csat-suneung

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
2007 csat__math-humanities

30 maths questions

Q1 2 marks Laws of Logarithms Simplify or Evaluate a Logarithmic Expression View
What is the value of $\left( \log _ { 3 } 27 \right) \times 8 ^ { \frac { 1 } { 3 } }$? [2 points]
(1) 12
(2) 10
(3) 8
(4) 6
(5) 4
Q2 2 marks Matrices Linear System and Inverse Existence View
For two matrices $A = \left( \begin{array} { r r } - 1 & 0 \\ 0 & 1 \end{array} \right) , B = \left( \begin{array} { l l } 2 & 1 \\ 3 & 3 \end{array} \right)$, what is the sum of all components of the matrix $( A + B ) ^ { - 1 }$? [2 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5
Q3 2 marks Sequences and series, recurrence and convergence Multiple-choice on sequence properties View
What is the value of $\lim _ { n \rightarrow \infty } \frac { 3 + \left( \frac { 1 } { 3 } \right) ^ { n } } { 2 + \left( \frac { 1 } { 2 } \right) ^ { n } }$? [2 points]
(1) 1
(2) $\frac { 3 } { 2 }$
(3) 2
(4) $\frac { 5 } { 2 }$
(5) 3
Q4 3 marks Exponential Functions Ordering and Comparing Exponential Values View
For two exponential functions $f ( x ) = 4 ^ { x }$, $g ( x ) = \left( \frac { 1 } { 2 } \right) ^ { x }$ with domain $\{ x \mid - 1 \leqq x \leqq 3 \}$, let $M$ be the maximum value of $f ( x )$ and $m$ be the minimum value of $g ( x )$. What is the value of $M m$? [3 points]
(1) 8
(2) 6
(3) 4
(4) 2
(5) 1
Q5 3 marks Conditional Probability Direct Conditional Probability Computation from Definitions View
For two events $A , B$, $$\mathrm { P } ( A ) = \frac { 1 } { 4 } , \quad \mathrm { P } ( B ) = \frac { 2 } { 3 } , \quad A \subset B$$ What is the value of $\mathrm { P } ( A \mid B )$? [3 points]
(1) $\frac { 1 } { 8 }$
(2) $\frac { 1 } { 4 }$
(3) $\frac { 3 } { 8 }$
(4) $\frac { 1 } { 2 }$
(5) $\frac { 5 } { 8 }$
Q6 3 marks Arithmetic Sequences and Series Arithmetic-Geometric Hybrid Problem View
Three numbers $a , 0 , b$ form an arithmetic sequence in this order, and three numbers $2 b , a , - 7$ form a geometric sequence in this order. What is the value of $a$? [3 points]
(1) 10
(2) 12
(3) 14
(4) 16
(5) 18
Q7 3 marks Binomial Theorem (positive integer n) Determine Parameters from Conditions on Coefficients or Terms View
In the expansion of the polynomial $( x - a ) ^ { 5 }$, when the sum of the coefficient of $x$ and the constant term is 0, what is the value of the positive constant $a$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5
Q8 3 marks Laws of Logarithms Simplify or Evaluate a Logarithmic Expression View
For three real numbers $a , b , c$ greater than 1, when $\log _ { a } c : \log _ { b } c = 2 : 1$, what is the value of $\log _ { a } b + \log _ { b } a$? [3 points]
(1) 1
(2) $\frac { 3 } { 2 }$
(3) 2
(4) $\frac { 5 } { 2 }$
(5) 3
Q9 3 marks Normal Distribution Direct Probability Calculation from Given Normal Distribution View
At a certain car wash, the time required to wash one car follows a normal distribution with mean 30 minutes and standard deviation 2 minutes. When washing one car at this car wash, what is the probability that the washing time is 33 minutes or more, using the standard normal distribution table below? [3 points]
$z$$\mathrm { P } ( 0 \leqq Z \leqq z )$
0.50.1915
1.00.3413
1.50.4332
2.00.4772

(1) 0.0228
(2) 0.0668
(3) 0.1587
(4) 0.2708
(5) 0.3085
Q10 3 marks Normal Distribution Sampling Distribution of the Mean View
At a certain factory, table tennis balls are dropped onto a steel floor from a fixed height, and the height to which the table tennis ball bounces follows a normal distribution. When 100 table tennis balls produced by this factory were randomly sampled and the bounce height was measured, the mean was 245 and the standard deviation was 20. What is the number of integers in the 95\% confidence interval for the mean bounce height of all table tennis balls produced by this factory? (Note: The unit of height is mm, and when $Z$ follows a standard normal distribution, $\mathrm { P } ( 0 \leqq Z \leqq 1.96 ) = 0.4750$.) [3 points]
(1) 5
(2) 6
(3) 7
(4) 8
(5) 9
Q11 3 marks Exponential Functions Applied/Contextual Exponential Modeling View
Even when the surroundings suddenly become dark, the human eye perceives the change gradually. After the light intensity suddenly changes from 1000 to 10 and $t$ seconds have elapsed, the light intensity $I ( t )$ perceived by a person is $$I ( t ) = 10 + 990 \times a ^ { - 5 t } ( \text { where } a \text { is a constant with } a > 1 )$$ After the light intensity suddenly changes from 1000 to 10, let $s$ seconds elapse until the person perceives the light intensity as 21. What is the value of $s$? (Note: The unit of light intensity is Td (troland).) [3 points]
(1) $\frac { 1 + 2 \log 3 } { 5 \log a }$
(2) $\frac { 1 + 3 \log 3 } { 5 \log a }$
(3) $\frac { 2 + \log 3 } { 5 \log a }$
(4) $\frac { 2 + 2 \log 3 } { 5 \log a }$
(5) $\frac { 2 + 3 \log 3 } { 5 \log a }$
Q12 3 marks Matrices True/False or Multiple-Select Conceptual Reasoning View
Two $2 \times 2$ square matrices $A , B$ satisfy $A ^ { 2 } = E , B ^ { 2 } = B$. Which of the following statements in the given options are always true? (Note: $E$ is the identity matrix.) [3 points]
Given Options ㄱ. If matrix $B$ has an inverse matrix, then $B = E$. ㄴ. $( E - A ) ^ { 5 } = 2 ^ { 4 } ( E - A )$ ㄷ. $( E - A B A ) ^ { 2 } = E - A B A$
(1) ㄱ
(2) ㄷ
(3) ㄱ, ㄴ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ
Q13 4 marks Laws of Logarithms Verify Truth of Logarithmic Statements View
For an integer $n$, two sets $A ( n ) , B ( n )$ are defined as $$\begin{aligned} & A ( n ) = \left\{ x \mid \log _ { 2 } x \leqq n \right\} \\ & B ( n ) = \left\{ x \mid \log _ { 4 } x \leqq n \right\} \end{aligned}$$ Which of the following statements in the given options are correct? [4 points] Given Options ㄱ. $A ( 1 ) = \{ x \mid 0 < x \leqq 1 \}$ ㄴ. $A ( 4 ) = B ( 2 )$ ㄷ. When $A ( n ) \subset B ( n )$, then $B ( - n ) \subset A ( - n )$.
(1) ㄱ
(2) ㄴ
(3) ㄷ
(4) ㄱ, ㄷ
(5) ㄴ, ㄷ
Q14 4 marks Combinations & Selection Distribution of Objects to Positions or Containers View
Five balls labeled with the numbers $1,2,3,4,5$ are to be placed into 3 boxes $\mathrm { A } , \mathrm { B } , \mathrm { C }$. How many ways are there to place the balls in the boxes such that no box has a sum of the numbers on the balls that is 13 or more? (Note: For an empty box, the sum of the numbers on the balls is considered to be 0.) [4 points]
(1) 233
(2) 228
(3) 222
(4) 215
(5) 211
Q15 4 marks Probability Definitions Finite Equally-Likely Probability Computation View
Among $3n$ cards labeled with the numbers $1,2,3 , \cdots , 3 n$ ($n$ is a natural number), two cards are randomly drawn, and the two numbers on them are denoted as $a , b$ ($a < b$) respectively. Let $\mathrm { P } _ { n }$ be the probability that $3 a < b$. The following is the process of finding $\lim _ { n \rightarrow \infty } \mathrm { P } _ { n }$. The number of ways to draw 2 cards from $3 n$ cards is ${ } _ { 3 n } \mathrm { C } _ { 2 }$. When $3 a < b$, we have $b \leqq 3 n$, so $1 \leqq a < n$. Therefore, if $a = k$, the number of cases of $b$ satisfying $3 a < b$ is (A), so $$\mathrm { P } _ { n } = \frac { \text { (B) } } { { } _ { 3 n } \mathrm { C } _ { 2 } } \text { . }$$ Therefore, $\lim _ { n \rightarrow \infty } \mathrm { P } _ { n } =$ (C).
What are the correct values for (A), (B), and (C)? [4 points]
(A)(B)(C)
(1)$3 ( n - k )$$\frac { 3 } { 2 } n ( n - 1 )$$\frac { 1 } { 3 }$
(2)$3 ( n - k )$$\frac { 3 } { 2 } n ( n - 1 )$$\frac { 2 } { 3 }$
(3)$3 ( n - k )$$3 n ( n - 1 )$$\frac { 2 } { 3 }$
(4)$3 ( n - k + 1 )$$3 n ( n - 1 )$$\frac { 1 } { 3 }$
(5)$3 ( n - k + 1 )$$3 n ( n - 1 )$$\frac { 2 } { 3 }$
Q16 4 marks Curve Sketching Lattice Points and Counting via Graph Geometry View
In the coordinate plane, for a natural number $n$, let $A _ { n }$ be a square with vertices at the four points $$\left( n ^ { 2 } , n ^ { 2 } \right) , \left( 4 n ^ { 2 } , n ^ { 2 } \right) , \left( 4 n ^ { 2 } , 4 n ^ { 2 } \right) , \left( n ^ { 2 } , 4 n ^ { 2 } \right)$$ Let $a _ { n }$ be the number of natural numbers $k$ such that the square $A _ { n }$ and the graph of the function $y = k \sqrt { x }$ intersect. Which of the following statements in the given options are correct? [4 points] Given Options ㄱ. $a _ { 5 } = 15$ ㄴ. $a _ { n + 2 } - a _ { n } = 7$ ㄷ. $\sum _ { k = 1 } ^ { 10 } a _ { k } = 200$
(1) ㄴ
(2) ㄷ
(3) ㄱ, ㄴ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ
Q17 4 marks Geometric Sequences and Series Fractal/Iterative Geometric Construction (Area, Length, or Perimeter Series) View
There is a right isosceles triangle with the two legs forming the right angle each having length 1. A square is drawn such that 2 of its vertices lie on the hypotenuse and the other 2 vertices lie on the two legs forming the right angle. The figure obtained by coloring this square is called $R _ { 1 }$. In figure $R _ { 1 }$, 2 congruent right isosceles triangles are formed. In each of these triangles, a square is drawn such that 2 of its vertices lie on the hypotenuse and the other 2 vertices lie on the two legs forming the right angle. The figure obtained by coloring these 2 squares is called $R _ { 2 }$. In figure $R _ { 2 }$, 4 congruent right isosceles triangles are formed. In each of these triangles, a square is drawn such that 2 of its vertices lie on the hypotenuse and the other 2 vertices lie on the two legs forming the right angle. The figure obtained by coloring these 4 squares is called $R _ { 3 }$. Continuing this process, let $S _ { n }$ be the sum of the areas of all colored squares in the $n$-th figure $R _ { n }$. What is the value of $\lim _ { n \rightarrow \infty } S _ { n }$? [4 points]
(1) $\frac { 3 \sqrt { 2 } } { 20 }$
(2) $\frac { \sqrt { 2 } } { 5 }$
(3) $\frac { 3 } { 10 }$
(4) $\frac { \sqrt { 3 } } { 5 }$
(5) $\frac { 2 } { 5 }$
Q18 3 marks Geometric Sequences and Series Finite Geometric Sum and Term Relationships View
For a geometric sequence $\left\{ a _ { n } \right\}$, when $a _ { 3 } = 2 , a _ { 6 } = 16$, find the value of $a _ { 9 }$. [3 points]
Q19 3 marks Laws of Logarithms Solve a Logarithmic Equation View
For the logarithmic equation $\left( \log _ { 2 } x \right) ^ { 2 } - 4 \log _ { 2 } x = 0$, let the two roots be $\alpha , \beta$ respectively. Find the value of $\alpha + \beta$. [3 points]
Q20 3 marks Geometric Sequences and Series Determine the Limit of a Sequence via Geometric Series View
For the sequence $\left\{ \left( \frac { 2 x - 1 } { 4 } \right) ^ { n } \right\}$ to converge, let $k$ be the number of integers $x$. Find the value of $10 k$. [3 points]
Q21 3 marks Matrices Linear System and Inverse Existence View
For a $2 \times 2$ square matrix $A$ satisfying $( A + E ) ^ { 2 } = A$ and a matrix $\binom { p } { q }$, $$\left( A + A ^ { - 1 } \right) \binom { p } { q } = \binom { 3 } { - 7 }$$ holds. Find the value of $p ^ { 2 } + q ^ { 2 }$. (Note: $E$ is the identity matrix.) [3 points]
Q22 4 marks Arithmetic Sequences and Series Summation of Derived Sequence from AP View
For an arithmetic sequence $\left\{ a _ { n } \right\}$ with first term 0 and common difference not equal to 0, a sequence $\left\{ b _ { n } \right\}$ satisfies $a _ { n + 1 } b _ { n } = \sum _ { k = 1 } ^ { n } a _ { k }$. Find the value of $b _ { 27 }$. [4 points]
Q23 4 marks Permutations & Arrangements Linear Arrangement with Constraints View
Two adults and three children go to an amusement park to ride a certain ride. This ride has 2 chairs in the front row and 3 chairs in the back row. When each child must sit in the same row as an adult, find the number of ways for all 5 people to sit in the chairs of the ride. [4 points]
Q24 4 marks Continuous Probability Distributions and Random Variables PDF Graph Interpretation and Probability Computation View
For two positive numbers $a , b$, a continuous random variable $X$ has a range of $0 \leqq X \leqq a$, and the graph of the probability density function is as shown. When $\mathrm { P } \left( 0 \leqq X \leqq \frac { a } { 2 } \right) = \frac { b } { 2 }$, find the value of $a ^ { 2 } + 4 b ^ { 2 }$. [4 points]
Q25 4 marks Exponential Equations & Modelling Geometric Properties of Exponential/Logarithmic Curves View
The graph of the function $y = k \cdot 3 ^ { x } ( 0 < k < 1 )$ intersects the graphs of the two functions $y = 3 ^ { - x }$ and $y = - 4 \cdot 3 ^ { x } + 8$ at points $\mathrm { P }$ and $\mathrm { Q }$ respectively. When the ratio of the $x$-coordinates of points P and Q is $1 : 2$, find the value of $35 k$. [4 points]
Q26 3 marks Sequences and series, recurrence and convergence Direct term computation from recurrence View
For a sequence $\left\{ a _ { n } \right\}$ with $a _ { 1 } = 2$ and $a _ { n + 1 } = 2 a _ { n } + 2$, what is the value of $a _ { 10 }$? [3 points]
(1) 1022
(2) 1024
(3) 2021
(4) 2046
(5) 2082
Q27 4 marks Laws of Logarithms Characteristic and Mantissa of Common Logarithms View
For $a$ with $0 < a < 1$, when $10 ^ { a }$ is divided by 3, the quotient is an integer and the remainder is 2. What is the sum of all values of $a$? [4 points]
(1) $3 \log 2$
(2) $6 \log 2$
(3) $1 + 3 \log 2$
(4) $1 + 6 \log 2$
(5) $2 + 3 \log 2$
Q28 4 marks Independent Events View
When 3 coins are tossed simultaneously, let $A$ be the event that at most 1 coin shows heads, and let $B$ be the event that all 3 coins show the same face. Which of the following statements in the given options are correct? [4 points] Given Options ㄱ. $\mathrm { P } ( A ) = \frac { 1 } { 2 }$ ㄴ. $\mathrm { P } ( A \cap B ) = \frac { 1 } { 8 }$ ㄷ. Events $A$ and $B$ are independent of each other.
(1) ㄱ
(2) ㄷ
(3) ㄱ, ㄴ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ
Q29 4 marks Binomial Distribution Compute Exact Binomial Probability View
There is a television with channels set from 1 to 100. The currently active channel is 50. When one button is randomly pressed six times, either the channel increase button or the channel decrease button, what is the probability that the channel returns to 50? (Note: Each time a button is pressed, the channel changes by 1.) [4 points]
(1) $\frac { 1 } { 4 }$
(2) $\frac { 5 } { 16 }$
(3) $\frac { 3 } { 8 }$
(4) $\frac { 7 } { 16 }$
(5) $\frac { 1 } { 2 }$
Q30 4 marks Matrices Determinant and Rank Computation View
For a $2 \times 2$ square matrix $X = \left( \begin{array} { l l } a & b \\ c & d \end{array} \right)$, $$D ( X ) = a d - b c$$ is defined. For a $2 \times 2$ square matrix $A = \left( \begin{array} { l l } 1 & 1 \\ 0 & p \end{array} \right)$, $$D \left( A ^ { 2 } \right) = D ( 5 A )$$ Find the sum of all constants $p$ that satisfy this condition. [4 points]