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
2019 csat__math-humanities

30 maths questions

Q1 2 marks Indices and Surds Evaluating Expressions Using Index Laws View
What is the value of $2 ^ { - 1 } \times 16 ^ { \frac { 1 } { 2 } }$? [2 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5
Q2 2 marks Probability Definitions Set Operations View
Two sets $$A = \{ 3,5,7,9 \} , B = \{ 3,7 \}$$ For the sets above, when $A - B = \{ a , 9 \}$, what is the value of $a$? [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 $\lim _ { n \rightarrow \infty } \frac { 6 n ^ { 2 } - 3 } { 2 n ^ { 2 } + 5 n }$? [2 points]
(1) 5
(2) 4
(3) 3
(4) 2
(5) 1
Q4 3 marks Composite & Inverse Functions Evaluate Composition from Diagram or Mapping View
The figure shows a function $f : X \rightarrow X$. [Figure] What is the value of $f ( 4 ) + ( f \circ f ) ( 2 )$? [3 points]
(1) 3
(2) 4
(3) 5
(4) 6
(5) 7
Q5 3 marks Arithmetic Sequences and Series Find Specific Term from Given Conditions View
For an arithmetic sequence $\left\{ a _ { n } \right\}$ with first term 4, $$a _ { 10 } - a _ { 7 } = 6$$ What is the value of $a _ { 4 }$? [3 points]
(1) 10
(2) 11
(3) 12
(4) 13
(5) 14
Q6 3 marks Binomial Theorem (positive integer n) Find a Specific Coefficient in a Single Binomial Expansion View
What is the coefficient of $x ^ { 4 }$ in the expansion of the polynomial $( 1 + x ) ^ { 7 }$? [3 points]
(1) 42
(2) 35
(3) 28
(4) 21
(5) 14
Q7 3 marks Curve Sketching Limit Reading from Graph View
The graph of the function $y = f ( x )$ is shown in the figure. [Figure] What is the value of $\lim _ { x \rightarrow - 1 - } f ( x ) - \lim _ { x \rightarrow 1 + } f ( x )$? [3 points]
(1) - 2
(2) - 1
(3) 0
(4) 1
(5) 2
Q8 3 marks Principle of Inclusion/Exclusion View
For two events $A$ and $B$, $A$ and $B ^ { C }$ are mutually exclusive events, and $$\mathrm { P } ( A ) = \frac { 1 } { 3 } , \mathrm { P } \left( A ^ { C } \cap B \right) = \frac { 1 } { 6 }$$ What is the value of $\mathrm { P } ( B )$? (Here, $A ^ { C }$ is the complement of $A$.) [3 points]
(1) $\frac { 5 } { 12 }$
(2) $\frac { 1 } { 2 }$
(3) $\frac { 7 } { 12 }$
(4) $\frac { 2 } { 3 }$
(5) $\frac { 3 } { 4 }$
Q9 Stationary points and optimisation Determine parameters from given extremum conditions View
For the function $f ( x ) = x ^ { 3 } - 3 x + a$, when the local maximum value is 7, what is the value of the constant $a$?
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5
Q10 3 marks Continuous Probability Distributions and Random Variables PDF Graph Interpretation and Probability Computation View
A continuous random variable $X$ has range $0 \leq X \leq 2$, and the graph of the probability density function of $X$ is shown in the figure. What is the value of $\mathrm { P } \left( \frac { 1 } { 3 } \leq X \leq a \right)$? (Here, $a$ is a constant.) [3 points] [Figure]
(1) $\frac { 11 } { 16 }$
(2) $\frac { 5 } { 8 }$
(3) $\frac { 9 } { 16 }$
(4) $\frac { 1 } { 2 }$
(5) $\frac { 7 } { 16 }$
Q11 3 marks Inequalities Sufficient/Necessary Conditions Between Inequality Conditions View
For two conditions $p$ and $q$ on real numbers $x$: $$\begin{aligned} & p : x ^ { 2 } - 4 x + 3 > 0 , \\ & q : x \leq a \end{aligned}$$ What is the minimum value of the real number $a$ such that $\sim p$ is a sufficient condition for $q$? [3 points]
(1) 5
(2) 4
(3) 3
(4) 2
(5) 1
Q12 3 marks Confidence intervals Find a specific bound or margin of error from the CI formula View
The weight of watermelons harvested in a certain village follows a normal distribution with mean $m$ kg and standard deviation 1.4 kg. When 49 watermelons are randomly sampled from this village and a 95\% confidence interval for the mean weight $m$ is constructed using the sample mean, the interval is $a \leq m \leq 7.992$. What is the value of $a$? (Here, when $Z$ is a random variable following the standard normal distribution, calculate using $\mathrm { P } ( | Z | \leq 1.96 ) = 0.95$.) [3 points]
(1) 7.198
(2) 7.208
(3) 7.218
(4) 7.228
(5) 7.238
Q13 3 marks Sequences and Series Recurrence Relations and Sequence Properties View
A sequence $\left\{ a _ { n } \right\}$ has $a _ { 1 } = 2$ and satisfies for all natural numbers $n$: $$a _ { n + 1 } = \left\{ \begin{array} { c c } \frac { a _ { n } } { 2 - 3 a _ { n } } & ( \text{when } n \text{ is odd} ) \\ 1 + a _ { n } & ( \text{when } n \text{ is even} ) \end{array} \right.$$ What is the value of $\sum _ { n = 1 } ^ { 40 } a _ { n }$? [3 points]
(1) 30
(2) 35
(3) 40
(4) 45
(5) 50
Q14 4 marks Indefinite & Definite Integrals Finding a Function from an Integral Equation View
A polynomial function $f ( x )$ satisfies for all real numbers $x$: $$\int _ { 1 } ^ { x } \left\{ \frac { d } { d t } f ( t ) \right\} d t = x ^ { 3 } + a x ^ { 2 } - 2$$ What is the value of $f ^ { \prime } ( a )$? (Here, $a$ is a constant.) [4 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5
Q15 4 marks Laws of Logarithms Solve a Logarithmic Equation View
For natural numbers $n \geq 2$, what is the sum of all values of $n$ such that $5 \log _ { n } 2$ is a natural number? [4 points]
(1) 34
(2) 38
(3) 42
(4) 46
(5) 50
Q16 4 marks Geometric Sequences and Series Fractal/Iterative Geometric Construction (Area, Length, or Perimeter Series) View
As shown in the figure, there is a right triangle $\mathrm { OA } _ { 1 } \mathrm {~B} _ { 1 }$ with $\overline { \mathrm { OA } _ { 1 } } = 4$ and $\overline { \mathrm { OB } _ { 1 } } = 4 \sqrt { 3 }$. Let $\mathrm { B } _ { 2 }$ be the point where the circle with center O and radius $\overline { \mathrm { OA } _ { 1 } }$ meets the line segment $\mathrm { OB } _ { 1 }$. The figure $R _ { 1 }$ is obtained by shading the region that is inside triangle $\mathrm { OA } _ { 1 } \mathrm {~B} _ { 1 }$ but outside the sector $\mathrm { OA } _ { 1 } \mathrm {~B} _ { 2 }$. In figure $R _ { 1 }$, let $\mathrm { A } _ { 2 }$ be the point where the line passing through $\mathrm { B } _ { 2 }$ and parallel to segment $\mathrm { A } _ { 1 } \mathrm {~B} _ { 1 }$ meets segment $\mathrm { OA } _ { 1 }$, and let $\mathrm { B } _ { 3 }$ be the point where the circle with center O and radius $\overline { \mathrm { OA } _ { 2 } }$ meets segment $\mathrm { OB } _ { 2 }$. The figure $R _ { 2 }$ is obtained by shading the region that is inside triangle $\mathrm { OA } _ { 2 } \mathrm {~B} _ { 2 }$ but outside the sector $\mathrm { OA } _ { 2 } \mathrm {~B} _ { 3 }$. 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] [Figure] [Figure]
(1) $\frac { 3 } { 2 } \pi$
(2) $\frac { 5 } { 3 } \pi$
(3) $\frac { 11 } { 6 } \pi$
(4) $2 \pi$
(5) $\frac { 13 } { 6 } \pi$
Q17 4 marks Areas by integration View
An increasing continuous function $f ( x )$ on the set of all real numbers satisfies the following conditions. (가) For all real numbers $x$, $f ( x ) = f ( x - 3 ) + 4$. (나) $\int _ { 0 } ^ { 6 } f ( x ) d x = 0$ What is the area enclosed by the graph of $y = f ( x )$, the $x$-axis, and the two lines $x = 6$ and $x = 9$? [4 points]
(1) 9
(2) 12
(3) 15
(4) 18
(5) 21
Q18 4 marks Tree Diagrams Multi-Stage Sequential Process View
Point A is at the origin of the coordinate plane. The following trial is performed using one coin. Flip the coin once. If heads appears, move point A by 1 in the positive direction of the $x$-axis; if tails appears, move point A by 1 in the positive direction of the $y$-axis. Repeat this trial until the $x$-coordinate or $y$-coordinate of point A becomes 3 for the first time, then stop. What is the probability that when the $y$-coordinate of point A becomes 3 for the first time, the $x$-coordinate of point A is 1? [4 points]
(1) $\frac { 1 } { 4 }$
(2) $\frac { 5 } { 16 }$
(3) $\frac { 3 } { 8 }$
(4) $\frac { 7 } { 16 }$
(5) $\frac { 1 } { 2 }$
Q19 4 marks Combinations & Selection Counting Functions or Mappings with Constraints View
The following is a process for finding the number of functions $f : X \rightarrow X$ where $X = \{ 1,2,3,4,5,6 \}$ such that the range of the composite function $f \circ f$ has 5 elements. Let the ranges of functions $f$ and $f \circ f$ be $A$ and $B$, respectively. If $n ( A ) = 6$, then $f$ is a bijection, and $f \circ f$ is also a bijection, so $n ( B ) = 6$. Also, if $n ( A ) \leq 4$, then $B \subset A$, so $n ( B ) \leq 4$. Therefore, we only need to consider the case where $n ( A ) = 5$, that is, $B = A$.
(i) The number of ways to choose a subset $A$ of $X$ with $n ( A ) = 5$ is (가).
(ii) For the set $A$ chosen in (i), let $k$ be the element of $X$ that does not belong to $A$. Since $n ( A ) = 5$, the number of ways to choose $f ( k )$ from set $A$ is (나).
(iii) For $A = \left\{ a _ { 1 } , a _ { 2 } , a _ { 3 } , a _ { 4 } , a _ { 5 } \right\}$ chosen in (i) and $f ( k )$ chosen in (ii), since $f ( k ) \in A$ and $A = B$, we have $A = \left\{ f \left( a _ { 1 } \right) , f \left( a _ { 2 } \right) , f \left( a _ { 3 } \right) , f \left( a _ { 4 } \right) , f \left( a _ { 5 } \right) \right\} \cdots ( * )$. The number of cases satisfying (*) is equal to the number of bijections from set $A$ to set $A$, so it is (다). Therefore, by (i), (ii), and (iii), the number of functions $f$ we seek is (가) $\times$ (나) $\times$ (다). When the numbers corresponding to (가), (나), and (다) are $p$, $q$, and $r$, respectively, what is the value of $p + q + r$? [4 points]
(1) 131
(2) 136
(3) 141
(4) 146
(5) 151
Q20 4 marks Curve Sketching Multi-Statement Verification (Remarks/Options) View
As shown in the figure, let $\mathrm { A }$ and $\mathrm { B }$ be the $x$-intercept and $y$-intercept, respectively, of the graph of the function $y = \frac { k } { x - 1 } + 3$ where $0 < k < 3$. [Figure] Let $\mathrm { P }$ be the point (other than $\mathrm { B }$) where the line passing through the intersection of the two asymptotes of this graph and point $\mathrm { B }$ meets the graph, and let $\mathrm { Q }$ be the foot of the perpendicular from point $\mathrm { P }$ to the $x$-axis. Which of the following statements are correct? [4 points] ㄱ. When $k = 1$, the coordinates of point $\mathrm { P }$ are $( 2,4 )$. ㄴ. For real numbers $0 < k < 3$, the sum of the slope of line AB and the slope of line AP is 0. ㄷ. When the area of quadrilateral PBAQ is a natural number, the slope of line BP is between 0 and 1.
(1) ㄱ
(2) ㄱ, ㄴ
(3) ㄱ, ㄷ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ
Q21 4 marks Factor & Remainder Theorem Polynomial Construction from Root/Value Conditions View
For a cubic function $f ( x )$ with leading coefficient 1 and a function $g ( x )$ that is continuous on the set of all real numbers, the following conditions are satisfied. (가) For all real numbers $x$, $f ( x ) g ( x ) = x ( x + 3 )$. (나) $g ( 0 ) = 1$ When $f ( 1 )$ is a natural number, what is the minimum value of $g ( 2 )$? [4 points]
(1) $\frac { 5 } { 13 }$
(2) $\frac { 5 } { 14 }$
(3) $\frac { 1 } { 3 }$
(4) $\frac { 5 } { 16 }$
(5) $\frac { 5 } { 17 }$
Q22 3 marks Permutations & Arrangements Factorial and Combinatorial Expression Simplification View
Find the value of ${ } _ { 6 } \mathrm { P } _ { 2 } - { } _ { 6 } \mathrm { C } _ { 2 }$. [3 points]
Q23 3 marks Applied differentiation MCQ on derivative and graph interpretation View
For the function $f ( x ) = x ^ { 4 } - 3 x ^ { 2 } + 8$, find the value of $f ^ { \prime } ( 2 )$. [3 points]
Q24 3 marks Geometric Sequences and Series Finite Geometric Sum and Term Relationships View
For a geometric sequence $\left\{ a _ { n } \right\}$ with first term 7, let $S _ { n }$ denote the sum of the first $n$ terms. $$\frac { S _ { 9 } - S _ { 5 } } { S _ { 6 } - S _ { 2 } } = 3$$ Find the value of $a _ { 7 }$. [3 points]
Q25 3 marks Indefinite & Definite Integrals Piecewise/Periodic Function Integration View
Find the value of $\int _ { 1 } ^ { 4 } ( x + | x - 3 | ) d x$. [3 points]
Q26 4 marks Inequalities Optimization Subject to an Algebraic Constraint View
Find the maximum value of the real number $k$ such that the graphs of $y = \sqrt { x + 3 }$ and $y = \sqrt { 1 - x } + k$ intersect. [4 points]
Q27 4 marks Variable acceleration (1D) Determine velocity at zero acceleration View
The position $x$ at time $t$ ($t \geq 0$) of a point P moving on a number line is $$x = - \frac { 1 } { 3 } t ^ { 3 } + 3 t ^ { 2 } + k \quad ( k \text{ is a constant} )$$ When the acceleration of point P is 0, the position of point P is 40. Find the value of $k$. [4 points]
Q28 4 marks Permutations & Arrangements Probability via Permutation Counting View
There are 4 white balls with the numbers $1, 2, 3, 4$ written on them and 3 black balls with the numbers $4, 5, 6$ written on them. When these 7 balls are randomly arranged in a line, the probability that the balls with the same number do not lie adjacent to each other is $\frac { q } { p }$. Find the value of $p + q$. (Here, $p$ and $q$ are coprime natural numbers.) [4 points]
Q29 4 marks Arithmetic Sequences and Series Arithmetic-Geometric Hybrid Problem View
An arithmetic sequence $\left\{ a _ { n } \right\}$ with first term a natural number and common difference a negative integer, and a geometric sequence $\left\{ b _ { n } \right\}$ with first term a natural number and common ratio a negative integer, satisfy the following conditions. Find the value of $a _ { 7 } + b _ { 7 }$. [4 points] (가) $\sum _ { n = 1 } ^ { 5 } \left( a _ { n } + b _ { n } \right) = 27$ (나) $\sum _ { n = 1 } ^ { 5 } \left( a _ { n } + \left| b _ { n } \right| \right) = 67$ (다) $\sum _ { n = 1 } ^ { 5 } \left( \left| a _ { n } \right| + \left| b _ { n } \right| \right) = 81$
Q30 4 marks Stationary points and optimisation Geometric or applied optimisation problem View
A cubic function $f ( x )$ with leading coefficient 1 and a quadratic function $g ( x )$ with leading coefficient $-1$ satisfy the following conditions. (가) The tangent line to the curve $y = f ( x )$ at the point $( 0,0 )$ and the tangent line to the curve $y = g ( x )$ at the point $( 2,0 )$ are both the $x$-axis. (나) The number of tangent lines to the curve $y = f ( x )$ drawn from the point $( 2,0 )$ is 2. (다) The equation $f ( x ) = g ( x )$ has exactly one real root. For all real numbers $x > 0$, $$g ( x ) \leq k x - 2 \leq f ( x )$$ Let $\alpha$ and $\beta$ be the maximum and minimum values of the real number $k$ satisfying the above inequality, respectively. When $\alpha - \beta = a + b \sqrt { 2 }$, find the value of $a ^ { 2 } + b ^ { 2 }$. (Here, $a$ and $b$ are rational numbers.) [4 points]