csat-suneung

Papers (39)
2026
csat__math 29
2025
csat__math 43
2024
csat__math 43
2023
csat__math 29
2022
csat__math 38
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 29 csat__math-science 30
2016
csat__math-A 28 csat__math-B 29
2015
csat__math-A 30 csat__math-B 30
2014
csat__math-A 26 csat__math-B 30
2013
csat__math-humanities 27 csat__math-science 29
2012
csat__math-humanities 29 csat__math-science 30
2011
csat__math-humanities 30 csat__math-science 33
2010
csat__math-humanities 24 csat__math-science 32
2009
csat__math-humanities 27 csat__math-science 32
2008
csat__math-humanities 29 csat__math-science 26
2007
csat__math-humanities 29 csat__math-science 32
2006
csat__math-humanities 30 csat__math-science 26
2005
csat__math-humanities 22 csat__math-science 31
2007 csat__math-science

32 maths questions

Q1 2 marks Laws of Logarithms Simplify or Evaluate a Logarithmic Expression View
The value of $\left( \log _ { 3 } 27 \right) \times 8 ^ { \frac { 1 } { 3 } }$ is? [2 points]
(1) 12
(2) 10
(3) 8
(4) 6
(5) 4
Q2 2 marks Matrices Linear System and Inverse Existence View
For two matrices $A = \left( \begin{array} { r r } - 1 & 0 \\ 0 & 1 \end{array} \right) , B = \left( \begin{array} { l l } 2 & 1 \\ 3 & 3 \end{array} \right)$, what is the sum of all components of the matrix $( A + B ) ^ { - 1 }$? [2 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5
Q3 2 marks Chain Rule Limit Evaluation Involving Composition or Substitution View
The value of $\lim _ { x \rightarrow 1 } \frac { x ^ { 2 } - 1 } { \sqrt { x + 3 } - 2 }$ is? [2 points]
(1) 7
(2) 8
(3) 9
(4) 10
(5) 11
Q4 3 marks Inequalities Integer Solutions of an Inequality View
System of inequalities $$\left\{ \begin{array} { l } x ( x - 4 ) ( x - 5 ) \geqq 0 \\ \frac { x - 3 } { x ^ { 2 } - 3 x + 2 } \leqq 0 \end{array} \right.$$ What is the number of integers $x$ that satisfy the system? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5
Q5 3 marks Circles Focal Distance and Point-on-Conic Metric Computation View
On a parabola $y ^ { 2 } = x$ with focus F, there is a point P such that $\overline { \mathrm { FP } } = 4$. As shown in the figure, point Q is taken on the extension of segment FP such that $\overline { \mathrm { FP } } = \overline { \mathrm { PQ } }$. What is the $x$-coordinate of point Q? [3 points]
(1) $\frac { 29 } { 4 }$
(2) 7
(3) $\frac { 27 } { 4 }$
(4) $\frac { 13 } { 2 }$
(5) $\frac { 25 } { 4 }$
Q6 3 marks Vectors 3D & Lines Dihedral Angle or Angle Between Planes/Lines View
For a regular hexahedron (cube) $\mathrm { ABCD } - \mathrm { EFGH }$, let $\theta$ be the angle between plane AFG and plane AGH. What is the value of $\cos ^ { 2 } \theta$? [3 points]
(1) $\frac { 1 } { 6 }$
(2) $\frac { 1 } { 5 }$
(3) $\frac { 1 } { 4 }$
(4) $\frac { 1 } { 3 }$
(5) $\frac { 1 } { 2 }$
Q7 3 marks Chain Rule Piecewise Function Differentiability Analysis View
When the function $f ( x )$ is $$f ( x ) = \begin{cases} 1 - x & ( x < 0 ) \\ x ^ { 2 } - 1 & ( 0 \leqq x < 1 ) \\ \frac { 2 } { 3 } \left( x ^ { 3 } - 1 \right) & ( x \geqq 1 ) \end{cases}$$ which of the following statements in are correct? [3 points]
Remarks ㄱ. $f ( x )$ is differentiable at $x = 1$. ㄴ. $| f ( x ) |$ is differentiable at $x = 0$. ㄷ. The minimum natural number $k$ such that $x ^ { k } f ( x )$ is differentiable at $x = 0$ is 2.
(1) ㄱ
(2) ㄴ
(3) ㄱ, ㄷ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ
Q8 3 marks Indefinite & Definite Integrals Net Change from Rate Functions (Applied Context) View
The following is a graph showing the velocity $v ( t )$ at time $t$ ( $0 \leqq t \leqq d$ ) of a point P moving on a number line starting from the origin.
When $\int _ { 0 } ^ { a } | v ( t ) | d t = \int _ { a } ^ { d } | v ( t ) | d t$, which of the following statements in are correct? (Here, $0 < a < b < c < d$.) [3 points]
Remarks ㄱ. Point P passes through the origin again after starting. ㄴ. $\int _ { 0 } ^ { c } v ( t ) d t = \int _ { c } ^ { d } v ( t ) d t$ ㄷ. $\int _ { 0 } ^ { b } v ( t ) d t = \int _ { b } ^ { d } | v ( t ) | d t$
(1) ㄴ
(2) ㄷ
(3) ㄱ, ㄴ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ
Q9 4 marks Curve Sketching Multi-Statement Verification (Remarks/Options) View
In the coordinate plane, let $C$ be a circle with center $(0,3)$ and radius 1. For a positive number $r$, let $f ( r )$ be the number of circles with radius $r$ that meet circle $C$ at exactly one point and are tangent to the $x$-axis. Which of the following statements in are correct? [4 points]
Remarks ㄱ. $f ( 2 ) = 3$ ㄴ. $\lim _ { r \rightarrow 1 + 0 } f ( r ) = f ( 1 )$ ㄷ. The number of discontinuity points of the function $f ( r )$ on the interval $( 0,4 )$ is 2.
(1) ㄱ
(2) ㄴ
(3) ㄷ
(4) ㄱ, ㄷ
(5) ㄱ, ㄴ, ㄷ
Q10 3 marks Confidence intervals Count integers or determine length of a confidence interval View
A factory produces table tennis balls. When dropped from a certain height onto a steel floor, the height to which the table tennis ball bounces follows a normal distribution. From the table tennis balls produced by this factory, 100 balls were randomly sampled and the bounce height was measured, resulting in a mean of 245 and a standard deviation of 20.
What is the number of integers in the 95\% confidence interval for the mean bounce height of all table tennis balls produced by this factory? (Here, the unit of height is mm, and when $Z$ follows the standard normal distribution, $\mathrm { P } ( 0 \leqq Z \leqq 1.96 ) = 0.4750$.) [3 points]
(1) 5
(2) 6
(3) 7
(4) 8
(5) 9
Q11 3 marks Exponential Functions Applied/Contextual Exponential Modeling View
Even when the surroundings suddenly become dark, the human eye perceives the change gradually. After the light intensity suddenly changes from 1000 to 10, and $t$ seconds have elapsed, the light intensity $I ( t )$ perceived by a person is $$I ( t ) = 10 + 990 \times a ^ { - 5 t } \text{ (where } a \text{ is a constant with } a > 1 \text{)}$$ After the light intensity suddenly changes from 1000 to 10, let $s$ seconds elapse until the person perceives the light intensity as 21. What is the value of $s$? (Here, the unit of light intensity is Td (troland).) [3 points]
(1) $\frac { 1 + 2 \log 3 } { 5 \log a }$
(2) $\frac { 1 + 3 \log 3 } { 5 \log a }$
(3) $\frac { 2 + \log 3 } { 5 \log a }$
(4) $\frac { 2 + 2 \log 3 } { 5 \log a }$
(5) $\frac { 2 + 3 \log 3 } { 5 \log a }$
Q12 3 marks Matrices True/False or Multiple-Select Conceptual Reasoning View
Two square matrices $A , B$ of order 2 satisfy $A ^ { 2 } = E , B ^ { 2 } = B$. Which of the following statements in are always correct? (Here, $E$ is the identity matrix.) [3 points]
Remarks ㄱ. If matrix $B$ has an inverse, then $B = E$. ㄴ. $( E - A ) ^ { 5 } = 2 ^ { 4 } ( E - A )$ ㄷ. $( E - A B A ) ^ { 2 } = E - A B A$
(1) ㄱ
(2) ㄷ
(3) ㄱ, ㄴ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ
Q13 4 marks Laws of Logarithms Verify Truth of Logarithmic Statements View
For an integer $n$, two sets $A ( n ) , B ( n )$ are defined as $$\begin{aligned} & A ( n ) = \left\{ x \mid \log _ { 2 } x \leqq n \right\} \\ & B ( n ) = \left\{ x \mid \log _ { 4 } x \leqq n \right\} \end{aligned}$$ Which of the following statements in are correct? [4 points]
Remarks ㄱ. $A ( 1 ) = \{ x \mid 0 < x \leqq 1 \}$ ㄴ. $A ( 4 ) = B ( 2 )$ ㄷ. When $A ( n ) \subset B ( n )$, we have $B ( - n ) \subset A ( - n )$.
(1) ㄱ
(2) ㄴ
(3) ㄷ
(4) ㄱ, ㄷ
(5) ㄴ, ㄷ
Q14 4 marks Combinations & Selection Distribution of Objects to Positions or Containers View
Five balls labeled with the numbers $1,2,3,4,5$ are to be placed into 3 boxes A, B, C. In how many ways can the balls be placed so that no box contains balls whose numbers sum to 13 or more? (Here, for an empty box, the sum of the numbers on the balls is taken as 0.) [4 points]
(1) 233
(2) 228
(3) 222
(4) 215
(5) 211
Q15 4 marks Probability Definitions Finite Equally-Likely Probability Computation View
From $3n$ cards labeled with the numbers $1,2,3 , \cdots , 3 n$ ($n$ is a natural number), two cards are randomly drawn. Let the two numbers on the cards be $a , b$ ($a < b$) respectively. Let $\mathrm { P } _ { n }$ be the probability that $3 a < b$. The following is the process of finding $\lim _ { n \rightarrow \infty } \mathrm { P } _ { n }$.
The number of ways to draw 2 cards from $3n$ cards is ${ } _ { 3 n } \mathrm { C } _ { 2 }$. For $3 a < b$, we have $b \leqq 3 n$, so $1 \leqq a < n$. Therefore, if $a = k$, the number of cases of $b$ satisfying $3 a < b$ is (A), so $$\mathrm { P } _ { n } = \frac { \text{(B)} } { { } _ { 3 n } \mathrm { C } _ { 2 } } \text{.}$$ Therefore, $\lim _ { n \rightarrow \infty } \mathrm { P } _ { n } =$ (C).
What are the correct values for (A), (B), and (C) in the above process? [4 points]
(1) (A) $3(n-k)$, (B) $\frac{3}{2}n(n-1)$, (C) $\frac{1}{3}$
(2) (A) $3(n-k)$, (B) $\frac{3}{2}n(n-1)$, (C) $\frac{2}{3}$
(3) (A) $3(n-k)$, (B) $3n(n-1)$, (C) $\frac{2}{3}$
(4) (A) $3(n-k+1)$, (B) $3n(n-1)$, (C) $\frac{1}{3}$
(5) (A) $3(n-k+1)$, (B) $3n(n-1)$, (C) $\frac{2}{3}$
Q16 4 marks Curve Sketching Lattice Points and Counting via Graph Geometry View
In the coordinate plane, for a natural number $n$, let $A _ { n }$ be a square with vertices at the four points $$\left( n ^ { 2 } , n ^ { 2 } \right) , \left( 4 n ^ { 2 } , n ^ { 2 } \right) , \left( 4 n ^ { 2 } , 4 n ^ { 2 } \right) , \left( n ^ { 2 } , 4 n ^ { 2 } \right)$$ Let $a _ { n }$ be the number of natural numbers $k$ such that the square $A _ { n }$ and the graph of the function $y = k \sqrt { x }$ intersect. Which of the following statements in are correct? [4 points]
Remarks ㄱ. $a _ { 5 } = 15$ ㄴ. $a _ { n + 2 } - a _ { n } = 7$ ㄷ. $\sum _ { k = 1 } ^ { 10 } a _ { k } = 200$
(1) ㄴ
(2) ㄷ
(3) ㄱ, ㄴ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ
Q17 4 marks Geometric Probability View
As shown below, there is a right isosceles triangle with the two legs forming the right angle each having length 1. Let $R _ { 1 }$ be the figure obtained by coloring a square that has 2 vertices on the hypotenuse and the remaining 2 vertices on the two legs forming the right angle.
Let $R _ { 2 }$ be the figure obtained by coloring 2 congruent squares in the figure $R _ { 1 }$, where each square has 2 vertices on the hypotenuse of each of 2 congruent right isosceles triangles and the remaining 2 vertices on the two legs forming the right angle.
Let $R _ { 3 }$ be the figure obtained by coloring 4 congruent squares in the figure $R _ { 2 }$, where each square has 2 vertices on the hypotenuse of each of 4 congruent right isosceles triangles and the remaining 2 vertices on the two legs forming the right angle.
Continuing this process, let $S _ { n }$ be the sum of the areas of all colored squares in the figure $R _ { n }$ obtained at the $n$-th step. What is the value of $\lim _ { n \rightarrow \infty } S _ { n }$? [4 points]
(1) $\frac { 3 \sqrt { 2 } } { 20 }$
(2) $\frac { \sqrt { 2 } } { 5 }$
(3) $\frac { 3 } { 10 }$
(4) $\frac { \sqrt { 3 } } { 5 }$
(5) $\frac { 2 } { 5 }$
Q18 3 marks Tangents, normals and gradients Find tangent line with a specified slope or from an external point View
For the quartic function $f ( x ) = x ^ { 4 } - 4 x ^ { 3 } + 6 x ^ { 2 } + 4$, the slope of the tangent line at the point $( a , b )$ on the graph is 4. Find the value of $a ^ { 2 } + b ^ { 2 }$. [3 points]
Q19 3 marks Indefinite & Definite Integrals Finding a Function from an Integral Equation View
A polynomial function $f ( x )$ satisfies $$\int _ { 1 } ^ { x } f ( t ) d t = x ^ { 3 } - 2 a x ^ { 2 } + a x$$ for all real numbers $x$. Find the value of $f ( 3 )$. (Here, $a$ is a constant.) [3 points]
Q20 Conic sections Focal Distance and Point-on-Conic Metric Computation View
Let $\mathrm { F } , \mathrm { F } ^ { \prime }$ be the two foci of the ellipse $\frac { x ^ { 2 } } { 4 } + y ^ { 2 } = 1$. For a point P on this ellipse satisfying $| \overrightarrow { \mathrm { OP } } + \overrightarrow { \mathrm { OF } } | = 1$, the length of segment PF is $k$. Find the value of $5k$. (Here, O is the origin.)
Q21 4 marks Vectors 3D & Lines Vector Algebra and Triple Product Computation View
In coordinate space, let B be the foot of the perpendicular from the point $\mathrm { A } ( 3,6,0 )$ to the plane $\sqrt { 3 } y - z = 0$. Find the value of $\overrightarrow { \mathrm { OA } } \cdot \overrightarrow { \mathrm { OB } }$. (Here, O is the origin.) [4 points]
Q22 4 marks Arithmetic Sequences and Series Summation of Derived Sequence from AP View
For an arithmetic sequence $\left\{ a _ { n } \right\}$ with first term 0 and nonzero common difference, the sequence $\left\{ b _ { n } \right\}$ satisfies $a _ { n + 1 } b _ { n } = \sum _ { k = 1 } ^ { n } a _ { k }$. Find the value of $b _ { 27 }$. [4 points]
Q23 4 marks Vectors 3D & Lines Volume of a 3D Solid View
In coordinate space, consider the triangle ABC with vertices $\mathrm { A } ( 54,0,0 ) , \mathrm { B } ( 0,27,0 ) , \mathrm { C } ( 0,0,27 )$ on the plane $x + 2 y + 2 z = 54$. A point $\mathrm { P } ( x , y , z )$ is in the interior of triangle ABC. Let Q be the orthogonal projection of P onto the $xy$-plane, R be the orthogonal projection of P onto the $yz$-plane, and S be the orthogonal projection of P onto the $zx$-plane. When $\overline { \mathrm { QR } } = \overline { \mathrm { QS } }$, find the maximum volume of the tetrahedron QPRS. [4 points]
Q24 4 marks Vectors 3D & Lines Magnitude of Vector Expression View
As shown in the figure, on a plane $\alpha$ there is an equilateral triangle ABC with side length 3, and a sphere $S$ with radius 2 is tangent to the plane $\alpha$ at point A. For a point D on the sphere $S$ such that the segment AD passes through the center O of the sphere $S$, find the value of $| \overrightarrow { \mathrm { AB } } + \overrightarrow { \mathrm { DC } } | ^ { 2 }$. [4 points]
Q25 4 marks Exponential Functions Geometric Properties of Exponential/Logarithmic Curves View
The graph of the function $y = k \cdot 3 ^ { x }$ ($0 < k < 1$) meets the graphs of the two functions $y = 3 ^ { - x }$ and $y = - 4 \cdot 3 ^ { x } + 8$ at points P and Q, respectively. When the ratio of the $x$-coordinates of points P and Q is $1 : 2$, find the value of $35k$. [4 points]
Q26 (Calculus) 3 marks Chain Rule Limit Involving Derivative Definition of Composed Functions View
$\lim _ { x \rightarrow a } \frac { 2 ^ { x } - 1 } { 3 \sin ( x - a ) } = b \ln 2$ is satisfied by two constants $a , b$. What is the value of $a + b$? [3 points]
(1) $\frac { 1 } { 6 }$
(2) $\frac { 1 } { 5 }$
(3) $\frac { 1 } { 4 }$
(4) $\frac { 1 } { 3 }$
(5) $\frac { 1 } { 2 }$
Q26 (Probability and Statistics) 3 marks Independent Events View
For two independent events $A$ and $B$, $$\mathrm { P } ( A \cap B ) = 2 \mathrm { P } \left( A \cap B ^ { c } \right) , \quad \mathrm { P } \left( A ^ { c } \cap B \right) = \frac { 1 } { 12 }$$ find the value of $\mathrm { P } ( A )$. (Given that $\mathrm { P } ( A ) \neq 0$.) [3 points]
(1) $\frac { 1 } { 2 }$
(2) $\frac { 5 } { 8 }$
(3) $\frac { 3 } { 4 }$
(4) $\frac { 7 } { 8 }$
(5) $\frac { 15 } { 16 }$
Q26 (Discrete Mathematics) 3 marks Combinations & Selection Distribution of Objects into Bins/Groups View
In how many ways can 9 identical candies be distributed into 5 identical bags such that no bag is empty? [3 points]
(1) 8
(2) 7
(3) 6
(4) 5
(5) 4
Q27 (Calculus) 3 marks Tangents, normals and gradients Geometric properties of tangent lines (intersections, lengths, areas) View
As shown in the figure, let $\mathrm { Q } _ { 1 }$ be the point where the tangent line to the circle $x ^ { 2 } + y ^ { 2 } = 1$ at point $\mathrm { P } _ { 1 }$ meets the $x$-axis. The area of triangle $\mathrm { P } _ { 1 } \mathrm { OQ } _ { 1 }$ is $\frac { 1 } { 4 }$. Let $\mathrm { P } _ { 2 }$ be the point obtained by rotating $\mathrm { P } _ { 1 }$ about the origin O by $\frac { \pi } { 4 }$, and let $\mathrm { Q } _ { 2 }$ be the point where the tangent line at $\mathrm { P } _ { 2 }$ meets the $x$-axis. What is the area of triangle $\mathrm { P } _ { 2 } \mathrm { OQ } _ { 2 }$? (Here, point $\mathrm { P } _ { 1 }$ is in the first quadrant.) [3 points]
(1) 1
(2) $\frac { 5 } { 4 }$
(3) $\frac { 3 } { 2 }$
(4) $\frac { 7 } { 4 }$
(5) 2
Q28 (Probability and Statistics) 3 marks Approximating Binomial to Normal Distribution View
It is known that $10\%$ of the notebooks displayed in a certain stationery store are products from Company A. When a customer randomly purchases 100 notebooks from this store, find the probability that at least 13 notebooks from Company A are included using the standard normal distribution table below. [3 points]
$z$$\mathrm { P } ( 0 \leqq Z \leqq z )$
0.750.2734
1.000.3413
1.250.3944
1.500.4332

(1) 0.0668
(2) 0.1056
(3) 0.1587
(4) 0.2266
(5) 0.2734
Q29 (Probability and Statistics) 4 marks Confidence intervals Conceptual reasoning about confidence level and sample size effects View
A population follows a normal distribution $\mathrm { N } \left( m , 2 ^ { 2 } \right)$. Let $\overline { X _ { A } }$ and $\overline { X _ { B } }$ be the sample means of samples of size 7 and size 10, respectively, drawn from this population. Using the distributions of $\overline { X _ { A } }$ and $\overline { X _ { B } }$, the 95\% confidence intervals for the population mean $m$ are $[a, b]$ and $[c, d]$, respectively. Choose all correct statements from the given options. [4 points]
Options ㄱ. The variance of $\overline { X _ { A } }$ is greater than the variance of $\overline { X _ { B } }$. ㄴ. $\mathrm { P } \left( \overline { X _ { A } } \leqq m + 2 \right) < \mathrm { P } \left( \overline { X _ { B } } \leqq m + 2 \right)$ ㄷ. $d - c < b - a$
(1) ㄱ
(2) ㄷ
(3) ㄱ, ㄴ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ
Q30 (Probability and Statistics) 4 marks Binomial Distribution Binomial Distribution Identification and Application View
A factory produces products that are sold with 50 items per box. The number of defective items in a box follows a binomial distribution with mean $m$ and variance $\frac { 48 } { 25 }$. Before selling a box, all 50 products are inspected to find defective items, which costs a total of 60,000 won. If a box is sold without inspection, an after-sales service cost of $a$ won is required for each defective item.
When the expected value of the cost of inspecting all products in a box equals the expected cost of after-sales service, find the value of $\frac { a } { 1000 }$. (Given that $a$ is a constant and $m$ is a natural number not exceeding 5.) [4 points]