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
2009 csat__math-science

23 maths questions

Q1 2 marks Indices and Surds Evaluating Expressions Using Index Laws View
What is the value of $9 ^ { \frac { 3 } { 2 } } \times 27 ^ { - \frac { 2 } { 3 } }$? [2 points]
(1) $\frac { 1 } { 3 }$
(2) 1
(3) $\sqrt { 3 }$
(4) 3
(5) $3 \sqrt { 3 }$
Q2 2 marks Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View
When the function $f ( x ) = 6 x ^ { 2 } + 2 a x$ satisfies $\int _ { 0 } ^ { 1 } f ( x ) d x = f ( 1 )$, what is the value of the constant $a$? [2 points]
(1) $- 4$
(2) $- 2$
(3) 0
(4) 2
(5) 4
Q3 3 marks Solving quadratics and applications Solving an equation via substitution to reduce to quadratic form View
What is the sum of all real roots of the equation $\sqrt { x ^ { 2 } - 2 x + 1 } - \sqrt { x ^ { 2 } - 2 x } = \frac { 1 } { 2 }$? [3 points]
(1) 5
(2) 4
(3) 3
(4) 2
(5) 1
Q4 2 marks Matrices Matrix Algebra and Product Properties View
For two matrices $A = \left( \begin{array} { l l } 2 & 1 \\ 1 & 1 \end{array} \right) , B = \left( \begin{array} { r r } - 1 & - 2 \\ 1 & 0 \end{array} \right)$, what is the sum of all entries of the matrix $( A + B ) A$? [2 points]
(1) 9
(2) 10
(3) 11
(4) 12
(5) 13
Q5 3 marks Curve Sketching Number of Solutions / Roots via Curve Analysis View
The figure on the right shows a circle with center at the origin O and radius 1, and the graph of a quadratic function $y = f ( x )$ passing through the point $( 0 , - 1 )$ on the coordinate plane. The equation $$\frac { 1 } { f ( x ) + 1 } - \frac { 1 } { f ( x ) - 1 } = \frac { 2 } { x ^ { 2 } }$$ has how many distinct real roots $x$? [3 points]
(1) 2
(2) 3
(3) 4
(4) 5
(5) 6
Q6 3 marks Curve Sketching Finding Parameters for Continuity View
For the function $f ( x ) = x ^ { 2 } - 4 x + a$ and the function $g ( x ) = \lim _ { n \rightarrow \infty } \frac { 2 | x - b | ^ { n } + 1 } { | x - b | ^ { n } + 1 }$, let $h ( x ) = f ( x ) g ( x )$. What is the value of $a + b$, the sum of the two constants $a , b$ such that the function $h ( x )$ is continuous for all real numbers $x$? [3 points]
(1) 3
(2) 4
(3) 5
(4) 6
(5) 7
Q16 4 marks Probability Definitions Conditional Probability and Bayes' Theorem View
Pouches A and B each contain five marbles with the numbers $1,2,3,4,5$ written on them, one number per marble. Chulsu draws one marble from pouch A, and Younghee draws one marble from pouch B. They check the numbers on the two marbles and do not put them back. This trial is repeated. What is the probability that the numbers on the two marbles drawn the first time are different, and the numbers on the two marbles drawn the second time are the same? [4 points]
(1) $\frac { 3 } { 20 }$
(2) $\frac { 1 } { 5 }$
(3) $\frac { 1 } { 4 }$
(4) $\frac { 3 } { 10 }$
(5) $\frac { 7 } { 20 }$
Q17 4 marks Probability Definitions Verifying Statements About Probability Properties View
In information theory, when an event $E$ occurs, the information content $I ( E )$ of the event $E$ is defined as follows. $$I ( E ) = - \log _ { 2 } \mathrm { P } ( E )$$ Which of the following are correct? Select all that apply from . (Note: the probability that event $E$ occurs, $\mathrm { P } ( E )$, is positive, and the unit of information content is bits.) [4 points]
ㄱ. If event $E$ is rolling one die and getting an odd number, then $I ( E ) = 1$. ㄴ. If two events $A , B$ are independent and $\mathrm { P } ( A \cap B ) > 0$, then $I ( A \cap B ) = I ( A ) + I ( B )$. ㄷ. For two events $A , B$ with $\mathrm { P } ( A ) > 0 , \mathrm { P } ( B ) > 0$, we have $2 I ( A \cup B ) \leqq I ( A ) + I ( B )$.
(1) ㄱ
(2) ㄱ, ㄴ
(3) ㄱ, ㄷ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ
Q18 3 marks Differentiation from First Principles View
For a polynomial function $f ( x )$, if $\lim _ { x \rightarrow 2 } \frac { f ( x + 1 ) - 8 } { x ^ { 2 } - 4 } = 5$, find the value of $f ( 3 ) + f ^ { \prime } ( 3 )$. [3 points]
Q19 3 marks Vectors Introduction & 2D Dot Product Computation View
As shown in the figure, in a rectangular parallelepiped $\mathrm { ABCD } - \mathrm { EFGH }$ with $\overline { \mathrm { AB } } = \overline { \mathrm { AD } } = 4$ and $\overline { \mathrm { AE } } = 8$, let P be the point that divides the edge AE in the ratio $1 : 3$, and let Q, R, S be the midpoints of edges $\mathrm { AB }$, $\mathrm { AD }$, and $\mathrm { FG }$, respectively. Let T be the midpoint of segment QR. Find the value of the dot product $\overrightarrow { \mathrm { TP } } \cdot \overrightarrow { \mathrm { QS } }$ of vectors $\overrightarrow { \mathrm { TP } }$ and $\overrightarrow { \mathrm { QS } }$. [3 points]
Q20 3 marks Conic sections Equation Determination from Geometric Conditions View
An ellipse $\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$ is inscribed in a quadrilateral formed by connecting the four vertices of the ellipse $\frac { x ^ { 2 } } { 4 } + y ^ { 2 } = 1$. When the two foci of the ellipse $\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$ are $\mathrm { F } ( b , 0 ) , \mathrm { F } ^ { \prime } ( - b , 0 )$, find the value of $a^2 + b^2$ (or the relevant quantity as stated in the problem). [3 points]
Q21 4 marks Sequences and series, recurrence and convergence Direct term computation from recurrence View
Let $a_n$ denote the sum of all natural numbers such that when divided by a natural number $n$ ($n \geqq 2$), the quotient and remainder are equal. For example, when divided by 4, the natural numbers for which the quotient and remainder are equal are $5, 10, 15$, so $a_4 = 5 + 10 + 15 = 30$. Find the minimum value of the natural number $n$ satisfying $a_n > 500$. [4 points]
Q22 4 marks Vectors Introduction & 2D Angle or Cosine Between Vectors View
As shown in the figure, three cylinders with radius $\sqrt{3}$ and different heights are mutually externally tangent and placed on a plane $\alpha$. Let $\mathrm{P}$, $\mathrm{Q}$, $\mathrm{R}$ be the centers of the bases of the three cylinders that do not meet plane $\alpha$. Triangle $\mathrm{QPR}$ is an isosceles triangle, and the angle between plane $\mathrm{QPR}$ and plane $\alpha$ is $60°$. If the heights of the three cylinders are $8$, $a$, and $b$ respectively, find the value of $a + b$. (Given: $8 < a < b$) [4 points]
Q23 4 marks Vectors 3D & Lines MCQ: Cross-Section or Surface Area of a Solid View
In coordinate space, let $C$ be the circle formed by the intersection of the sphere $S : x^2 + y^2 + z^2 = 4$ and the plane $\alpha : y - \sqrt{3}z = 2$. For point $\mathrm{A}(0, 2, 0)$ on circle $C$, let $\mathrm{P}$ and $\mathrm{Q}$ be the endpoints of a diameter of circle $C$ such that $\overline{\mathrm{AP}} = \overline{\mathrm{AQ}}$. Let $\mathrm{R}$ be another point where the line passing through $\mathrm{P}$ and perpendicular to plane $\alpha$ meets sphere $S$. If the area of triangle $\mathrm{ARQ}$ is $s$, find the value of $s^2$. [4 points]
Q26 3 marks Quadratic trigonometric equations View
(Calculus) For $0 \leqq x < 2\pi$, the sum of all distinct values of $x$ satisfying the equation $\sin 2x = 2\cos x - 2\cos^2 x$ is? [3 points]
(1) $\pi$
(2) $\frac{5}{4}\pi$
(3) $\frac{3}{2}\pi$
(4) $\frac{7}{4}\pi$
(5) $2\pi$
Q26b 3 marks Measures of Location and Spread View
(Probability and Statistics) A certain basketball player practices free throws 40 times every day. The stem-and-leaf plot below shows the number of successful free throws each day for the first 10 days, with the tens digit as the stem and the units digit as the leaf. On the 11th day, the number of successful free throws was $n$, and the average number of successful free throws for the first 11 days was equal to the mode in the stem-and-leaf plot below. What is the value of $n$? [3 points]
Stem\multicolumn{4}{|c|}{Leaf}
09
179
21446
3013

(1) 24
(2) 26
(3) 28
(4) 30
(5) 32
Q27 3 marks Indefinite & Definite Integrals Definite Integral as a Limit of Riemann Sums View
(Calculus) A continuous function $f(x)$ defined on the closed interval $[0, 1]$ satisfies $f(0) = 0$, $f(1) = 1$, has a second derivative on the open interval $(0, 1)$, and $f'(x) > 0$, $f''(x) > 0$. Which of the following is equal to $\int_0^1 \{f^{-1}(x) - f(x)\} dx$? [3 points]
(1) $\lim_{n \rightarrow \infty} \sum_{k=1}^{n} \left\{\frac{k}{n} - f\left(\frac{k}{n}\right)\right\} \frac{1}{2n}$
(2) $\lim_{n \rightarrow \infty} \sum_{k=1}^{n} \left\{\frac{k}{n} - f\left(\frac{k}{n}\right)\right\} \frac{2}{n}$
(3) $\lim_{n \rightarrow \infty} \sum_{k=1}^{n} \left\{\frac{k}{n} - f\left(\frac{k}{n}\right)\right\} \frac{1}{n}$
(4) $\lim_{n \rightarrow \infty} \sum_{k=1}^{n} \left\{\frac{k}{2n} - f\left(\frac{k}{n}\right)\right\} \frac{1}{n}$
(5) $\lim_{n \rightarrow \infty} \sum_{k=1}^{n} \left\{\frac{2k}{n} - f\left(\frac{k}{n}\right)\right\} \frac{1}{n}$
Q27b 3 marks Discrete Probability Distributions Probability Distribution Table Completion and Expectation Calculation View
(Probability and Statistics) A coin is tossed three times, and based on the results, a score is obtained as a random variable $X$ according to the following rules. (가) If the same face does not appear consecutively, the score is 0 points. (나) If the same face appears consecutively exactly twice, the score is 1 point. (다) If the same face appears consecutively three times, the score is 3 points.
What is the variance $\mathrm{V}(X)$ of the random variable $X$? [3 points]
(1) $\frac{9}{8}$
(2) $\frac{19}{16}$
(3) $\frac{5}{4}$
(4) $\frac{21}{16}$
(5) $\frac{11}{8}$
Q28 3 marks Differentiating Transcendental Functions Full function study with transcendental functions View
(Calculus) For the function $f(x) = 4\ln x + \ln(10 - x)$, which of the following statements in are correct? [3 points]
ㄱ. The maximum value of function $f(x)$ is $13\ln 2$. ㄴ. The equation $f(x) = 0$ has two distinct real roots. ㄷ. The graph of function $y = e^{f(x)}$ is concave downward on the interval $(4, 8)$.
(1) ㄱ
(2) ㄷ
(3) ㄱ, ㄴ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ
Q28b 3 marks Combinations & Selection Combinatorial Probability View
(Probability and Statistics) There are 9 balls in a bag, each labeled with a natural number from 1 to 9. When 4 balls are randomly drawn simultaneously from the bag, what is the probability that the sum of the largest and smallest numbers on the drawn balls is at least 7 and at most 9? [3 points]
(1) $\frac{5}{9}$
(2) $\frac{1}{2}$
(3) $\frac{4}{9}$
(4) $\frac{7}{18}$
(5) $\frac{1}{3}$
Q29 4 marks Chain Rule Derivative of Inverse Functions View
(Calculus) Let the function $f(x)$ be defined as $$f(x) = \int_a^x \{2 + \sin(t^2)\} dt$$ If $f''(a) = \sqrt{3}a$, find the value of $(f^{-1})'(0)$. (Given: $a$ is a constant satisfying $0 < a < \sqrt{\frac{\pi}{2}}$) [4 points]
(1) $\frac{1}{10}$
(2) $\frac{1}{5}$
(3) $\frac{3}{10}$
(4) $\frac{2}{5}$
(5) $\frac{1}{2}$
Q29b 4 marks Linear combinations of normal random variables View
(Probability and Statistics) Random variables $X$ and $Y$ follow normal distributions with mean 0 and variances $\sigma^2$ and $\frac{\sigma^2}{4}$ respectively, and random variable $Z$ follows the standard normal distribution. For two positive numbers $a$ and $b$, $$\mathrm{P}(|X| \leqq a) = \mathrm{P}(|Y| \leqq b)$$ Which of the following statements in are correct? [4 points] ㄱ. $a > b$ ㄴ. $\mathrm{P}\left(Z > \frac{2b}{\sigma}\right) = \mathrm{P}\left(Y > \frac{a}{2}\right)$ ㄷ. If $\mathrm{P}(Y \leqq b) = 0.7$, then $\mathrm{P}(|X| \leqq a) = 0.3$.
(1) ㄱ
(2) ㄴ
(3) ㄱ, ㄴ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ
Q30 4 marks Small angle approximation View
(Calculus) Point $\mathrm{A}$ is on circle $\mathrm{O}$ with radius 1. As shown in the figure, for a positive angle $\theta$, two points $\mathrm{B}$ and $\mathrm{C}$ on circle $\mathrm{O}$ are chosen such that $\angle \mathrm{BAC} = \theta$ and $\overline{\mathrm{AB}} = \overline{\mathrm{AC}}$. Let $r(\theta)$ denote the radius of the inscribed circle of triangle $\mathrm{ABC}$. If $\lim_{\theta \rightarrow \pi - 0} \frac{r(\theta)}{(\pi - \theta)^2} = \frac{q}{p}$, find the value of $p^2 + q^2$. (Given: $p$ and $q$ are coprime natural numbers.) [4 points]