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
2012 csat__math-humanities

29 maths questions

Q1 2 marks Matrices Linear System and Inverse Existence View
For the matrix $A = \left( \begin{array} { r r } 1 & - 2 \\ 0 & 1 \end{array} \right)$, what is the sum of all components of the inverse matrix $A ^ { - 1 }$? [2 points]
(1) 5
(2) 4
(3) 3
(4) 2
(5) 1
Q2 2 marks Sequences and series, recurrence and convergence Multiple-choice on sequence properties View
What is the value of $\lim _ { n \rightarrow \infty } \frac { 5 ^ { n + 1 } + 2 } { 5 ^ { n } + 3 ^ { n } }$? [2 points]
(1) 2
(2) 3
(3) 4
(4) 5
(5) 6
Q3 2 marks Differentiation from First Principles View
For the function $f ( x ) = x ^ { 2 } + 5$, what is the value of $\lim _ { h \rightarrow 0 } \frac { f ( 1 + h ) - f ( 1 ) } { h }$? [2 points]
(1) 2
(2) 3
(3) 4
(4) 5
(5) 6
Q5 3 marks Arithmetic Sequences and Series Sequence Defined by Recurrence with AP Connection View
For a sequence $\left\{ a _ { n } \right\}$ with $a _ { 1 } = 1$ and satisfying
$$a _ { n + 1 } = \frac { 2 n } { n + 1 } a _ { n }$$
for all natural numbers $n$, what is the value of $a _ { 4 }$? [3 points]
(1) $\frac { 3 } { 2 }$
(2) 2
(3) $\frac { 5 } { 2 }$
(4) 3
(5) $\frac { 7 } { 2 }$
Q6 3 marks Discrete Probability Distributions Probability Distribution Table Completion and Expectation Calculation View
The probability distribution of a random variable $X$ is shown in the table below.
$X$012Total
$\mathrm { P } ( X = x )$$\frac { 1 } { 4 }$$a$$2a$1

What is the value of $\mathrm { E } ( 4X + 10 )$? [3 points]
(1) 11
(2) 12
(3) 13
(4) 14
(5) 15
Q7 3 marks Laws of Logarithms Logarithmic Formula Application (Modeling) View
The female silkworm moth secretes pheromone to attract males. When $t$ seconds have passed since the female silkworm moth secreted pheromone, the concentration $y$ of pheromone measured at a distance $x$ from the secretion site satisfies the following equation.
$$\log y = A - \frac { 1 } { 2 } \log t - \frac { K x ^ { 2 } } { t } \text { (where } A \text { and } K \text { are positive constants.) }$$
When 1 second has passed since the female silkworm moth secreted pheromone, the pheromone concentration measured at a distance of 2 from the secretion site is $a$, and when 4 seconds have passed, the pheromone concentration measured at a distance of $d$ from the secretion site is $\frac { a } { 2 }$. What is the value of $d$? [3 points]
(1) 7
(2) 6
(3) 5
(4) 4
(5) 3
Q8 3 marks Binomial Theorem (positive integer n) Determine Parameters from Conditions on Coefficients or Terms View
In the expansion of the polynomial $( x + a ) ^ { 7 }$, when the coefficient of $x ^ { 4 }$ is 280, what is the coefficient of $x ^ { 5 }$? (where $a$ is a constant) [3 points]
(1) 84
(2) 91
(3) 98
(4) 105
(5) 112
Q9 3 marks Indefinite & Definite Integrals Finding a Function from an Integral Equation View
For the function $F ( x ) = \int _ { 0 } ^ { x } \left( t ^ { 3 } - 1 \right) d t$, what is the value of $F ^ { \prime } ( 2 )$? [3 points]
(1) 11
(2) 9
(3) 7
(4) 5
(5) 3
Q10 3 marks Independent Events View
Two events $A$ and $B$ are independent, and
$$\mathrm { P } ( A \cup B ) = \frac { 1 } { 2 } , \quad \mathrm { P } ( A \mid B ) = \frac { 3 } { 8 }$$
What is the value of $\mathrm { P } \left( A \cap B ^ { C } \right)$? (where $B ^ { C }$ is the complement of $B$) [3 points]
(1) $\frac { 1 } { 10 }$
(2) $\frac { 3 } { 20 }$
(3) $\frac { 1 } { 5 }$
(4) $\frac { 1 } { 4 }$
(5) $\frac { 3 } { 10 }$
Q11 3 marks Arithmetic Sequences and Series Compute Partial Sum of an Arithmetic Sequence View
For an arithmetic sequence $\left\{ a _ { n } \right\}$ with first term $- 5$ and common difference 2, what is the value of $\sum _ { k = 11 } ^ { 20 } a _ { k }$? [3 points]
(1) 260
(2) 255
(3) 250
(4) 245
(5) 240
Q12 3 marks Stationary points and optimisation Geometric or applied optimisation problem View
As shown in the figure, there are two points $\mathrm { A } ( - 1,0 )$ and $\mathrm { P } ( t , t + 1 )$ on the line $y = x + 1$. Let Q be the point where the line passing through P and perpendicular to the line $y = x + 1$ meets the $y$-axis. What is the value of $\lim _ { t \rightarrow \infty } \frac { \overline { \mathrm { AQ } } ^ { 2 } } { \overline { \mathrm { AP } } ^ { 2 } }$? [3 points]
(1) 1
(2) $\frac { 3 } { 2 }$
(3) 2
(4) $\frac { 5 } { 2 }$
(5) 3
Q13 3 marks Conditional Probability Bayes' Theorem with Production/Source Identification View
Bag A contains 5 cards with the numbers $1,2,3,4,5$ written on them, and Bag B contains 6 cards with the numbers $1,2,3,4,5,6$ written on them. A die is rolled once. If the result is a multiple of 3, a card is randomly drawn from Bag A; otherwise, a card is randomly drawn from Bag B. Given that the number on the card drawn from the bag is even, what is the probability that the card was drawn from Bag A? [3 points]
(1) $\frac { 1 } { 5 }$
(2) $\frac { 2 } { 9 }$
(3) $\frac { 1 } { 4 }$
(4) $\frac { 2 } { 7 }$
(5) $\frac { 1 } { 3 }$
Q14 4 marks Geometric Sequences and Series Fractal/Iterative Geometric Construction (Area, Length, or Perimeter Series) View
There is a circle with radius 1. As shown in the figure, a rectangle with the ratio of width to height of $3 : 1$ is inscribed in this circle, and the common part of the interior of the circle and the exterior of the rectangle is colored to obtain the figure $R _ { 1 }$. In figure $R _ { 1 }$, 2 circles are drawn tangent to three sides of the rectangle. A rectangle is drawn in each of the newly drawn circles using the same method as obtaining figure $R _ { 1 }$, and the colored figure obtained is $R _ { 2 }$. In figure $R _ { 2 }$, 4 circles are drawn tangent to three sides of the newly drawn rectangle. A rectangle is drawn in each of the newly drawn circles using the same method as obtaining figure $R _ { 1 }$, and the colored figure obtained is $R _ { 3 }$. Continuing this process, let $S _ { n }$ be the area of the colored part in the $n$-th obtained figure $R _ { n }$. What is the value of $\lim _ { n \rightarrow \infty } S _ { n }$? [4 points]
(1) $\frac { 5 } { 4 } \pi - \frac { 5 } { 3 }$
(2) $\frac { 5 } { 4 } \pi - \frac { 3 } { 2 }$
(3) $\frac { 4 } { 3 } \pi - \frac { 8 } { 5 }$
(4) $\frac { 5 } { 4 } \pi - 1$
(5) $\frac { 4 } { 3 } \pi - \frac { 16 } { 15 }$
Q15 4 marks Matrices True/False or Multiple-Select Conceptual Reasoning View
Two $2 \times 2$ square matrices $A , B$ satisfy
$$A ^ { 2 } + B = 3 E , \quad A ^ { 4 } + B ^ { 2 } = 7 E$$
Which of the following statements are correct? (where $E$ is the identity matrix) [4 points]
ㄱ. $A B = B A$ ㄴ. $B ^ { - 1 } = A ^ { 2 }$ ㄷ. $A ^ { 6 } + B ^ { 3 } = 18 E$
(1) ㄱ
(2) ㄴ
(3) ㄱ, ㄴ
(4) ㄱ, ㄷ
(5) ㄱ, ㄴ, ㄷ
Q16 4 marks Normal Distribution Sampling Distribution of the Mean View
The length $X$ of products manufactured at a certain factory follows a normal distribution with mean $m$ and standard deviation 4. When $\mathrm { P } ( m \leq X \leq a ) = 0.3413$, what is the probability that the sample mean of the lengths of 16 products randomly selected from this factory is at least $a - 2$, using the standard normal distribution table on the right? (where $a$ is a constant and the unit of length is cm) [4 points]
$z$$\mathrm { P } ( 0 \leq Z \leq z )$
1.00.3413
1.50.4332
2.00.4772

(1) 0.0228
(2) 0.0668
(3) 0.0919
(4) 0.1359
(5) 0.1587
Q17 Sequences and series, recurrence and convergence Auxiliary sequence transformation View
For a sequence $\left\{ a _ { n } \right\}$ with first term 1, let $S _ { n } = \sum _ { k = 1 } ^ { n } a _ { k }$. The following holds:
$$n S _ { n + 1 } = ( n + 2 ) S _ { n } + ( n + 1 ) ^ { 3 } \quad ( n \geq 1 )$$
The following is part of the process of finding the general term of the sequence $\left\{ a _ { n } \right\}$.
Since $S _ { n + 1 } = S _ { n } + a _ { n + 1 }$ for natural numbers $n$,
$$n a _ { n + 1 } = 2 S _ { n } + ( n + 1 ) ^ { 3 } \quad \cdots (\text{ㄱ})$$
For natural numbers $n \geq 2$,
$$( n - 1 ) a _ { n } = 2 S _ { n - 1 } + n ^ { 3 } \quad \cdots (\text{ㄴ})$$
Subtracting (ㄴ) from (ㄱ), we obtain
$$n a _ { n + 1 } = ( n + 1 ) a _ { n } + \text{ (A) }$$
Dividing both sides by $n ( n + 1 )$,
$$\frac { a _ { n + 1 } } { n + 1 } = \frac { a _ { n } } { n } + \frac { \text{ (A) } } { n ( n + 1 ) }$$
Let $b _ { n } = \frac { a _ { n } } { n }$. Then
$$b _ { n + 1 } = b _ { n } + 3 + \text{ (B) } \quad ( n \geq 2 )$$
Therefore
$$b _ { n } = b _ { 2 } + \text{ (C) } \quad ( n \geq 3 )$$
holds.
What are the correct expressions for (A), (B), and (C)?
Q18 4 marks Composite & Inverse Functions Graphical Interpretation of Inverse or Composition View
When the graph of the function $y = f ( x )$ is as shown in the figure, which of the following statements are correct? [4 points]
ㄱ. $\lim _ { x \rightarrow +0 } f ( x ) = 1$ ㄴ. $\lim _ { x \rightarrow 1 } f ( x ) = f ( 1 )$ ㄷ. The function $( x - 1 ) f ( x )$ is continuous at $x = 1$.
(1) ㄱ
(2) ㄱ, ㄴ
(3) ㄱ, ㄷ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ
Q19 4 marks Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View
A quadratic function $f ( x )$ satisfies $f ( 0 ) = - 1$ and
$$\int _ { - 1 } ^ { 1 } f ( x ) d x = \int _ { 0 } ^ { 1 } f ( x ) d x = \int _ { - 1 } ^ { 0 } f ( x ) d x$$
What is the value of $f ( 2 )$? [4 points]
(1) 11
(2) 10
(3) 9
(4) 8
(5) 7
Q20 4 marks Laws of Logarithms Characteristic and Mantissa of Common Logarithms View
For a positive number $x$, let the characteristic and mantissa of $\log x$ be $f ( x )$ and $g ( x )$, respectively. The number of natural numbers $n$ satisfying the two inequalities
$$f ( n ) \leq f ( 54 ) , \quad g ( n ) \leq g ( 54 )$$
is? [4 points]
(1) 42
(2) 44
(3) 46
(4) 48
(5) 50
Q21 4 marks Curve Sketching Number of Solutions / Roots via Curve Analysis View
A cubic function $f ( x )$ with leading coefficient 1 satisfies $f ( - x ) = - f ( x )$ for all real numbers $x$. When the equation $| f ( x ) | = 2$ has exactly 4 distinct real roots, what is the value of $f ( 3 )$? [4 points]
(1) 12
(2) 14
(3) 16
(4) 18
(5) 20
Q22 3 marks Chain Rule Limit Evaluation Involving Composition or Substitution View
Find the value of $\lim _ { x \rightarrow 1 } \frac { ( x - 1 ) \left( x ^ { 2 } + 3 x + 7 \right) } { x - 1 }$. [3 points]
Q23 3 marks Laws of Logarithms Solve a Logarithmic Equation View
Find the value of $x$ that satisfies the equation $\log _ { 3 } ( x - 11 ) = 3 \log _ { 3 } 2$. [3 points]
Q24 3 marks Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View
Find the value of $\int _ { 0 } ^ { 5 } ( 4 x - 3 ) d x$. [3 points]
Q25 3 marks Arithmetic Sequences and Series Arithmetic-Geometric Hybrid Problem View
Three numbers $a , a + b , 2 a - b$ form an arithmetic sequence in this order, and three numbers $1 , a - 1, 3 b + 1$ form a geometric sequence with positive common ratio in this order. Find the value of $a ^ { 2 } + b ^ { 2 }$. [3 points]
Q26 4 marks Tangents, normals and gradients Find tangent line equation at a given point View
The equation of the tangent line to the curve $y = - x ^ { 3 } + 4 x$ at the point $( 1,3 )$ is $y = a x + b$. Find the value of $10 a + b$. (where $a , b$ are constants) [4 points]
Q27 4 marks Geometric Sequences and Series Sum of an Infinite Geometric Series (Direct Computation) View
On the coordinate plane, for a natural number $n$, the coordinates of point $\mathrm { P } _ { n }$ are $\left( n , 3 ^ { n } \right)$ and the coordinates of point $\mathrm { Q } _ { n }$ are $( n , 0 )$. Let $a _ { n }$ be the area of the quadrilateral $\mathrm { P } _ { n } \mathrm { Q } _ { n + 1 } \mathrm { Q } _ { n + 2 } \mathrm { P } _ { n + 1 }$. When $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { a _ { n } } = \frac { q } { p }$, find the value of $p ^ { 2 } + q ^ { 2 }$. (where $p$ and $q$ are coprime natural numbers) [4 points]
Q28 4 marks Continuous Probability Distributions and Random Variables Change of Variable and Integral Evaluation View
A continuous random variable $X$ defined on the interval $[ 0,1 ]$ has probability density function $f ( x )$. The mean of $X$ is $\frac { 1 } { 4 }$ and $\int _ { 0 } ^ { 1 } ( a x + 5 ) f ( x ) d x = 10$. Find the value of the constant $a$. [4 points]
Q29 4 marks Matrices Linear System and Inverse Existence View
A $2 \times 2$ square matrix $A$ satisfies the following conditions. (where $E$ is the identity matrix and $O$ is the zero matrix)
(A) $A ^ { 2 } + 2 A - E = O$
(B) $A \binom { 1 } { - 1 } = \binom { 3 } { 4 }$ Find the value of $x + y$ for real numbers $x , y$ satisfying $( A + 2 E ) \binom { x } { y } = \binom { 3 } { - 3 }$. [4 points]
Q30 4 marks Exponential Functions Intersection and Distance between Curves View
For natural numbers $a , b$, let P and Q be the points where the curve $y = a ^ { x + 1 }$ and the curve $y = b ^ { x }$ meet the line $x = t ( t \geq 1 )$, respectively. Find the number of all ordered pairs $( a , b )$ of $a , b$ satisfying the following condition. For example, $a = 4 , b = 5$ satisfies the following condition. [4 points]
(A) $2 \leq a \leq 10, 2 \leq b \leq 10$
(B) For some real number $t \geq 1$, $\overline { \mathrm { PQ } } \leq 10$.