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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
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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
2021 csat__math-science

29 maths questions

Q1 2 marks Indices and Surds Evaluating Expressions Using Index Laws View
What is the value of $\sqrt [ 3 ] { 9 } \times 3 ^ { \frac { 1 } { 3 } }$? [2 points]
(1) 1
(2) $3 ^ { \frac { 1 } { 2 } }$
(3) 3
(4) $3 ^ { \frac { 3 } { 2 } }$
(5) 9
Q2 2 marks Trig Graphs & Exact Values View
For $\theta$ satisfying $\frac { \pi } { 2 } < \theta < \pi$ and $\sin \theta = \frac { \sqrt { 21 } } { 7 }$, what is the value of $\tan \theta$? [2 points]
(1) $- \frac { \sqrt { 3 } } { 2 }$
(2) $- \frac { \sqrt { 3 } } { 4 }$
(3) 0
(4) $\frac { \sqrt { 3 } } { 4 }$
(5) $\frac { \sqrt { 3 } } { 2 }$
Q3 2 marks Sequences and Series Limit Evaluation Involving Sequences View
What is the value of $\lim _ { n \rightarrow \infty } \frac { 1 } { \sqrt { 4 n ^ { 2 } + 2 n + 1 } - 2 n }$? [2 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5
Q4 3 marks Conditional Probability Direct Conditional Probability Computation from Definitions View
For two events $A$ and $B$, $$\mathrm { P } ( B \mid A ) = \frac { 1 } { 4 } , \quad \mathrm { P } ( A \mid B ) = \frac { 1 } { 3 } , \quad \mathrm { P } ( A ) + \mathrm { P } ( B ) = \frac { 7 } { 10 }$$ what is the value of $\mathrm { P } ( A \cap B )$? [3 points]
(1) $\frac { 1 } { 7 }$
(2) $\frac { 1 } { 8 }$
(3) $\frac { 1 } { 9 }$
(4) $\frac { 1 } { 10 }$
(5) $\frac { 1 } { 11 }$
Q5 3 marks Exponential Functions Exponential Equation Solving View
How many natural numbers $x$ satisfy the inequality $\left( \frac { 1 } { 9 } \right) ^ { x } < 3 ^ { 21 - 4 x }$? [3 points]
(1) 6
(2) 7
(3) 8
(4) 9
(5) 10
Q6 3 marks Linear combinations of normal random variables View
For a population following the normal distribution $\mathrm { N } \left( 20,5 ^ { 2 } \right)$, a sample of size 16 is randomly extracted and the sample mean is denoted by $\bar { X }$. What is the value of $\mathrm { E } ( \bar { X } ) + \sigma ( \bar { X } )$? [3 points]
(1) $\frac { 83 } { 4 }$
(2) $\frac { 85 } { 4 }$
(3) $\frac { 87 } { 4 }$
(4) $\frac { 89 } { 4 }$
(5) $\frac { 91 } { 4 }$
Q7 3 marks Differentiating Transcendental Functions Monotonicity or convexity of transcendental functions View
For the function $f ( x ) = \left( x ^ { 2 } - 2 x - 7 \right) e ^ { x }$, let the local maximum value and local minimum value be $a$ and $b$ respectively. What is the value of $a \times b$? [3 points]
(1) - 32
(2) - 30
(3) - 28
(4) - 26
(5) - 24
Q8 3 marks Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View
What is the area of the region enclosed by the curve $y = e ^ { 2 x }$, the $x$-axis, and the two lines $x = \ln \frac { 1 } { 2 }$ and $x = \ln 2$? [3 points]
(1) $\frac { 5 } { 3 }$
(2) $\frac { 15 } { 8 }$
(3) $\frac { 15 } { 7 }$
(4) $\frac { 5 } { 2 }$
(5) 3
Q9 3 marks Probability Definitions Finite Equally-Likely Probability Computation View
There are 5 cards with letters $\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E }$ written on them and 4 cards with numbers $1,2,3,4$ written on them. When all 9 cards are arranged in a line in random order using each card once, what is the probability that the card with letter A has number cards on both sides? [3 points]
(1) $\frac { 5 } { 12 }$
(2) $\frac { 1 } { 3 }$
(3) $\frac { 1 } { 4 }$
(4) $\frac { 1 } { 6 }$
(5) $\frac { 1 } { 12 }$
Q10 3 marks Sine and Cosine Rules Circumradius or incircle radius computation View
In triangle ABC with $\angle \mathrm { A } = \frac { \pi } { 3 }$ and $\overline { \mathrm { AB } } : \overline { \mathrm { AC } } = 3 : 1$, the radius of the circumcircle of triangle ABC is 7. What is the length of segment AC? [3 points]
(1) $2 \sqrt { 5 }$
(2) $\sqrt { 21 }$
(3) $\sqrt { 22 }$
(4) $\sqrt { 23 }$
(5) $2 \sqrt { 6 }$
Q11 3 marks Indefinite & Definite Integrals Definite Integral as a Limit of Riemann Sums View
What is the value of $\lim _ { n \rightarrow \infty } \frac { 1 } { n } \sum _ { k = 1 } ^ { n } \sqrt { \frac { 3 n } { 3 n + k } }$? [3 points]
(1) $4 \sqrt { 3 } - 6$
(2) $\sqrt { 3 } - 1$
(3) $5 \sqrt { 3 } - 8$
(4) $2 \sqrt { 3 } - 3$
(5) $3 \sqrt { 3 } - 5$
Q12 3 marks Normal Distribution Algebraic Relationship Between Normal Parameters and Probability View
The random variable $X$ follows a normal distribution with mean 8 and standard deviation 3, and the random variable $Y$ follows a normal distribution with mean $m$ and standard deviation $\sigma$. The two random variables $X$ and $Y$ satisfy $$\mathrm { P } ( 4 \leq X \leq 8 ) + \mathrm { P } ( Y \geq 8 ) = \frac { 1 } { 2 }$$ Find the value of $\mathrm { P } \left( Y \leq 8 + \frac { 2 \sigma } { 3 } \right)$ using the standard normal distribution table on the right.
$z$$\mathrm { P } ( 0 \leq Z \leq z )$
1.00.3413
1.50.4332
2.00.4772
2.50.4938

[3 points]
(1) 0.8351
(2) 0.8413
(3) 0.9332
(4) 0.9772
(5) 0.9938
Q13 3 marks Laws of Logarithms Logarithmic Function Graph Intersection or Geometric Analysis View
For a real number $a$ with $\frac { 1 } { 4 } < a < 1$, let A and B be the points where the line $y = 1$ meets the curves $y = \log _ { a } x$ and $y = \log _ { 4 a } x$ respectively, and let C and D be the points where the line $y = - 1$ meets the curves $y = \log _ { a } x$ and $y = \log _ { 4 a } x$ respectively. Choose all correct statements from the following. [3 points]
$\langle$Statements$\rangle$
ㄱ. The point that divides segment AB externally in the ratio $1 : 4$ has coordinates $( 0,1 )$. ㄴ. If quadrilateral ABCD is a rectangle, then $a = \frac { 1 } { 2 }$. ㄷ. If $\overline { \mathrm { AB } } < \overline { \mathrm { CD } }$, then $\frac { 1 } { 2 } < a < 1$.
(1) ㄱ
(2) ㄷ
(3) ㄱ, ㄴ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ
Q14 4 marks Geometric Sequences and Series Fractal/Iterative Geometric Construction (Area, Length, or Perimeter Series) View
As shown in the figure, there is a rectangle $\mathrm { AB } _ { 1 } \mathrm { C } _ { 1 } \mathrm { D } _ { 1 }$ with $\overline { \mathrm { AB } _ { 1 } } = 2$ and $\overline { \mathrm { AD } _ { 1 } } = 4$. Let $\mathrm { E } _ { 1 }$ be the point that divides segment $\mathrm { AD } _ { 1 }$ internally in the ratio $3 : 1$, and let $\mathrm { F } _ { 1 }$ be a point inside rectangle $\mathrm { AB } _ { 1 } \mathrm { C } _ { 1 } \mathrm { D } _ { 1 }$ such that $\overline { \mathrm { F } _ { 1 } \mathrm { E } _ { 1 } } = \overline { \mathrm { F } _ { 1 } \mathrm { C } _ { 1 } }$ and $\angle \mathrm { E } _ { 1 } \mathrm {~F} _ { 1 } \mathrm { C } _ { 1 } = \frac { \pi } { 2 }$. Triangle $\mathrm { E } _ { 1 } \mathrm {~F} _ { 1 } \mathrm { C } _ { 1 }$ is drawn. The figure obtained by shading quadrilateral $\mathrm { E } _ { 1 } \mathrm {~F} _ { 1 } \mathrm { C } _ { 1 } \mathrm { D } _ { 1 }$ is called $R _ { 1 }$. In figure $R _ { 1 }$, a rectangle $\mathrm { AB } _ { 2 } \mathrm { C } _ { 2 } \mathrm { D } _ { 2 }$ is drawn with vertices at point $\mathrm { B } _ { 2 }$ on segment $\mathrm { AB } _ { 1 }$, point $\mathrm { C } _ { 2 }$ on segment $\mathrm { E } _ { 1 } \mathrm {~F} _ { 1 }$, point $\mathrm { D } _ { 2 }$ on segment $\mathrm { AE } _ { 1 }$, and point A, such that $\overline { \mathrm { AB } _ { 2 } } : \overline { \mathrm { AD } _ { 2 } } = 1 : 2$. Using the same method as for obtaining figure $R _ { 1 }$, triangle $\mathrm { E } _ { 2 } \mathrm {~F} _ { 2 } \mathrm { C } _ { 2 }$ is drawn in rectangle $\mathrm { AB } _ { 2 } \mathrm { C } _ { 2 } \mathrm { D } _ { 2 }$ and quadrilateral $\mathrm { E } _ { 2 } \mathrm {~F} _ { 2 } \mathrm { C } _ { 2 } \mathrm { D } _ { 2 }$ is shaded to obtain figure $R _ { 2 }$. Continuing this process, let $S _ { n }$ be the area of the shaded region in the $n$-th figure $R _ { n }$. What is the value of $\lim _ { n \rightarrow \infty } S _ { n }$? [4 points]
(1) $\frac { 441 } { 103 }$
(2) $\frac { 441 } { 109 }$
(3) $\frac { 441 } { 115 }$
(4) $\frac { 441 } { 121 }$
(5) $\frac { 441 } { 127 }$
Q15 4 marks Standard Integrals and Reverse Chain Rule Antiderivative with Initial Condition View
For a function $f ( x )$ that is differentiable on $x > 0$, $$f ^ { \prime } ( x ) = 2 - \frac { 3 } { x ^ { 2 } } , \quad f ( 1 ) = 5$$ For a function $g ( x )$ that is differentiable on $x < 0$ and satisfies the following conditions, what is the value of $g ( - 3 )$? [4 points] (가) For all real numbers $x < 0$, $g ^ { \prime } ( x ) = f ^ { \prime } ( - x )$. (나) $f ( 2 ) + g ( - 2 ) = 9$
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5
Q16 4 marks Geometric Sequences and Series Geometric Sequence from Recurrence Identification View
For a constant $k$ with $k > 1$, there is a sequence $\left\{ a _ { n } \right\}$ satisfying the following conditions.
For all natural numbers $n$, $a _ { n } < a _ { n + 1 }$ and the slope of the line passing through two points $\mathrm { P } _ { n } \left( a _ { n } , 2 ^ { a _ { n } } \right)$ and $\mathrm { P } _ { n + 1 } \left( a _ { n + 1 } , 2 ^ { a _ { n + 1 } } \right)$ on the curve $y = 2 ^ { x }$ is $k \times 2 ^ { a _ { n } }$.
Let $\mathrm { Q } _ { n }$ be the point where the line passing through $\mathrm { P } _ { n }$ parallel to the $x$-axis and the line passing through $\mathrm { P } _ { n + 1 }$ parallel to the $y$-axis meet, and let $A _ { n }$ be the area of triangle $\mathrm { P } _ { n } \mathrm { Q } _ { n } \mathrm { P } _ { n + 1 }$. The following is the process of finding $A _ { n }$ when $a _ { 1 } = 1$ and $\frac { A _ { 3 } } { A _ { 1 } } = 16$.
Since the slope of the line passing through two points $\mathrm { P } _ { n }$ and $\mathrm { P } _ { n + 1 }$ is $k \times 2 ^ { a _ { n } }$, $$2 ^ { a _ { n + 1 } - a _ { n } } = k \left( a _ { n + 1 } - a _ { n } \right) + 1$$ Thus, for all natural numbers $n$, $a _ { n + 1 } - a _ { n }$ is a solution of the equation $2 ^ { x } = k x + 1$. Since $k > 1$, the equation $2 ^ { x } = k x + 1$ has exactly one positive real root $d$. Therefore, for all natural numbers $n$, $a _ { n + 1 } - a _ { n } = d$, and the sequence $\left\{ a _ { n } \right\}$ is an arithmetic sequence with common difference $d$. Since the coordinates of point $\mathrm { Q } _ { n }$ are $\left( a _ { n + 1 } , 2 ^ { a _ { n } } \right)$, $$A _ { n } = \frac { 1 } { 2 } \left( a _ { n + 1 } - a _ { n } \right) \left( 2 ^ { a _ { n + 1 } } - 2 ^ { a _ { n } } \right)$$ Since $\frac { A _ { 3 } } { A _ { 1 } } = 16$, the value of $d$ is (가), and the general term of the sequence $\left\{ a _ { n } \right\}$ is $$a _ { n } = \text { (나) }$$ Therefore, for all natural numbers $n$, $A _ { n } =$ (다).
When the number corresponding to (가) is $p$, and the expressions corresponding to (나) and (다) are $f ( n )$ and $g ( n )$ respectively, what is the value of $p + \frac { g ( 4 ) } { f ( 2 ) }$? [4 points]
(1) 118
(2) 121
(3) 124
(4) 127
(5) 130
Q17 4 marks Discrete Probability Distributions Expectation and Variance from Context-Based Random Variables View
Point P is at the origin of the coordinate plane. The following trial is performed using one die.
When the die is rolled and the number shown is
2 or less, point P is moved 3 units in the positive direction of the $x$-axis,
3 or more, point P is moved 1 unit in the positive direction of the $y$-axis.
This trial is repeated 15 times, and the distance between the moved point P and the line $3 x + 4 y = 0$ is the random variable $X$. What is the value of $\mathrm { E } ( X )$? [4 points]
(1) 13
(2) 15
(3) 17
(4) 19
(5) 21
Q18 4 marks Curve Sketching Continuity and Discontinuity Analysis of Piecewise Functions View
For a constant $a$, define the function $f ( x )$ as $$f ( x ) = \lim _ { n \rightarrow \infty } \frac { ( a - 2 ) x ^ { 2 n + 1 } + 2 x } { 3 x ^ { 2 n } + 1 }$$ What is the sum of all values of $a$ such that $( f \circ f ) ( 1 ) = \frac { 5 } { 4 }$? [4 points]
(1) $\frac { 11 } { 2 }$
(2) $\frac { 13 } { 2 }$
(3) $\frac { 15 } { 2 }$
(4) $\frac { 17 } { 2 }$
(5) $\frac { 19 } { 2 }$
Q19 4 marks Discrete Probability Distributions Probability Computation for Compound or Multi-Stage Random Experiments View
There is a bag containing 5 balls with the numbers $3,3,4,4,4$ written on them, one each. Using this bag and one die, a trial is performed to obtain a score according to the following rule.
A ball is randomly drawn from the bag. If the number on the drawn ball is 3, the die is rolled 3 times and the sum of the three numbers shown is the score. If the number on the drawn ball is 4, the die is rolled 4 times and the sum of the four numbers shown is the score.
What is the probability that the score obtained from one trial is 10 points? [4 points]
(1) $\frac { 13 } { 180 }$
(2) $\frac { 41 } { 540 }$
(3) $\frac { 43 } { 540 }$
(4) $\frac { 1 } { 12 }$
(5) $\frac { 47 } { 540 }$
Q20 4 marks Indefinite & Definite Integrals Piecewise/Periodic Function Integration View
For the function $f ( x ) = \pi \sin 2 \pi x$, a function $g ( x )$ with domain being the set of all real numbers and range being the set $\{ 0,1 \}$, and a natural number $n$ satisfy the following conditions. What is the value of $n$? [4 points]
The function $h ( x ) = f ( n x ) g ( x )$ is continuous on the set of all real numbers and $$\int _ { - 1 } ^ { 1 } h ( x ) d x = 2 , \quad \int _ { - 1 } ^ { 1 } x h ( x ) d x = - \frac { 1 } { 32 }$$
(1) 8
(2) 10
(3) 12
(4) 14
(5) 16
Q21 4 marks Sequences and series, recurrence and convergence Direct term computation from recurrence View
The sequence $\left\{ a _ { n } \right\}$ satisfies $0 < a _ { 1 } < 1$ and the following conditions for all natural numbers $n$. (가) $a _ { 2 n } = a _ { 2 } \times a _ { n } + 1$ (나) $a _ { 2 n + 1 } = a _ { 2 } \times a _ { n } - 2$ When $a _ { 8 } - a _ { 15 } = 63$, what is the value of $\frac { a _ { 8 } } { a _ { 1 } }$? [4 points]
(1) 91
(2) 92
(3) 93
(4) 94
(5) 95
Q22 3 marks Binomial Theorem (positive integer n) Find a Specific Coefficient in a Single Binomial Expansion View
Find the coefficient of $x ^ { 2 }$ in the expansion of $\left( x + \frac { 3 } { x ^ { 2 } } \right) ^ { 5 }$. [3 points]
Q23 Product & Quotient Rules View
For the function $f ( x ) = \frac { x ^ { 2 } - 2 x - 6 } { x - 1 }$, find the value of $f ^ { \prime } ( 0 )$.
Q24 3 marks Applied differentiation Limit evaluation involving derivatives or asymptotic analysis View
As shown in the figure, in a right triangle ABC with $\overline { \mathrm { AB } } = 2$ and $\angle \mathrm {~B} = \frac { \pi } { 2 }$, let D and E be the points where the circle with center A and radius 1 meets the two segments $\mathrm { AB }$ and $\mathrm { AC }$ respectively. Let F be the trisection point of arc DE closer to point D, and let G be the point where line AF meets segment BC. Let $\angle \mathrm { BAG } = \theta$. Let $f ( \theta )$ be the area of the common part of the interior of triangle ABG and the exterior of sector ADF, and let $g ( \theta )$ be the area of sector AFE. Find the value of $40 \times \lim _ { \theta \rightarrow 0 + } \frac { f ( \theta ) } { g ( \theta ) }$. (where $0 < \theta < \frac { \pi } { 6 }$) [3 points]
Q25 3 marks Arithmetic Sequences and Series Summation of Derived Sequence from AP View
For an arithmetic sequence $\left\{ a _ { n } \right\}$ with first term 3, if $\sum _ { k = 1 } ^ { 5 } a _ { k } = 55$, find the value of $\sum _ { k = 1 } ^ { 5 } k \left( a _ { k } - 3 \right)$. [3 points]
Q26 4 marks Permutations & Arrangements Circular Arrangement View
There are 6 students including three students $\mathrm { A } , \mathrm { B } , \mathrm { C }$.
Find the number of ways these 6 students can all sit around a circular table with equal spacing satisfying the following conditions. (Note: rotations that coincide are considered the same.) [4 points] (가) A and B are adjacent. (나) B and C are not adjacent.
Q27 4 marks Laws of Logarithms Simplify or Evaluate a Logarithmic Expression View
Find the number of natural numbers $n$ such that $\log _ { 4 } 2 n ^ { 2 } - \frac { 1 } { 2 } \log _ { 2 } \sqrt { n }$ is a natural number not exceeding 40. [4 points]
Q28 4 marks Stationary points and optimisation Determine parameters from given extremum conditions View
For two constants $a$ and $b$ with $a < b$, define the function $f ( x )$ as $$f ( x ) = ( x - a ) ( x - b ) ^ { 2 }$$ For the inverse function $g ^ { - 1 } ( x )$ of the function $g ( x ) = x ^ { 3 } + x + 1$, the composite function $h ( x ) = \left( f \circ g ^ { - 1 } \right) ( x )$ satisfies the following conditions. Find the value of $f ( 8 )$. [4 points] (가) The function $( x - 1 ) | h ( x ) |$ is differentiable on the set of all real numbers. (나) $h ^ { \prime } ( 3 ) = 2$
Q29 4 marks Combinations & Selection Counting Integer Solutions to Equations View
Find the number of ways to distribute 6 black hats and 6 white hats among four students $\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D }$ according to the following rules without remainder. (Note: hats of the same color are not distinguished from each other.) [4 points] (가) Each student receives at least 1 hat. (나) The number of black hats each student receives is different from one another.