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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
2020 csat__math-science

30 maths questions

Q1 2 marks Vectors Introduction & 2D Magnitude of Vector Expression View
For two vectors $\vec { a } = ( 3,1 ) , \vec { b } = ( - 2,4 )$, what is the sum of all components of the vector $\vec { a } + \frac { 1 } { 2 } \vec { b }$? [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 $\mathrm { A } ( 2,0,1 ) , \mathrm { B } ( 3,2,0 )$ in coordinate space, if the coordinates of a point on the $y$-axis that is equidistant from both points is $( 0 , a , 0 )$, what is the value of $a$? [2 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5
Q3 2 marks Differentiating Transcendental Functions Limit involving transcendental functions View
What is the value of $\lim _ { x \rightarrow 0 } \frac { 6 x } { e ^ { 4 x } - e ^ { 2 x } }$? [2 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5
Q4 3 marks Binomial Theorem (positive integer n) Find a Specific Coefficient in a Single Binomial Expansion View
In the expansion of $\left( 2 x + \frac { 1 } { x ^ { 2 } } \right) ^ { 4 }$, what is the coefficient of $x$? [3 points]
(1) 16
(2) 20
(3) 24
(4) 28
(5) 32
Q5 3 marks Implicit equations and differentiation Compute slope at a point via implicit differentiation (single-step) View
What is the slope of the tangent line to the curve $x ^ { 2 } - 3 x y + y ^ { 2 } = x$ at the point $( 1,0 )$? [3 points]
(1) $\frac { 1 } { 12 }$
(2) $\frac { 1 } { 6 }$
(3) $\frac { 1 } { 4 }$
(4) $\frac { 1 } { 3 }$
(5) $\frac { 5 } { 12 }$
Q6 3 marks Standard trigonometric equations Solve trigonometric inequality View
For $0 < x < 2 \pi$, what is the sum of all values of $x$ that simultaneously satisfy the equation $4 \cos ^ { 2 } x - 1 = 0$ and the inequality $\sin x \cos x < 0$? [3 points]
(1) $\frac { 10 } { 3 } \pi$
(2) $3 \pi$
(3) $\frac { 8 } { 3 } \pi$
(4) $\frac { 7 } { 3 } \pi$
(5) $2 \pi$
Q7 3 marks Combinations & Selection Combinatorial Probability View
A bag contains 3 white balls and 4 black balls. When drawing 4 balls simultaneously at random from the bag, what is the probability of drawing 2 white balls and 2 black balls? [3 points]
(1) $\frac { 2 } { 5 }$
(2) $\frac { 16 } { 35 }$
(3) $\frac { 18 } { 35 }$
(4) $\frac { 4 } { 7 }$
(5) $\frac { 22 } { 35 }$
Q8 3 marks Integration by Substitution Substitution to Evaluate a Definite Integral (Numerical Answer) View
What is the value of $\int _ { e } ^ { e ^ { 2 } } \frac { \ln x - 1 } { x ^ { 2 } } d x$? [3 points]
(1) $\frac { e - 2 } { e ^ { 2 } }$
(2) $\frac { e - 1 } { e ^ { 2 } }$
(3) $\frac { 1 } { e }$
(4) $\frac { e + 1 } { e ^ { 2 } }$
(5) $\frac { e + 2 } { e ^ { 2 } }$
Q9 3 marks Addition & Double Angle Formulae Addition/Subtraction Formula Evaluation View
In an isosceles triangle ABC with $\overline { \mathrm { AB } } = \overline { \mathrm { AC } }$, let $\angle \mathrm { A } = \alpha , \angle \mathrm { B } = \beta$. If $\tan ( \alpha + \beta ) = - \frac { 3 } { 2 }$, what is the value of $\tan \alpha$? [3 points]
(1) $\frac { 5 } { 2 }$
(2) $\frac { 12 } { 5 }$
(3) $\frac { 23 } { 10 }$
(4) $\frac { 11 } { 5 }$
(5) $\frac { 21 } { 10 }$
Q10 3 marks Applied differentiation Kinematics via differentiation View
A point P moving on the coordinate plane has position $( x , y )$ at time $t \left( 0 < t < \frac { \pi } { 2 } \right)$ given by
$$x = t + \sin t \cos t , \quad y = \tan t$$
What is the minimum speed of point P for $0 < t < \frac { \pi } { 2 }$? [3 points]
(1) 1
(2) $\sqrt { 3 }$
(3) 2
(4) $2 \sqrt { 2 }$
(5) $2 \sqrt { 3 }$
Q11 3 marks Stationary points and optimisation Find concavity, inflection points, or second derivative properties View
How many integers $a$ are there such that the curve $y = a x ^ { 2 } - 2 \sin 2 x$ has an inflection point? [3 points]
(1) 4
(2) 5
(3) 6
(4) 7
(5) 8
Q12 3 marks Volumes of Revolution Volume by Cross Sections with Known Geometry View
As shown in the figure, for a positive number $k$, the region enclosed by the curve $y = \sqrt { \frac { e ^ { x } } { e ^ { x } + 1 } }$, the $x$-axis, the $y$-axis, and the line $x = k$ is the base of a solid figure. When the cross-section perpendicular to the $x$-axis is always a square and the volume is $\ln 7$, what is the value of $k$? [3 points]
(1) $\ln 11$
(2) $\ln 13$
(3) $\ln 15$
(4) $\ln 17$
(5) $\ln 19$
Q13 3 marks Conic sections Triangle or Quadrilateral Area and Perimeter with Foci View
As shown in the figure, an ellipse $\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { 25 } = 1$ has foci at $\mathrm { F } ( 0 , c ) , \mathrm { F } ^ { \prime } ( 0 , - c )$. Let A be the point with positive $x$-coordinate where the ellipse meets the $x$-axis. Let B be the intersection of the line $y = c$ and the line $\mathrm { AF } ^ { \prime}$, and let P be the point with positive $x$-coordinate where the line $y = c$ meets the ellipse. If the difference between the perimeter of triangle $\mathrm { BPF } ^ { \prime}$ and the perimeter of triangle BFA is 4, what is the area of triangle $\mathrm { AFF } ^ { \prime}$? (Given: $0 < a < 5 , c > 0$) [3 points]
(1) $3 \sqrt { 6 }$
(2) $\frac { 7 \sqrt { 6 } } { 2 }$
(3) $4 \sqrt { 6 }$
(4) $\frac { 9 \sqrt { 6 } } { 2 }$
(5) $5 \sqrt { 6 }$
Q14 4 marks Measures of Location and Spread View
A bag contains 10 balls labeled with the number 1, 20 balls labeled with the number 2, and 30 balls labeled with the number 3. A ball is drawn at random from the bag, the number on the ball is noted, and the ball is returned. This procedure is repeated 10 times, and let $Y$ be the sum of the 10 numbers observed. The following is the process of finding the mean $\mathrm { E } ( Y )$ and variance $\mathrm { V } ( Y )$ of the random variable $Y$.
Consider the 60 balls in the bag as a population. When a ball is drawn at random from this population, let $X$ be the random variable representing the number on the ball. The probability distribution of $X$, which is the probability distribution of the population, is shown in the following table.
$X$123Total
$\mathrm { P } ( X = x )$$\frac { 1 } { 6 }$$\frac { 1 } { 3 }$$\frac { 1 } { 2 }$1

Therefore, the population mean $m$ and population variance $\sigma ^ { 2 }$ are
$$m = \mathrm { E } ( X ) = \frac { 7 } { 3 } , \quad \sigma ^ { 2 } = \mathrm { V } ( X ) = \text { (가) }$$
When a sample of size 10 is randomly extracted from the population and the sample mean is $\bar { X }$,
$$\mathrm { E } ( \bar { X } ) = \frac { 7 } { 3 } , \quad \mathrm {~V} ( \bar { X } ) = \text { (나) }$$
If the number on the $n$-th ball drawn from the bag is $X _ { n }$, then
$$Y = \sum _ { n = 1 } ^ { 10 } X _ { n } = 10 \bar { X }$$
so
$$\mathrm { E } ( Y ) = \frac { 70 } { 3 } , \quad \mathrm {~V} ( Y ) = \text { (다) }$$
If the numbers that fit (가), (나), and (다) are $p , q , r$ respectively, what is the value of $p + q + r$? [4 points]
(1) $\frac { 31 } { 6 }$
(2) $\frac { 11 } { 2 }$
(3) $\frac { 35 } { 6 }$
(4) $\frac { 37 } { 6 }$
(5) $\frac { 13 } { 2 }$
Q15 4 marks Exponential Functions Intersection and Distance between Curves View
Let A be the point where the graph of the exponential function $y = a ^ { x } ( a > 1 )$ meets the line $y = \sqrt { 3 }$. For the point $\mathrm { B } ( 4,0 )$, if the line OA and the line AB are perpendicular to each other, what is the product of all values of $a$? (Here, O is the origin.) [4 points]
(1) $3 ^ { \frac { 1 } { 3 } }$
(2) $3 ^ { \frac { 2 } { 3 } }$
(3) 3
(4) $3 ^ { \frac { 4 } { 3 } }$
(5) $3 ^ { \frac { 5 } { 3 } }$
Q16 4 marks Permutations & Arrangements Distribution of Objects into Bins/Groups View
How many ordered pairs $( a , b , c , d )$ of non-negative integers satisfy the following conditions? [4 points] (가) $a + b + c - d = 9$ (나) $d \leq 4$ and $c \geq d$
(1) 265
(2) 270
(3) 275
(4) 280
(5) 285
Q17 4 marks Conic sections Optimization on Conics View
In a plane, there is an equilateral triangle ABC with side length 10. For a point P satisfying $\overline { \mathrm { PB } } - \overline { \mathrm { PC } } = 2$, when the length of segment PA is minimized, what is the area of triangle PBC? [4 points]
(1) $20 \sqrt { 3 }$
(2) $21 \sqrt { 3 }$
(3) $22 \sqrt { 3 }$
(4) $23 \sqrt { 3 }$
(5) $24 \sqrt { 3 }$
Q18 4 marks Normal Distribution Algebraic Relationship Between Normal Parameters and Probability View
The random variable $X$ follows the normal distribution $\mathrm { N } \left( 10,2 ^ { 2 } \right)$, and the random variable $Y$ follows the normal distribution $\mathrm { N } \left( m , 2 ^ { 2 } \right)$. The probability density functions of $X$ and $Y$ are $f ( x )$ and $g ( x )$ respectively.
$$f ( 12 ) \leq g ( 20 )$$
For $m$ satisfying this condition, what is the maximum value of $\mathrm { P } ( 21 \leq Y \leq 24 )$ using the standard normal distribution table on the right? [4 points]
$z$$\mathrm { P } ( 0 \leq Z \leq z )$
0.50.1915
1.00.3413
1.50.4332
2.00.4772

(1) 0.5328
(2) 0.6247
(3) 0.7745
(4) 0.8185
(5) 0.9104
Q19 4 marks Vectors Introduction & 2D Magnitude of Vector Expression View
Four distinct points $\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D }$ on a circle satisfy the following conditions. What is the value of $| \overrightarrow { \mathrm { AD } } | ^ { 2 }$? [4 points] (가) $| \overrightarrow { \mathrm { AB } } | = 8 , \overrightarrow { \mathrm { AC } } \cdot \overrightarrow { \mathrm { BC } } = 0$ (나) $\overrightarrow { \mathrm { AD } } = \frac { 1 } { 2 } \overrightarrow { \mathrm { AB } } - 2 \overrightarrow { \mathrm { BC } }$
(1) 32
(2) 34
(3) 36
(4) 38
(5) 40
Q20 4 marks Binomial Distribution Contextual Probability Requiring Binomial Modeling Setup View
A coin is tossed 7 times. What is the probability of satisfying the following conditions? [4 points] (가) Heads appears at least 3 times. (나) There is a case where heads appears consecutively.
(1) $\frac { 11 } { 16 }$
(2) $\frac { 23 } { 32 }$
(3) $\frac { 3 } { 4 }$
(4) $\frac { 25 } { 32 }$
(5) $\frac { 13 } { 16 }$
Q21 4 marks Applied differentiation Tangent line computation and geometric consequences View
For a real number $t$, let the equation of the tangent line to the curve $y = e ^ { x }$ at the point $\left( t , e ^ { t } \right)$ be $y = f ( x )$. Let $g ( t )$ be the minimum value of the real number $k$ such that the function $y = | f ( x ) + k - \ln x |$ is differentiable on the entire set of positive real numbers. For two real numbers $a , b ( a < b )$, let $\int _ { a } ^ { b } g ( t ) d t = m$. Which of the following statements in the given options are correct? [4 points]
$\langle$Options$\rangle$
ㄱ. There exist two real numbers $a , b ( a < b )$ such that $m < 0$. ㄴ. If $g ( c ) = 0$ for a real number $c$, then $g ( - c ) = 0$. ㄷ. If $m$ is minimized when $a = \alpha , b = \beta ( \alpha < \beta )$, then
$$\frac { 1 + g ^ { \prime } ( \beta ) } { 1 + g ^ { \prime } ( \alpha ) } < - e ^ { 2 }$$
(1) ㄱ
(2) ㄴ
(3) ㄱ, ㄴ
(4) ㄱ, ㄷ
(5) ㄱ, ㄴ, ㄷ
Q22 3 marks Product & Quotient Rules View
For the function $f ( x ) = x ^ { 3 } \ln x$, find the value of $\frac { f ^ { \prime } ( e ) } { e ^ { 2 } }$. [3 points]
Q23 3 marks Binomial Distribution Compute Expectation, Variance, or Standard Deviation View
The random variable $X$ follows the binomial distribution $\mathrm { B } ( 80 , p )$ and $\mathrm { E } ( X ) = 20$. Find the value of $\mathrm { V } ( X )$. [3 points]
Q24 3 marks Small angle approximation View
On the coordinate plane, let P $( t , \sin t ) ( 0 < t < \pi )$ be a point on the curve $y = \sin x$. Let circle $C$ be centered at P and tangent to the $x$-axis. Let Q be the point where circle $C$ is tangent to the $x$-axis, and let R be the point where circle $C$ meets segment OP.
If $\lim _ { t \rightarrow 0 + } \frac { \overline { \mathrm { OQ } } } { \overline { \mathrm { OR } } } = a + b \sqrt { 2 }$, find the value of $a + b$. (Here, O is the origin, and $a , b$ are integers.) [3 points]
Q25 4 marks Composite & Inverse Functions Derivative of an Inverse Function View
For the function $f ( x ) = \left( x ^ { 2 } + 2 \right) e ^ { - x }$, the function $g ( x )$ is differentiable and satisfies
$$g \left( \frac { x + 8 } { 10 } \right) = f ^ { - 1 } ( x ) , \quad g ( 1 ) = 0$$
Find the value of $\left| g ^ { \prime } ( 1 ) \right|$. [4 points]
Q26 3 marks Discrete Probability Distributions Probability Computation for Compound or Multi-Stage Random Experiments View
A die is rolled 5 times, and let $a$ be the number of times an odd number appears. A coin is tossed 4 times, and let $b$ be the number of times heads appears. If the probability that $a - b = 3$ is $\frac { q } { p }$, find the value of $p + q$. (Here, $p$ and $q$ are coprime natural numbers.) [3 points]
Q27 4 marks Vectors: Lines & Planes Perpendicular/Orthogonal Projection onto a Plane View
As shown in the figure, there is a rhombus-shaped piece of paper ABCD with side length 4 and $\angle \mathrm { BAD } = \frac { \pi } { 3 }$. Let M and N be the midpoints of sides BC and CD respectively. The paper is folded along the three line segments $\mathrm { AM } , \mathrm { AN } , \mathrm { MN }$ to form a tetrahedron PAMN. The area of the orthogonal projection of triangle AMN onto the plane PAM is $\frac { q } { p } \sqrt { 3 }$. Find the value of $p + q$. (Here, the thickness of the paper is neglected, P is the point where the three points $\mathrm { B } , \mathrm { C } , \mathrm { D }$ coincide when the paper is folded, and $p$ and $q$ are coprime natural numbers.) [4 points]
Q28 4 marks Permutations & Arrangements Forming Numbers with Digit Constraints View
From the numbers $1,2,3,4,5,6$, select five numbers with repetition allowed and arrange them in a line to form a five-digit natural number, satisfying the following conditions. How many such five-digit natural numbers can be formed? [4 points] (가) Each odd number is either not selected or selected exactly once. (나) Each even number is either not selected or selected exactly twice.
Q29 4 marks Vectors: Lines & Planes Volume of Pyramid/Tetrahedron Using Planes and Lines View
In coordinate space, for two points $\mathrm { A } ( 3 , - 3,3 ) , \mathrm { B } ( - 2,7 , - 2 )$, let $\alpha , \beta$ be the two planes that contain segment AB and are tangent to the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 1$. Let C and D be the points of tangency of the two planes $\alpha , \beta$ with the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 1$ respectively. If the volume of tetrahedron ABCD is $\frac { q } { p } \sqrt { 3 }$, find the value of $p + q$. (Here, $p$ and $q$ are coprime natural numbers.) [4 points]
Q30 4 marks Applied differentiation Finding parameter values from differentiability or equation constraints View
For a positive real number $t$, let $f ( t )$ be the value of the real number $a$ such that the curve $y = t ^ { 3 } \ln ( x - t )$ meets the curve $y = 2 e ^ { x - a }$ at exactly one point. Find the value of $\left\{ f ^ { \prime } \left( \frac { 1 } { 3 } \right) \right\} ^ { 2 }$. [4 points]