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
2017 csat__math-science

30 maths questions

Q1 2 marks Vectors Introduction & 2D Magnitude of Vector Expression View
For two vectors $\vec { a } = ( 1,3 ) , \vec { b } = ( 5 , - 6 )$, what is the sum of all components of the vector $\vec { a } - \vec { b }$? [2 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5
Q2 2 marks Differentiating Transcendental Functions Limit involving transcendental functions View
What is the value of $\lim _ { x \rightarrow 0 } \frac { e ^ { 6 x } - 1 } { \ln ( 1 + 3 x ) }$? [2 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5
Q3 2 marks Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View
What is the value of $\int _ { 0 } ^ { \frac { \pi } { 2 } } 2 \sin x \, d x$? [2 points]
(1) 0
(2) $\frac { 1 } { 2 }$
(3) 1
(4) $\frac { 3 } { 2 }$
(5) 2
Q4 3 marks Independent Events View
Two events $A$ and $B$ are mutually independent and $$\mathrm { P } \left( B ^ { C } \right) = \frac { 1 } { 3 } , \mathrm { P } ( A \mid B ) = \frac { 1 } { 2 }$$ What is the value of $\mathrm { P } ( A ) \mathrm { P } ( B )$? (Here, $B ^ { C }$ is the complement of $B$.) [3 points]
(1) $\frac { 5 } { 6 }$
(2) $\frac { 2 } { 3 }$
(3) $\frac { 1 } { 2 }$
(4) $\frac { 1 } { 3 }$
(5) $\frac { 1 } { 6 }$
Q5 3 marks Permutations & Arrangements Forming Numbers with Digit Constraints View
Among the numbers $1,2,3,4,5$, if we select four numbers with repetition allowed and arrange them in a row to form a four-digit natural number that is a multiple of 5, how many cases are there? [3 points]
(1) 115
(2) 120
(3) 125
(4) 130
(5) 135
Q6 3 marks Differentiation from First Principles View
For the function $f ( x ) = x ^ { 3 } + x + 1$, let $g ( x )$ be its inverse function. What is the value of $g ^ { \prime } ( 1 )$? [3 points]
(1) $\frac { 1 } { 3 }$
(2) $\frac { 2 } { 5 }$
(3) $\frac { 2 } { 3 }$
(4) $\frac { 4 } { 5 }$
(5) 1
Q7 3 marks Binomial Distribution Compute Exact Binomial Probability View
When rolling a die three times, what is the probability that the number 4 appears exactly once? [3 points]
(1) $\frac { 25 } { 72 }$
(2) $\frac { 13 } { 36 }$
(3) $\frac { 3 } { 8 }$
(4) $\frac { 7 } { 18 }$
(5) $\frac { 29 } { 72 }$
Q8 3 marks Vectors Introduction & 2D Section Ratios and Intersection via Vectors View
In coordinate space, for two points $\mathrm { A } ( 1 , a , - 6 ) , \mathrm { B } ( - 3,2 , b )$, when the point that externally divides the line segment AB in the ratio $3 : 2$ lies on the $x$-axis, what is the value of $a + b$? [3 points]
(1) $-1$
(2) $-2$
(3) $-3$
(4) $-4$
(5) $-5$
Q9 3 marks Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View
What is the value of $\int _ { 1 } ^ { e } \ln \frac { x } { e } \, d x$? [3 points]
(1) $\frac { 1 } { e } - 1$
(2) $2 - e$
(3) $\frac { 1 } { e } - 2$
(4) $1 - e$
(5) $\frac { 1 } { 2 } - e$
Q10 3 marks Variable acceleration (1D) Find velocity/speed by differentiating position View
A point P moving on the coordinate plane has position $( x , y )$ at time $t$ $(t > 0)$ given by $$x = t - \frac { 2 } { t } , \quad y = 2 t + \frac { 1 } { t }$$ What is the speed of point P at time $t = 1$? [3 points]
(1) $2 \sqrt { 2 }$
(2) 3
(3) $\sqrt { 10 }$
(4) $\sqrt { 11 }$
(5) $2 \sqrt { 3 }$
Q11 3 marks Volumes of Revolution Volume by Cross Sections with Known Geometry View
As shown in the figure, there is a solid figure with base formed by the curve $y = \sqrt { x } + 1$, the $x$-axis, the $y$-axis, and the line $x = 1$. When the cross-section of this solid figure cut by a plane perpendicular to the $x$-axis is always a square, what is the volume of this solid figure? [3 points]
(1) $\frac { 7 } { 3 }$
(2) $\frac { 5 } { 2 }$
(3) $\frac { 8 } { 3 }$
(4) $\frac { 17 } { 6 }$
(5) 3
Q12 3 marks Vectors: Lines & Planes Dihedral Angle or Angle Between Planes/Lines View
In coordinate space, let $\theta$ be the acute angle between the plane $2 x + 2 y - z + 5 = 0$ and the $xy$-plane. What is the value of $\cos \theta$? [3 points]
(1) $\frac { 1 } { 12 }$
(2) $\frac { 1 } { 6 }$
(3) $\frac { 1 } { 4 }$
(4) $\frac { 1 } { 3 }$
(5) $\frac { 5 } { 12 }$
Q13 3 marks Linear combinations of normal random variables View
Let $\bar { X }$ be the sample mean obtained by randomly sampling 9 items from a population following the normal distribution $\mathrm { N } \left( 0,4 ^ { 2 } \right)$, and let $\bar { Y }$ be the sample mean obtained by randomly sampling 16 items from a population following the normal distribution $\mathrm { N } \left( 3,2 ^ { 2 } \right)$. What is the value of the constant $a$ satisfying $\mathrm { P } ( \bar { X } \geq 1 ) = \mathrm { P } ( \bar { Y } \leq a )$? [3 points]
(1) $\frac { 19 } { 8 }$
(2) $\frac { 5 } { 2 }$
(3) $\frac { 21 } { 8 }$
(4) $\frac { 11 } { 4 }$
(5) $\frac { 23 } { 8 }$
Q14 4 marks Connected Rates of Change Geometric Related Rates with Distance or Angle View
As shown in the figure, there is a sector OAB with radius 1 and central angle $\frac { \pi } { 2 }$. Let H be the foot of the perpendicular from point P on arc AB to line segment OA, and let Q be the intersection of line segment PH and line segment AB. When $\angle \mathrm { POH } = \theta$, let $S ( \theta )$ be the area of triangle AQH. What is the value of $\lim _ { \theta \rightarrow 0 + } \frac { S ( \theta ) } { \theta ^ { 4 } }$? (Here, $0 < \theta < \frac { \pi } { 2 }$) [4 points]
(1) $\frac { 1 } { 8 }$
(2) $\frac { 1 } { 4 }$
(3) $\frac { 3 } { 8 }$
(4) $\frac { 1 } { 2 }$
(5) $\frac { 5 } { 8 }$
Q15 4 marks Applied differentiation Tangent line computation and geometric consequences View
Let P be a point on the curve $y = 2 e ^ { - x }$ at $\mathrm { P } \left( t , 2 e ^ { - t } \right)$ $(t > 0)$. Let A be the foot of the perpendicular from P to the $y$-axis, and let B be the point where the tangent line at P intersects the $y$-axis. What is the value of $t$ that maximizes the area of triangle APB? [4 points]
(1) 1
(2) $\frac { e } { 2 }$
(3) $\sqrt { 2 }$
(4) 2
(5) $e$
Q16 4 marks Vectors 3D & Lines Vector Algebra and Triple Product Computation View
In coordinate space, let $\vec { a } , \vec { b } , \vec { c }$ be the position vectors of three points $\mathrm { A } , \mathrm { B } , \mathrm { C }$ with respect to the origin. The dot products between these vectors are shown in the following table.
$\cdot$$\vec { a }$$\vec { b }$$\vec { c }$
$\vec { a }$21$- \sqrt { 2 }$
$\vec { b }$120
$\vec { c }$$- \sqrt { 2 }$02

For example, $\vec { a } \cdot \vec { c } = - \sqrt { 2 }$. Which of the following correctly shows the order of the distances between the three points $\mathrm { A } , \mathrm { B } , \mathrm { C }$? [4 points]
(1) $\overline { \mathrm { AB } } < \overline { \mathrm { AC } } < \overline { \mathrm { BC } }$
(2) $\overline { \mathrm { AB } } < \overline { \mathrm { BC } } < \overline { \mathrm { AC } }$
(3) $\overline { \mathrm { AC } } < \overline { \mathrm { AB } } < \overline { \mathrm { BC } }$
(4) $\overline { \mathrm { BC } } < \overline { \mathrm { AB } } < \overline { \mathrm { AC } }$
(5) $\overline { \mathrm { BC } } < \overline { \mathrm { AC } } < \overline { \mathrm { AB } }$
Q17 4 marks Combinations & Selection Lattice Path Counting View
A jump is defined as moving from a point $( x , y )$ on the coordinate plane to one of the three points $( x + 1 , y )$, $( x , y + 1 ) , ( x + 1 , y + 1 )$. Let $X$ be the random variable representing the number of jumps when randomly selecting one case from all possible ways to move from point $( 0,0 )$ to point $( 4,3 )$ by repeating jumps. The following is the process of finding the expected value $\mathrm { E } ( X )$ of the random variable $X$. (Here, each case is selected with equal probability.)
Let $N$ be the total number of ways to move from point $( 0,0 )$ to point $( 4,3 )$ by repeating jumps. If the smallest value that the random variable $X$ can take is $k$, then $k =$ (a), and the largest value is $k + 3$.
$$\begin{aligned} & \mathrm { P } ( X = k ) = \frac { 1 } { N } \times \frac { 4 ! } { 3 ! } = \frac { 4 } { N } \\ & \mathrm { P } ( X = k + 1 ) = \frac { 1 } { N } \times \frac { 5 ! } { 2 ! 2 ! } = \frac { 30 } { N } \\ & \mathrm { P } ( X = k + 2 ) = \frac { 1 } { N } \times \text { (b) } \\ & \mathrm { P } ( X = k + 3 ) = \frac { 1 } { N } \times \frac { 7 ! } { 3 ! 4 ! } = \frac { 35 } { N } \end{aligned}$$
and $$\sum _ { i = k } ^ { k + 3 } \mathrm { P } ( X = i ) = 1$$ so $N =$ (c). Therefore, the expected value $\mathrm { E } ( X )$ of the random variable $X$ is as follows. $$\mathrm { E } ( X ) = \sum _ { i = k } ^ { k + 3 } \{ i \times \mathrm { P } ( X = i ) \} = \frac { 257 } { 43 }$$
When the numbers corresponding to (a), (b), (c) are $a , b , c$ respectively, what is the value of $a + b + c$? [4 points]
(1) 190
(2) 193
(3) 196
(4) 199
(5) 202
Q18 4 marks Normal Distribution Finding Unknown Mean from a Given Probability Condition View
The random variable $X$ follows a normal distribution with mean $m$ and standard deviation 5, and the probability density function $f ( x )$ of the random variable $X$ satisfies the following conditions.
(a) $f ( 10 ) > f ( 20 )$
(b) $f ( 4 ) < f ( 22 )$ When $m$ is a natural number, what is the value of $\mathrm { P } ( 17 \leq X \leq 18 )$ obtained using the standard normal distribution table on the right? [4 points]
$z$$\mathrm { P } ( 0 \leq Z \leq z )$
0.60.226
0.80.288
1.00.341
1.20.385
1.40.419

(1) 0.044
(2) 0.053
(3) 0.062
(4) 0.078
(5) 0.097
Q19 4 marks Conic sections Equation Determination from Geometric Conditions View
For two positive numbers $k , p$, two tangent lines are drawn from point $\mathrm { A } ( - k , 0 )$ to the parabola $y ^ { 2 } = 4 p x$. Let $\mathrm { F } , \mathrm { F } ^ { \prime }$ be the two points where these tangent lines meet the $y$-axis, and let $\mathrm { P } , \mathrm { Q }$ be the two points where they meet the parabola. When $\angle \mathrm { PAQ } = \frac { \pi } { 3 }$, if the length of the major axis of the ellipse with foci at $\mathrm { F } , \mathrm { F } ^ { \prime }$ and passing through points $\mathrm { P } , \mathrm { Q }$ is $4 \sqrt { 3 } + 12$, what is the value of $k + p$? [4 points]
(1) 8
(2) 10
(3) 12
(4) 14
(5) 16
Q20 4 marks Differentiating Transcendental Functions Differentiation under the integral sign with transcendental kernels View
For the function $f ( x ) = e ^ { - x } \int _ { 0 } ^ { x } \sin \left( t ^ { 2 } \right) d t$, which of the following statements are correct? [4 points]
ㄱ. $f ( \sqrt { \pi } ) > 0$ ㄴ. There exists at least one $a$ in the open interval $( 0 , \sqrt { \pi } )$ such that $f ^ { \prime } ( a ) > 0$. ㄷ. There exists at least one $b$ in the open interval $( 0 , \sqrt { \pi } )$ such that $f ^ { \prime } ( b ) = 0$.
(1) ㄱ
(2) ㄷ
(3) ㄱ, ㄴ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ
Q21 4 marks Integration by Parts Multiple-Choice Primitive Identification View
A continuous increasing function $f ( x )$ on the closed interval $[ 0,1 ]$ satisfies $$\int _ { 0 } ^ { 1 } f ( x ) d x = 2 , \quad \int _ { 0 } ^ { 1 } | f ( x ) | d x = 2 \sqrt { 2 }$$ When the function $F ( x )$ is defined as $$F ( x ) = \int _ { 0 } ^ { x } | f ( t ) | d t \quad ( 0 \leq x \leq 1 )$$ what is the value of $\int _ { 0 } ^ { 1 } f ( x ) F ( x ) d x$? [4 points]
(1) $4 - \sqrt { 2 }$
(2) $2 + \sqrt { 2 }$
(3) $5 - \sqrt { 2 }$
(4) $1 + 2 \sqrt { 2 }$
(5) $2 + 2 \sqrt { 2 }$
Q22 3 marks Combinations & Selection Basic Combination Computation View
Find the value of ${}_{4}\mathrm{H}_{2}$. [3 points]
Q23 3 marks Exponential Functions Exponential Equation Solving View
Find the sum of all natural numbers $x$ satisfying the inequality $\left( \frac { 1 } { 2 } \right) ^ { x - 5 } \geq 4$. [3 points]
Q24 3 marks Circles Sphere and 3D Circle Problems View
In coordinate space, find the sum of all real numbers $k$ such that the plane $x + 8 y - 4 z + k = 0$ is tangent to the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } + 2 y - 3 = 0$. [3 points]
Q25 3 marks Standard trigonometric equations Solve trigonometric equation for solutions in an interval View
When $0 < x < 2 \pi$, the sum of all real roots of the equation $\cos ^ { 2 } x - \sin x = 1$ is $\frac { q } { p } \pi$. Find the value of $p + q$. (Here, $p , q$ are coprime natural numbers.) [3 points]
Q26 4 marks Probability Definitions Finite Equally-Likely Probability Computation View
There are two bags A and B, each containing 4 cards with the numbers $1,2,3,4$ written on them. Person 甲 draws two cards from bag A, and person 乙 draws two cards from bag B, each randomly. The probability that the sum of the numbers on the two cards held by 甲 equals the sum of the numbers on the two cards held by 乙 is $\frac { q } { p }$. Find the value of $p + q$. (Here, $p , q$ are coprime natural numbers.) [4 points]
Q27 4 marks Permutations & Arrangements Distribution of Objects into Bins/Groups View
Find the total number of ordered triples $( a , b , c )$ of non-negative integers satisfying the following conditions. [4 points]
(a) $a + b + c = 7$
(b) $2 ^ { a } \times 4 ^ { b }$ is a multiple of 8.
Q28 4 marks Conic sections Equation Determination from Geometric Conditions View
A hyperbola has asymptotes with equations $y = \pm \frac { 4 } { 3 } x$ and two foci at $\mathrm { F } ( c , 0 )$, $\mathrm { F } ^ { \prime } ( - c , 0 )$ $(c > 0)$, and satisfies the following conditions.
(a) For a point P on the hyperbola, $\overline { \mathrm { PF } ^ { \prime } } = 30$ and $16 \leq \overline { \mathrm { PF } } \leq 20$.
(b) For the vertex A with positive $x$-coordinate, the length of segment AF is a natural number. Find the length of the major axis of this hyperbola. [4 points]
Q29 4 marks Vectors 3D & Lines Vector Algebra and Triple Product Computation View
In a regular tetrahedron ABCD with edge length 4, let O be the centroid of triangle ABC and P be the midpoint of segment AD. For a point Q on face BCD of the regular tetrahedron ABCD, when the two vectors $\overrightarrow { \mathrm { OQ } }$ and $\overrightarrow { \mathrm { OP } }$ are perpendicular to each other, the maximum value of $| \overrightarrow { \mathrm { PQ } } |$ is $\frac { q } { p }$. Find the value of $p + q$. (Here, $p , q$ are coprime natural numbers.) [4 points]
Q30 4 marks Stationary points and optimisation Existence or properties of extrema via abstract/theoretical argument View
A function $f ( x )$ defined for $x > a$ and a quartic function $g ( x )$ with leading coefficient $-1$ satisfy the following conditions. (Here, $a$ is a constant.)
(a) For all real numbers $x > a$, $$( x - a ) f ( x ) = g ( x ).$$ (b) For two distinct real numbers $\alpha , \beta$, the function $f ( x )$ has the same local maximum value $M$ at $x = \alpha$ and $x = \beta$. (Here, $M > 0$)
(c) The number of values of $x$ where the function $f ( x )$ has a local maximum or minimum is greater than the number of values of $x$ where the function $g ( x )$ has a local maximum or minimum. When $\beta - \alpha = 6 \sqrt { 3 }$, find the minimum value of $M$. [4 points]