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Papers (39)
2026
csat__math 30
2025
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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
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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
2018 csat__math-humanities

30 maths questions

Q1 2 marks Indices and Surds Evaluating Expressions Using Index Laws View
Find the value of $2 \times 16 ^ { \frac { 1 } { 4 } }$. [2 points]
(1) 2
(2) 4
(3) 6
(4) 8
(5) 10
Q2 2 marks Sequences and series, recurrence and convergence Multiple-choice on sequence properties View
Find the value of $\lim _ { n \rightarrow \infty } \frac { 5 ^ { n } - 3 } { 5 ^ { n + 1 } }$. [2 points]
(1) $\frac { 1 } { 5 }$
(2) $\frac { 1 } { 4 }$
(3) $\frac { 1 } { 3 }$
(4) $\frac { 1 } { 2 }$
(5) 1
Q3 2 marks Probability Definitions Set Operations View
Two sets $A = \{ 2 , a + 1,5 \} , B = \{ 2,3 , b \}$ satisfy $A = B$. Find the value of $a + b$. (Here, $a$ and $b$ are real numbers.) [2 points]
(1) 4
(2) 5
(3) 6
(4) 7
(5) 8
Q4 3 marks Composite & Inverse Functions Evaluate Composition from Diagram or Mapping View
The figure shows two functions $f : X \rightarrow Y , g : Y \rightarrow Z$. Find the value of $( g \circ f ) ( 2 )$. [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5
Q5 3 marks Curve Sketching Limit Reading from Graph View
The graph of the function $y = f ( x )$ is shown in the figure. Find the value of $\lim _ { x \rightarrow 0 - } f ( x ) + \lim _ { x \rightarrow 1 + } f ( x )$. [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5
Q6 3 marks Conditional Probability Direct Conditional Probability Computation from Definitions View
A survey was conducted on 500 students at a high school regarding their desire to visit regions A and B for cultural exploration. The results are as follows. (Unit: students)
Region BWishDo not wishTotal
Wish140310450
Do not wish401050
Total180320500

When one student is randomly selected from this high school and is found to wish to visit region A, what is the probability that this student also wishes to visit region B? [3 points]
(1) $\frac { 19 } { 45 }$
(2) $\frac { 23 } { 45 }$
(3) $\frac { 3 } { 5 }$
(4) $\frac { 31 } { 45 }$
(5) $\frac { 7 } { 9 }$
Q7 3 marks Inequalities Sufficient/Necessary Conditions Between Inequality Conditions View
For two conditions on real number $x$: $$\begin{aligned} & p : ( x - 1 ) ( x - 4 ) = 0 , \\ & q : 1 < 2 x \leq a \end{aligned}$$ Find the minimum value of the natural number $a$ such that $p$ is a sufficient condition for $q$. [3 points]
(1) 4
(2) 5
(3) 6
(4) 7
(5) 8
Q8 3 marks Arithmetic Sequences and Series Counting or Combinatorial Problems on APs View
Find the number of ways to partition the natural number 11 into natural numbers between 3 and 7 (inclusive). [3 points]
(1) 2
(2) 4
(3) 6
(4) 8
(5) 10
Q9 3 marks Indefinite & Definite Integrals Maximizing or Optimizing a Definite Integral View
Find the positive value of $a$ that satisfies $\int _ { 0 } ^ { a } \left( 3 x ^ { 2 } - 4 \right) d x = 0$. [3 points]
(1) 2
(2) $\frac { 9 } { 4 }$
(3) $\frac { 5 } { 2 }$
(4) $\frac { 11 } { 4 }$
(5) 3
Q10 3 marks Independent Events View
Two events $A$ and $B$ are independent, and $$\mathrm { P } ( A ) = \frac { 2 } { 3 } , \quad \mathrm { P } ( A \cup B ) = \frac { 5 } { 6 }$$ Find the value of $\mathrm { P } ( B )$. [3 points]
(1) $\frac { 1 } { 3 }$
(2) $\frac { 5 } { 12 }$
(3) $\frac { 1 } { 2 }$
(4) $\frac { 7 } { 12 }$
(5) $\frac { 2 } { 3 }$
Q11 3 marks Curve Sketching Lattice Points and Counting via Graph Geometry View
In the coordinate plane, find the number of points with both natural number coordinates that are contained in the interior of the region enclosed by the curve $y = \frac { 1 } { 2 x - 8 } + 3$ and the $x$-axis and $y$-axis. [3 points]
(1) 3
(2) 4
(3) 5
(4) 6
(5) 7
Q12 3 marks Binomial Theorem (positive integer n) Find a Specific Coefficient in a Single Binomial Expansion View
In the expansion of $\left( x + \frac { 2 } { x } \right) ^ { 8 }$, find the coefficient of $x ^ { 4 }$. [3 points]
(1) 128
(2) 124
(3) 120
(4) 116
(5) 112
Q13 3 marks Sequences and series, recurrence and convergence Direct term computation from recurrence View
The sequence $\left\{ a _ { n } \right\}$ has $a _ { 1 } = 2$, and for all natural numbers $n$, $$a _ { n + 1 } = \begin{cases} a _ { n } - 1 & \text{(when } a _ { n } \text{ is even)} \\ a _ { n } + n & \text{(when } a _ { n } \text{ is odd)} \end{cases}$$ Find the value of $a _ { 7 }$. [3 points]
(1) 7
(2) 9
(3) 11
(4) 13
(5) 15
Q14 4 marks Arithmetic Sequences and Series Find Specific Term from Given Conditions View
An arithmetic sequence $\left\{ a _ { n } \right\}$ satisfies $$a _ { 5 } + a _ { 13 } = 3 a _ { 9 } , \quad \sum _ { k = 1 } ^ { 18 } a _ { k } = \frac { 9 } { 2 }$$ Find the value of $a _ { 13 }$. [4 points]
(1) 2
(2) 1
(3) 0
(4) $-1$
(5) $-2$
Q15 4 marks Normal Distribution Sampling Distribution of the Mean View
The content volume of a cosmetic product produced by a factory follows a normal distribution with mean 201.5 g and standard deviation 1.8 g. Find the probability that the sample mean of 9 randomly selected cosmetic products from this factory is at least 200 g using the standard normal distribution table on the right. [4 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) 0.8413
(3) 0.9332
(4) 0.9772
(5) 0.9938
Q16 4 marks Laws of Logarithms Solve a Logarithmic Equation View
For two real numbers $a , b$ greater than 1, $$\log _ { \sqrt { 3 } } a = \log _ { 9 } a b$$ holds. Find the value of $\log _ { a } b$. [4 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5
Q17 4 marks Discrete Probability Distributions Probability Distribution Table Completion and Expectation Calculation View
The probability distribution of the random variable $X$ is shown in the table below.
$X$0.1210.2210.321Total
$\mathrm { P } ( X = x )$$a$$b$$\frac { 2 } { 3 }$1

The following is the process of finding $\mathrm { V } ( X )$ when $\mathrm { E } ( X ) = 0.271$. Let $Y = 10 X - 2.21$. The probability distribution of the random variable $Y$ is shown in the table below.
$Y$$-1$01Total
$\mathrm { P } ( Y = y )$$a$$b$$\frac { 2 } { 3 }$1

Since $\mathrm { E } ( Y ) = 10 \mathrm { E } ( X ) - 2.21 = 0.5$, $a =$ (가), $b =$ (나) and $\mathrm { V } ( Y ) = \frac { 7 } { 12 }$. On the other hand, since $Y = 10 X - 2.21$, we have $\mathrm { V } ( Y ) =$ (다) $\times \mathrm { V } ( X )$. Therefore, $\mathrm { V } ( X ) = \frac { 1 } { \text{(다)} } \times \frac { 7 } { 12 }$. When the values in (가), (나), and (다) are $p$, $q$, and $r$ respectively, find the value of $pqr$. (Here, $a$ and $b$ are constants.) [4 points]
(1) $\frac { 13 } { 9 }$
(2) $\frac { 16 } { 9 }$
(3) $\frac { 19 } { 9 }$
(4) $\frac { 22 } { 9 }$
(5) $\frac { 25 } { 9 }$
Q18 4 marks Stationary points and optimisation Determine parameters from given extremum conditions View
A cubic function $f ( x )$ with leading coefficient 1 and $f ( 1 ) = 0$ satisfies $$\lim _ { x \rightarrow 2 } \frac { f ( x ) } { ( x - 2 ) \left\{ f ^ { \prime } ( x ) \right\} ^ { 2 } } = \frac { 1 } { 4 }$$ Find the value of $f ( 3 )$. [4 points]
(1) 4
(2) 6
(3) 8
(4) 10
(5) 12
Q19 4 marks Radians, Arc Length and Sector Area View
As shown in the figure, there is an equilateral triangle $\mathrm { A } _ { 1 } \mathrm {~B} _ { 1 } \mathrm { C } _ { 1 }$ with side length 1. Let $\mathrm { D } _ { 1 }$ be the midpoint of segment $\mathrm { A } _ { 1 } \mathrm {~B} _ { 1 }$, and let $\mathrm { B } _ { 2 }$ be a point on segment $\mathrm { B } _ { 1 } \mathrm { C } _ { 1 }$ such that $\overline { \mathrm { C } _ { 1 } \mathrm { D } _ { 1 } } = \overline { \mathrm { C } _ { 1 } \mathrm {~B} _ { 2 } }$. Draw a sector $\mathrm { C } _ { 1 } \mathrm { D } _ { 1 } \mathrm {~B} _ { 2 }$ with center $\mathrm { C } _ { 1 }$. Let $\mathrm { A } _ { 2 }$ be the foot of the perpendicular from $\mathrm { B } _ { 2 }$ to segment $\mathrm { C } _ { 1 } \mathrm { D } _ { 1 }$, and let $\mathrm { C } _ { 2 }$ be the midpoint of segment $\mathrm { C } _ { 1 } \mathrm {~B} _ { 2 }$. The figure $R _ { 1 }$ is obtained by shading the region enclosed by two segments $\mathrm { B } _ { 1 } \mathrm {~B} _ { 2 } , \mathrm {~B} _ { 1 } \mathrm { D } _ { 1 }$ and arc $\mathrm { D } _ { 1 } \mathrm {~B} _ { 2 }$, and the interior of triangle $\mathrm { C } _ { 1 } \mathrm {~A} _ { 2 } \mathrm { C } _ { 2 }$. In figure $R _ { 1 }$, let $\mathrm { D } _ { 2 }$ be the midpoint of segment $\mathrm { A } _ { 2 } \mathrm {~B} _ { 2 }$, and let $\mathrm { B } _ { 3 }$ be a point on segment $\mathrm { B } _ { 2 } \mathrm { C } _ { 2 }$ such that $\overline { \mathrm { C } _ { 2 } \mathrm { D } _ { 2 } } = \overline { \mathrm { C } _ { 2 } \mathrm {~B} _ { 3 } }$. Draw a sector $\mathrm { C } _ { 2 } \mathrm { D } _ { 2 } \mathrm {~B} _ { 3 }$ with center $\mathrm { C } _ { 2 }$. Let $\mathrm { A } _ { 3 }$ be the foot of the perpendicular from $\mathrm { B } _ { 3 }$ to segment $\mathrm { C } _ { 2 } \mathrm { D } _ { 2 }$, and let $\mathrm { C } _ { 3 }$ be the midpoint of segment $\mathrm { C } _ { 2 } \mathrm {~B} _ { 3 }$. The figure $R _ { 2 }$ is obtained by shading the region enclosed by two segments $\mathrm { B } _ { 2 } \mathrm {~B} _ { 3 }$, $\mathrm { B } _ { 2 } \mathrm { D } _ { 2 }$ and arc $\mathrm { D } _ { 2 } \mathrm {~B} _ { 3 }$, and the interior of triangle $\mathrm { C } _ { 2 } \mathrm {~A} _ { 3 } \mathrm { C } _ { 3 }$. Continuing this process, let $S _ { n }$ be the area of the shaded part in the $n$-th figure $R _ { n }$. Find the value of $\lim _ { n \rightarrow \infty } S _ { n }$. [4 points]
(1) $\frac { 11 \sqrt { 3 } - 4 \pi } { 56 }$
(2) $\frac { 11 \sqrt { 3 } - 4 \pi } { 52 }$
(3) $\frac { 15 \sqrt { 3 } - 6 \pi } { 56 }$
(4) $\frac { 15 \sqrt { 3 } - 6 \pi } { 52 }$
(5) $\frac { 15 \sqrt { 3 } - 4 \pi } { 52 }$
Q20 4 marks Stationary points and optimisation Find critical points and classify extrema of a given function View
A quartic function $f ( x )$ with leading coefficient 1 satisfies the following conditions. (가) $f ^ { \prime } ( 0 ) = 0 , f ^ { \prime } ( 2 ) = 16$ (나) For some positive number $k$, $f ^ { \prime } ( x ) < 0$ on the two open intervals $( - \infty , 0 ) , ( 0 , k )$. Choose all correct statements from the following. [4 points]
$\langle$Statements$\rangle$ ㄱ. The equation $f ^ { \prime } ( x ) = 0$ has exactly one real root in the open interval $( 0,2 )$. ㄴ. The function $f ( x )$ has a local maximum value. ㄷ. If $f ( 0 ) = 0$, then $f ( x ) \geq - \frac { 1 } { 3 }$ for all real numbers $x$.
(1) ㄱ
(2) ㄴ
(3) ㄱ, ㄷ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ
Q21 4 marks Composite & Inverse Functions Evaluate Composition from Algebraic Definitions View
As shown in the figure, the graph of the function $f ( x )$ defined on the closed interval $[ 0,4 ]$ is formed by connecting the points $( 0,0 ) , ( 1,4 ) , ( 2,1 ) , ( 3,4 ) , ( 4,3 )$ in order with line segments. Find the number of sets $X = \{ a , b \}$ satisfying the following condition. (Here, $0 \leq a < b \leq 4$) [4 points]
A function $g ( x ) = f ( f ( x ) )$ from $X$ to $X$ exists and satisfies $g ( a ) = f ( a ) , g ( b ) = f ( b )$.
(1) 11
(2) 13
(3) 15
(4) 17
(5) 19
Q22 3 marks Combinations & Selection Basic Combination Computation View
Find the value of ${}_{5}\mathrm{C}_{3}$. [3 points]
Q23 3 marks Chain Rule Straightforward Polynomial or Basic Differentiation View
For the function $f ( x ) = 2 x ^ { 3 } + x + 1$, find the value of $f ^ { \prime } ( 1 )$. [3 points]
Q24 3 marks Principle of Inclusion/Exclusion View
For the universal set $U = \{ 1,2,3,4,5,6,7,8 \}$ and two subsets $$A = \{ 1,2,3 \} , \quad B = \{ 2,4,6,8 \}$$ Find the value of $n \left( A \cup B ^ { C } \right)$. [3 points]
Q25 3 marks Sequences and series, recurrence and convergence Convergence proof and limit determination View
A function $f ( x )$ satisfies $\lim _ { x \rightarrow 1 } ( x + 1 ) f ( x ) = 1$. When $\lim _ { x \rightarrow 1 } \left( 2 x ^ { 2 } + 1 \right) f ( x ) = a$, find the value of $20 a$. [3 points]
Q26 4 marks Areas Between Curves Compute Area Directly (Numerical Answer) View
The area enclosed by the curve $y = - 2 x ^ { 2 } + 3 x$ and the line $y = x$ is $\frac { q } { p }$. Find the value of $p + q$. (Here, $p$ and $q$ are coprime natural numbers.) [4 points]
Q27 4 marks Sequences and Series Evaluation of a Finite or Infinite Sum View
For the sequence $\left\{ a _ { n } \right\}$, $$\sum _ { k = 1 } ^ { 10 } \left( a _ { k } + 1 \right) ^ { 2 } = 28 , \sum _ { k = 1 } ^ { 10 } a _ { k } \left( a _ { k } + 1 \right) = 16$$ Find the value of $\sum _ { k = 1 } ^ { 10 } \left( a _ { k } \right) ^ { 2 }$. [4 points]
Q28 4 marks Binomial Distribution Compute Cumulative or Complement Binomial Probability View
When a coin is tossed 6 times, the probability that the number of heads is greater than the number of tails is $\frac { q } { p }$. Find the value of $p + q$. (Here, $p$ and $q$ are coprime natural numbers.) [4 points]
Q29 4 marks Stationary points and optimisation Composite or piecewise function extremum analysis View
For two real numbers $a$ and $k$, two functions $f ( x )$ and $g ( x )$ are defined as $$\begin{aligned} & f ( x ) = \begin{cases} 0 & ( x \leq a ) \\ ( x - 1 ) ^ { 2 } ( 2 x + 1 ) & ( x > a ) \end{cases} \\ & g ( x ) = \begin{cases} 0 & ( x \leq k ) \\ 12 ( x - k ) & ( x > k ) \end{cases} \end{aligned}$$ and satisfy the following conditions. (가) The function $f ( x )$ is differentiable on the entire set of real numbers. (나) For all real numbers $x$, $f ( x ) \geq g ( x )$. When the minimum value of $k$ is $\frac { q } { p }$, find the value of $a + p + q$. (Here, $p$ and $q$ are coprime natural numbers.) [4 points]
Q30 4 marks Indefinite & Definite Integrals Piecewise/Periodic Function Integration View
For the quadratic function $f ( x ) = \frac { 3 x - x ^ { 2 } } { 2 }$, a function $g ( x )$ defined on the interval $[ 0 , \infty )$ satisfies the following conditions. (가) When $0 \leq x < 1$, $g ( x ) = f ( x )$. (나) When $n \leq x < n + 1$, $$\begin{aligned} & g ( x ) = \frac { 1 } { 2 ^ { n } } \{ f ( x - n ) - ( x - n ) \} + x \\ & \text{(Here, } n \text{ is a natural number.)} \end{aligned}$$ For some natural number $k ( k \geq 6 )$, the function $h ( x )$ is defined as $$h ( x ) = \begin{cases} g ( x ) & ( 0 \leq x < 5 \text{ or } x \geq k ) \\ 2 x - g ( x ) & ( 5 \leq x < k ) \end{cases}$$ When the sequence $\left\{ a _ { n } \right\}$ is defined by $a _ { n } = \int _ { 0 } ^ { n } h ( x ) d x$ and $\lim _ { n \rightarrow \infty } \left( 2 a _ { n } - n ^ { 2 } \right) = \frac { 241 } { 768 }$, find the value of $k$. [4 points]