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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
2026 csat__math

30 maths questions

Q1 2 marks Indices and Surds Evaluating Expressions Using Index Laws View
What is the value of $9 ^ { \frac { 1 } { 4 } } \times 3 ^ { - \frac { 1 } { 2 } }$? [2 points]
(1) 1
(2) $\sqrt { 3 }$
(3) 3
(4) $3 \sqrt { 3 }$
(5) 9
Q2 3 marks Arithmetic Sequences and Series Compute Partial Sum of an Arithmetic Sequence View
For the sequence $\left\{ a _ { n } \right\}$, when $\sum _ { k = 1 } ^ { 4 } \left( 2 a _ { k } - k \right) = 0$, what is the value of $\sum _ { k = 1 } ^ { 4 } a _ { k }$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5
Q3 3 marks Curve Sketching Finding Parameters for Continuity View
The function $$f ( x ) = \begin{cases} 3 x - 2 & ( x < 1 ) \\ x ^ { 2 } - 3 x + a & ( x \geq 1 ) \end{cases}$$ is continuous on the set of all real numbers. What is the value of the constant $a$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5
Q4 2 marks Differentiation from First Principles View
For the function $f ( x ) = 3 x ^ { 3 } + 4 x + 1$, what is the value of $\lim _ { h \rightarrow 0 } \frac { f ( 1 + h ) - f ( 1 ) } { h }$? [2 points]
(1) 7
(2) 9
(3) 11
(4) 13
(5) 15
Q5 3 marks Product & Quotient Rules View
For the function $f ( x ) = ( x + 2 ) \left( 2 x ^ { 2 } - x - 2 \right)$, what is the value of $f ^ { \prime } ( 1 )$? [3 points]
(1) 6
(2) 7
(3) 8
(4) 9
(5) 10
Q6 3 marks Laws of Logarithms Simplify or Evaluate a Logarithmic Expression View
Two real numbers $a , b$ greater than 1 satisfy $$\log _ { a } b = 3 , \quad \log _ { 3 } \frac { b } { a } = \frac { 1 } { 2 }$$ What is the value of $\log _ { 9 } a b$? [3 points]
(1) $\frac { 3 } { 8 }$
(2) $\frac { 1 } { 2 }$
(3) $\frac { 5 } { 8 }$
(4) $\frac { 3 } { 4 }$
(5) $\frac { 7 } { 8 }$
Q7 3 marks Areas Between Curves Compute Area Directly (Numerical Answer) View
What is the area of the region enclosed by the two curves $y = x ^ { 2 } + 3$, $y = - \frac { 1 } { 5 } x ^ { 2 } + 3$ and the line $x = 2$? [3 points]
(1) $\frac { 18 } { 5 }$
(2) $\frac { 7 } { 2 }$
(3) $\frac { 17 } { 5 }$
(4) $\frac { 33 } { 10 }$
(5) $\frac { 16 } { 5 }$
Q8 3 marks Standard trigonometric equations Evaluate trigonometric expression given a constraint View
When $\sin \theta + 3 \cos \theta = 0$ and $\cos ( \pi - \theta ) > 0$, what is the value of $\sin \theta$? [3 points]
(1) $\frac { 3 \sqrt { 10 } } { 10 }$
(2) $\frac { \sqrt { 10 } } { 5 }$
(3) 0
(4) $- \frac { \sqrt { 10 } } { 5 }$
(5) $- \frac { 3 \sqrt { 10 } } { 10 }$
Q9 4 marks Tangents, normals and gradients Find tangent line with a specified slope or from an external point View
For a positive number $a$, let the function $f ( x )$ be $$f ( x ) = x ^ { 3 } + 3 a x ^ { 2 } - 9 a ^ { 2 } x + 4$$ When the line $y = 5$ is tangent to the curve $y = f ( x )$, what is the value of $f ( 2 )$? [4 points]
(1) 11
(2) 12
(3) 13
(4) 14
(5) 15
Q10 4 marks Exponential Functions Intersection and Distance between Curves View
For a constant $a$ ($a > 1$), let A be a point in the first quadrant on the curve $y = a ^ { x } - 2$. Let B be the point where the line passing through A and parallel to the $y$-axis meets the $x$-axis, and let C be the point where this line meets the asymptote of the curve $y = a ^ { x } - 2$. If $\overline { \mathrm { AB } } = \overline { \mathrm { BC } }$ and the area of triangle AOC is 8, what is the value of $a \times \overline { \mathrm { OB } }$? (Here, O is the origin.) [4 points]
(1) $2 ^ { \frac { 13 } { 6 } }$
(2) $2 ^ { \frac { 7 } { 3 } }$
(3) $2 ^ { \frac { 5 } { 2 } }$
(4) $2 ^ { \frac { 8 } { 3 } }$
(5) $2 ^ { \frac { 17 } { 6 } }$
Q11 4 marks Variable acceleration (1D) True/false or multiple-statement verification View
There is a point P that starts from the origin at time $t = 0$ and moves on a number line. For a real number $k$, the velocity $v ( t )$ of point P at time $t$ ($t \geq 0$) is $$v ( t ) = t ^ { 2 } - k t + 4$$ Which of the following in are correct? [4 points]
ᄀ. If $k = 0$, then the position of point P at time $t = 1$ is $\frac { 13 } { 3 }$. ㄴ. If $k = 3$, then the direction of motion of point P changes once after departure. ㄷ. If $k = 5$, then the distance traveled by point P from time $t = 0$ to $t = 2$ is 3.
(1) ᄀ
(2) ᄀ, ᄂ
(3) ᄀ, ᄃ
(4) ㄴ, ㄱ
(5) ᄀ, ᄂ, ᄃ
Q12 4 marks Geometric Sequences and Series Finite Geometric Sum and Term Relationships View
A geometric sequence $\left\{ a _ { n } \right\}$ satisfies $$2 \left( a _ { 1 } + a _ { 4 } + a _ { 7 } \right) = a _ { 4 } + a _ { 7 } + a _ { 10 } = 6$$ What is the value of $a _ { 10 }$? [4 points]
(1) $\frac { 22 } { 7 }$
(2) $\frac { 24 } { 7 }$
(3) $\frac { 26 } { 7 }$
(4) $\frac { 30 } { 7 }$
(5) $\frac { 32 } { 7 }$
Q13 4 marks Tangents, normals and gradients Geometric properties of tangent lines (intersections, lengths, areas) View
For the function $f ( x ) = x ^ { 2 } - 4 x - 3$, let $l$ be the tangent line to the curve $y = f ( x )$ at the point $( 1 , - 6 )$, and for the function $g ( x ) = \left( x ^ { 3 } - 2 x \right) f ( x )$, let $m$ be the tangent line to the curve $y = g ( x )$ at the point $( 1,6 )$. What is the area of the figure enclosed by the two lines $l , m$ and the $y$-axis? [4 points]
(1) 21
(2) 28
(3) 35
(4) 42
(5) 49
Q14 4 marks Sine and Cosine Rules Multi-step composite figure problem View
As shown in the figure, there is a right triangle ABC with $\overline { \mathrm { AB } } = 3$, $\overline { \mathrm { BC } } = 4$, and $\angle \mathrm { B } = \frac { \pi } { 2 }$. Let D be the point that divides segment AB internally in the ratio $2 : 1$, let E be the point where the circle centered at A with radius $\overline { \mathrm { AD } }$ meets segment AC, let F be the point where the line AB meets this circle other than D, and let G be a point on arc EF such that $\overline { \mathrm { CG } } = 2 \sqrt { 6 }$. When point H on the circle passing through the three points C, E, G satisfies $\angle \mathrm { HCG } = \angle \mathrm { BAC }$, what is the length of segment GH? [4 points]
(1) $\frac { 6 \sqrt { 15 } } { 5 }$
(2) $\frac { 38 \sqrt { 10 } } { 25 }$
(3) $\frac { 14 \sqrt { 3 } } { 5 }$
(4) $\frac { 32 \sqrt { 15 } } { 25 }$
(5) $\frac { 8 \sqrt { 10 } } { 5 }$
Q15 4 marks Indefinite & Definite Integrals Accumulation Function Analysis View
The function $f ( x )$ is $$f ( x ) = \begin{cases} - x ^ { 2 } & ( x < 0 ) \\ x ^ { 2 } - x & ( x \geq 0 ) \end{cases}$$ and for a positive number $a$, the function $g ( x )$ is $$g ( x ) = \left\{ \begin{array} { c l } a x + a & ( x < - 1 ) \\ 0 & ( - 1 \leq x < 1 ) \\ a x - a & ( x \geq 1 ) \end{array} \right.$$ Let $k$ be the maximum value of $a$ such that the function $h ( x ) = \int _ { 0 } ^ { x } ( g ( t ) - f ( t ) ) d t$ has exactly one extremum. When $a = k$, what is the value of $k + h ( 3 )$? [4 points]
(1) $\frac { 9 } { 2 }$
(2) $\frac { 11 } { 2 }$
(3) $\frac { 13 } { 2 }$
(4) $\frac { 15 } { 2 }$
(5) $\frac { 17 } { 2 }$
Q16 3 marks Sequences and series, recurrence and convergence Direct term computation from recurrence View
The sequence $\left\{ a _ { n } \right\}$ has $a _ { 1 } = 1$ and satisfies $$a _ { n + 1 } = n ^ { 2 } a _ { n } + 1$$ for all natural numbers $n$. Find the value of $a _ { 3 }$. [3 points]
Q17 3 marks Indefinite & Definite Integrals Antiderivative Verification and Construction View
For the function $f ( x ) = 4 x ^ { 3 } - 2 x$, let $F ( x )$ be an antiderivative with $F ( 0 ) = 4$. Find the value of $F ( 2 )$. [3 points]
Q18 3 marks Sine and Cosine Rules Compute area of a triangle or related figure View
In triangle ABC, $\overline { \mathrm { AB } } = 5$, $\overline { \mathrm { AC } } = 6$, and $\cos ( \angle \mathrm { BAC } ) = - \frac { 3 } { 5 }$. Find the area of triangle ABC. [3 points]
Q19 3 marks Stationary points and optimisation Find absolute extrema on a closed interval or domain View
For all real numbers $x$ with $- 2 \leq x \leq 2$, the inequality $$- k \leq 2 x ^ { 3 } + 3 x ^ { 2 } - 12 x - 8 \leq k$$ holds. Find the minimum value of the positive number $k$. [3 points]
Q20 4 marks Sequences and Series Recurrence Relations and Sequence Properties View
The sequence $\left\{ a _ { n } \right\}$ satisfies the following conditions.
  • $a _ { 1 } = 7$
  • For natural numbers $n \geq 2$,
$$\sum _ { k = 1 } ^ { n } a _ { k } = \frac { 2 } { 3 } a _ { n } + \frac { 1 } { 6 } n ^ { 2 } - \frac { 1 } { 6 } n + 10$$
The following is the process of finding the value of $\sum _ { k = 1 } ^ { 12 } a _ { k } + \sum _ { k = 1 } ^ { 5 } a _ { 2 k + 1 }$.
For natural numbers $n \geq 2$, since $a _ { n + 1 } = \sum _ { k = 1 } ^ { n + 1 } a _ { k } - \sum _ { k = 1 } ^ { n } a _ { k }$, $$a _ { n + 1 } = \frac { 2 } { 3 } \left( a _ { n + 1 } - a _ { n } \right) + \text { (가) }$$ and simplifying this equation gives $$2 a _ { n } + a _ { n + 1 } = 3 \times \text { (가) } \quad \cdots \cdots \text { (ㄱ) }$$ From $$\sum _ { k = 1 } ^ { n } a _ { k } = \frac { 2 } { 3 } a _ { n } + \frac { 1 } { 6 } n ^ { 2 } - \frac { 1 } { 6 } n + 10 \quad ( n \geq 2 )$$ substituting $n = 2$ gives $$a _ { 2 } = \text { (나) }$$ By (ㄱ) and (ㄴ), $$\begin{aligned} \sum _ { k = 1 } ^ { 12 } a _ { k } + \sum _ { k = 1 } ^ { 5 } a _ { 2 k + 1 } & = a _ { 1 } + a _ { 2 } + \sum _ { k = 1 } ^ { 5 } \left( 2 a _ { 2 k + 1 } + a _ { 2 k + 2 } \right) \\ & = \text { (다) } \end{aligned}$$ Let $f ( n )$ be the expression that fits in (가), and let $p$ and $q$ be the numbers that fit in (나) and (다), respectively. Find the value of $\frac { p \times q } { f ( 12 ) }$. [4 points]
Q21 4 marks Curve Sketching Continuity and Discontinuity Analysis of Piecewise Functions View
Let $f ( x )$ be a cubic function with positive leading coefficient, and for a real number $t$, let the function $$g ( x ) = \left\{ \begin{array} { r r } - f ( x ) & ( x < t ) \\ f ( x ) & ( x \geq t ) \end{array} \right.$$ be continuous on the set of all real numbers and satisfy the following conditions. (가) For all real numbers $a$, the value of $\lim _ { x \rightarrow a + } \frac { g ( x ) } { x ( x - 2 ) }$ exists. (나) The set of natural numbers $m$ such that $\lim _ { x \rightarrow m + } \frac { g ( x ) } { x ( x - 2 ) }$ is negative is $\left\{ g ( - 1 ) , - \frac { 7 } { 2 } g ( 1 ) \right\}$. Find the value of $g ( - 5 )$. (Given that $g ( - 1 ) \neq - \frac { 7 } { 2 } g ( 1 )$) [4 points]
Q22 4 marks Exponential Equations & Modelling Geometric Properties of Exponential/Logarithmic Curves View
Point A$(a, b)$ is on the curve $y = \log _ { 16 } ( 8 x + 2 )$ and point B is on the curve $y = 4 ^ { x - 1 } - \frac { 1 } { 2 }$, both in the first quadrant. The point obtained by reflecting A across the line $y = x$ lies on the line OB, and the midpoint of segment AB has coordinates $\left( \frac { 77 } { 8 } , \frac { 133 } { 8 } \right)$. When $a \times b = \frac { q } { p }$, find the value of $p + q$. (Here, O is the origin, and $p$ and $q$ are coprime natural numbers.) [4 points]
Q23 2 marks Permutations & Arrangements Linear Arrangement with Constraints View
How many ways are there to select 3 letters from the four letters $a , b , c , d$ with repetition allowed and arrange them in a row? [2 points]
(1) 56
(2) 60
(3) 64
(4) 68
(5) 72
Q24 3 marks Conditional Probability Direct Conditional Probability Computation from Definitions View
For two events $A , B$, $$\mathrm { P } ( A ) = \frac { 2 } { 5 } , \quad \mathrm { P } ( B \mid A ) = \frac { 1 } { 4 } , \quad \mathrm { P } ( A \cup B ) = 1$$ What is the value of $\mathrm { P } ( B )$? [3 points]
(1) $\frac { 7 } { 10 }$
(2) $\frac { 3 } { 4 }$
(3) $\frac { 4 } { 5 }$
(4) $\frac { 17 } { 20 }$
(5) $\frac { 9 } { 10 }$
Q25 3 marks Principle of Inclusion/Exclusion View
A bag contains 5 white balls with the numbers $1,2,3,4,5$ written on them one each, and 5 black balls with the numbers $2,3,4,5,6$ written on them one each. When 2 balls are drawn simultaneously at random from the bag, what is the probability that the 2 balls drawn are either the same color or have the same number written on them? [3 points]
(1) $\frac { 7 } { 15 }$
(2) $\frac { 8 } { 15 }$
(3) $\frac { 3 } { 5 }$
(4) $\frac { 2 } { 3 }$
(5) $\frac { 11 } { 15 }$
Q26 3 marks Confidence intervals Find a specific bound or margin of error from the CI formula View
A population with mean $m$ and standard deviation 5 follows a normal distribution. A sample of size 36 is randomly extracted, and the 99\% confidence interval for the population mean $m$ obtained using the sample mean is $1.2 \leq m \leq a$. What is the value of $a$? (When $Z$ is a random variable following the standard normal distribution, calculate using $\mathrm { P } ( | Z | \leq 2.58 ) = 0.99$.) [3 points]
(1) 5.1
(2) 5.2
(3) 5.3
(4) 5.4
(5) 5.5
Q27 3 marks Discrete Probability Distributions Probability Distribution Table Completion and Expectation Calculation View
A discrete random variable $X$ takes values that are integers from 0 to 4, and $$\mathrm { P } ( X = x ) = \left\{ \begin{array} { c l } \frac { | 2 x - 1 | } { 12 } & ( x = 0,1,2,3 ) \\ a & ( x = 4 ) \end{array} \right.$$ What is the value of $\mathrm { V } \left( \frac { 1 } { a } X \right)$? (Here, $a$ is a nonzero constant.) [3 points]
(1) 36
(2) 39
(3) 42
(4) 45
(5) 48
Q28 4 marks Conditional Probability Sequential/Multi-Stage Conditional Probability View
There are 16 balls and six empty boxes with the natural numbers 1 through 6 written on them. A trial is performed using one die.
When the die is rolled and the result is $k$: If $k$ is odd, place 1 ball each in the boxes labeled $1, 3, 5$, and if $k$ is even, place 1 ball each in the boxes labeled with the divisors of $k$.
After repeating this trial 4 times, given that the sum of all balls in the six boxes is odd, what is the probability that the number of balls in the box labeled 3 is 1 more than the number of balls in the box labeled 2? [4 points]
(1) $\frac { 1 } { 8 }$
(2) $\frac { 3 } { 16 }$
(3) $\frac { 1 } { 4 }$
(4) $\frac { 5 } { 16 }$
(5) $\frac { 3 } { 8 }$
Q29 4 marks Approximating Binomial to Normal Distribution View
For a natural number $a$ not exceeding 6, a trial is performed using one die and one coin.
Roll the die once. If the result is less than or equal to $a$, flip the coin 5 times and record the number of heads. If the result is greater than $a$, flip the coin 3 times and record the number of heads.
This trial is repeated 19200 times, and let $X$ be the number of times the recorded number is 3. When $\mathrm { E } ( X ) = 4800$, find the value of $\mathrm { P } ( X \leq 4800 + 30 a )$ using the standard normal distribution table below, which equals $k$.
$z$$\mathrm { P } ( 0 \leq Z \leq z )$
0.50.191
1.00.341
1.50.433
2.00.477
2.50.494
3.00.499

Find the value of $1000 \times k$. [4 points]
Q30 4 marks Permutations & Arrangements Distribution of Objects into Bins/Groups View
There are 10 empty bags arranged in a row, and 8 balls. Distribute the balls into the bags so that each bag contains at most 2 balls. Find the number of cases satisfying the following conditions. (Here, the balls are indistinguishable from each other.) [4 points] (가) The number of bags containing 1 ball is either 4 or 6. (나) Bags adjacent to a bag containing 2 balls contain no balls.