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
2013 csat__math-humanities

30 maths questions

Q1 2 marks Matrices Matrix Algebra and Product Properties View
For two matrices $A = \left( \begin{array} { l l } 0 & 0 \\ 1 & 1 \end{array} \right) , B = \left( \begin{array} { l l } 1 & 0 \\ 1 & 1 \end{array} \right)$, what is the sum of all entries of the matrix $2A + B$? [2 points]
(1) 10
(2) 9
(3) 8
(4) 7
(5) 6
Q2 2 marks Sequences and series, recurrence and convergence Multiple-choice on sequence properties View
What is the value of $\lim _ { n \rightarrow \infty } \frac { 5n ^ { 2 } + 1 } { 3n ^ { 2 } - 1 }$? [2 points]
(1) $\frac { 1 } { 3 }$
(2) $\frac { 2 } { 3 }$
(3) 1
(4) $\frac { 4 } { 3 }$
(5) $\frac { 5 } { 3 }$
Q3 2 marks Laws of Logarithms Simplify or Evaluate a Logarithmic Expression View
What is the value of $\log _ { 2 } 40 - \log _ { 2 } 5$? [2 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5
Q4 3 marks Matrices Structured Matrix Characterization View
What is the sum of all entries of the matrix representing the connection relationships between vertices of the following graph? [3 points]
(1) 6
(2) 8
(3) 10
(4) 12
(5) 14
Q5 3 marks Curve Sketching Limit Reading from Graph View
The graph of the function $y = f(x)$ is shown in the figure. What is the value of $\lim _ { x \rightarrow - 1 - 0 } f ( x ) + \lim _ { x \rightarrow + 0 } f ( x )$? [3 points]
(1) $-2$
(2) $-1$
(3) 0
(4) 1
(5) 2
Q6 3 marks Laws of Logarithms Logarithmic Formula Application (Modeling) View
The temperature of a fire room changes over time. For a certain fire room, let the initial temperature be $T _ { 0 } \left( {}^{\circ}\mathrm{C} \right)$ and the temperature $t$ minutes after the fire starts be $T \left( {}^{\circ}\mathrm{C} \right)$. The following equation holds. $T = T _ { 0 } + k \log ( 8t + 1 ) \quad ($ where $k$ is a constant.$)$ In this fire room with an initial temperature of $20^{\circ}\mathrm{C}$, the temperature was $365^{\circ}\mathrm{C}$ after $\frac{9}{8}$ minutes from the start of the fire, and the temperature was $710^{\circ}\mathrm{C}$ after $a$ minutes from the start of the fire. What is the value of $a$? [3 points]
(1) $\frac{99}{8}$
(2) $\frac{109}{8}$
(3) $\frac{119}{8}$
(4) $\frac{129}{8}$
(5) $\frac{139}{8}$
Q7 3 marks Geometric Sequences and Series Finite Geometric Sum and Term Relationships View
For a geometric sequence $\left\{ a _ { n } \right\}$ with all positive terms, $$\frac { a _ { 1 } a _ { 2 } } { a _ { 3 } } = 2 , \quad \frac { 2a _ { 2 } } { a _ { 1 } } + \frac { a _ { 4 } } { a _ { 2 } } = 8$$ what is the value of $a _ { 3 }$? [3 points]
(1) 16
(2) 18
(3) 20
(4) 22
(5) 24
Q8 3 marks Conditional Probability Direct Conditional Probability Computation from Definitions View
For two events $A, B$, $$\mathrm{P}(A \cap B) = \frac{1}{8}, \quad \mathrm{P}\left(B^{C} \mid A\right) = 2\mathrm{P}(B \mid A)$$ what is the value of $\mathrm{P}(A)$? (Here, $B^{C}$ is the complement of $B$.) [3 points]
(1) $\frac{5}{12}$
(2) $\frac{3}{8}$
(3) $\frac{1}{3}$
(4) $\frac{7}{24}$
(5) $\frac{1}{4}$
Q9 Matrices Linear System and Inverse Existence View
For the system of equations in $x, y$: $$\left( \begin{array} { c c } a + 1 & a \\ 1 & 1 \end{array} \right) \binom{x}{y} = \binom{-4}{1}$$ the solution satisfies the equation $x + 2y - 4a = 0$. What is the value of the constant $a$?
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5
Q10 3 marks Binomial Distribution Find Parameters from Moment Conditions View
A random variable $X$ follows a binomial distribution $\mathrm{B}(n, p)$. If the mean and standard deviation of the random variable $2X - 5$ are 175 and 12, respectively, what is the value of $n$? [3 points]
(1) 130
(2) 135
(3) 140
(4) 145
(5) 150
Q11 3 marks Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View
For the function $f(x) = x + 1$, $$\int _ { -1 } ^ { 1 } \{ f(x) \} ^ { 2 } dx = k \left( \int _ { -1 } ^ { 1 } f(x) dx \right) ^ { 2 }$$ what is the value of the constant $k$? [3 points]
(1) $\frac{1}{6}$
(2) $\frac{1}{3}$
(3) $\frac{1}{2}$
(4) $\frac{2}{3}$
(5) $\frac{5}{6}$
Q12 3 marks Combinations & Selection Distribution of Objects to Positions or Containers View
In how many ways can 4 bottles of the same type of juice, 2 bottles of the same type of water, and 1 bottle of milk be distributed to 3 people without remainder? (Note: Some people may not receive any bottles.) [3 points]
(1) 330
(2) 315
(3) 300
(4) 285
(5) 270
Q13 3 marks Normal Distribution Direct Probability Calculation from Given Normal Distribution View
The test scores of all students at a certain school follow a normal distribution with mean 500 and standard deviation 25. When one student is randomly selected from this school, what is the probability that the student's test score is at least 475 and at most 550, using the standard normal distribution table below? [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) 0.8185
(3) 0.9104
(4) 0.9270
(5) 0.9710
Q14 4 marks Geometric Sequences and Series Fractal/Iterative Geometric Construction (Area, Length, or Perimeter Series) View
As shown in the figure, there is a circle O with diameter AB of length 2. Let C be one of the two points where the line passing through the center of circle O and perpendicular to segment AB meets the circle. The figure obtained by shading the region that is outside the circle centered at C passing through points A and B and inside circle O is called $R_1$. In figure $R_1$, circles are inscribed in each of the 2 quarter-circles obtained by bisecting the semicircle of circle O that does not include the shaded part. In these 2 circles, 2 triangular figures are created using the same method as for figure $R_1$ and shaded to obtain figure $R_2$. In figure $R_2$, circles are inscribed in each of the 4 quarter-circles obtained by bisecting the semicircles of the 2 newly created circles that do not include the shaded parts. In these 4 circles, 4 triangular figures are created using the same method as for figure $R_1$ and shaded to obtain figure $R_3$. Continuing this process, let $S_n$ be the area of the shaded part in the $n$-th figure $R_n$. What is the value of $\lim_{n \rightarrow \infty} S_n$? [4 points]
(1) $\frac{5 + 2\sqrt{2}}{7}$
(2) $\frac{5 + 3\sqrt{2}}{7}$
(3) $\frac{5 + 4\sqrt{2}}{7}$
(4) $\frac{5 + 5\sqrt{2}}{7}$
(5) $\frac{5 + 6\sqrt{2}}{7}$
Q15 4 marks Tangents, normals and gradients Determine unknown parameters from tangent conditions View
The equation of the tangent line to the graph of the cubic function $f(x) = x^3 + ax^2 + 9x + 3$ at the point $(1, f(1))$ is $y = 2x + b$. What is the value of $a + b$? (Here, $a, b$ are constants.) [4 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5
Q16 4 marks Matrices Matrix Algebra and Product Properties View
Two $2 \times 2$ square matrices $A, B$ satisfy $$2A^2 + AB = E, \quad AB + BA = 2A + E$$ Which of the following statements are correct? Choose all that apply from $\langle$Remarks$\rangle$. (Here, $E$ is the identity matrix.) [4 points]
Remarks ᄀ. $A^{-1} = 2A + B$ ㄴ. $B = 2A + 2E$ ㄷ. $(B - E)^2 = O$ (Here, $O$ is the zero matrix.)
(1) ㄴ
(2) ㄷ
(3) ᄀ, ㄴ
(4) ᄀ, ㄷ
(5) ᄀ, ㄴ, ㄷ
Q17 4 marks Sequences and Series Recurrence Relations and Sequence Properties View
The sequence $\left\{ a_n \right\}$ satisfies $a_1 = 4$ and $$a_{n+1} = n \cdot 2^n + \sum_{k=1}^{n} \frac{a_k}{k} \quad (n \geq 1)$$ The following is the process of finding the general term $a_n$.
From the given equation, $$a_n = (n-1) \cdot 2^{n-1} + \sum_{k=1}^{n-1} \frac{a_k}{k} \quad (n \geq 2)$$ Therefore, for natural numbers $n \geq 2$, $$a_{n+1} - a_n = \text{(가)} + \frac{a_n}{n}$$ so $$a_{n+1} = \frac{(n+1)a_n}{n} + \text{(가)}$$ If $b_n = \frac{a_n}{n}$, then $$b_{n+1} = b_n + \frac{(\text{가})}{n+1} \quad (n \geq 2)$$ and since $b_2 = 3$, $$b_n = \text{(나)} \quad (n \geq 2)$$ Therefore, $$a_n = \begin{cases} 4 & (n = 1) \\ n \times (\text{나}) & (n \geq 2) \end{cases}$$ If the expressions for (가) and (나) are $f(n)$ and $g(n)$, respectively, what is the value of $f(4) + g(7)$? [4 points]
(1) 90
(2) 95
(3) 100
(4) 105
(5) 110
Q18 4 marks Implicit equations and differentiation Piecewise differentiability and continuity conditions View
The function $$f(x) = \begin{cases} x^3 + ax & (x < 1) \\ bx^2 + x + 1 & (x \geq 1) \end{cases}$$ is differentiable at $x = 1$. What is the value of $a + b$? (Here, $a, b$ are constants.) [4 points]
(1) 5
(2) 6
(3) 7
(4) 8
(5) 9
Q19 4 marks Sequences and series, recurrence and convergence Convergence proof and limit determination View
For the sequence $\left\{ a_n \right\}$, $$\sum_{n=1}^{\infty} \left( na_n - \frac{n^2 + 1}{2n + 1} \right) = 3$$ What is the value of $\lim_{n \rightarrow \infty} \left( a_n^2 + 2a_n + 2 \right)$? [4 points]
(1) $\frac{13}{4}$
(2) 3
(3) $\frac{11}{4}$
(4) $\frac{5}{2}$
(5) $\frac{9}{4}$
Q20 4 marks Curve Sketching Multi-Statement Verification (Remarks/Options) View
For the two functions $$f(x) = \begin{cases} -1 & (|x| \geq 1) \\ 1 & (|x| < 1) \end{cases}, \quad g(x) = \begin{cases} 1 & (|x| \geq 1) \\ -x & (|x| < 1) \end{cases}$$ which of the following statements are correct? Choose all that apply from $\langle$Remarks$\rangle$. [4 points]
Remarks ᄀ. $\lim_{x \rightarrow 1} f(x)g(x) = -1$ ㄴ. The function $g(x+1)$ is continuous at $x = 0$. ㄷ. The function $f(x)g(x+1)$ is continuous at $x = -1$.
(1) ᄀ
(2) ㄴ
(3) ᄀ, ㄴ
(4) ᄀ, ㄷ
(5) ᄀ, ㄴ, ㄷ
Q21 4 marks Indefinite & Definite Integrals Accumulation Function Analysis View
For the cubic function $f(x) = x^3 - 3x + a$, the function $$F(x) = \int_{0}^{x} f(t)\, dt$$ has exactly one extremum. What is the minimum value of the positive number $a$? [4 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5
Q22 3 marks Composite & Inverse Functions Evaluate Composition from Algebraic Definitions View
Find the value of $\lim_{x \rightarrow 2} \frac{(x-2)(x+3)}{x-2}$. [3 points]
Q23 3 marks Arithmetic Sequences and Series Find Specific Term from Given Conditions View
For an arithmetic sequence $\left\{ a_n \right\}$, $$a_2 = 16, \quad a_5 = 10$$ Find the value of $k$ that satisfies $a_k = 0$. [3 points]
Q24 3 marks Differentiation from First Principles View
For the function $f(x) = x^3 + 9x + 2$, find the value of $\lim_{x \rightarrow 1} \frac{f(x) - f(1)}{x - 1}$. [3 points]
Q25 3 marks Confidence intervals Find a specific bound or margin of error from the CI formula View
The lifespan of monitors produced by a certain company follows a normal distribution. From a random sample of 100 monitors produced by this company, the sample mean is $\bar{x}$ and the sample standard deviation is 500. Using this result, the confidence interval for the mean lifespan of monitors produced by this company at a confidence level of $95\%$ is $[\bar{x} - c, \bar{x} + c]$. Find the value of $c$. (Here, $Z$ is a random variable following the standard normal distribution, and $\mathrm{P}(0 \leq Z \leq 1.96) = 0.4750$.) [3 points]
Q26 4 marks Indices and Surds Number-Theoretic Reasoning with Indices View
For natural numbers $n$ with $2 \leq n \leq 100$, find the number of values of $n$ such that $\left( \sqrt[3]{3^5} \right)^{\frac{1}{2}}$ is the $n$-th root of some natural number. [4 points]
Q27 4 marks Arithmetic Sequences and Series Sequence Defined by Recurrence with AP Connection View
For natural numbers $n$, the point $\mathrm{P}_n$ on the coordinate plane is determined according to the following rules. (가) The coordinates of the three points $\mathrm{P}_1, \mathrm{P}_2, \mathrm{P}_3$ are $(-1, 0)$, $(1, 0)$, and $(-1, 2)$, respectively. (나) The midpoint of segment $\mathrm{P}_n \mathrm{P}_{n+1}$ and the midpoint of segment $\mathrm{P}_{n+2} \mathrm{P}_{n+3}$ are the same. For example, the coordinates of point $\mathrm{P}_4$ are $(1, -2)$. If the coordinates of point $\mathrm{P}_{25}$ are $(a, b)$, find the value of $a + b$. [4 points]
Q28 4 marks Areas by integration View
A quadratic function $f(x)$ with leading coefficient 1 satisfies $f(3) = 0$ and $$\int_{0}^{2013} f(x)\, dx = \int_{3}^{2013} f(x)\, dx$$ If the area enclosed by the curve $y = f(x)$ and the $x$-axis is $S$, find the value of $30S$. [4 points]
Q29 4 marks Combinations & Selection Combinatorial Probability View
In the following seating chart, 4 female students and 4 male students are randomly assigned to 8 seats excluding the seat at row 2, column 2, with one person per seat. Find the value of $70p$, where $p$ is the probability that at least 2 male students are seated adjacent to each other. (Two people are considered adjacent if they are next to each other in the same row or directly in front or behind each other in the same column.) [4 points]
Q30 4 marks Laws of Logarithms Logarithmic Function Graph Intersection or Geometric Analysis View
On the coordinate plane, for natural numbers $n$, consider the region $$\left\{ (x, y) \mid 2^x - n \leq y \leq \log_2(x + n) \right\}$$ Let $a_n$ be the number of points in this region satisfying the following conditions. (가) The $x$-coordinate and $y$-coordinate are equal. (나) Both the $x$-coordinate and $y$-coordinate are integers. For example, $a_1 = 2, a_2 = 4$. Find the value of $\sum_{n=1}^{30} a_n$. [4 points]