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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-humanities

30 maths questions

Q1 2 marks Indices and Surds Evaluating Expressions Using Index Laws View
What is the value of $16 \times 2 ^ { - 3 }$? [2 points]
(1) 1
(2) 2
(3) 4
(4) 8
(5) 16
Q2 2 marks Sequences and series, recurrence and convergence Multiple-choice on sequence properties View
What is the value of $\lim _ { n \rightarrow \infty } \frac { \sqrt { 9 n ^ { 2 } + 4 } } { 5 n - 2 }$? [2 points]
(1) $\frac { 1 } { 5 }$
(2) $\frac { 2 } { 5 }$
(3) $\frac { 3 } { 5 }$
(4) $\frac { 4 } { 5 }$
(5) 1
Q3 2 marks Simultaneous equations View
For two sets $A = \{ a + 2,6 \} , B = \{ 3 , b - 1 \}$, when $A = B$, what is the value of $a + b$? (Note: $a , b$ are real numbers.) [2 points]
(1) 5
(2) 6
(3) 7
(4) 8
(5) 9
Q4 3 marks Composite & Inverse Functions Evaluate Composition from Diagram or Mapping View
The figure shows two functions $f : X \rightarrow X , g : X \rightarrow X$. What is the value of $( g \circ f ) ( 1 )$? [3 points]
(1) 1
(2) 3
(3) 5
(4) 7
(5) 9
Q5 3 marks Principle of Inclusion/Exclusion View
For two events $A , B$, $$\mathrm { P } \left( A ^ { C } \right) = \frac { 2 } { 3 } , \quad \mathrm { P } \left( A ^ { C } \cap B \right) = \frac { 1 } { 4 }$$ What is the value of $\mathrm { P } ( A \cup B )$? (Note: $A ^ { C }$ is the complement of $A$.) [3 points]
(1) $\frac { 1 } { 2 }$
(2) $\frac { 7 } { 12 }$
(3) $\frac { 2 } { 3 }$
(4) $\frac { 3 } { 4 }$
(5) $\frac { 5 } { 6 }$
Q6 3 marks Solving quadratics and applications Sufficient/necessary condition or logical relationship involving a quadratic View
For two conditions on real number $x$: $$\begin{aligned} & p : x = a , \\ & q : 3 x ^ { 2 } - a x - 32 = 0 \end{aligned}$$ What is the value of positive $a$ such that $p$ is a sufficient condition for $q$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5
Q7 3 marks Composite & Inverse Functions Find or Apply an Inverse Function Formula View
For the function $f ( x ) = \frac { k } { x - 3 } + 1$, when $f ^ { - 1 } ( 7 ) = 4$, what is the value of the constant $k$? (Note: $k \neq 0$) [3 points]
(1) 2
(2) 4
(3) 6
(4) 8
(5) 10
Q8 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 $\lim _ { x \rightarrow 0 + } f ( x ) - \lim _ { x \rightarrow 1 - } f ( x )$? [3 points]
(1) $-2$
(2) $-1$
(3) 0
(4) 1
(5) 2
Q9 3 marks Conditional Probability Direct Conditional Probability Computation from Definitions View
A survey was conducted on 200 students at a school regarding their preferences for experiential activities. The students who participated in this survey chose one of cultural experience or ecological research, and the number of students who chose each activity is as follows.
ClassificationCultural ExperienceEcological ResearchTotal
Male Students4060100
Female Students5050100
Total90110200

When one student is randomly selected from the 200 students who participated in this survey and is a student who chose ecological research, what is the probability that this student is a female student? [3 points]
(1) $\frac { 5 } { 11 }$
(2) $\frac { 1 } { 2 }$
(3) $\frac { 6 } { 11 }$
(4) $\frac { 5 } { 9 }$
(5) $\frac { 3 } { 5 }$
Q10 3 marks Composite & Inverse Functions Graphical Interpretation of Inverse or Composition View
For the function $y = \sqrt { 4 - 2 x } + 3$, what is the minimum value of the real number $k$ such that the graph of its inverse function and the line $y = - x + k$ intersect at two distinct points? [3 points]
(1) 1
(2) 3
(3) 5
(4) 7
(5) 9
Q11 3 marks Indefinite & Definite Integrals Definite Integral as a Limit of Riemann Sums View
For the function $f ( x ) = 4 x ^ { 3 } + x$, what is the value of $\lim _ { n \rightarrow \infty } \sum _ { k = 1 } ^ { n } \frac { 1 } { n } f \left( \frac { 2 k } { n } \right)$? [3 points]
(1) 6
(2) 7
(3) 8
(4) 9
(5) 10
Q12 3 marks Stationary points and optimisation Determine parameters from given extremum conditions View
The function $f ( x ) = - x ^ { 4 } + 8 a ^ { 2 } x ^ { 2 } - 1$ has local maxima at $x = b$ and $x = 2 - 2 b$. What is the value of $a + b$? (Note: $a , b$ are constants with $a > 0 , b > 1$.) [3 points]
(1) 3
(2) 5
(3) 7
(4) 9
(5) 11
Q13 3 marks Normal Distribution Direct Probability Calculation from Given Normal Distribution View
The weight of one paprika harvested at a certain farm follows a normal distribution with mean 180 g and standard deviation 20 g. Using the standard normal distribution table below, what is the probability that the weight of one randomly selected paprika from this farm is at least 190 g and at most 210 g? [3 points]
$z$$\mathrm { P } ( 0 \leq Z \leq z )$
0.50.1915
1.00.3413
1.50.4332
2.00.4772

(1) 0.0440
(2) 0.0919
(3) 0.1359
(4) 0.1498
(5) 0.2417
Q14 4 marks Polynomial Division & Manipulation View
Two polynomial functions $f ( x ) , g ( x )$ with integer constant terms and coefficients satisfy the following conditions. What is the maximum value of $f ( 2 )$? [4 points] (가) $\lim _ { x \rightarrow \infty } \frac { f ( x ) g ( x ) } { x ^ { 3 } } = 2$ (나) $\lim _ { x \rightarrow 0 } \frac { f ( x ) g ( x ) } { x ^ { 2 } } = - 4$
(1) 4
(2) 6
(3) 8
(4) 10
(5) 12
Q15 4 marks Arithmetic Sequences and Series Optimization Involving an Arithmetic Sequence View
For an arithmetic sequence with first term 50 and common difference $- 4$, let $S _ { n }$ denote the sum of the first $n$ terms. What is the value of the natural number $m$ that maximizes $\sum _ { k = m } ^ { m + 4 } S _ { k }$? [4 points]
(1) 8
(2) 9
(3) 10
(4) 11
(5) 12
Q16 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. One ball is randomly drawn from the bag, the number on the ball is confirmed, and then the ball is returned. This procedure is repeated 10 times, and the sum of the 10 numbers confirmed is the random variable $Y$. The following is the process of finding the mean $\mathrm { E } ( Y )$ and variance $\mathrm { V } ( Y )$ of the random variable $Y$.
Let the 60 balls in the bag be the population. When one ball is randomly drawn from this population, let the number on the ball be the random variable $X$. The probability distribution of $X$, that 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 $\bar { X }$, $$\mathrm { E } ( \bar { X } ) = \frac { 7 } { 3 } , \quad \mathrm {~V} ( \bar { X } ) = \text { (나) }$$
Let the number on the $n$-th ball drawn from the bag be $X _ { n }$. Then $$Y = \sum _ { n = 1 } ^ { 10 } X _ { n } = 10 \bar { X }$$ so $$\mathrm { E } ( Y ) = \frac { 70 } { 3 } , \quad \mathrm {~V} ( Y ) = \text { (다) }$$
When 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) $\frac { 13 } { 2 }$
Q17 4 marks Number Theory Arithmetic Functions and Multiplicative Number Theory View
Let $f ( n )$ denote the number of positive divisors of a natural number $n$, and let $a _ { 1 } , a _ { 2 } , a _ { 3 } , \cdots , a _ { 9 }$ be all positive divisors of 36. What is the value of $\sum _ { k = 1 } ^ { 9 } \left\{ ( - 1 ) ^ { f \left( a _ { k } \right) } \times \log a _ { k } \right\}$? [4 points]
(1) $\log 2 + \log 3$
(2) $2 \log 2 + \log 3$
(3) $\log 2 + 2 \log 3$
(4) $2 \log 2 + 2 \log 3$
(5) $3 \log 2 + 2 \log 3$
Q18 4 marks Volumes of Revolution Volume by Displacement or Composite Solid View
As shown in the figure, a sector ABD with center A and central angle $90 ^ { \circ }$ is drawn in a square ABCD with side length 5. Let $\mathrm { A } _ { 1 }$ be the point that divides segment AD in the ratio $3 : 2$, and let $\mathrm { B } _ { 1 }$ be the point where the line passing through $\mathrm { A } _ { 1 }$ and parallel to segment AB meets arc BD. A square $\mathrm { A } _ { 1 } \mathrm {~B} _ { 1 } \mathrm { C } _ { 1 } \mathrm { D } _ { 1 }$ is drawn with segment $\mathrm { A } _ { 1 } \mathrm {~B} _ { 1 }$ as one side and meeting segment DC, and then a sector $\mathrm { D } _ { 1 } \mathrm {~A} _ { 1 } \mathrm { C } _ { 1 }$ with center $\mathrm { D } _ { 1 }$ and central angle $90 ^ { \circ }$ is drawn. Let $\mathrm { E } _ { 1 } , \mathrm {~F} _ { 1 }$ be the points where segment DC meets arc $\mathrm { A } _ { 1 } \mathrm { C } _ { 1 }$ and segment $\mathrm { B } _ { 1 } \mathrm { C } _ { 1 }$ respectively. The region enclosed by two segments $\mathrm { DA } _ { 1 } , \mathrm { DE } _ { 1 }$ and arc $\mathrm { A } _ { 1 } \mathrm { E } _ { 1 }$, and the region enclosed by two segments $\mathrm { E } _ { 1 } \mathrm {~F} _ { 1 } , \mathrm {~F} _ { 1 } \mathrm { C } _ { 1 }$ and arc $\mathrm { E } _ { 1 } \mathrm { C } _ { 1 }$ are shaded to obtain the figure $R _ { 1 }$.
In figure $R _ { 1 }$, a sector $\mathrm { A } _ { 1 } \mathrm {~B} _ { 1 } \mathrm { D } _ { 1 }$ with center $\mathrm { A } _ { 1 }$ and central angle $90 ^ { \circ }$ is drawn in square $\mathrm { A } _ { 1 } \mathrm {~B} _ { 1 } \mathrm { C } _ { 1 } \mathrm { D } _ { 1 }$. Let $\mathrm { A } _ { 2 }$ be the point that divides segment $\mathrm { A } _ { 1 } \mathrm { D } _ { 1 }$ in the ratio $3 : 2$, and let $\mathrm { B } _ { 2 }$ be the point where the line passing through $\mathrm { A } _ { 2 }$ and parallel to segment $\mathrm { A } _ { 1 } \mathrm {~B} _ { 1 }$ meets arc $\mathrm { B } _ { 1 } \mathrm { D } _ { 1 }$. A square $\mathrm { A } _ { 2 } \mathrm {~B} _ { 2 } \mathrm { C } _ { 2 } \mathrm { D } _ { 2 }$ is drawn with segment $\mathrm { A } _ { 2 } \mathrm {~B} _ { 2 }$ as one side and meeting segment $\mathrm { D } _ { 1 } \mathrm { C } _ { 1 }$, and then a shaded figure is drawn and colored in square $\mathrm { A } _ { 2 } \mathrm {~B} _ { 2 } \mathrm { C } _ { 2 } \mathrm { D } _ { 2 }$ in the same way as obtaining figure $R _ { 1 }$ to obtain the figure $R _ { 2 }$. Continuing this process, let $S _ { n }$ denote the area of the shaded part in the $n$-th obtained figure $R _ { n }$. What is the value of $\lim _ { n \rightarrow \infty } S _ { n }$? [4 points]
(1) $\frac { 50 } { 3 } \left( 3 - \sqrt { 3 } + \frac { \pi } { 6 } \right)$
(2) $\frac { 100 } { 9 } \left( 3 - \sqrt { 3 } + \frac { \pi } { 3 } \right)$
(3) $\frac { 50 } { 3 } \left( 2 - \sqrt { 3 } + \frac { \pi } { 3 } \right)$
(4) $\frac { 100 } { 9 } \left( 3 - \sqrt { 3 } + \frac { \pi } { 6 } \right)$
(5) $\frac { 100 } { 9 } \left( 2 - \sqrt { 3 } + \frac { \pi } { 3 } \right)$
Q19 4 marks Permutations & Arrangements Word Permutations with Repeated Letters View
From the numbers $1,2,3,4,5,6$, five numbers are selected with repetition allowed to satisfy the following conditions, and then all five-digit natural numbers that can be formed by arranging them in a line are counted. What is the number of such natural numbers? [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.
(1) 450
(2) 445
(3) 440
(4) 435
(5) 430
Q20 4 marks Applied differentiation Finding parameter values from differentiability or equation constraints View
For the function $$f ( x ) = \begin{cases} - x & ( x \leq 0 ) \\ x - 1 & ( 0 < x \leq 2 ) \\ 2 x - 3 & ( x > 2 ) \end{cases}$$ and a non-constant polynomial $p ( x )$, which of the following statements are correct? [4 points]
ㄱ. If the function $p ( x ) f ( x )$ is continuous on the entire set of real numbers, then $p ( 0 ) = 0$. ㄴ. If the function $p ( x ) f ( x )$ is differentiable on the entire set of real numbers, then $p ( 2 ) = 0$. ㄷ. If the function $p ( x ) \{ f ( x ) \} ^ { 2 }$ is differentiable on the entire set of real numbers, then $p ( x )$ is divisible by $x ^ { 2 } ( x - 2 ) ^ { 2 }$.
(1) ㄱ
(2) ㄱ, ㄴ
(3) ㄱ, ㄷ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ
Q21 4 marks Sequences and series, recurrence and convergence Summation of sequence terms View
The sequence $\left\{ a _ { n } \right\}$ satisfies the following conditions for all natural numbers $n$. (가) $a _ { 2 n } = a _ { n } - 1$ (나) $a _ { 2 n + 1 } = 2 a _ { n } + 1$ When $a _ { 20 } = 1$, what is the value of $\sum _ { n = 1 } ^ { 63 } a _ { n }$? [4 points]
(1) 704
(2) 712
(3) 720
(4) 728
(5) 736
Q22 3 marks Permutations & Arrangements Factorial and Combinatorial Expression Simplification View
Find the value of ${ } _ { 7 } \mathrm { P } _ { 2 } + { } _ { 7 } \mathrm { C } _ { 2 }$. [3 points]
Q23 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, $$\frac { a _ { 16 } } { a _ { 14 } } + \frac { a _ { 8 } } { a _ { 7 } } = 12$$ Find the value of $\frac { a _ { 3 } } { a _ { 1 } } + \frac { a _ { 6 } } { a _ { 3 } }$. [3 points]
Q24 3 marks Binomial Distribution Compute Expectation, Variance, or Standard Deviation View
The random variable $X$ follows a binomial distribution $\mathrm { B } ( 80 , p )$ and $\mathrm { E } ( X ) = 20$. Find the value of $\mathrm { V } ( X )$. [3 points]
Q25 3 marks Factor & Remainder Theorem Remainder by Linear Divisor View
For a natural number $n$, let $a _ { n }$ be the remainder when the polynomial $2 x ^ { 2 } - 3 x + 1$ is divided by $x - n$. Find the value of $\sum _ { n = 1 } ^ { 7 } \left( a _ { n } - n ^ { 2 } + n \right)$. [3 points]
Q26 4 marks Areas Between Curves Area Involving Piecewise or Composite Functions View
For two functions $$f ( x ) = \frac { 1 } { 3 } x ( 4 - x ) , \quad g ( x ) = | x - 1 | - 1$$ let $S$ denote the area enclosed by their graphs. Find the value of $4 S$. [4 points]
Q27 4 marks Variable acceleration (1D) Two-particle comparison problem View
Two points P and Q move on a number line. Their positions $x _ { 1 } , x _ { 2 }$ at time $t ( t \geq 0 )$ are $$x _ { 1 } = t ^ { 3 } - 2 t ^ { 2 } + 3 t , \quad x _ { 2 } = t ^ { 2 } + 12 t$$ Find the distance between points P and Q at the moment when their velocities are equal. [4 points]
Q28 4 marks Indefinite & Definite Integrals Finding a Function from an Integral Equation View
A polynomial function $f ( x )$ satisfies the following conditions. (가) For all real numbers $x$, $$\int _ { 1 } ^ { x } f ( t ) d t = \frac { x - 1 } { 2 } \{ f ( x ) + f ( 1 ) \}$$ (나) $\int _ { 0 } ^ { 2 } f ( x ) d x = 5 \int _ { - 1 } ^ { 1 } x f ( x ) d x$ When $f ( 0 ) = 1$, find the value of $f ( 4 )$. [4 points]
Q29 4 marks Combinations & Selection Counting Integer Solutions to Equations View
Three students A, B, and C are given 6 identical candies and 5 identical chocolates to be distributed completely according to the following rules. Find the number of ways to do this. [4 points] (가) The number of candies that student A receives is at least 1. (나) The number of chocolates that student B receives is at least 1. (다) The sum of the number of candies and chocolates that student C receives is at least 1.
Q30 4 marks Stationary points and optimisation Determine parameters from given extremum conditions View
A cubic function $f ( x )$ with positive leading coefficient satisfies the following conditions. (가) The equation $f ( x ) - x = 0$ has exactly 2 distinct real roots. (나) The equation $f ( x ) + x = 0$ has exactly 2 distinct real roots. When $f ( 0 ) = 0$ and $f ^ { \prime } ( 1 ) = 1$, find the value of $f ( 3 )$. [4 points]