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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
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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
2025 csat__math

43 maths questions

Q1 2 marks Indices and Surds Evaluating Expressions Using Index Laws View
What is the value of $\sqrt[3]{5} \times 25^{\frac{1}{3}}$? [2 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5
Q2 3 marks Geometric Sequences and Series Finite Geometric Sum and Term Relationships View
A geometric sequence $\left\{ a_{n} \right\}$ with first term and common ratio both equal to a positive number $k$ satisfies $$\frac{a_{4}}{a_{2}} + \frac{a_{2}}{a_{1}} = 30$$ What is the value of $k$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5
Q3 2 marks Differentiation from First Principles View
For the function $f(x) = x^{3} - 8x + 7$, what is the value of $\lim_{h \rightarrow 0} \frac{f(2+h) - f(2)}{h}$? [2 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5
Q4 3 marks Composite & Inverse Functions Existence or Properties of Functions and Inverses (Proof-Based) View
The function $$f(x) = \left\{ \begin{array}{cc} 5x + a & (x < -2) \\ x^{2} - a & (x \geq -2) \end{array} \right.$$ is continuous on the set of all real numbers. What is the value of the constant $a$? [3 points]
(1) 6
(2) 7
(3) 8
(4) 9
(5) 10
Q5 3 marks Product & Quotient Rules View
For the function $f(x) = \left(x^{2} + 1\right)\left(3x^{2} - x\right)$, what is the value of $f'(1)$? [3 points]
(1) 8
(2) 10
(3) 12
(4) 14
(5) 16
Q6 3 marks Reciprocal Trig & Identities View
When $\cos\left(\frac{\pi}{2} + \theta\right) = -\frac{1}{5}$, what is the value of $\frac{\sin\theta}{1 - \cos^{2}\theta}$? [3 points]
(1) $-5$
(2) $-\sqrt{5}$
(3) $0$
(4) $\sqrt{5}$
(5) $5$
Q7 3 marks Indefinite & Definite Integrals Finding a Function from an Integral Equation View
A polynomial function $f(x)$ satisfies $$\int_{0}^{x} f(t)\, dt = 3x^{3} + 2x$$ for all real numbers $x$. What is the value of $f(1)$? [3 points]
(1) 7
(2) 9
(3) 11
(4) 13
(5) 15
Q8 3 marks Laws of Logarithms Simplify or Evaluate a Logarithmic Expression View
For two real numbers $a = 2\log\frac{1}{\sqrt{10}} + \log_{2}20$ and $b = \log 2$, what is the value of $a \times b$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5
Q9 4 marks Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View
For the function $f(x) = 3x^{2} - 16x - 20$, $$\int_{-2}^{a} f(x)\, dx = \int_{-2}^{0} f(x)\, dx$$ When this condition is satisfied, what is the value of the positive number $a$? [4 points]
(1) 16
(2) 14
(3) 12
(4) 10
(5) 8
Q10 4 marks Trig Graphs & Exact Values View
The function $f(x) = a\cos bx + 3$ is defined on the closed interval $[0, 2\pi]$ and has a maximum value of 13 at $x = \frac{\pi}{3}$. For the ordered pair $(a, b)$ of two natural numbers $a$ and $b$ satisfying this condition, what is the minimum value of $a + b$? [4 points]
(1) 12
(2) 14
(3) 16
(4) 18
(5) 20
Q11 4 marks Applied differentiation Kinematics via differentiation View
A point P starts at time $t = 0$ and moves on a number line. At time $t$ ($t \geq 0$), its position $x$ is given by $$x = t^{3} - \frac{3}{2}t^{2} - 6t$$ What is the acceleration of point P at the time when its direction of motion changes after starting? [4 points]
(1) 6
(2) 9
(3) 12
(4) 15
(5) 18
Q12 4 marks Arithmetic Sequences and Series Summation of Derived Sequence from AP View
A sequence $\left\{ a_{n} \right\}$ with $a_{1} = 2$ and an arithmetic sequence $\left\{ b_{n} \right\}$ with $b_{1} = 2$ satisfy $$\sum_{k=1}^{n} \frac{a_{k}}{b_{k+1}} = \frac{1}{2}n^{2}$$ for all natural numbers $n$. What is the value of $\sum_{k=1}^{5} a_{k}$? [4 points]
(1) 120
(2) 125
(3) 130
(4) 135
(5) 140
Q13 4 marks Areas Between Curves Area Ratio or Comparative Area View
A cubic function $f(x)$ with leading coefficient 1 satisfies $$f(1) = f(2) = 0, \quad f'(0) = -7$$ Let Q be the point where the line segment OP intersects the curve $y = f(x)$ other than P, where O is the origin and $\mathrm{P}(3, f(3))$. Let $A$ be the area enclosed by the curve $y = f(x)$, the $y$-axis, and the line segment OQ, and let $B$ be the area enclosed by the curve $y = f(x)$ and the line segment PQ. What is the value of $B - A$? [4 points]
(1) $\frac{37}{4}$
(2) $\frac{39}{4}$
(3) $\frac{41}{4}$
(4) $\frac{43}{4}$
(5) $\frac{45}{4}$
Q14 4 marks Sine and Cosine Rules Multi-step composite figure problem View
As shown in the figure, in triangle ABC, point D is taken on segment AB such that $\overline{\mathrm{AD}} : \overline{\mathrm{DB}} = 3 : 2$, and a circle $O$ centered at A passing through D intersects segment AC at point E. $\sin A : \sin C = 8 : 5$, and the ratio of the areas of triangles ADE and ABC is $9 : 35$. When the circumradius of triangle ABC is 7, what is the maximum area of triangle PBC for a point P on circle $O$? (Given: $\overline{\mathrm{AB}} < \overline{\mathrm{AC}}$) [4 points]
(1) $18 + 15\sqrt{3}$
(2) $24 + 20\sqrt{3}$
(3) $30 + 25\sqrt{3}$
(4) $36 + 30\sqrt{3}$
(5) $42 + 35\sqrt{3}$
Q15 4 marks Stationary points and optimisation Determine parameters from given extremum conditions View
For a constant $a$ ($a \neq 3\sqrt{5}$) and a quadratic function $f(x)$ with negative leading coefficient, the function $$g(x) = \begin{cases} x^{3} + ax^{2} + 15x + 7 & (x \leq 0) \\ f(x) & (x > 0) \end{cases}$$ satisfies the following conditions. (가) The function $g(x)$ is differentiable on the set of all real numbers. (나) The equation $g'(x) \times g'(x - 4) = 0$ has exactly 4 distinct real roots. What is the value of $g(-2) + g(2)$? [4 points]
(1) 30
(2) 32
(3) 34
(4) 36
(5) 38
Q16 3 marks Laws of Logarithms Solve a Logarithmic Equation View
Solve the equation $$\log_{2}(x - 3) = \log_{4}(3x - 5)$$ for the real number $x$. [3 points]
Q17 3 marks Indefinite & Definite Integrals Recovering Function Values from Derivative Information View
For a polynomial function $f(x)$, $f'(x) = 9x^{2} + 4x$ and $f(1) = 6$. What is the value of $f(2)$? [3 points]
Q18 3 marks Arithmetic Sequences and Series Compute Partial Sum of an Arithmetic Sequence View
A sequence $\left\{ a_{n} \right\}$ satisfies $$a_{n} + a_{n+4} = 12$$ for all natural numbers $n$. What is the value of $\sum_{n=1}^{16} a_{n}$? [3 points]
Q19 3 marks Stationary points and optimisation Determine parameters from given extremum conditions View
For a positive number $a$, let the function $f(x)$ be $$f(x) = 2x^{3} - 3ax^{2} - 12a^{2}x$$ When the local maximum value of $f(x)$ is $\frac{7}{27}$, what is the value of $f(3)$? [3 points]
Q20 4 marks Exponential Equations & Modelling Evaluate Expression Given Exponential/Logarithmic Conditions View
Let $k$ be the $x$-coordinate of the intersection point of the curve $y = \left(\frac{1}{5}\right)^{x-3}$ and the line $y = x$. A function $f(x)$ defined on the set of all real numbers satisfies the following conditions. For all real numbers $x > k$, $f(x) = \left(\frac{1}{5}\right)^{x-3}$ and $f(f(x)) = 3x$. What is the value of $f\left(\frac{1}{k^{3} \times 5^{3k}}\right)$? [4 points]
Q21 4 marks Roots of polynomials Determine coefficients or parameters from root conditions View
A function $f(x) = x^{3} + ax^{2} + bx + 4$ satisfies the following condition for two integers $a$ and $b$. What is the maximum value of $f(1)$? [4 points] For all real numbers $\alpha$, the limit $\lim_{x \rightarrow \alpha} \frac{f(2x+1)}{f(x)}$ exists.
Q22 4 marks Sequences and Series Recurrence Relations and Sequence Properties View
All terms of a sequence $\left\{ a_{n} \right\}$ are integers and satisfy the following conditions. What is the sum of the values of $\left| a_{1} \right|$? [4 points] (가) For all natural numbers $n$, $$a_{n+1} = \begin{cases} a_{n} - 3 & \left(\left| a_{n} \right| \text{ is odd}\right) \\ \frac{1}{2}a_{n} & \left(a_{n} = 0 \text{ or } \left| a_{n} \right| \text{ is even}\right) \end{cases}$$ (나) The minimum value of the natural number $m$ such that $\left| a_{m} \right| = \left| a_{m+2} \right|$ is 3.
Q23 2 marks Binomial Theorem (positive integer n) Find a Specific Coefficient in a Single Binomial Expansion View
What is the coefficient of $x^{6}$ in the expansion of $\left(x^{3} + 2\right)^{5}$? [2 points]
(1) 40
(2) 50
(3) 60
(4) 70
(5) 80
Q23C 2 marks Small angle approximation View
What is the value of $\lim_{x \rightarrow 0} \frac{3x^{2}}{\sin^{2} x}$? [2 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5
Q23G 3 marks Conic sections Tangent and Normal Line Problems View
For a natural number $n$ ($n \geq 2$), let the line $x = \frac{1}{n}$ meet the two ellipses $$C_{1} : \frac{x^{2}}{2} + y^{2} = 1, \quad C_{2} : 2x^{2} + \frac{y^{2}}{2} = 1$$ at points P and Q respectively in the first quadrant. Let $\alpha$ be the $x$-intercept of the tangent line to ellipse $C_{1}$ at point P, and let $\beta$ be the $x$-intercept of the tangent line to ellipse $C_{2}$ at point Q. How many values of $n$ satisfy $6 \leq \alpha - \beta \leq 15$? [3 points]
(1) 7
(2) 9
(3) 11
(4) 13
(5) 15
Q24 3 marks Conditional Probability Direct Conditional Probability Computation from Definitions View
For two events $A$ and $B$, $$\mathrm{P}(A \mid B) = \mathrm{P}(A) = \frac{1}{2}, \quad \mathrm{P}(A \cap B) = \frac{1}{5}$$ What is the value of $\mathrm{P}(A \cup B)$? [3 points]
(1) $\frac{1}{2}$
(2) $\frac{3}{5}$
(3) $\frac{7}{10}$
(4) $\frac{4}{5}$
(5) $\frac{9}{10}$
Q24C 3 marks Standard Integrals and Reverse Chain Rule Definite Integral Evaluation via Substitution or Standard Forms View
What is the value of $\int_{0}^{10} \frac{x+2}{x+1}\, dx$? [3 points]
(1) $10 + \ln 5$
(2) $10 + \ln 7$
(3) $10 + 2\ln 3$
(4) $10 + \ln 11$
(5) $10 + \ln 13$
Q25 3 marks Confidence intervals Count integers or determine length of a confidence interval View
A sample of size 256 is randomly extracted from a population following a normal distribution $\mathrm{N}\left(m, 2^{2}\right)$. The 95\% confidence interval for $m$ obtained using the sample mean is $a \leq m \leq b$. What is the value of $b - a$? (Given: When $Z$ is a random variable following the standard normal distribution, $\mathrm{P}(|Z| \leq 1.96) = 0.95$.) [3 points]
(1) 0.49
(2) 0.52
(3) 0.55
(4) 0.58
(5) 0.61
Q25C 3 marks Sequences and series, recurrence and convergence Convergence proof and limit determination View
For a sequence $\left\{ a_{n} \right\}$, $\lim_{n \rightarrow \infty} \frac{n a_{n}}{n^{2} + 3} = 1$. What is the value of $\lim_{n \rightarrow \infty} \left(\sqrt{a_{n}^{2} + n} - a_{n}\right)$? [3 points]
(1) $\frac{1}{3}$
(2) $\frac{1}{2}$
(3) $1$
(4) $2$
(5) $3$
Q26 3 marks Combinations & Selection Combinatorial Probability View
A survey was conducted on the preferences for Subject A and Subject B among 16 students in a class. Each student who participated in the survey chose one of the two subjects. 9 students chose Subject A and 7 students chose Subject B. When 3 students are randomly selected from the 16 students who participated in the survey, what is the probability that at least one of the 3 selected students chose Subject B? [3 points]
(1) $\frac{3}{4}$
(2) $\frac{4}{5}$
(3) $\frac{17}{20}$
(4) $\frac{9}{10}$
(5) $\frac{19}{20}$
Q26C 3 marks Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View
As shown in the figure, a solid figure has as its base the region enclosed by the curve $y = \sqrt{\frac{x+1}{x(x + \ln x)}}$, the $x$-axis, and the two lines $x = 1$ and $x = e$. When the cross-section of this solid figure cut by a plane perpendicular to the $x$-axis is a square, what is the volume of this solid figure? [3 points]
(1) $\ln(e+1)$
(2) $\ln(e+2)$
(3) $\ln(e+3)$
(4) $\ln(2e+1)$
(5) $\ln(2e+2)$
Q27 3 marks Discrete Probability Distributions Probability Distribution Table Completion and Expectation Calculation View
There is a bag containing 5 cards with the numbers $1, 3, 5, 7, 9$ written on them, one number per card. A trial is performed by randomly drawing one card from the bag, confirming the number on the card, and putting it back. This trial is repeated 3 times, and let $\bar{X}$ be the average of the three numbers confirmed. When $\mathrm{V}(a\bar{X} + 6) = 24$, what is the value of the positive number $a$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5
Q27C 3 marks Composite & Inverse Functions Derivative of an Inverse Function View
For a cubic function $f(x)$ with leading coefficient 1, let the function $g(x)$ be $$g(x) = f\left(e^{x}\right) + e^{x}$$ The tangent line to the curve $y = g(x)$ at the point $(0, g(0))$ is the $x$-axis, and the function $g(x)$ has an inverse function $h(x)$. What is the value of $h'(8)$? [3 points]
(1) $\frac{1}{36}$
(2) $\frac{1}{18}$
(3) $\frac{1}{12}$
(4) $\frac{1}{9}$
(5) $\frac{5}{36}$
Q27G 3 marks Vectors: Lines & Planes Sphere-Plane Intersection and Projection of Circles View
As shown in the figure, for a tetrahedron ABCD with $\overline{\mathrm{AB}} = 6$, $\overline{\mathrm{BC}} = 4\sqrt{5}$, let M be the midpoint of segment BC. Triangle AMD is equilateral and line BC is perpendicular to plane AMD. Find the area of the orthogonal projection of the circle inscribed in triangle ACD onto plane BCD. [3 points]
(1) $\frac{\sqrt{10}}{4}\pi$
(2) $\frac{\sqrt{10}}{6}\pi$
(3) $\frac{\sqrt{10}}{8}\pi$
(4) $\frac{\sqrt{10}}{10}\pi$
(5) $\frac{\sqrt{10}}{12}\pi$
Q28 4 marks Permutations & Arrangements Counting Functions with Constraints View
For the set $X = \{1, 2, 3, 4, 5, 6\}$, how many functions $f : X \rightarrow X$ satisfy the following conditions? [4 points] (가) The value of $f(1) \times f(6)$ is a divisor of 6. (나) $2f(1) \leq f(2) \leq f(3) \leq f(4) \leq f(5) \leq 2f(6)$
(1) 166
(2) 171
(3) 176
(4) 181
(5) 186
Q28C 4 marks Areas Between Curves Multi-Part Free Response with Area, Volume, and Additional Calculus View
The derivative $f'(x)$ of a function $f(x)$ that is differentiable on the set of all real numbers is $$f'(x) = -x + e^{1 - x^{2}}$$ For a positive number $t$, let $g(t)$ be the area of the region enclosed by the tangent line to the curve $y = f(x)$ at the point $(t, f(t))$, the curve $y = f(x)$, and the $y$-axis. What is the value of $g(1) + g'(1)$? [4 points]
(1) $\frac{1}{2}e + \frac{1}{2}$
(2) $\frac{1}{2}e + \frac{2}{3}$
(3) $\frac{1}{2}e + \frac{5}{6}$
(4) $\frac{2}{3}e + \frac{1}{2}$
(5) $\frac{2}{3}e + \frac{2}{3}$
Q28G 4 marks Vectors 3D & Lines MCQ: Cross-Section or Surface Area of a Solid View
In coordinate space, there is a right triangle ABC with $\overline{\mathrm{AB}} = 8$, $\overline{\mathrm{BC}} = 6$, $\angle\mathrm{ABC} = \frac{\pi}{2}$ and a sphere $S$ with diameter AC. Let $O$ be the circle formed by the intersection of sphere $S$ with the plane that contains line AB and is perpendicular to plane ABC. Let P and Q be two distinct points on circle $O$ such that the distance from each to line AC is 4. Find the length of segment PQ. [4 points]
(1) $\sqrt{43}$
(2) $\sqrt{47}$
(3) $\sqrt{51}$
(4) $\sqrt{55}$
(5) $\sqrt{59}$
Q29 4 marks Normal Distribution Algebraic Relationship Between Normal Parameters and Probability View
A random variable $X$ follows a normal distribution $\mathrm{N}\left(m_{1}, \sigma_{1}^{2}\right)$ and a random variable $Y$ follows a normal distribution $\mathrm{N}\left(m_{2}, \sigma_{2}^{2}\right)$, satisfying the following conditions. For all real numbers $x$, $\mathrm{P}(X \leq x) = \mathrm{P}(X \geq 40 - x)$ and $\mathrm{P}(Y \leq x) = \mathrm{P}(X \leq x + 10)$. When $\mathrm{P}(15 \leq X \leq 20) + \mathrm{P}(15 \leq Y \leq 20) = 0.4772$ using the standard normal distribution table below, what is the value of $m_{1} + \sigma_{2}$? (Given: $\sigma_{1}$ and $\sigma_{2}$ are positive.) [4 points]
$z$$\mathrm{P}(0 \leq Z \leq z)$
0.50.1915
1.00.3413
1.50.4332
2.00.4772
Q29C 4 marks Geometric Sequences and Series Sum of an Infinite Geometric Series (Direct Computation) View
A geometric sequence $\left\{ a_{n} \right\}$ satisfies $$\sum_{n=1}^{\infty} \left(\left| a_{n} \right| + a_{n}\right) = \frac{40}{3}, \quad \sum_{n=1}^{\infty} \left(\left| a_{n} \right| - a_{n}\right) = \frac{20}{3}$$ The inequality $$\lim_{n \rightarrow \infty} \sum_{k=1}^{2n} \left((-1)^{\frac{k(k+1)}{2}} \times a_{m+k}\right) > \frac{1}{700}$$ is satisfied. What is the sum of all natural numbers $m$ satisfying this inequality? [4 points]
Q29G 4 marks Conic sections Triangle or Quadrilateral Area and Perimeter with Foci View
There is a hyperbola $x^{2} - \frac{y^{2}}{35} = 1$ with foci at $\mathrm{F}(c, 0)$, $\mathrm{F}'(-c, 0)$ ($c > 0$). For a point P on this hyperbola in the first quadrant, let Q be a point on line $\mathrm{PF}'$ such that $\overline{\mathrm{PQ}} = \overline{\mathrm{PF}}$. When triangle $\mathrm{QF'F}$ and triangle $\mathrm{FF'P}$ are similar, the area of triangle PFQ is $\frac{q}{p}\sqrt{5}$. Find the value of $p + q$. (Here, $\overline{\mathrm{PF}'} < \overline{\mathrm{QF}'}$ and $p$ and $q$ are coprime natural numbers.) [4 points]
Q30 4 marks Tree Diagrams Multi-Stage Sequential Process View
Five coins are placed in a line on a table. At the start, the coins in the 1st and 2nd positions show heads, and the coins in the remaining 3 positions show tails. Using these 5 coins and one die, the following trial is performed. Roll the die once. If the result is $k$, if $k \leq 5$, flip the coin in the $k$-th position once and place it back, if $k = 6$, flip all coins once and place them back. After repeating this trial 3 times, what is the probability that all 5 coins show heads? Express the answer as $\frac{q}{p}$. What is the value of $p + q$? (Given: $p$ and $q$ are coprime natural numbers.) [4 points]
Q30C 4 marks Trigonometric equations in context View
For two constants $a$ ($1 \leq a \leq 2$) and $b$, the function $f(x) = \sin(ax + b + \sin x)$ satisfies the following conditions. (가) $f(0) = 0$ and $f(2\pi) = 2\pi a + b$ (나) The minimum positive value of $t$ such that $f'(0) = f'(t)$ is $4\pi$. Let $A$ be the set of all values of $\alpha$ in the open interval $(0, 4\pi)$ where the function $f(x)$ has a local maximum. If $n$ is the number of elements in set $A$ and $\alpha_{1}$ is the smallest element in set $A$, then $n\alpha_{1} - ab = \frac{q}{p}\pi$. What is the value of $p + q$? [4 points]
Q30G 4 marks Vectors Introduction & 2D Optimization of a Vector Expression View
In the coordinate plane, there is a square ABCD with side length 4. $$|\overrightarrow{\mathrm{XB}} + \overrightarrow{\mathrm{XC}}| = |\overrightarrow{\mathrm{XB}} - \overrightarrow{\mathrm{XC}}|$$ Let $S$ be the figure formed by points X satisfying this condition. For a point P on figure $S$, $$4\overrightarrow{\mathrm{PQ}} = \overrightarrow{\mathrm{PB}} + 2\overrightarrow{\mathrm{PD}}$$ Let Q be the point satisfying this condition. If the maximum and minimum values of $\overrightarrow{\mathrm{AC}} \cdot \overrightarrow{\mathrm{AQ}}$ are $M$ and $m$ respectively, find the value of $M \times m$. [4 points]