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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
2023 csat__math

30 maths questions

Q1 2 marks Indices and Surds Evaluating Expressions Using Index Laws View
What is the value of $\left( \frac { 4 } { 2 ^ { \sqrt { 2 } } } \right) ^ { 2 + \sqrt { 2 } }$? [2 points]
(1) $\frac { 1 } { 4 }$
(2) $\frac { 1 } { 2 }$
(3) 1
(4) 2
(5) 4
Q2 2 marks Curve Sketching Limit Computation from Algebraic Expressions View
What is the value of $\lim _ { x \rightarrow \infty } \frac { \sqrt { x ^ { 2 } - 2 } + 3 x } { x + 5 }$? [2 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5
Q3 3 marks Geometric Sequences and Series Finite Geometric Sum and Term Relationships View
A geometric sequence $\left\{ a _ { n } \right\}$ with positive common ratio satisfies $$a _ { 2 } + a _ { 4 } = 30 , \quad a _ { 4 } + a _ { 6 } = \frac { 15 } { 2 }$$ What is the value of $a _ { 1 }$? [3 points]
(1) 48
(2) 56
(3) 64
(4) 72
(5) 80
Q4 3 marks Product & Quotient Rules View
For a polynomial function $f ( x )$, define the function $g ( x )$ as $$g ( x ) = x ^ { 2 } f ( x )$$ If $f ( 2 ) = 1$ and $f ^ { \prime } ( 2 ) = 3$, what is the value of $g ^ { \prime } ( 2 )$? [3 points]
(1) 12
(2) 14
(3) 16
(4) 18
(5) 20
Q5 3 marks Reciprocal Trig & Identities View
If $\tan \theta < 0$ and $\cos \left( \frac { \pi } { 2 } + \theta \right) = \frac { \sqrt { 5 } } { 5 }$, what is the value of $\cos \theta$? [3 points]
(1) $- \frac { 2 \sqrt { 5 } } { 5 }$
(2) $- \frac { \sqrt { 5 } } { 5 }$
(3) 0
(4) $\frac { \sqrt { 5 } } { 5 }$
(5) $\frac { 2 \sqrt { 5 } } { 5 }$
Q6 3 marks Stationary points and optimisation Determine parameters from given extremum conditions View
The function $f ( x ) = 2 x ^ { 3 } - 9 x ^ { 2 } + a x + 5$ has a local maximum at $x = 1$ and a local minimum at $x = b$. What is the value of $a + b$? (Here, $a$ and $b$ are constants.) [3 points]
(1) 12
(2) 14
(3) 16
(4) 18
(5) 20
Q7 3 marks Arithmetic Sequences and Series Telescoping or Non-Standard Summation Involving an AP View
An arithmetic sequence $\left\{ a _ { n } \right\}$ with all positive terms and equal first term and common difference satisfies $$\sum _ { k = 1 } ^ { 15 } \frac { 1 } { \sqrt { a _ { k } } + \sqrt { a _ { k + 1 } } } = 2$$ What is the value of $a _ { 4 }$? [3 points]
(1) 6
(2) 7
(3) 8
(4) 9
(5) 10
Q8 3 marks Tangents, normals and gradients Find tangent line with a specified slope or from an external point View
What is the $x$-intercept of the tangent line drawn from the point $(0, 4)$ to the curve $y = x ^ { 3 } - x + 2$? [3 points]
(1) $- \frac { 1 } { 2 }$
(2) $- 1$
(3) $- \frac { 3 } { 2 }$
(4) $- 2$
(5) $- \frac { 5 } { 2 }$
Q9 4 marks Harmonic Form View
The function $$f ( x ) = a - \sqrt { 3 } \tan 2 x$$ has a maximum value of 7 and a minimum value of 3 on the closed interval $\left[ - \frac { \pi } { 6 } , b \right]$. What is the value of $a \times b$? (Here, $a$ and $b$ are constants.) [4 points]
(1) $\frac { \pi } { 2 }$
(2) $\frac { 5 \pi } { 12 }$
(3) $\frac { \pi } { 3 }$
(4) $\frac { \pi } { 4 }$
(5) $\frac { \pi } { 6 }$
Q10 4 marks Areas by integration View
Let $A$ be the area enclosed by the two curves $y = x ^ { 3 } + x ^ { 2 }$, $y = - x ^ { 2 } + k$, and the $y$-axis, and let $B$ be the area enclosed by the two curves $y = x ^ { 3 } + x ^ { 2 }$, $y = - x ^ { 2 } + k$, and the line $x = 2$. When $A = B$, what is the value of the constant $k$? (Here, $4 < k < 5$) [4 points]
(1) $\frac { 25 } { 6 }$
(2) $\frac { 13 } { 3 }$
(3) $\frac { 9 } { 2 }$
(4) $\frac { 14 } { 3 }$
(5) $\frac { 29 } { 6 }$
Q11 4 marks Sine and Cosine Rules Circumradius or incircle radius computation View
As shown in the figure, quadrilateral ABCD is inscribed in a circle and $$\overline { \mathrm { AB } } = 5 , \overline { \mathrm { AC } } = 3 \sqrt { 5 } , \overline { \mathrm { AD } } = 7 , \angle \mathrm { BAC } = \angle \mathrm { CAD }$$ What is the radius of this circle? [4 points]
(1) $\frac { 5 \sqrt { 2 } } { 2 }$
(2) $\frac { 8 \sqrt { 5 } } { 5 }$
(3) $\frac { 5 \sqrt { 5 } } { 3 }$
(4) $\frac { 8 \sqrt { 2 } } { 3 }$
(5) $\frac { 9 \sqrt { 3 } } { 4 }$
Q12 4 marks Indefinite & Definite Integrals Piecewise/Periodic Function Integration View
A function $f ( x )$ that is continuous on the set of all real numbers satisfies the following condition. When $n - 1 \leq x < n$, $| f ( x ) | = | 6 ( x - n + 1 ) ( x - n ) |$. (Here, $n$ is a natural number.)
For the function $$g ( x ) = \int _ { 0 } ^ { x } f ( t ) d t - \int _ { x } ^ { 4 } f ( t ) d t$$ defined on the open interval $(0, 4)$, when $g ( x )$ has a minimum value of 0 at $x = 2$, what is the value of $\int _ { \frac { 1 } { 2 } } ^ { 4 } f ( x ) d x$? [4 points]
(1) $- \frac { 3 } { 2 }$
(2) $- \frac { 1 } { 2 }$
(3) $\frac { 1 } { 2 }$
(4) $\frac { 3 } { 2 }$
(5) $\frac { 5 } { 2 }$
Q13 4 marks Indices and Surds Number-Theoretic Reasoning with Indices View
For a natural number $m$ ($m \geq 2$), let $f ( m )$ be the number of natural numbers $n \geq 2$ such that an integer exists among the $n$-th roots of $m ^ { 12 }$. What is the value of $\sum _ { m = 2 } ^ { 9 } f ( m )$? [4 points]
(1) 37
(2) 42
(3) 47
(4) 52
(5) 57
Q14 4 marks Composite & Inverse Functions Symmetry, Periodicity, and Parity from Composition Conditions View
For a polynomial function $f ( x )$, define the function $g ( x )$ as follows: $$g ( x ) = \begin{cases} x & ( x < - 1 \text{ or } x > 1 ) \\ f ( x ) & ( - 1 \leq x \leq 1 ) \end{cases}$$ For the function $h ( x ) = \lim _ { t \rightarrow 0 + } g ( x + t ) \times \lim _ { t \rightarrow 2 + } g ( x + t )$, which of the following statements in the given options are correct? [4 points]
ㄱ. $h ( 1 ) = 3$ ㄴ. The function $h ( x )$ is continuous on the set of all real numbers. ㄷ. If the function $g ( x )$ is decreasing on the closed interval $[ - 1, 1 ]$ and $g ( - 1 ) = - 2$, then the function $h ( x )$ has a minimum value on the set of all real numbers.
(1) ㄱ
(2) ㄴ
(3) ㄱ, ㄴ
(4) ㄱ, ㄷ
(5) ㄴ, ㄷ
Q15 4 marks Sequences and series, recurrence and convergence Direct term computation from recurrence View
For all sequences $\left\{ a _ { n } \right\}$ with all natural number terms satisfying the following conditions, let $M$ and $m$ be the maximum and minimum values of $a _ { 9 }$, respectively. What is the value of $M + m$? [4 points] (가) $a _ { 7 } = 40$ (나) For all natural numbers $n$, $$a _ { n + 2 } = \begin{cases} a _ { n + 1 } + a _ { n } & ( \text{when } a _ { n + 1 } \text{ is not a multiple of } 3 ) \\ \frac { 1 } { 3 } a _ { n + 1 } & ( \text{when } a _ { n + 1 } \text{ is a multiple of } 3 ) \end{cases}$$ (1) 216
(2) 218
(3) 220
(4) 222
(5) 224
Q16 3 marks Laws of Logarithms Solve a Logarithmic Equation View
Solve the equation $$\log _ { 2 } ( 3 x + 2 ) = 2 + \log _ { 2 } ( x - 2 )$$ for the real number $x$. [3 points]
Q17 3 marks Standard Integrals and Reverse Chain Rule Antiderivative with Initial Condition View
For a function $f ( x )$, if $f ^ { \prime } ( x ) = 4 x ^ { 3 } - 2 x$ and $f ( 0 ) = 3$, what is the value of $f ( 2 )$? [3 points]
Q18 3 marks Arithmetic Sequences and Series Compute Partial Sum of an Arithmetic Sequence View
For two sequences $\left\{ a _ { n } \right\}$ and $\left\{ b _ { n } \right\}$, $$\sum _ { k = 1 } ^ { 5 } \left( 3 a _ { k } + 5 \right) = 55 , \quad \sum _ { k = 1 } ^ { 5 } \left( a _ { k } + b _ { k } \right) = 32$$ What is the value of $\sum _ { k = 1 } ^ { 5 } b _ { k }$? [3 points]
Q19 3 marks Curve Sketching Number of Solutions / Roots via Curve Analysis View
Find the number of integers $k$ such that the equation $2 x ^ { 3 } - 6 x ^ { 2 } + k = 0$ has exactly 2 distinct positive real roots. [3 points]
Q20 4 marks Variable acceleration (1D) Compute total distance traveled over an interval View
The velocity $v ( t )$ and acceleration $a ( t )$ of a point P moving on a number line at time $t$ ($t \geq 0$) satisfy the following conditions. (가) When $0 \leq t \leq 2$, $v ( t ) = 2 t ^ { 3 } - 8 t$. (나) When $t \geq 2$, $a ( t ) = 6 t + 4$.
Find the distance traveled by point P from time $t = 0$ to $t = 3$. [4 points]
Q21 4 marks Exponential Functions Exponential Equation Solving View
For a natural number $n$, define the function $f ( x )$ as $$f ( x ) = \begin{cases} \left| 3 ^ { x + 2 } - n \right| & ( x < 0 ) \\ \left| \log _ { 2 } ( x + 4 ) - n \right| & ( x \geq 0 ) \end{cases}$$ Let $g ( t )$ be the number of distinct real roots of the equation $f ( x ) = t$ for a real number $t$. Find the sum of all natural numbers $n$ such that the maximum value of the function $g ( t )$ is 4. [4 points]
Q22 4 marks Differential equations Qualitative Analysis of DE Solutions View
A cubic function $f ( x )$ with leading coefficient 1 and a function $g ( x )$ that is continuous on the set of all real numbers satisfy the following conditions. Find the value of $f ( 4 )$. [4 points] (가) For all real numbers $x$, $$f ( x ) = f ( 1 ) + ( x - 1 ) f ^ { \prime } ( g ( x ) )$$ (나) The minimum value of the function $g ( x )$ is $\frac { 5 } { 2 }$. (다) $f ( 0 ) = - 3$, $f ( g ( 1 ) ) = 6$
Q23 2 marks Binomial Theorem (positive integer n) Find a Specific Coefficient in a Single Binomial Expansion View
What is the coefficient of $x ^ { 9 }$ in the expansion of $\left( x ^ { 3 } + 3 \right) ^ { 5 }$? [2 points]
(1) 30
(2) 60
(3) 90
(4) 120
(5) 150
Q24 3 marks Permutations & Arrangements Forming Numbers with Digit Constraints View
Among four-digit natural numbers that can be formed by selecting 4 numbers from the digits 1, 2, 3, 4, 5 with repetition allowed and arranging them in a line, how many are odd numbers greater than or equal to 4000? [3 points]
(1) 125
(2) 150
(3) 175
(4) 200
(5) 225
Q25 3 marks Probability Definitions Probability Using Set/Event Algebra View
A box contains 5 white masks and 9 black masks. When 3 masks are randomly drawn simultaneously from the box, what is the probability that at least one of the 3 masks is white? [3 points]
(1) $\frac { 8 } { 13 }$
(2) $\frac { 17 } { 26 }$
(3) $\frac { 9 } { 13 }$
(4) $\frac { 19 } { 26 }$
(5) $\frac { 10 } { 13 }$
Q26 3 marks Conditional Probability Combinatorial Conditional Probability (Counting-Based) View
A bag contains 1 white ball marked with 1, 1 white ball marked with 2, 1 black ball marked with 1, and 3 black balls marked with 2. We perform a trial of simultaneously drawing 3 balls from the bag. Let $A$ be the event that among the 3 balls drawn, 1 is white and 2 are black, and let $B$ be the event that the product of the numbers on the 3 balls is 8. What is the value of $\mathrm { P } ( A \cup B )$? [3 points]
(1) $\frac { 11 } { 20 }$
(2) $\frac { 3 } { 5 }$
(3) $\frac { 13 } { 20 }$
(4) $\frac { 7 } { 10 }$
(5) $\frac { 3 } { 4 }$
Q27 3 marks Confidence intervals Determine minimum sample size for a desired interval width View
A company produces shampoo where the volume of 1 bottle follows a normal distribution $\mathrm { N } \left( m , \sigma ^ { 2 } \right)$. A sample of 16 bottles was randomly selected from the company's shampoo, and the 95\% confidence interval for $m$ using the sample mean is $746.1 \leq m \leq 755.9$. When a sample of $n$ bottles is randomly selected and a 99\% confidence interval for $m$ is constructed as $a \leq m \leq b$, what is the minimum natural number $n$ such that $b - a \leq 6$? (Here, the unit of volume is mL, and when $Z$ is a random variable following the standard normal distribution, $\mathrm { P } ( | Z | \leq 1.96 ) = 0.95$ and $\mathrm { P } ( | Z | \leq 2.58 ) = 0.99$.) [3 points]
(1) 70
(2) 74
(3) 78
(4) 82
(5) 86
Q28 4 marks Continuous Probability Distributions and Random Variables PDF Graph Interpretation and Probability Computation View
A continuous random variable $X$ has a range of $0 \leq X \leq a$, and the graph of the probability density function of $X$ is as shown in the figure. When $\mathrm { P } ( X \leq b ) - \mathrm { P } ( X \geq b ) = \frac { 1 } { 4 }$ and $\mathrm { P } ( X \leq \sqrt { 5 } ) = \frac { 1 } { 2 }$, what is the value of $a + b + c$? (Here, $a$, $b$, and $c$ are constants.) [4 points]
(1) $\frac { 11 } { 2 }$
(2) 6
(3) $\frac { 13 } { 2 }$
(4) 7
(5) $\frac { 15 } { 2 }$
Q29 4 marks Conditional Probability Conditional Probability with Discrete Random Variable View
There are 6 cards with natural numbers 1 through 6 written on the front and 0 written on the back. These 6 cards are placed so that the natural number $k$ is visible in the $k$-th position for natural numbers $k$ not exceeding 6.
Using these 6 cards and one die, we perform the following trial:
Roll the die once. If the result is $k$, flip the card in the $k$-th position and place it back in its original position.
After repeating this trial 3 times, given that the sum of all numbers visible on the 6 cards is even, what is the probability that the die shows 1 exactly once? The probability is $\frac { q } { p }$. Find the value of $p + q$. (Here, $p$ and $q$ are coprime natural numbers.) [4 points]
Q30 4 marks Permutations & Arrangements Counting Functions with Constraints View
For the set $X = \{ x \mid x \text{ is a natural number not exceeding } 10 \}$, find the number of functions $f : X \rightarrow X$ satisfying the following conditions. [4 points] (가) For all natural numbers $x$ not exceeding 9, $f ( x ) \leq f ( x + 1 )$. (나) When $1 \leq x \leq 5$, $f ( x ) \leq x$, and when $6 \leq x \leq 10$, $f ( x ) \geq x$. (다) $f ( 6 ) = f ( 5 ) + 6$