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
2018 csat__math-science

30 maths questions

Q1 2 marks Vectors Introduction & 2D Vector Properties and Identities (Conceptual) View
For two vectors $\vec { a } = ( 3 , - 1 ) , \vec { b } = ( 1,2 )$, what is the sum of all components of vector $\vec { a } + \vec { b }$? [2 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5
Q2 2 marks Vectors 3D & Lines Section Division and Coordinate Computation View
For two points $\mathrm { A } ( 1,6,4 ) , \mathrm { B } ( a , 2 , - 4 )$ in coordinate space, the point that divides segment AB internally in the ratio $1 : 3$ has coordinates $( 2,5,2 )$. What is the value of $a$? [2 points]
(1) 1
(2) 3
(3) 5
(4) 7
(5) 9
Q3 2 marks Laws of Logarithms Simplify or Evaluate a Logarithmic Expression View
What is the value of $\lim _ { x \rightarrow 0 } \frac { \ln ( 1 + 5 x ) } { e ^ { 2 x } - 1 }$? [2 points]
(1) 1
(2) $\frac { 3 } { 2 }$
(3) 2
(4) $\frac { 5 } { 2 }$
(5) 3
Q4 3 marks Independent Events View
Two events $A$ and $B$ are mutually independent and $$\mathrm { P } ( A ) = \frac { 2 } { 3 } , \quad \mathrm { P } ( A \cup B ) = \frac { 5 } { 6 }$$ What is the value of $\mathrm { P } ( B )$? [3 points]
(1) $\frac { 1 } { 3 }$
(2) $\frac { 5 } { 12 }$
(3) $\frac { 1 } { 2 }$
(4) $\frac { 7 } { 12 }$
(5) $\frac { 2 } { 3 }$
Q5 3 marks Exponential Functions Variation and Monotonicity Analysis View
What is the maximum value of the function $f ( x ) = 1 + \left( \frac { 1 } { 3 } \right) ^ { x - 1 }$ on the closed interval $[ 1,3 ]$? [3 points]
(1) $\frac { 5 } { 3 }$
(2) 2
(3) $\frac { 7 } { 3 }$
(4) $\frac { 8 } { 3 }$
(5) 3
Q6 3 marks Binomial Theorem (positive integer n) Find a Specific Coefficient in a Single Binomial Expansion View
What is the coefficient of $x ^ { 4 }$ in the expansion of $\left( x + \frac { 2 } { x } \right) ^ { 8 }$? [3 points]
(1) 108
(2) 112
(3) 116
(4) 120
(5) 124
Q7 3 marks Quadratic trigonometric equations View
When $0 \leq x < 2 \pi$, what is the sum of all solutions to the equation $$\cos ^ { 2 } x = \sin ^ { 2 } x - \sin x$$ ? [3 points]
(1) $2 \pi$
(2) $\frac { 5 } { 2 } \pi$
(3) $3 \pi$
(4) $\frac { 7 } { 2 } \pi$
(5) $4 \pi$
Q8 3 marks Conic sections Equation Determination from Geometric Conditions View
For the ellipse $\frac { ( x - 2 ) ^ { 2 } } { a } + \frac { ( y - 2 ) ^ { 2 } } { 4 } = 1$, the coordinates of the two foci are $( 6 , b ) , ( - 2 , b )$. What is the value of $ab$? (Here, $a$ is positive.) [3 points]
(1) 40
(2) 42
(3) 44
(4) 46
(5) 48
Q9 3 marks Chain Rule Limit Involving Derivative Definition of Composed Functions View
For a function $f ( x )$ differentiable on the set of all real numbers, let the function $g ( x )$ be defined as $$g ( x ) = \frac { f ( x ) } { e ^ { x - 2 } }$$ If $\lim _ { x \rightarrow 2 } \frac { f ( x ) - 3 } { x - 2 } = 5$, what is the value of $g ^ { \prime } ( 2 )$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5
Q10 3 marks Normal Distribution Sampling Distribution of the Mean View
A cosmetic product produced at a certain factory has a content weight that follows a normal distribution with mean 201.5 g and standard deviation 1.8 g. Using the standard normal distribution table on the right, what is the probability that the sample mean of 9 randomly selected cosmetic products from this factory is at least 200 g? [3 points]
$z$$\mathrm { P } ( 0 \leq Z \leq z )$
1.00.3413
1.50.4332
2.00.4772
2.50.4938

(1) 0.7745
(2) 0.8413
(3) 0.9332
(4) 0.9772
(5) 0.9938
Q11 3 marks Composite & Inverse Functions Derivative of an Inverse Function View
There are two functions $f ( x ) , g ( x )$ differentiable on the set of all real numbers. $f ( x )$ is the inverse function of $g ( x )$ and $f ( 1 ) = 2 , f ^ { \prime } ( 1 ) = 3$. If the function $h ( x ) = x g ( x )$, what is the value of $h ^ { \prime } ( 2 )$? [3 points]
(1) 1
(2) $\frac { 4 } { 3 }$
(3) $\frac { 5 } { 3 }$
(4) 2
(5) $\frac { 7 } { 3 }$
Q12 3 marks Areas by integration View
Let region $A$ be enclosed by the curve $y = e ^ { 2 x }$, the $y$-axis, and the line $y = - 2 x + a$, and let region $B$ be enclosed by the curve $y = e ^ { 2 x }$ and the two lines $y = - 2 x + a , x = 1$. When the area of $A$ equals the area of $B$, what is the value of the constant $a$? (Here, $1 < a < e ^ { 2 }$) [3 points]
(1) $\frac { e ^ { 2 } + 1 } { 2 }$
(2) $\frac { 2 e ^ { 2 } + 1 } { 4 }$
(3) $\frac { e ^ { 2 } } { 2 }$
(4) $\frac { 2 e ^ { 2 } - 1 } { 4 }$
(5) $\frac { e ^ { 2 } - 1 } { 2 }$
Q13 3 marks Conditional Probability Combinatorial Conditional Probability (Counting-Based) View
A die is rolled twice. Given that the number 6 does not appear at all, what is the probability that the sum of the two numbers is a multiple of 4? [3 points]
(1) $\frac { 4 } { 25 }$
(2) $\frac { 1 } { 5 }$
(3) $\frac { 6 } { 25 }$
(4) $\frac { 7 } { 25 }$
(5) $\frac { 8 } { 25 }$
Q14 4 marks Addition & Double Angle Formulae Addition/Subtraction Formula Evaluation View
As shown in the figure, in triangle ABC with $\overline { \mathrm { AB } } = 5 , \overline { \mathrm { AC } } = 2 \sqrt { 5 }$, let D be the foot of the perpendicular from vertex A to segment BC.
For point E that divides segment AD internally in the ratio $3 : 1$, we have $\overline { \mathrm { EC } } = \sqrt { 5 }$. If $\angle \mathrm { ABD } = \alpha , \angle \mathrm { DCE } = \beta$, what is the value of $\cos ( \alpha - \beta )$? [4 points]
(1) $\frac { \sqrt { 5 } } { 5 }$
(2) $\frac { \sqrt { 5 } } { 4 }$
(3) $\frac { 3 \sqrt { 5 } } { 10 }$
(4) $\frac { 7 \sqrt { 5 } } { 20 }$
(5) $\frac { 2 \sqrt { 5 } } { 5 }$
Q15 4 marks Indefinite & Definite Integrals Finding a Function from an Integral Equation View
When the function $f ( x )$ is $$f ( x ) = \int _ { 0 } ^ { x } \frac { 1 } { 1 + e ^ { - t } } d t$$ what is the value of the real number $a$ that satisfies $( f \circ f ) ( a ) = \ln 5$? [4 points]
(1) $\ln 11$
(2) $\ln 13$
(3) $\ln 15$
(4) $\ln 17$
(5) $\ln 19$
Q16 4 marks Variable acceleration (vectors) View
The position $\mathrm { P } ( x , y )$ of a point P moving on the coordinate plane at time $t ( 0 < t < \pi )$ is given by $$x = \sqrt { 3 } \sin t , \quad y = 2 \cos t - 5$$ At time $t = \alpha ( 0 < \alpha < \pi )$, the velocity $\vec { v }$ of point P and $\overrightarrow { \mathrm { OP } }$ are parallel. What is the value of $\cos \alpha$? (Here, O is the origin.) [4 points]
(1) $\frac { 1 } { 10 }$
(2) $\frac { 1 } { 5 }$
(3) $\frac { 3 } { 10 }$
(4) $\frac { 2 } { 5 }$
(5) $\frac { 1 } { 2 }$
Q17 4 marks Small angle approximation View
As shown in the figure, there is a rhombus ABCD with side length 1. Let E be the foot of the perpendicular from point C to the extension of segment AB, let F be the foot of the perpendicular from point E to segment AC, and let G be the intersection of segment EF and segment BC. If $\angle \mathrm { DAB } = \theta$, let the area of triangle CFG be $S ( \theta )$.
What is the value of $\lim _ { \theta \rightarrow 0 + } \frac { S ( \theta ) } { \theta ^ { 5 } }$? (Here, $0 < \theta < \frac { \pi } { 2 }$) [4 points]
(1) $\frac { 1 } { 24 }$
(2) $\frac { 1 } { 20 }$
(3) $\frac { 1 } { 16 }$
(4) $\frac { 1 } { 12 }$
(5) $\frac { 1 } { 8 }$
Q18 4 marks Permutations & Arrangements Distribution of Objects into Bins/Groups View
When distributing 4 distinct balls into 4 distinct boxes without remainder, how many ways are there to distribute them such that there is at least one box containing exactly 1 ball? (Here, there may be boxes with no balls.) [4 points]
(1) 220
(2) 216
(3) 212
(4) 208
(5) 204
Q19 4 marks Discrete Probability Distributions Deriving or Identifying a Probability Distribution from a Random Process View
There are 6 weights of 1 unit, 3 weights of 2 units, and 1 empty bag. Using one die, the following trial is performed. (Here, the unit of weight is g.)
Roll the die once. If the number shown is 2 or less, put one weight of 1 unit into the bag. If the number shown is 3 or more, put one weight of 2 units into the bag.
Repeat this trial until the total weight of the weights in the bag is first greater than or equal to 6. Let $X$ be the random variable representing the number of weights in the bag. The following is the process of finding the probability mass function $\mathrm { P } ( X = x ) ( x = 3,4,5,6 )$ of $X$.
(i) The event $X = 3$ is the case where 3 weights of 2 units are in the bag, so $$\mathrm { P } ( X = 3 ) = \text{ (a) }$$ (ii) The event $X = 4$ can be divided into the case where the total weight of weights put in by the third trial is 4 and a weight of 2 units is put in on the fourth trial, and the case where the total weight of weights put in by the third trial is 5. Therefore, $$\mathrm { P } ( X = 4 ) = \left( \text{ (b) } + { } _ { 3 } \mathrm { C } _ { 1 } \left( \frac { 1 } { 3 } \right) ^ { 1 } \left( \frac { 2 } { 3 } \right) ^ { 2 } \right) \times \frac { 2 } { 3 }$$ (iii) The event $X = 5$ can be divided into the case where the total weight of weights put in by the fourth trial is 4 and a weight of 2 units is put in on the fifth trial, and the case where the total weight of weights put in by the fourth trial is 5. Therefore, $$\mathrm { P } ( X = 5 ) = { } _ { 4 } \mathrm { C } _ { 4 } \left( \frac { 1 } { 3 } \right) ^ { 4 } \left( \frac { 2 } { 3 } \right) ^ { 0 } \times \frac { 2 } { 3 } + \text{ (c) }$$ (iv) The event $X = 6$ is the case where the total weight of weights put in by the fifth trial is 5, so $$\mathrm { P } ( X = 6 ) = \left( \frac { 1 } { 3 } \right) ^ { 5 }$$ If the values corresponding to (a), (b), (c) are $a , b , c$ respectively, what is the value of $\frac { a b } { c }$? [4 points]
(1) $\frac { 4 } { 9 }$
(2) $\frac { 7 } { 9 }$
(3) $\frac { 10 } { 9 }$
(4) $\frac { 13 } { 9 }$
(5) $\frac { 16 } { 9 }$
Q20 4 marks Vectors: Lines & Planes Distance Computation (Point-to-Plane or Line-to-Line) View
In coordinate space, there are three points $\mathrm { A } , \mathrm { B } , \mathrm { C }$ not on the same line. For a plane $\alpha$ satisfying the following conditions, let $d ( \alpha )$ be the minimum distance among the distances from each point $\mathrm { A } , \mathrm { B } , \mathrm { C }$ to plane $\alpha$.
(a) Plane $\alpha$ intersects segment AC and also intersects segment BC.
(b) Plane $\alpha$ does not intersect segment AB.
Among planes $\alpha$ satisfying the above conditions, let $\beta$ be the plane where $d ( \alpha )$ is maximized. Which of the following statements in are correct? [4 points]
$\text{ㄱ}$. Plane $\beta$ is perpendicular to the plane passing through the three points $\mathrm { A } , \mathrm { B } , \mathrm { C }$. $\text{ㄴ}$. Plane $\beta$ passes through the midpoint of segment AC or the midpoint of segment BC. $\text{ㄷ}$. When the three points are $\mathrm { A } ( 2,3,0 ) , \mathrm { B } ( 0,1,0 ) , \mathrm { C } ( 2 , - 1,0 )$, $d ( \beta )$ equals the distance between point B and plane $\beta$.
(1) ㄱ
(2) ㄷ
(3) ㄱ, ㄴ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ
Q21 4 marks Applied differentiation Full function study (variation table, limits, asymptotes) View
For a positive number $t$, the function $f ( x )$ defined on the interval $[ 1 , \infty )$ is $$f ( x ) = \begin{cases} \ln x & ( 1 \leq x < e ) \\ - t + \ln x & ( x \geq e ) \end{cases}$$ Among linear functions $g ( x )$ satisfying the following condition, let $h ( t )$ be the minimum value of the slope of the line $y = g ( x )$.
For all real numbers $x \geq 1$, $( x - e ) \{ g ( x ) - f ( x ) \} \geq 0$.
For a differentiable function $h ( t )$, a positive number $a$ satisfies $h ( a ) = \frac { 1 } { e + 2 }$. What is the value of $h ^ { \prime } \left( \frac { 1 } { 2 e } \right) \times h ^ { \prime } ( a )$? [4 points]
(1) $\frac { 1 } { ( e + 1 ) ^ { 2 } }$
(2) $\frac { 1 } { e ( e + 1 ) }$
(3) $\frac { 1 } { e ^ { 2 } }$
(4) $\frac { 1 } { ( e - 1 ) ( e + 1 ) }$
(5) $\frac { 1 } { e ( e - 1 ) }$
Q22 3 marks Combinations & Selection Basic Combination Computation View
Find the value of ${ } _ { 5 } \mathrm { C } _ { 3 }$. [3 points]
Q23 3 marks Differentiating Transcendental Functions Evaluate derivative at a point or find tangent slope View
For the function $f ( x ) = \ln \left( x ^ { 2 } + 1 \right)$, find the value of $f ^ { \prime } ( 1 )$. [3 points]
Q24 3 marks Implicit equations and differentiation Compute slope at a point via implicit differentiation (single-step) View
Find the slope of the tangent line to the curve $2 x + x ^ { 2 } y - y ^ { 3 } = 2$ at the point $( 1,1 )$. [3 points]
Q25 3 marks Straight Lines & Coordinate Geometry Line Equation and Parametric Representation View
On the coordinate plane, a line passes through the point $( 4,1 )$ and is perpendicular to the vector $\vec { n } = ( 1,2 )$. Let the coordinates of the points where this line meets the $x$-axis and $y$-axis be $( a , 0 ) , ( 0 , b )$ respectively. Find the value of $a + b$. [3 points]
Q26 4 marks Normal Distribution Algebraic Relationship Between Normal Parameters and Probability View
A random variable $X$ follows a normal distribution with mean $m$ and standard deviation $\sigma$, and $$\mathrm { P } ( X \leq 3 ) = \mathrm { P } ( 3 \leq X \leq 80 ) = 0.3$$ Find the value of $m + \sigma$. (Here, when $Z$ is a random variable following the standard normal distribution, $\mathrm { P } ( 0 \leq Z \leq 0.25 ) = 0.1 , \mathrm { P } ( 0 \leq Z \leq 0.52 ) = 0.2$ for calculation purposes.) [4 points]
Q27 4 marks Conic sections Circle-Conic Interaction with Tangency or Intersection View
As shown in the figure, for a point P on the hyperbola $\frac { x ^ { 2 } } { 8 } - \frac { y ^ { 2 } } { 17 } = 1$ with foci $\mathrm { F } , \mathrm { F } ^ { \prime }$, there is a circle $C$ that is tangent to both line FP and line $\mathrm { F } ^ { \prime } \mathrm { P }$ simultaneously and has its center on the $y$-axis. For point Q, the point of tangency of line $\mathrm { F } ^ { \prime } \mathrm { P }$ with circle $C$, we have $\overline { \mathrm { F } ^ { \prime } \mathrm { Q } } = 5 \sqrt { 2 }$. Find the value of $\overline { \mathrm { FP } } ^ { 2 } + { \overline { \mathrm { F } ^ { \prime } \mathrm { P } } } ^ { 2 }$. (Here, $\overline { \mathrm { F } ^ { \prime } \mathrm { P } } < \overline { \mathrm { FP } }$) [4 points]
Q28 4 marks Combinations & Selection Combinatorial Probability View
Among all ordered pairs $( x , y , z )$ of non-negative integers satisfying the equation $x + y + z = 10$, one is randomly selected. Find the probability that the selected ordered pair $( x , y , z )$ satisfies $( x - y ) ( y - z ) ( z - x ) \neq 0$. If this probability is $\frac { q } { p }$, find the value of $p + q$. (Here, $p$ and $q$ are coprime natural numbers.) [4 points]
Q29 4 marks Vectors: Lines & Planes Sphere-Plane Intersection and Projection of Circles View
In coordinate space, there is a circle $C$ formed by the intersection of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 6$ and the plane $x + 2 z - 5 = 0$. Let P be the point on circle $C$ with the minimum $y$-coordinate, and let Q be the foot of the perpendicular from point P to the $xy$-plane. For a point X moving on circle $C$, the maximum value of $| \overrightarrow { \mathrm { PX } } + \overrightarrow { \mathrm { QX } } | ^ { 2 }$ is $a + b \sqrt { 30 }$.
Find the value of $10 ( a + b )$. (Here, $a$ and $b$ are rational numbers.) [4 points]
Q30 4 marks Integration by Parts Differentiation Under the Integral Sign Combined with Parts View
For a real number $t$, define the function $f ( x )$ as $$f ( x ) = \left\{ \begin{array} { c c } 1 - | x - t | & ( | x - t | \leq 1 ) \\ 0 & ( | x - t | > 1 ) \end{array} \right.$$ For a certain odd number $k$, the function $$g ( t ) = \int _ { k } ^ { k + 8 } f ( x ) \cos ( \pi x ) d x$$ satisfies the following condition.
When all $\alpha$ for which the function $g ( t )$ has a local minimum at $t = \alpha$ and $g ( \alpha ) < 0$ are listed in increasing order as $\alpha _ { 1 } , \alpha _ { 2 } , \cdots , \alpha _ { m }$ (where $m$ is a natural number), we have $\sum _ { i = 1 } ^ { m } \alpha _ { i } = 45$. Find the value of $k - \pi ^ { 2 } \sum _ { i = 1 } ^ { m } g \left( \alpha _ { i } \right)$. [4 points]