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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
2014 csat__math-A

30 maths questions

Q1 2 marks Indices and Surds Evaluating Expressions Using Index Laws View
What is the value of $8 ^ { \frac { 2 } { 3 } } \times 9 ^ { \frac { 1 } { 2 } }$? [2 points]
(1) 4
(2) 6
(3) 8
(4) 10
(5) 12
Q2 2 marks Sequences and series, recurrence and convergence Convergence proof and limit determination View
What is the value of $\lim _ { n \rightarrow \infty } \frac { 2 \times 3 ^ { n + 1 } + 5 } { 3 ^ { n } }$? [2 points]
(1) 6
(2) 7
(3) 8
(4) 9
(5) 10
Q3 2 marks Matrices Matrix Algebra and Product Properties View
For two matrices $A = \left( \begin{array} { l l } 2 & 0 \\ 1 & 1 \end{array} \right) , B = \left( \begin{array} { l l } 1 & 1 \\ 0 & 1 \end{array} \right)$, what is the sum of all components of matrix $A + 2B$? [2 points]
Q4 Matrices Structured Matrix Characterization View
A graph and the matrix representing the connection relationships between each vertex of the graph are as follows. What is the value of $a + b + c + d + e$?
A B C D E
$$\left( \begin{array} { l l l l l } 0 & 1 & 1 & 0 & a \\ 1 & 0 & 1 & b & 1 \\ 1 & 1 & c & 1 & 0 \\ 0 & d & 1 & 0 & 1 \\ e & 1 & 0 & 1 & 0 \end{array} \right)$$
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5
Q5 3 marks Differentiation from First Principles View
For the function $f ( x ) = 2 x ^ { 2 } + a x$, when $\lim _ { h \rightarrow 0 } \frac { f ( 1 + h ) - f ( 1 ) } { h } = 6$, what is the value of the constant $a$? [3 points]
(1) $-4$
(2) $-2$
(3) $0$
(4) $2$
(5) $4$
Q6 3 marks Arithmetic Sequences and Series Find Common Difference from Given Conditions View
For an arithmetic sequence $\left\{ a _ { n } \right\}$ with first term 6 and common difference $d$, let $S _ { n }$ denote the sum of the first $n$ terms. When
$$\frac { a _ { 8 } - a _ { 6 } } { S _ { 8 } - S _ { 6 } } = 2$$
holds, what is the value of $d$? [3 points]
(1) $-1$
(2) $-2$
(3) $-3$
(4) $-4$
(5) $-5$
Q7 3 marks Independent Events View
Two events $A , B$ are independent, and $\mathrm { P } ( A ) = \frac { 1 } { 3 } , \mathrm { P } ( B ) = \frac { 1 } { 3 }$. What is the value of $\mathrm { P } \left( A \cap B ^ { C } \right)$? (Here, $B ^ { C }$ is the complement of $B$.) [3 points]
(1) $\frac { 5 } { 27 }$
(2) $\frac { 2 } { 9 }$
(3) $\frac { 7 } { 27 }$
(4) $\frac { 8 } { 27 }$
(5) $\frac { 1 } { 3 }$
Q8 3 marks Areas by integration View
What is the area of the region enclosed by the curve $y = x ^ { 2 } - 4 x + 3$ and the line $y = 3$? [3 points]
(1) 10
(2) $\frac { 31 } { 3 }$
(3) $\frac { 32 } { 3 }$
(4) 11
(5) $\frac { 34 } { 3 }$
Q9 3 marks Binomial Distribution Find Parameters from Moment Conditions View
A random variable $X$ follows a binomial distribution $\mathrm { B } ( 9 , p )$, and $\{ \mathrm { E } ( X ) \} ^ { 2 } = \mathrm { V } ( X )$. What is the value of $p$? (Here, $0 < p < 1$) [3 points]
(1) $\frac { 1 } { 13 }$
(2) $\frac { 1 } { 12 }$
(3) $\frac { 1 } { 11 }$
(4) $\frac { 1 } { 10 }$
(5) $\frac { 1 } { 9 }$
Q10 3 marks Laws of Logarithms Logarithmic Formula Application (Modeling) View
Water flows completely through a cylindrical water pipe with cross-sectional radius $R ( R < 1 )$. Let $v _ { c }$ be the speed of water at the center of the cross-section, and let $v$ be the speed of water at a point $x ( 0 < x \leq R )$ away from the wall toward the center. The following relationship holds:
$$\frac { v _ { c } } { v } = 1 - k \log \frac { x } { R }$$
(Here, $k$ is a positive constant, the unit of length is m, and the unit of speed is m/s.)
When the speed of water at a point $R ^ { \frac { 27 } { 23 } }$ away from the wall toward the center is $\frac { 1 } { 2 }$ of the speed at the center, the speed of water at a point $R ^ { a }$ away from the wall toward the center is $\frac { 1 } { 3 }$ of the speed at the center. What is the value of $a$? [3 points]
(1) $\frac { 39 } { 23 }$
(2) $\frac { 37 } { 23 }$
(3) $\frac { 35 } { 23 }$
(4) $\frac { 33 } { 23 }$
(5) $\frac { 31 } { 23 }$
Q11 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
Q12 3 marks Linear combinations of normal random variables View
A pharmaceutical company produces medicine bottles with capacity following a normal distribution with mean $m$ and standard deviation 10. When a random sample of 25 bottles is taken from the company's production, the probability that the sample mean capacity is at least 2000 is 0.9772. Using the standard normal distribution table below, what is the value of $m$? (Here, the unit of capacity is mL.) [3 points]
$z$$\mathrm { P } ( 0 \leq Z \leq z )$
1.50.4332
2.00.4772
2.50.4938
3.00.4987

(1) 2003
(2) 2004
(3) 2005
(4) 2006
(5) 2007
Q13 3 marks Arithmetic Sequences and Series Compute Partial Sum of an Arithmetic Sequence View
For a natural number $n$, $f ( n )$ is defined as follows:
$$f ( n ) = \begin{cases} \log _ { 3 } n & ( n \text{ is odd} ) \\ \log _ { 2 } n & ( n \text{ is even} ) \end{cases}$$
For the sequence $\left\{ a _ { n } \right\}$ where $a _ { n } = f \left( 6 ^ { n } \right) - f \left( 3 ^ { n } \right)$, what is the value of $\sum _ { n = 1 } ^ { 15 } a _ { n }$? [3 points]
(1) $120 \left( \log _ { 2 } 3 - 1 \right)$
(2) $105 \log _ { 3 } 2$
(3) $105 \log _ { 2 } 3$
(4) $120 \log _ { 2 } 3$
(5) $120 \left( \log _ { 3 } 2 + 1 \right)$
Q14 Permutations & Arrangements Counting Functions with Constraints View
For a natural number $n$, $f ( n )$ is defined as follows:
$$f ( n ) = \begin{cases} \log _ { 3 } n & ( n \text{ is odd} ) \\ \log _ { 2 } n & ( n \text{ is even} ) \end{cases}$$
For two natural numbers $m , n$ with $m, n \leq 20$, how many ordered pairs $( m , n )$ satisfy $f ( m n ) = f ( m ) + f ( n )$?
(1) 220
(2) 230
(3) 240
(4) 250
(5) 260
Q15 4 marks Probability Definitions Conditional Probability and Bayes' Theorem View
Bag A contains 2 white balls and 3 black balls, and Bag B contains 1 white ball and 3 black balls. One ball is randomly drawn from Bag A. If it is white, 2 white balls are put into Bag B; if it is black, 2 black balls are put into Bag B. Then one ball is randomly drawn from Bag B. What is the probability that the drawn ball is white? [4 points]
(1) $\frac { 1 } { 6 }$
(2) $\frac { 1 } { 5 }$
(3) $\frac { 7 } { 30 }$
(4) $\frac { 4 } { 15 }$
(5) $\frac { 3 } { 10 }$
Q16 4 marks Sequences and series, recurrence and convergence Auxiliary sequence transformation View
A sequence $\left\{ a _ { n } \right\}$ with all positive terms has $a _ { 1 } = 10$ and satisfies
$$\left( a _ { n + 1 } \right) ^ { n } = 10 \left( a _ { n } \right) ^ { n + 1 } \quad ( n \geq 1 )$$
The following is the process of finding the general term $a _ { n }$.
Taking the common logarithm of both sides of the given equation:
$$n \log a _ { n + 1 } = ( n + 1 ) \log a _ { n } + 1$$
Dividing both sides by $n ( n + 1 )$:
$$\frac { \log a _ { n + 1 } } { n + 1 } = \frac { \log a _ { n } } { n } + ( \text{(가)} )$$
Let $b _ { n } = \frac { \log a _ { n } } { n }$. Then $b _ { 1 } = 1$ and
$$b _ { n + 1 } = b _ { n } + \text{(가)}$$
Finding the general term of the sequence $\left\{ b _ { n } \right\}$:
$$b _ { n } = \text{(나)}$$
Therefore,
$$\log a _ { n } = n \times \text{(나)}$$
Thus $a _ { n } = 10 ^ { n \times \text{(나)} }$.
Let $f ( n )$ and $g ( n )$ be the expressions that fit in (가) and (나), respectively. What is the value of $\frac { g ( 10 ) } { f ( 4 ) }$? [4 points]
(1) 38
(2) 40
(3) 42
(4) 44
(5) 46
Q17 4 marks Radians, Arc Length and Sector Area View
In rectangle $\mathrm { A } _ { 1 } \mathrm {~B} _ { 1 } \mathrm { C } _ { 1 } \mathrm { D } _ { 1 }$, $\overline { \mathrm { A } _ { 1 } \mathrm {~B} _ { 1 } } = 1$ and $\overline { \mathrm {~A} _ { 1 } \mathrm { D } _ { 1 } } = 2$. As shown in the figure, let $\mathrm { M } _ { 1 }$ and $\mathrm {~N} _ { 1 }$ be the midpoints of segments $\mathrm { A } _ { 1 } \mathrm { D } _ { 1 }$ and $\mathrm { B } _ { 1 } \mathrm { C } _ { 1 }$, respectively.
Draw a circular sector $\mathrm { N } _ { 1 } \mathrm { M } _ { 1 } \mathrm {~B} _ { 1 }$ with center $\mathrm { N } _ { 1 }$, radius $\overline { \mathrm { B } _ { 1 } \mathrm {~N} _ { 1 } }$, and central angle $\frac { \pi } { 2 }$, and draw a circular sector $\mathrm { D } _ { 1 } \mathrm { M } _ { 1 } \mathrm { C } _ { 1 }$ with center $\mathrm { D } _ { 1 }$, radius $\overline { \mathrm { C } _ { 1 } \mathrm { D } _ { 1 } }$, and central angle $\frac { \pi } { 2 }$. The region bounded by the arc $\mathrm { M } _ { 1 } \mathrm {~B} _ { 1 }$ and segment $\mathrm { M } _ { 1 } \mathrm {~B} _ { 1 }$ of sector $\mathrm { N } _ { 1 } \mathrm { M } _ { 1 } \mathrm {~B} _ { 1 }$ and the region bounded by the arc $\mathrm { M } _ { 1 } \mathrm { C } _ { 1 }$ and segment $\mathrm { M } _ { 1 } \mathrm { C } _ { 1 }$ of sector $\mathrm { D } _ { 1 } \mathrm { M } _ { 1 } \mathrm { C } _ { 1 }$ form a checkmark shape. Color this shape to obtain figure $R _ { 1 }$.
In figure $R _ { 1 }$, construct a rectangle $\mathrm { A } _ { 2 } \mathrm {~B} _ { 2 } \mathrm { C } _ { 2 } \mathrm { D } _ { 2 }$ with vertices at point $\mathrm { A } _ { 2 }$ on segment $\mathrm { M } _ { 1 } \mathrm {~B} _ { 1 }$, point $\mathrm { D } _ { 2 }$ on arc $\mathrm { M } _ { 1 } \mathrm { C } _ { 1 }$, and two points $\mathrm { B } _ { 2 } , \mathrm { C } _ { 2 }$ on side $\mathrm { B } _ { 1 } \mathrm { C } _ { 1 }$, such that $\overline { \mathrm { A } _ { 2 } \mathrm {~B} _ { 2 } } : \overline { \mathrm { A } _ { 2 } \mathrm { D } _ { 2 } } = 1 : 2$. Color the shape created in the same way as for figure $R _ { 1 }$ to obtain figure $R _ { 2 }$.
Continue this process. Let $S _ { n }$ be the area of the colored region in figure $R _ { n }$ obtained at the $n$-th step. What is the value of $\lim _ { n \rightarrow \infty } S _ { n }$? [4 points]
(1) $\frac { 25 } { 19 } \left( \frac { \pi } { 2 } - 1 \right)$
(2) $\frac { 5 } { 4 } \left( \frac { \pi } { 2 } - 1 \right)$
(3) $\frac { 25 } { 21 } \left( \frac { \pi } { 2 } - 1 \right)$
(4) $\frac { 25 } { 22 } \left( \frac { \pi } { 2 } - 1 \right)$
(5) $\frac { 25 } { 23 } \left( \frac { \pi } { 2 } - 1 \right)$
Q18 4 marks Combinations & Selection Counting Integer Solutions to Equations View
There are 8 white ping-pong balls and 7 orange ping-pong balls to be distributed entirely among 3 students. In how many ways can the balls be distributed so that each student receives at least one white ball and at least one orange ball? [4 points]
(1) 295
(2) 300
(3) 305
(4) 310
(5) 315
Q19 4 marks Matrices True/False or Multiple-Select Conceptual Reasoning View
Two square matrices $A , B$ satisfy
$$A B + A ^ { 2 } B = E , \quad ( A - E ) ^ { 2 } + B ^ { 2 } = O$$
Which of the following statements in the given options are correct? (Here, $E$ is the identity matrix and $O$ is the zero matrix.) [4 points]
Options
$\text{ᄀ}$. The inverse matrix of $B$ exists. $\text{ㄴ}$. $A B = B A$ $\text{ㄷ}$. $\left( A ^ { 3 } - A \right) ^ { 2 } + E = O$
(1) ㄴ
(2) ㄷ
(3) ᄀ, ㄴ
(4) ᄀ, ㄷ
(5) ᄀ, ㄴ, ㄷ
Q20 4 marks Laws of Logarithms Characteristic and Mantissa of Common Logarithms View
For a positive real number $x$, let $f ( x )$ and $g ( x )$ be the characteristic and mantissa of $\log x$, respectively. For a natural number $n$, let $a _ { n }$ be the product of all values of $x$ satisfying $f ( x ) - ( n + 1 ) g ( x ) = n$. What is the value of $\lim _ { n \rightarrow \infty } \frac { \log a _ { n } } { n ^ { 2 } }$? [4 points]
(1) 1
(2) $\frac { 3 } { 2 }$
(3) 2
(4) $\frac { 5 } { 2 }$
(5) 3
Q21 4 marks Stationary points and optimisation Determine parameters from given extremum conditions View
On the coordinate plane, for a cubic function $f ( x ) = x ^ { 3 } + a x ^ { 2 } + b x$ and a real number $t$, let P be the point where the tangent line to the curve $y = f ( x )$ at the point $( t , f ( t ) )$ intersects the $y$-axis. Let $g ( t )$ be the distance from the origin to point P. The function $f ( x )$ and the function $g ( t )$ satisfy the following conditions. (가) $f ( 1 ) = 2$ (나) The function $g ( t )$ is differentiable on the entire set of real numbers. What is the value of $f ( 3 )$? (Here, $a , b$ are constants.) [4 points]
(1) 21
(2) 24
(3) 27
(4) 30
(5) 33
Q22 3 marks Sign Change & Interval Methods View
Find the value of $\lim _ { x \rightarrow 0 } \sqrt { 2 x + 9 }$. [3 points]
Q23 3 marks Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View
For a real number $a$, when $\int _ { - a } ^ { a } \left( 3 x ^ { 2 } + 2 x \right) d x = \frac { 1 } { 4 }$, find the value of $50 a$. [3 points]
Q24 3 marks Arithmetic Sequences and Series Sequence Defined by Recurrence with AP Connection View
A sequence $\left\{ a _ { n } \right\}$ satisfies the following conditions. (가) $a _ { 1 } = a _ { 2 } + 3$ (나) $a _ { n + 1 } = - 2 a _ { n } ( n \geq 1 )$ Find the value of $a _ { 9 }$. [3 points]
Q25 4 marks Matrices Determinant and Rank Computation View
For the system of linear equations in $x , y$:
$$\left( \begin{array} { l l } 5 & a \\ a & 3 \end{array} \right) \binom { x } { y } = \binom { x + 5 y } { 6 x + y }$$
Find the sum of all real values of $a$ such that the system has a solution other than $x = 0 , y = 0$. [4 points]
Q26 3 marks Stationary points and optimisation Determine parameters from given extremum conditions View
The function $f ( x ) = 2 x ^ { 3 } - 12 x ^ { 2 } + a x - 4$ has a local maximum value $M$ at $x = 1$. Find the value of $a + M$. (Here, $a$ is a constant.) [3 points]
Q27 4 marks Discrete Probability Distributions Expectation and Variance from Context-Based Random Variables View
There are 5 drawers, each labeled with a natural number from 1 to 5. Two drawers are randomly assigned to Younghee. Let $X$ be the random variable representing the smaller of the two natural numbers on the assigned drawers. Find the value of $\mathrm { E } ( 10 X )$. [4 points]
Q28 4 marks Curve Sketching Finding Parameters for Continuity View
For the function
$$f ( x ) = \begin{cases} x + 1 & ( x \leq 0 ) \\ - \frac { 1 } { 2 } x + 7 & ( x > 0 ) \end{cases}$$
Find the sum of all real values of $a$ such that the function $f ( x ) f ( x - a )$ is continuous at $x = a$. [4 points]
Q29 4 marks Integration by Substitution Substitution to Evaluate a Definite Integral (Numerical Answer) View
The function $f ( x ) = 3 x ^ { 2 } - a x$ satisfies
$$\lim _ { n \rightarrow \infty } \frac { 1 } { n } \sum _ { k = 1 } ^ { n } f \left( \frac { 3 k } { n } \right) = f ( 1 )$$
Find the value of the constant $a$. [4 points]
Q30 4 marks Polar coordinates View
On the coordinate plane, for a natural number $a > 1$, consider the two curves $y = 4 ^ { x } , y = a ^ { - x + 4 }$ and the line $y = 1$. Find the number of values of $a$ such that the number of points with integer coordinates inside or on the boundary of the region enclosed by these curves and line is between 20 and 40 (inclusive). [4 points]