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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
2015 csat__math-A

30 maths questions

Q1 2 marks Indices and Surds Evaluating Expressions Using Index Laws View
What is the value of $5 \times 8 ^ { \frac { 1 } { 3 } }$? [2 points]
(1) 10
(2) 15
(3) 20
(4) 25
(5) 30
Q2 2 marks Matrices Matrix Algebra and Product Properties View
For two matrices $A = \left( \begin{array} { l l } 1 & 1 \\ 0 & 2 \end{array} \right) , B = \left( \begin{array} { l l } 1 & 1 \\ 3 & 0 \end{array} \right)$, what is the sum of all components of the matrix $A + B$? [2 points]
(1) 5
(2) 6
(3) 7
(4) 8
(5) 9
Q3 2 marks Sign Change & Interval Methods View
What is the value of $\lim _ { n \rightarrow \infty } \frac { 4 n ^ { 2 } + 6 } { n ^ { 2 } + 3 n }$? [2 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5
Q4 3 marks Matrices Structured Matrix Characterization View
In the following graph, how many zeros are there among the components of the matrix representing the connection relationships between vertices? [3 points]
(1) 9
(2) 11
(3) 13
(4) 15
(5) 17
Q5 3 marks Geometric Sequences and Series Finite Geometric Sum and Term Relationships View
For a geometric sequence $\left\{ a _ { n } \right\}$ with positive common ratio, if $a _ { 1 } = 3 , a _ { 5 } = 48$, what is the value of $a _ { 3 }$? [3 points]
(1) 18
(2) 16
(3) 14
(4) 12
(5) 10
Q6 3 marks Binomial Theorem (positive integer n) Determine Parameters from Conditions on Coefficients or Terms View
In the expansion of the polynomial $( x + a ) ^ { 6 }$, if the coefficient of $x ^ { 4 }$ is 60, what is the value of the positive number $a$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5
Q7 3 marks Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View
If $\int _ { 0 } ^ { 1 } ( 2 x + a ) d x = 4$, what is the value of the constant $a$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5
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 $\lim _ { x \rightarrow - 0 } f ( x ) + \lim _ { x \rightarrow 1 + 0 } f ( x )$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5
Q9 3 marks Laws of Logarithms Logarithmic Formula Application (Modeling) View
When compressing digital images, let $P$ denote the peak signal-to-noise ratio, which is an indicator of the difference between the original and compressed images, and let $E$ denote the mean squared error between the original and compressed images. The following relationship holds: $$P = 20 \log 255 - 10 \log E \quad ( E > 0 )$$ When two original images $A$ and $B$ are compressed, let $P _ { A }$ and $P _ { B }$ denote their peak signal-to-noise ratios, and let $E _ { A } \left( E _ { A } > 0 \right)$ and $E _ { B } \left( E _ { B } > 0 \right)$ denote their mean squared errors. If $E _ { B } = 100 E _ { A }$, what is the value of $P _ { A } - P _ { B }$? [3 points]
(1) 30
(2) 25
(3) 20
(4) 15
(5) 10
Q10 3 marks Arithmetic Sequences and Series Find Specific Term from Given Conditions View
For a sequence $\left\{ a _ { n } \right\}$, if the sum of the first $n$ terms $S _ { n } = \frac { n } { n + 1 }$, what is the value of $a _ { 4 }$? [3 points]
(1) $\frac { 1 } { 22 }$
(2) $\frac { 1 } { 20 }$
(3) $\frac { 1 } { 18 }$
(4) $\frac { 1 } { 16 }$
(5) $\frac { 1 } { 14 }$
Q11 3 marks Geometric Sequences and Series Sum of an Infinite Geometric Series (Direct Computation) View
For a geometric sequence $\left\{ a _ { n } \right\}$ with $a _ { 1 } = 3 , a _ { 2 } = 1$, what is the value of $\sum _ { n = 1 } ^ { \infty } \left( a _ { n } \right) ^ { 2 }$? [3 points]
(1) $\frac { 81 } { 8 }$
(2) $\frac { 83 } { 8 }$
(3) $\frac { 85 } { 8 }$
(4) $\frac { 87 } { 8 }$
(5) $\frac { 89 } { 8 }$
Q12 3 marks Normal Distribution Direct Probability Calculation from Given Normal Distribution View
A research institute investigated the length of tomato seedling stems 3 weeks after planting. The length of the tomato stems follows a normal distribution with mean 30 cm and standard deviation 2 cm. Using the standard normal distribution table on the right, find the probability that the length of a randomly selected tomato stem is at least 27 cm and at most 32 cm. [3 points]
$z$$\mathrm { P } ( 0 \leq Z \leq z )$
1.00.3413
1.50.4332
2.00.4772
2.50.4938

(1) 0.6826
(2) 0.7745
(3) 0.8185
(4) 0.9104
(5) 0.9270
Q13 3 marks Solving quadratics and applications Evaluating an algebraic expression given a constraint View
For the function $f ( x ) = x ( x + 1 ) ( x - 4 )$, answer the following. For the matrix $A = \left( \begin{array} { l l } 2 & 1 \\ 0 & 3 \end{array} \right)$, what is the sum of all constant values $a$ that satisfy $A \binom { 0 } { f ( a ) } = \binom { 0 } { 0 }$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5
Q14 4 marks Stationary points and optimisation Count or characterize roots using extremum values View
For the function $f ( x ) = x ( x + 1 ) ( x - 4 )$, answer the following. When the line $y = 5 x + k$ and the graph of the function $y = f ( x )$ intersect at two distinct points, what is the value of the positive number $k$? [4 points]
(1) 5
(2) $\frac { 11 } { 2 }$
(3) 6
(4) $\frac { 13 } { 2 }$
(5) 7
Q15 4 marks Indices and Surds Exponential Inequalities and Counting Solutions View
What is the sum of all natural numbers $x$ that satisfy the exponential inequality $\left( \frac { 1 } { 5 } \right) ^ { 1 - 2 x } \leq 5 ^ { x + 4 }$? [4 points]
(1) 11
(2) 12
(3) 13
(4) 14
(5) 15
Q16 4 marks Conditional Probability Direct Conditional Probability Computation from Definitions View
For two events $A$ and $B$, $$\mathrm { P } ( A ) = \frac { 1 } { 3 } , \quad \mathrm { P } ( A \cap B ) = \frac { 1 } { 8 }$$ what is the value of $\mathrm { P } \left( B ^ { C } \mid A \right)$? (Here, $B ^ { C }$ is the complement of $B$.) [4 points]
(1) $\frac { 11 } { 24 }$
(2) $\frac { 1 } { 2 }$
(3) $\frac { 13 } { 24 }$
(4) $\frac { 7 } { 12 }$
(5) $\frac { 5 } { 8 }$
Q17 4 marks Arithmetic Sequences and Series Find Specific Term from Given Conditions View
For an arithmetic sequence $\left\{ a _ { n } \right\}$ satisfying $\sum _ { k = 1 } ^ { n } a _ { 2 k - 1 } = 3 n ^ { 2 } + n$, what is the value of $a _ { 8 }$? [4 points]
(1) 16
(2) 19
(3) 22
(4) 25
(5) 28
Q18 4 marks Combinations & Selection Counting Integer Solutions to Equations View
How many ordered pairs $( x , y , z , w )$ of non-negative integers satisfy the system of equations $$\left\{ \begin{array} { l } x + y + z + 3 w = 14 \\ x + y + z + w = 10 \end{array} \right.$$ ? [4 points]
(1) 40
(2) 45
(3) 50
(4) 55
(5) 60
Q19 4 marks Matrices True/False or Multiple-Select Conceptual Reasoning View
Two $2 \times 2$ square matrices $A$ and $B$ satisfy $$A ^ { 2 } - A B = 3 E , \quad A ^ { 2 } B - B ^ { 2 } A = A + B$$ Among the statements in the given options, which are correct? (Here, $E$ is the identity matrix.) [4 points]
ᄀ. The inverse matrix of $A$ exists. ㄴ. $A B = B A$ ㄷ. $( A + 2 B ) ^ { 2 } = 24 E$
(1) ᄀ
(2) ᄃ
(3) ᄀ, ᄂ
(4) ᄂ, ᄃ
(5) ᄀ, ᄂ, ᄃ
Q20 4 marks Indefinite & Definite Integrals Piecewise/Periodic Function Integration View
The function $f ( x )$ satisfies $f ( x + 3 ) = f ( x )$ for all real numbers $x$, and $$f ( x ) = \begin{cases} x & ( 0 \leq x < 1 ) \\ 1 & ( 1 \leq x < 2 ) \\ - x + 3 & ( 2 \leq x < 3 ) \end{cases}$$ If $\int _ { - a } ^ { a } f ( x ) d x = 13$, what is the value of the constant $a$? [4 points]
(1) 10
(2) 12
(3) 14
(4) 16
(5) 18
Q21 4 marks Stationary points and optimisation Find absolute extrema on a closed interval or domain View
For all cubic functions $f ( x )$ satisfying the following conditions, what is the minimum value of $f ( 2 )$? [4 points] (가) The leading coefficient of $f ( x )$ is 1. (나) $f ( 0 ) = f ^ { \prime } ( 0 )$ (다) For all real numbers $x \geq - 1$, $f ( x ) \geq f ^ { \prime } ( x )$.
(1) 28
(2) 33
(3) 38
(4) 43
(5) 48
Q22 3 marks Curve Sketching Limit Computation from Algebraic Expressions View
Find the value of $\lim _ { x \rightarrow 0 } \frac { x ( x + 7 ) } { x }$. [3 points]
Q23 3 marks Curve Sketching Finding Parameters for Continuity View
For the function $$f ( x ) = \begin{cases} 2 x + 10 & ( x < 1 ) \\ x + a & ( x \geq 1 ) \end{cases}$$ find the value of the constant $a$ such that $f$ is continuous on the entire set of real numbers. [3 points]
Q24 3 marks Geometric Sequences and Series Sum of an Infinite Geometric Series (Direct Computation) View
For two sequences $\left\{ a _ { n } \right\}$ and $\left\{ b _ { n } \right\}$, $$\sum _ { n = 1 } ^ { \infty } a _ { n } = 4 , \quad \sum _ { n = 1 } ^ { \infty } b _ { n } = 10$$ find the value of $\sum _ { n = 1 } ^ { \infty } \left( a _ { n } + 5 b _ { n } \right)$. [3 points]
Q25 3 marks Binomial Distribution Find Parameters from Moment Conditions View
A random variable $X$ follows a binomial distribution $\mathrm { B } \left( n , \frac { 1 } { 3 } \right)$ and $\mathrm { V} ( 3 X ) = 40$. Find the value of $n$. [3 points]
Q26 4 marks Indefinite & Definite Integrals Recovering Function Values from Derivative Information View
The derivative $f ^ { \prime } ( x )$ of a polynomial function $f ( x )$ is $f ^ { \prime } ( x ) = 6 x ^ { 2 } + 4$. If the graph of $y = f ( x )$ passes through the point $( 0,6 )$, find the value of $f ( 1 )$. [4 points]
Q27 4 marks Continuous Probability Distributions and Random Variables PDF Graph Interpretation and Probability Computation View
For a continuous random variable $X$ that takes all real values in the interval $[ 0,3 ]$, the graph of the probability density function of $X$ is shown in the figure. If $\mathrm { P } ( 0 \leq X \leq 2 ) = \frac { q } { p }$, find the value of $p + q$. (Here, $k$ is a constant, and $p$ and $q$ are coprime natural numbers.) [4 points]
Q28 4 marks Sequences and series, recurrence and convergence Summation of sequence terms View
For a natural number $k$, $$a _ { k } = \lim _ { n \rightarrow \infty } \frac { \left( \frac { 6 } { k } \right) ^ { n + 1 } } { \left( \frac { 6 } { k } \right) ^ { n } + 1 }$$ Find the value of $\sum _ { k = 1 } ^ { 10 } k a _ { k }$. [4 points]
Q29 4 marks Stationary points and optimisation Determine parameters from given extremum conditions View
Two polynomial functions $f ( x )$ and $g ( x )$ satisfy $$g ( x ) = \left( x ^ { 3 } + 2 \right) f ( x )$$ for all real numbers $x$. If $g ( x )$ has a local minimum value of 24 at $x = 1$, find the value of $f ( 1 ) - f ^ { \prime } ( 1 )$. [4 points]
Q30 4 marks Curve Sketching Lattice Points and Counting via Graph Geometry View
In the coordinate plane, let $f ( n )$ denote the number of triangles OAB satisfying the following conditions for a natural number $n$. Find the value of $f ( 1 ) + f ( 2 ) + f ( 3 )$. (Here, O is the origin.) [4 points] (가) The coordinates of point A are $\left( - 2,3 ^ { n } \right)$. (나) If the coordinates of point B are $( a , b )$, then $a$ and $b$ are natural numbers and satisfy $b \leq \log _ { 2 } a$. (다) The area of triangle OAB is at most 50.