taiwan-gsat

Papers (18)
2025
ast__math-a 17 ast__math-b 18 gsat__math-a 20 gsat__math-b 19
2024
ast__math-a 17 gsat__math-a 20 gsat__math-b 20
2023
ast__math-a 17 gsat__math-a 20 gsat__math-b 20
2022
ast__math-a 15 gsat__math-a 19 gsat__math-b 20
2021
ast__math-a 13 ast__math-b 11 gsat__math 20
2020
ast__math-a 13 ast__math-b 12
2024 ast__math-a

17 maths questions

Q1 6 marks Sine and Cosine Rules Heights and distances / angle of elevation problem View
As shown in the diagram on the right, there is a $\triangle ABC$. It is known that the altitude $\overline{AD} = 12$ on side $\overline{BC}$, and $\tan \angle B = \frac{3}{2}$, $\tan \angle C = \frac{2}{3}$. What is the length of $\overline{BC}$?
(1) 20
(2) 21
(3) 24
(4) 25
(5) 26
Q2 6 marks Conic sections Equation Determination from Geometric Conditions View
On the coordinate plane, the equation of ellipse $\Gamma$ is $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{6^{2}} = 1$ (where $a$ is a positive real number). If $\Gamma$ is scaled by a factor of 2 in the $x$-axis direction and by a factor of 3 in the $y$-axis direction with the origin $O$ as the center, the resulting new figure passes through the point $(18, 0)$. Which of the following options is a focus of $\Gamma$?
(1) $(0, 3\sqrt{3})$
(2) $(-3\sqrt{5}, 0)$
(3) $(0, 6\sqrt{13})$
(4) $(-3\sqrt{13}, 0)$
(5) $(9, 0)$
Q3 6 marks Binomial Distribution Contextual Probability Requiring Binomial Modeling Setup View
It is desired to place 4 identical chess rooks on a $5 \times 5$ chessboard. Since rooks can capture pieces in the same row or column, the placement rule is that at most one rook can be placed in each row and each column. Given that rooks are not placed in the first, third, and fifth squares of the first row (as shown by the crossed squares in the diagram), how many ways are there to place the rooks?
(1) 216
(2) 240
(3) 288
(4) 312
(5) 360
Q4 8 marks Roots of polynomials Existence or counting of roots with specified properties View
A game company will hold a lottery activity. The company announces that each lottery draw requires using one token, and the winning probability for each draw is $\frac{1}{10}$. A certain person decides to save a certain number of tokens and start drawing after the activity begins, stopping only when all tokens are used. Select the correct options.
(1) The expected value of the number of draws needed for the person to win once is 10
(2) The probability that the person wins at least once in two draws is 0.2
(3) The probability that the person does not win in 10 draws is less than the probability of winning in 1 draw
(4) The person must save at least 22 tokens to guarantee a winning probability greater than 0.9
(5) If the person saves sufficiently many tokens, the winning probability can be guaranteed to be 1
Q5 8 marks Complex Numbers Arithmetic True/False or Property Verification Statements View
Let $f(x)$ be a cubic polynomial with real coefficients. It is known that $f(-2 - 3i) = 0$ (where $i = \sqrt{-1}$), and the remainder when $f(x)$ is divided by $x^{2} + x - 2$ is 18. Select the correct options.
(1) $f(2 + 3i) = 0$
(2) $f(-2) = 18$
(3) The coefficient of the cubic term of $f(x)$ is negative
(4) $f(x) = 0$ has exactly one positive real root
(5) The center of symmetry of the graph $y = f(x)$ is in the first quadrant
Q6 8 marks Vector Product and Surfaces View
In coordinate space, consider two vectors $\vec{u}$ and $\vec{v}$ satisfying the dot product $\vec{u} \cdot \vec{v} = \sqrt{15}$ and the cross product $\vec{u} \times \vec{v} = (-1, 0, 3)$. Select the correct options.
(1) The angle $\theta$ between $\vec{u}$ and $\vec{v}$ (where $0 \leq \theta \leq \pi$, $\pi$ is the circumference ratio) is greater than $\frac{\pi}{4}$
(2) $\vec{u}$ could be $(1, 0, -1)$
(3) $|\vec{u}| + |\vec{v}| \geq 2\sqrt{5}$
(4) If $\vec{v}$ is known, then $\vec{u}$ can be uniquely determined
(5) If $|\vec{u}| + |\vec{v}|$ is known, then $|\vec{v}|$ can be uniquely determined
Q7 8 marks Applied differentiation MCQ on derivative and graph interpretation View
On the coordinate plane, consider the graphs of two functions $f(x) = x^{5} - 5x^{3} + 5x^{2} + 5$ and $g(x) = \sin\left(\frac{\pi x}{3} + \frac{\pi}{2}\right)$ (where $\pi$ is the circumference ratio). Select the correct options.
(1) $f'(1) = 0$
(2) $y = f(x)$ is increasing on the closed interval $[0, 2]$
(3) $y = f(x)$ is concave up on the closed interval $[0, 2]$
(4) For any real number $x$, $g(x + 6\pi) = g(x)$
(5) Both $y = f(x)$ and $y = g(x)$ are increasing on the closed interval $[3, 4]$
Q8 8 marks Complex Numbers Arithmetic True/False or Property Verification Statements View
Let $z$ be a nonzero complex number, and let $\alpha = |z|$ and $\beta$ be the argument of $z$, where $0 \leq \beta < 2\pi$ (where $\pi$ is the circumference ratio). For any positive integer $n$, let the real numbers $x_{n}$ and $y_{n}$ be the real and imaginary parts of $z^{n}$, respectively. Select the correct options.
(1) If $\alpha = 1$ and $\beta = \frac{3\pi}{7}$, then $x_{10} = x_{3}$
(2) If $y_{3} = 0$, then $y_{6} = 0$
(3) If $x_{3} = 1$, then $x_{6} = 1$
(4) If the sequence $\langle y_{n} \rangle$ converges, then $\alpha \leq 1$
(5) If the sequence $\langle x_{n} \rangle$ converges, then the sequence $\langle y_{n} \rangle$ also converges
Q9 6 marks Simultaneous equations View
Let $a, b, c, d$ be real numbers. It is known that the augmented matrices of two systems of linear equations $\left\{\begin{array}{l} ax + by = 2 \\ cx + dy = 1 \end{array}\right.$ and $\left\{\begin{array}{l} ax + by = -1 \\ cx + dy = -1 \end{array}\right.$, after the same row operations, become $\left[\begin{array}{cc|c} 1 & -1 & 3 \\ 0 & 1 & 2 \end{array}\right]$ and $\left[\begin{array}{cc|c} 1 & -1 & 2 \\ 0 & 1 & -1 \end{array}\right]$ respectively. Then the solution to the system of equations $\left\{\begin{array}{l} ax + by = 0 \\ cx + dy = 1 \end{array}\right.$ is $x = $ 9-1, 9-2, $y = $ 9-3.
Q10 6 marks Circles Chord Length and Chord Properties View
On the coordinate plane, let $\Gamma$ be a circle with center at the origin, and $P$ be one of the intersection points of $\Gamma$ and the $x$-axis. It is known that the line passing through $P$ with slope $\frac{1}{2}$ intersects $\Gamma$ at another point $Q$, and $\overline{PQ} = 1$. Then the radius of $\Gamma$ is \hspace{2cm}. (Express as a simplified radical)
Q11 6 marks Sequences and series, recurrence and convergence Closed-form expression derivation View
Let the real numbers $a_{1}, a_{2}, \ldots, a_{9}$ form an arithmetic sequence with common difference 2, where $a_{1} \neq 0$ and $a_{3} > 0$. If $\log_{2} a_{3}$, $\log_{2} b$, $\log_{2} a_{9}$ form an arithmetic sequence in order, where $b$ is one of $a_{4}, a_{5}, a_{6}, a_{7}, a_{8}$, then $a_{9} = $ \hspace{2cm}. (Express as a simplified fraction)
Q12 4 marks Vectors: Lines & Planes Coplanarity and Relative Position of Planes View
In coordinate space, consider three planes $E_{1}: x + y + z = 7$, $E_{2}: x - y + z = 3$, $E_{3}: x - y - z = -5$. Let $L_{3}$ be the line of intersection of $E_{1}$ and $E_{2}$; $L_{1}$ be the line of intersection of $E_{2}$ and $E_{3}$; $L_{2}$ be the line of intersection of $E_{3}$ and $E_{1}$. It is known that the three lines $L_{1}, L_{2}, L_{3}$ have a common intersection point. Find the coordinates of this common intersection point $P$.
Q13 4 marks Vectors: Lines & Planes Dihedral Angle or Angle Between Planes/Lines View
In coordinate space, consider three planes $E_{1}: x + y + z = 7$, $E_{2}: x - y + z = 3$, $E_{3}: x - y - z = -5$. Let $L_{3}$ be the line of intersection of $E_{1}$ and $E_{2}$; $L_{1}$ be the line of intersection of $E_{2}$ and $E_{3}$; $L_{2}$ be the line of intersection of $E_{3}$ and $E_{1}$. Explain that among $L_{1}, L_{2}, L_{3}$, the acute angle between any two lines is $60^{\circ}$. (Note: Let the acute angle between $L_{1}$ and $L_{2}$ be $\alpha$, the acute angle between $L_{2}$ and $L_{3}$ be $\beta$, and the acute angle between $L_{3}$ and $L_{1}$ be $\gamma$)
Q14 4 marks Vectors: Lines & Planes Find Cartesian Equation of a Plane View
In coordinate space, consider three planes $E_{1}: x + y + z = 7$, $E_{2}: x - y + z = 3$, $E_{3}: x - y - z = -5$. Let $L_{3}$ be the line of intersection of $E_{1}$ and $E_{2}$; $L_{1}$ be the line of intersection of $E_{2}$ and $E_{3}$; $L_{2}$ be the line of intersection of $E_{3}$ and $E_{1}$. If a fourth plane $E_{4}$ together with $E_{1}, E_{2}, E_{3}$ encloses a regular tetrahedron with edge length $6\sqrt{2}$, find the equation of $E_{4}$ (write in the form $x + ay + bz = c$).
Q15 2 marks Chain Rule Straightforward Polynomial or Basic Differentiation View
On the coordinate plane, let $\Gamma$ be the graph of the cubic function $f(x) = x^{3} - 9x^{2} + 15x - 4$. Which of the following is the derivative of $f(x)$? (Single choice, 2 points)
(1) $x^{2} - 9x + 15$
(2) $3x^{3} - 18x^{2} + 15x - 4$
(3) $3x^{3} - 18x^{2} + 15x$
(4) $3x^{2} - 18x + 15$
(5) $x^{2} - 18x + 15$
Q16 4 marks Tangents, normals and gradients Find tangent line equation at a given point View
On the coordinate plane, let $\Gamma$ be the graph of the cubic function $f(x) = x^{3} - 9x^{2} + 15x - 4$. Show that $P(1, 3)$ is a point on $\Gamma$, and find the equation of the tangent line $L$ to $\Gamma$ at point $P$.
Q17 6 marks Areas by integration View
On the coordinate plane, let $\Gamma$ be the graph of the cubic function $f(x) = x^{3} - 9x^{2} + 15x - 4$. The tangent line $L$ to $\Gamma$ at point $P(1, 3)$ was found in question 16. Continuing from 16, find the area of the bounded region enclosed by $\Gamma$ and $L$.