3 points]
(1) $\frac { 3 } { 20 }$
(2) $\frac { 1 } { 5 }$
(3) $\frac { 1 } { 4 }$
(4) $\frac { 3 } { 10 }$
(5) $\frac { 7 } { 20 }$
Q9 3 marks Indefinite & Definite Integrals Definite Integral as a Limit of Riemann Sums View
For the function $f ( x ) = \frac { 1 } { x }$, what is the value of $\lim _ { n \rightarrow \infty } \sum _ { k = 1 } ^ { n } f \left( 1 + \frac { 2 k } { n } \right) \frac { 2 } { n }$? [3 points]
(1) $\ln 6$
(2) $\ln 5$
(3) $2 \ln 2$
(4) $\ln 3$
(5) $\ln 2$
Q10 3 marks Circles Chord Length and Chord Properties View
As shown in the figure, a line passes through the focus F of the parabola $y ^ { 2 } = 12 x$ and meets the parabola at two points $\mathrm { A } , \mathrm { B }$. Let C and D be the feet of the perpendiculars from A and B to the directrix $l$ respectively. When $\overline { \mathrm { AC } } = 4$, what is the length of segment BD? [3 points]
(1) 12
(2) $\frac { 25 } { 2 }$
(3) 13
(4) $\frac { 27 } { 2 }$
(5) 14
Q11 3 marks Normal Distribution Direct Probability Calculation from Given Normal Distribution View
A snack factory produces snacks where the weight of one package follows a normal distribution with mean 75 g and standard deviation 2 g. Using the standard normal distribution table below, what is the probability that the weight of a randomly selected package of snacks from this factory is at least 76 g and at most 78 g? [3 points]
$z$$\mathrm { P } ( 0 \leq Z \leq z )$
0.50.1915
1.00.3413
1.50.4332
2.00.4772

(1) 0.0440
(2) 0.0919
(3) 0.1359
(4) 0.1498
(5) 0.2417
Q12 3 marks Vectors 3D & Lines Distance from a Point to a Line (Show/Compute) View
Let $l$ be the line passing through two distinct points $\mathrm { A } , \mathrm { B }$ on plane $\alpha$, and let H be the foot of the perpendicular from point P (not on plane $\alpha$) to plane $\alpha$. When $\overline { \mathrm { AB } } = \overline { \mathrm { PA } } = \overline { \mathrm { PB } } = 6 , \overline { \mathrm { PH } } = 4$, what is the distance between point H and line $l$? [3 points]
(1) $\sqrt { 11 }$
(2) $2 \sqrt { 3 }$
(3) $\sqrt { 13 }$
(4) $\sqrt { 14 }$
(5) $\sqrt { 15 }$
Q13 3 marks Exponential Functions Limit Evaluation View
For a constant $a > 3$, two curves $y = a ^ { x - 1 }$ and $y = 3 ^ { x }$ meet at point P. Let the $x$-coordinate of point P be $k$.
What is the value of $\lim _ { n \rightarrow \infty } \frac { \left( \frac { a } { 3 } \right) ^ { n + k } } { \left( \frac { a } { 3 } \right) ^ { n + 1 } + 1 }$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5
Q14 4 marks Tangents, normals and gradients Geometric properties of tangent lines (intersections, lengths, areas) View
For a constant $a > 3$, two curves $y = a ^ { x - 1 }$ and $y = 3 ^ { x }$ meet at point P. Let the $x$-coordinate of point P be $k$.
Let A be the point where the tangent line to the curve $y = 3 ^ { x }$ at point P meets the $x$-axis, and let B be the point where the tangent line to the curve $y = a ^ { x - 1 }$ at point P meets the $x$-axis. For point $\mathrm { H } ( k , 0 )$, when $\overline { \mathrm { AH } } = 2 \overline { \mathrm { BH } }$, what is the value of $a$? [4 points]
(1) 6
(2) 7
(3) 8
(4) 9
(5) 10
Q15 4 marks Conditional Probability Reverse Inference / Determining Unknown Quantities from Conditional Probability Constraints View
A survey of 320 students at a school regarding membership in the mathematics club found that 60\% of male students and 50\% of female students joined the mathematics club. Let $p _ { 1 }$ be the probability that a randomly selected student from those who joined the mathematics club is male, and let $p _ { 2 }$ be the probability that a randomly selected student from those who joined the mathematics club is female. When $p _ { 1 } = 2 p _ { 2 }$, what is the number of male students at this school? [4 points]
(1) 170
(2) 180
(3) 190
(4) 200
(5) 210
Q16 4 marks Matrices True/False or Multiple-Select Conceptual Reasoning View
Two square matrices $A , B$ satisfy $$A ^ { 2 } - A B = 3 E , \quad A ^ { 2 } B - B ^ { 2 } A = A + B$$ From the statements below, select all correct ones. (Here, $E$ is the identity matrix.) [4 points]
Statements ᄀ. The inverse matrix of $A$ exists. ㄴ. $A B = B A$ ㄷ. $( A + 2 B ) ^ { 2 } = 24 E$
(1) ᄀ
(2) ㄷ
(3) ᄀ, ㄴ
(4) ㄴ, ㄷ
(5) ᄀ, ㄴ, ㄷ
Q17 4 marks Sequences and series, recurrence and convergence Auxiliary sequence transformation View
The sequence $\left\{ a _ { n } \right\}$ satisfies $a _ { 1 } = 1$, and with $S _ { n } = \sum _ { k = 1 } ^ { n } a _ { k }$, $$a _ { n + 1 } = ( n + 1 ) S _ { n } + n ! \quad ( n \geq 1 )$$ The following is the process of finding the general term $a _ { n }$.
For a natural number $n$, since $a _ { n + 1 } = S _ { n + 1 } - S _ { n }$, by the given equation, $$S _ { n + 1 } = ( n + 2 ) S _ { n } + n ! \quad ( n \geq 1 )$$ Dividing both sides by $( n + 2 ) !$, $$\frac { S _ { n + 1 } } { ( n + 2 ) ! } = \frac { S _ { n } } { ( n + 1 ) ! } + \frac { 1 } { ( n + 1 ) ( n + 2 ) }$$ Let $b _ { n } = \frac { S _ { n } } { ( n + 1 ) ! }$. Then $b _ { 1 } = \frac { 1 } { 2 }$ and $$b _ { n + 1 } = b _ { n } + \frac { 1 } { ( n + 1 ) ( n + 2 ) }$$ Finding the general term of the sequence $\left\{ b _ { n } \right\}$, $$b _ { n } = \frac { ( \text{(가)} ) } { n + 1 }$$ Therefore, $$S _ { n } = \text{(가)} \times n!$$ Thus, $$a _ { n } = \text{(나)} \times ( n - 1 ) ! \quad ( n \geq 1 )$$ When the expressions that fit (가) and (나) are $f ( n )$ and $g ( n )$ respectively, what is the value of $f ( 7 ) + g ( 6 )$? [4 points]
(1) 44
(2) 41
(3) 38
(4) 35
(5) 32
Q18 4 marks Discrete Probability Distributions Probability Computation for Compound or Multi-Stage Random Experiments View
A bag contains 1 ball with the number 1, 2 balls with the number 2, and 5 balls with the number 3. One ball is randomly drawn from the bag, the number on the ball is confirmed, and then it is returned. This trial is repeated 2 times. Let $\bar { X }$ be the average of the numbers on the drawn balls. What is the value of $\mathrm { P } ( \bar { X } = 2 )$? [4 points]
(1) $\frac { 5 } { 32 }$
(2) $\frac { 11 } { 64 }$
(3) $\frac { 3 } { 16 }$
(4) $\frac { 13 } { 64 }$
(5) $\frac { 7 } { 32 }$
Q19 4 marks Vectors 3D & Lines MCQ: Distance or Length Optimization on a Line View
In coordinate space, a line $l : \frac { x } { 2 } = 6 - y = z - 6$ and plane $\alpha$ meet perpendicularly at point $\mathrm { P } ( 2,5,7 )$. For a point $\mathrm { A } ( a , b , c )$ on line $l$ and a point Q on plane $\alpha$, when $\overrightarrow { \mathrm { AP } } \cdot \overrightarrow { \mathrm { AQ } } = 6$, what is the value of $a + b + c$? (Here, $a > 0$) [4 points]
(1) 15
(2) 16
(3) 17
(4) 18
(5) 19
Q20 4 marks Stationary points and optimisation Geometric or applied optimisation problem View
As shown in the figure, there is an isosceles triangle ABC with $\angle \mathrm { CAB } = \angle \mathrm { BCA } = \theta$ that is externally tangent to a circle of radius 1. On the extension of segment AB, a point D (not equal to A) is chosen such that $\angle \mathrm { DCB } = \theta$. Let the area of triangle BDC be $S ( \theta )$. What is the value of $\lim _ { \theta \rightarrow + 0 } \{ \theta \times S ( \theta ) \}$? (Here, $0 < \theta < \frac { \pi } { 4 }$) [4 points]
(1) $\frac { 2 } { 3 }$
(2) $\frac { 8 } { 9 }$
(3) $\frac { 10 } { 9 }$
(4) $\frac { 4 } { 3 }$
(5) $\frac { 14 } { 9 }$
Q21 4 marks Laws of Logarithms Logarithmic Function Graph Intersection or Geometric Analysis View
For a natural number $n$, let $a _ { n }$ be the smallest natural number $m$ satisfying the following conditions. What is the value of $\sum _ { n = 1 } ^ { 10 } a _ { n }$? [4 points] (가) The coordinates of point A are $\left( 2 ^ { n } , 0 \right)$. (나) Let D be the point on the line passing through two points $\mathrm { B } ( 1,0 )$ and $\mathrm { C } \left( 2 ^ { m } , m \right)$ whose $x$-coordinate is $2 ^ { n }$. The area of triangle ABD is less than or equal to $\frac { m } { 2 }$.
(1) 109
(2) 111
(3) 113
(4) 115
(5) 117
Q22 3 marks Laws of Logarithms Solve a Logarithmic Equation View
Solve the logarithmic equation $\log _ { 2 } ( x + 6 ) = 5$. [3 points]
Q23 3 marks Differentiating Transcendental Functions Evaluate derivative at a point or find tangent slope View
For the function $f ( x ) = \cos x + 4 e ^ { 2 x }$, find the value of $f ^ { \prime } ( 0 )$. [3 points]
Q24 3 marks Solving quadratics and applications Solving an equation via substitution to reduce to quadratic form View
For the irrational equation $x ^ { 2 } - 6 x - \sqrt { x ^ { 2 } - 6 x - 1 } = 3$, let $k$ be the product of all real roots. Find the value of $k ^ { 2 }$. [3 points]
Q25 4 marks Combinations & Selection Counting Integer Solutions to Equations View
Find the number of all ordered pairs $( a , b , c )$ of natural numbers satisfying the following conditions. [4 points] (가) $a \times b \times c$ is odd. (나) $a \leq b \leq c \leq 20$
Q26 3 marks Laws of Logarithms Logarithmic Formula Application (Modeling) View
When compressing digital images, let $P$ be the peak signal-to-noise ratio, which is an index indicating the degree of difference between the original and compressed images, and let $E$ be the mean squared error between the original and compressed images. The following relationship holds: $$P = 20 \log 255 - 10 \log E \quad ( E > 0 )$$ When two original images $A$ and $B$ are compressed, let the peak signal-to-noise ratios be $P _ { A }$ and $P _ { B }$ respectively, and the mean squared errors be $E _ { A } \left( E _ { A } > 0 \right)$ and $E _ { B } \left( E _ { B } > 0 \right)$ respectively. When $E _ { B } = 100 E _ { A }$, find the value of $P _ { A } - P _ { B }$. [3 points]
Q27 4 marks Conic sections Triangle or Quadrilateral Area and Perimeter with Foci View
For the ellipse $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 4 } = 1$, let F be the focus with positive $x$-coordinate and $\mathrm { F } ^ { \prime }$ be the focus with negative $x$-coordinate. A point P on this ellipse is chosen in the first quadrant such that $\angle \mathrm { FPF } ^ { \prime } = \frac { \pi } { 2 }$, and a point Q with positive $y$-coordinate is chosen on the extension of segment FP such that $\overline { \mathrm { FQ } } = 6$. Find the area of triangle $\mathrm { QF } ^ { \prime } \mathrm { F}$. [4 points]
Q28 4 marks Integration by Parts Area or Volume Computation Requiring Integration by Parts View
For a positive number $a$, the function $f ( x ) = \int _ { 0 } ^ { x } ( a - t ) e ^ { t } \, d t$ has a maximum value of 32. Find the area enclosed by the curve $y = 3 e ^ { x }$ and the two lines $x = a$ and $y = 3$. [4 points]
Q29 4 marks Vectors 3D & Lines Dihedral Angle Computation View
In coordinate space, there is a sphere $S : x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 50$ and a point $\mathrm { P } ( 0,5,5 )$. For all circles $C$ satisfying the following conditions, find the maximum area of the orthogonal projection of $C$ onto the $xy$-plane, expressed as $\frac { q } { p } \pi$. Find the value of $p + q$. (Here, $p$ and $q$ are coprime natural numbers.) [4 points] (가) Circle $C$ is formed by the intersection of a plane passing through point P and the sphere $S$. (나) The radius of circle $C$ is 1.
Q30 4 marks Differentiating Transcendental Functions Regularity and smoothness of transcendental functions View
For the function $f ( x ) = e ^ { x + 1 } - 1$ and a natural number $n$, let the function $g ( x )$ be defined as $$g ( x ) = 100 | f ( x ) | - \sum _ { k = 1 } ^ { n } \left| f \left( x ^ { k } \right) \right|$$ Find the sum of all natural numbers $n$ such that $g ( x )$ is differentiable on the entire set of real numbers. [4 points]