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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
2016 csat__math-A

30 maths questions

Q1 2 marks Matrices Matrix Algebra and Product Properties View
For two matrices $A = \left( \begin{array} { l l } 2 & 1 \\ 5 & 0 \end{array} \right) , B = \left( \begin{array} { l l } 1 & 0 \\ 1 & 1 \end{array} \right)$, what is the sum of all components of matrix $A - B$? [2 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5
Q2 2 marks Curve Sketching Limit Computation from Algebraic Expressions View
What is the value of $\lim _ { x \rightarrow - 2 } \frac { ( x + 2 ) \left( x ^ { 2 } + 5 \right) } { x + 2 }$? [2 points]
(1) 7
(2) 8
(3) 9
(4) 10
(5) 11
Q3 2 marks Indices and Surds Evaluating Expressions Using Index Laws View
What is the value of $8 ^ { \frac { 1 } { 3 } } + 27 ^ { \frac { 2 } { 3 } }$? [2 points]
(1) 8
(2) 9
(3) 10
(4) 11
(5) 12
Q4 3 marks Matrices Determinant and Rank Computation View
In the following graph, how many 1's are there among the components of the matrix representing the connection relationships between vertices? [3 points]
(1) 10
(2) 14
(3) 18
(4) 22
(5) 26
Q5 3 marks Chain Rule Straightforward Polynomial or Basic Differentiation View
For the function $f ( x ) = x ^ { 3 } + 7 x + 3$, what is the value of $f ^ { \prime } ( 1 )$? [3 points]
(1) 4
(2) 6
(3) 8
(4) 10
(5) 12
Q6 3 marks Geometric Sequences and Series Finite Geometric Sum and Term Relationships View
For a geometric sequence $\left\{ a _ { n } \right\}$ with a non-zero first term, $$a _ { 3 } = 4 a _ { 1 } , \quad a _ { 7 } = \left( a _ { 6 } \right) ^ { 2 }$$ what is the value of the first term $a _ { 1 }$? [3 points]
(1) $\frac { 1 } { 16 }$
(2) $\frac { 1 } { 8 }$
(3) $\frac { 3 } { 16 }$
(4) $\frac { 1 } { 4 }$
(5) $\frac { 5 } { 16 }$
Q7 3 marks Conditional Probability Direct Conditional Probability Computation from Definitions View
For two events $A , B$, $$\mathrm { P } ( A ) = \frac { 2 } { 5 } , \quad \mathrm { P } ( B \mid A ) = \frac { 5 } { 6 }$$ what is the value of $\mathrm { P } ( A \cap B )$? [3 points]
(1) $\frac { 1 } { 3 }$
(2) $\frac { 4 } { 15 }$
(3) $\frac { 1 } { 5 }$
(4) $\frac { 2 } { 15 }$
(5) $\frac { 1 } { 15 }$
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 $\lim _ { x \rightarrow - 1 - 0 } f ( x ) + \lim _ { x \rightarrow + 0 } f ( x )$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5
Q9 3 marks Measures of Location and Spread View
Let $\bar { X }$ be the sample mean obtained by randomly extracting a sample of size $n$ from a population with population standard deviation 14. When $\sigma ( \bar { X } ) = 2$, what is the value of $n$? [3 points]
(1) 9
(2) 16
(3) 25
(4) 36
(5) 49
Q10 3 marks Discriminant and conditions for roots Parameter range for specific root conditions (location/count) View
For a sequence $\left\{ a _ { n } \right\}$, the curve $y = x ^ { 2 } - ( n + 1 ) x + a _ { n }$ intersects the $x$-axis, and the curve $y = x ^ { 2 } - n x + a _ { n }$ does not intersect the $x$-axis. What is the value of $\lim _ { n \rightarrow \infty } \frac { a _ { n } } { n ^ { 2 } }$? [3 points]
(1) $\frac { 1 } { 20 }$
(2) $\frac { 1 } { 10 }$
(3) $\frac { 3 } { 20 }$
(4) $\frac { 1 } { 5 }$
(5) $\frac { 1 } { 4 }$
Q11 3 marks Inequalities Integer Solutions of an Inequality View
For the logarithmic inequality in $x$ $$\log _ { 5 } ( x - 1 ) \leq \log _ { 5 } \left( \frac { 1 } { 2 } x + k \right)$$ when the number of all integers $x$ satisfying this inequality is 3, what is the value of the natural number $k$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5
Q12 3 marks Normal Distribution Direct Probability Calculation from Given Normal Distribution View
At a rice collection event, the weight of rice donated by each student follows a normal distribution with mean 1.5 kg and standard deviation 0.2 kg. When one student is randomly selected from those who participated in the event, what is the probability that the weight of rice donated by this student is at least 1.3 kg and at most 1.8 kg, using the standard normal distribution table below? [3 points]
$z$$\mathrm { P } ( 0 \leq Z \leq z )$
1.000.3413
1.250.3944
1.500.4332
1.750.4599

(1) 0.8543
(2) 0.8012
(3) 0.7745
(4) 0.7357
(5) 0.6826
Q13 3 marks Areas by integration View
For a natural number $n$, let P be the point with coordinates $( 0,2 n + 1 )$, and let Q be the point on the graph of the function $f ( x ) = n x ^ { 2 }$ with $y$-coordinate 1 in the first quadrant. When $n = 1$, what is the area of the region enclosed by the line segment PQ, the curve $y = f ( x )$, and the $y$-axis? [3 points]
(1) $\frac { 3 } { 2 }$
(2) $\frac { 19 } { 12 }$
(3) $\frac { 5 } { 3 }$
(4) $\frac { 7 } { 4 }$
(5) $\frac { 11 } { 6 }$
Q14 4 marks Areas by integration View
For a natural number $n$, let P be the point with coordinates $( 0,2 n + 1 )$, and let Q be the point on the graph of the function $f ( x ) = n x ^ { 2 }$ with $y$-coordinate 1 in the first quadrant. For the point $\mathrm { R } ( 0,1 )$, let $S _ { n }$ be the area of triangle PRQ and $l _ { n }$ be the length of line segment PQ. What is the value of $\lim _ { n \rightarrow \infty } \frac { S _ { n } ^ { 2 } } { l _ { n } }$? [4 points]
(1) $\frac { 3 } { 2 }$
(2) $\frac { 5 } { 4 }$
(3) 1
(4) $\frac { 3 } { 4 }$
(5) $\frac { 1 } { 2 }$
Q15 4 marks Geometric Sequences and Series Fractal/Iterative Geometric Construction (Area, Length, or Perimeter Series) View
As shown in the figure, let $\mathrm { P } _ { 1 } , \mathrm { P } _ { 2 } , \mathrm { P } _ { 3 } , \mathrm { P } _ { 4 }$ be the five equal division points of the diagonal BD of a square ABCD with side length 5, in order from point B. Draw squares with line segments $\mathrm { BP } _ { 1 } , \mathrm { P } _ { 2 } \mathrm { P } _ { 3 } , \mathrm { P } _ { 4 } \mathrm { D }$ as diagonals respectively, and circles with line segments $\mathrm { P } _ { 1 } \mathrm { P } _ { 2 } , \mathrm { P } _ { 3 } \mathrm { P } _ { 4 }$ as diameters respectively. Color the figure-eight-shaped regions to obtain the figure $R _ { 1 }$. In figure $R _ { 1 }$, let $\mathrm { Q } _ { 1 }$ be the vertex of the square with diagonal $\mathrm { P } _ { 2 } \mathrm { P } _ { 3 }$ that is closest to point A, and let $\mathrm { Q } _ { 2 }$ be the vertex closest to point C. Draw a square with diagonal $\mathrm { AQ } _ { 1 }$ and a square with diagonal $\mathrm { CQ } _ { 2 }$, and in these 2 newly drawn squares, draw figure-eight-shaped regions using the same method as for obtaining figure $R _ { 1 }$, and color them to obtain figure $R _ { 2 }$. In figure $R _ { 2 }$, in the square with diagonal $\mathrm { AQ } _ { 1 }$ and the square with diagonal $\mathrm { CQ } _ { 2 }$, draw figure-eight-shaped regions using the same method as for obtaining figure $R _ { 2 }$ from figure $R _ { 1 }$, and color them to obtain figure $R _ { 3 }$. Continuing this process, let $S _ { n }$ be the area of the colored region in the $n$-th obtained figure $R _ { n }$. What is the value of $\lim _ { n \rightarrow \infty } S _ { n }$? [4 points]
(1) $\frac { 24 } { 17 } ( \pi + 3 )$
(2) $\frac { 25 } { 17 } ( \pi + 3 )$
(3) $\frac { 26 } { 17 } ( \pi + 3 )$
(4) $\frac { 24 } { 17 } ( 2 \pi + 1 )$
(5) $\frac { 25 } { 17 } ( 2 \pi + 1 )$
Q16 4 marks Exponential Equations & Modelling Evaluate Expression Given Exponential/Logarithmic Conditions View
For a certain financial product, the expected asset $W$ after $t$ years of investing an initial asset $W _ { 0 }$ is given as follows: $$W = \frac { W _ { 0 } } { 2 } 10 ^ { a t } \left( 1 + 10 ^ { a t } \right)$$ (where $W _ { 0 } > 0 , t \geq 0$, and $a$ is a constant.) When an initial asset $w _ { 0 }$ is invested in this financial product, the expected asset after 15 years is 3 times the initial asset. When an initial asset $w _ { 0 }$ is invested in this financial product, the expected asset after 30 years is $k$ times the initial asset. What is the value of the real number $k$? (where $w _ { 0 } > 0$) [4 points]
(1) 9
(2) 10
(3) 11
(4) 12
(5) 13
Q17 4 marks Combinations & Selection Counting Integer Solutions to Equations View
How many ordered pairs $( a , b , c , d , e )$ of non-negative integers satisfy the following conditions? [4 points] (가) Among $a , b , c , d , e$, the number of 0's is 2. (나) $a + b + c + d + e = 10$
(1) 240
(2) 280
(3) 320
(4) 360
(5) 400
Q18 4 marks Matrices True/False or Multiple-Select Conceptual Reasoning View
Two $2 \times 2$ square matrices $A , B$ satisfy $$A + B = ( B A ) ^ { 2 } , \quad A B A = B + E$$ In the following statements, which are correct? (where $E$ is the identity matrix.) [4 points]
Statements ㄱ. $A = B ^ { 2 }$ ㄴ. $B ^ { - 1 } = A ^ { 2 } + E$ ㄷ. $A ^ { 5 } - B ^ { 5 } = A + B$
(1) ㄱ
(2) ㄴ
(3) ㄱ, ㄷ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ
Q19 4 marks Sequences and series, recurrence and convergence Auxiliary sequence transformation View
A sequence $\left\{ a _ { n } \right\}$ with all positive terms has $a _ { 1 } = a _ { 2 } = 1$, and when $S _ { n } = \sum _ { k = 1 } ^ { n } a _ { k }$, $$a _ { n + 1 } = \frac { S _ { n } ^ { 2 } } { S _ { n - 1 } } + ( 2 n - 1 ) S _ { n } \quad ( n \geq 2 )$$ The following is the process of finding the general term $a _ { n }$.
Since $a _ { n + 1 } = S _ { n + 1 } - S _ { n }$, from the given equation we have $$S _ { n + 1 } = \frac { S _ { n } ^ { 2 } } { S _ { n - 1 } } + 2 n S _ { n } \quad ( n \geq 2 )$$ Dividing both sides by $S _ { n }$, $$\frac { S _ { n + 1 } } { S _ { n } } = \frac { S _ { n } } { S _ { n - 1 } } + 2 n$$ Let $b _ { n } = \frac { S _ { n + 1 } } { S _ { n } }$. Then $b _ { 1 } = 2$ and $$b _ { n } = b _ { n - 1 } + 2 n \quad ( n \geq 2 )$$ Finding the general term of the sequence $\left\{ b _ { n } \right\}$, $$b _ { n } = \text { (가) } \times ( n + 1 ) \quad ( n \geq 1 )$$ Therefore, $$S _ { n } = ( \text{가} ) \times \{ ( n - 1 ) ! \} ^ { 2 } \quad ( n \geq 1 )$$ Thus $a _ { 1 } = 1$, and for $n \geq 2$, $$\begin{aligned} a _ { n } & = S _ { n } - S _ { n - 1 } \\ & = \text { (나) } \times \{ ( n - 2 ) ! \} ^ { 2 } \end{aligned}$$ When the expressions for (가) and (나) are $f ( n )$ and $g ( n )$ respectively, what is the value of $f ( 10 ) + g ( 6 )$? [4 points]
(1) 110
(2) 125
(3) 140
(4) 155
(5) 170
Q20 4 marks Integration by Parts Definite Integral Evaluation by Parts View
Two polynomial functions $f ( x ) , g ( x )$ satisfy for all real numbers $x$ $$f ( - x ) = - f ( x ) , \quad g ( - x ) = g ( x )$$ For the function $h ( x ) = f ( x ) g ( x )$, $$\int _ { - 3 } ^ { 3 } ( x + 5 ) h ^ { \prime } ( x ) d x = 10$$ What is the value of $h ( 3 )$? [4 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5
Q21 4 marks Stationary points and optimisation Determine parameters from given extremum conditions View
For all cubic functions $f ( x )$ satisfying $f ( 0 ) = 0$ and the following conditions, let $M$ be the maximum value and $m$ be the minimum value of $\frac { f ^ { \prime } ( 0 ) } { f ( 0 ) }$. What is the value of $M m$? [4 points] (가) The function $| f ( x ) |$ is not differentiable only at $x = - 1$. (나) The equation $f ( x ) = 0$ has at least one real root in the closed interval $[ 3,5 ]$.
(1) $\frac { 1 } { 15 }$
(2) $\frac { 1 } { 10 }$
(3) $\frac { 2 } { 15 }$
(4) $\frac { 1 } { 6 }$
(5) $\frac { 1 } { 5 }$
Q22 3 marks Arithmetic Sequences and Series Find Common Difference from Given Conditions View
For an arithmetic sequence $\left\{ a _ { n } \right\}$, when $a _ { 8 } - a _ { 4 } = 28$, find the common difference of the sequence $\left\{ a _ { n } \right\}$. [3 points]
Q23 3 marks Geometric Sequences and Series Determine the Limit of a Sequence via Geometric Series View
Find the value of $\lim _ { n \rightarrow \infty } \frac { 3 \times 9 ^ { n } - 13 } { 9 ^ { n } }$. [3 points]
Q24 3 marks Invariant lines and eigenvalues and vectors Compute eigenvectors or eigenspaces View
For the system of linear equations in $x , y$ $$\left( \begin{array} { c c } 1 & a - 2 \\ 2 & - 1 \end{array} \right) \binom { x } { y } = 3 \binom { x } { y }$$ Find the value of the constant $a$ such that the system has a solution other than $x = 0 , y = 0$. [3 points]
Q25 3 marks Discrete Probability Distributions Probability Distribution Table Completion and Expectation Calculation View
The probability distribution of a discrete random variable $X$ is shown in the table below.
$X$- 505Total
$\mathrm { P } ( X = x )$$\frac { 1 } { 5 }$$\frac { 1 } { 5 }$$\frac { 3 } { 5 }$1

Find the value of $\mathrm { E } ( 4 X + 3 )$. [3 points]
Q26 4 marks Conditional Probability Total Probability via Tree Diagram (Two-Stage Partition) View
A company has a total of 60 employees, and each employee belongs to one of two departments, A or B. Department A has 20 employees and Department B has 40 employees. 50\% of the employees in Department A are women. 60\% of the women employees in the company belong to Department B. When one employee is randomly selected from the 60 employees and is found to belong to Department B, the probability that this employee is a woman is $p$. Find the value of $80p$. [4 points]
Q27 4 marks Curve Sketching Finding Parameters for Continuity View
Two functions $$f ( x ) = \left\{ \begin{array} { l l } x + 3 & ( x \leq a ) \\ x ^ { 2 } - x & ( x > a ) \end{array} , \quad g ( x ) = x - ( 2 a + 7 ) \right.$$ Find the product of all real values of $a$ such that the function $f ( x ) g ( x )$ is continuous on the entire set of real numbers. [4 points]
Q28 4 marks Tangents, normals and gradients Find tangent line equation at a given point View
Two polynomial functions $f ( x ) , g ( x )$ satisfy the following conditions. (가) $g ( x ) = x ^ { 3 } f ( x ) - 7$ (나) $\lim _ { x \rightarrow 2 } \frac { f ( x ) - g ( x ) } { x - 2 } = 2$
When the equation of the tangent line to the curve $y = g ( x )$ at the point $( 2 , g ( 2 ) )$ is $y = a x + b$, find the value of $a ^ { 2 } + b ^ { 2 }$. (where $a , b$ are constants.) [4 points]
Q29 4 marks Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View
A quadratic function $f ( x )$ satisfies $f ( 0 ) = 0$ and the following conditions. (가) $\int _ { 0 } ^ { 2 } | f ( x ) | d x = - \int _ { 0 } ^ { 2 } f ( x ) d x = 4$ (나) $\int _ { 2 } ^ { 3 } | f ( x ) | d x = \int _ { 2 } ^ { 3 } f ( x ) d x$ Find the value of $f ( 5 )$. [4 points]
Q30 4 marks Laws of Logarithms Logarithmic Function Graph Intersection or Geometric Analysis View
For a real number $x \geq \frac { 1 } { 100 }$, let $f ( x )$ be the mantissa of $\log x$. Let $R$ be the region representing the ordered pairs $( a , b )$ of two real numbers satisfying the following conditions on the coordinate plane. (가) $a < 0$ and $b > 10$. (나) The graph of the function $y = 9 f ( x )$ and the line $y = a x + b$ meet at exactly one point.
For a point $( a , b )$ in region $R$, the minimum value of $( a + 20 ) ^ { 2 } + b ^ { 2 }$ is $100 \times \frac { q } { p }$. Find the value of $p + q$. (where $p$ and $q$ are coprime natural numbers.) [4 points]