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
2023 gsat__math-a

20 maths questions

Q1 5 marks Indices and Surds Solving Exponential or Index Equations View
If a positive integer $N$ is entered into a calculator, and then the ``$\sqrt{ }$'' key (taking the positive square root) is pressed 3 times consecutively, the display shows the answer as 2. Then $N$ equals which of the following options?
(1) $2^{3}$
(2) $2^{4}$
(3) $2^{6}$
(4) $2^{8}$
(5) $2^{12}$
Q2 5 marks Trig Graphs & Exact Values View
On the coordinate plane, a circle with center at the origin $O$ and radius 1 intersects the positive directions of the coordinate axes at points $A$ and $B$ respectively. On the circular arc in the first quadrant, a point $C$ is taken to draw a tangent line to the circle that intersects the two axes at points $D$ and $E$ respectively, as shown in the figure. Let $\angle OEC = \theta$. Select the option that represents $\tan \theta$.
(1) $\overline{OE}$
(2) $\overline{OC}$
(3) $\overline{OD}$
(4) $\overline{CE}$
(5) $\overline{CD}$
Q3 5 marks Exponential Equations & Modelling Properties of Logarithmic Functions and Statement Verification View
A student derived an equation that two physical quantities $s$ and $t$ should satisfy. To verify the theory, he conducted an experiment and obtained 15 sets of data for the two physical quantities $(s_{k}, t_{k})$, $k = 1, \cdots, 15$. The teacher suggested that he first take the logarithm of $t_{k}$, and plot the corresponding points $\left(s_{k}, \log t_{k}\right)$, $k = 1, \cdots, 15$ on the coordinate plane; where the first data is the horizontal axis coordinate and the second data is the vertical axis coordinate. Using regression line analysis, the student verified his theory. The regression line passes through the origin with a positive slope less than 1. What is the relationship between $s$ and $t$ that the student obtained most likely to be which of the following options?
(1) $s = 2t$
(2) $s = 3t$
(3) $t = 10^{s}$
(4) $t^{2} = 10^{s}$
(5) $t^{3} = 10^{s}$
Q4 5 marks Permutations & Arrangements Linear Arrangement with Constraints View
Arrange the digits $1, 2, 3, \ldots, 9$ into a nine-digit number (digits cannot be repeated) such that the first 5 digits are increasing from left to right and the last 5 digits are decreasing from left to right. How many nine-digit numbers satisfy the conditions?
(1) $\frac{8!}{4!4!}$
(2) $\frac{8!}{5!3!}$
(3) $\frac{9!}{5!4!}$
(4) $\frac{8!}{5!}$
(5) $\frac{9!}{5!}$
Q5 5 marks Vector Product and Surfaces View
It is known that $P$, $Q$, $R$ are three non-collinear points on the plane $2x - 3y + 5z = \sqrt{7}$ in coordinate space. Let $\overrightarrow{PQ} = (a_{1}, b_{1}, c_{1})$, $\overrightarrow{PR} = (a_{2}, b_{2}, c_{2})$. Select the option in which the absolute value of the determinant is the largest.
(1) $\left|\begin{array}{ccc} -1 & 1 & 1 \\ a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \end{array}\right|$
(2) $\left|\begin{array}{ccc} 1 & -1 & 1 \\ a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \end{array}\right|$
(3) $\left|\begin{array}{ccc} 1 & 1 & -1 \\ a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \end{array}\right|$
(4) $\left|\begin{array}{ccc} -1 & -1 & 1 \\ a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \end{array}\right|$
(5) $\left|\begin{array}{ccc} -1 & -1 & -1 \\ a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \end{array}\right|$
Q6 5 marks Vectors Introduction & 2D Dot Product Computation View
In coordinate space, consider a unit cube with edge length 1, with one vertex $O$ fixed. From the seven vertices other than $O$, two distinct points are randomly selected, denoted as $P$ and $Q$. What is the expected value of the dot product $\overrightarrow{OP} \cdot \overrightarrow{OQ}$ among the following options?
(1) $\frac{4}{7}$
(2) $\frac{5}{7}$
(3) $\frac{6}{7}$
(4) 1
(5) $\frac{8}{7}$
Q7 5 marks Arithmetic Sequences and Series Compute Partial Sum of an Arithmetic Sequence View
A company has two new employees, A and B, who start at the same time with the same starting salary. The company promises the following salary adjustment methods for employees A and B:
Employee A: After 3 months of work, starting the next month, monthly salary increases by 200 yuan; thereafter, salary is adjusted in the same manner every 3 months.
Employee B: After 12 months of work, starting the next month, monthly salary increases by 1000 yuan; thereafter, salary is adjusted in the same manner every 12 months.
Based on the above description, select the correct options.
(1) After 8 months of work, the monthly salary in the 9th month is 600 yuan more than in the 1st month
(2) After one year of work, in the 13th month, employee A's monthly salary is higher than employee B's
(3) After 18 months of work, in the 19th month, employee A's monthly salary is higher than employee B's
(4) After 18 months of work, the total salary received by employee A is less than the total salary received by employee B
(5) After two years of work, in the 12 months of the 3rd year, there are exactly 3 months where employee A's monthly salary is higher than employee B's
Q8 5 marks Discrete Probability Distributions Properties of Named Discrete Distributions (Non-Binomial) View
A lottery game has a single-play winning probability of 0.1, and each play is an independent event. For each positive integer $n$, let $p_{n}$ be the probability of winning at least once in $n$ plays of this game. Select the correct options.
(1) $p_{n+1} > p_{n}$
(2) $p_{3} = 0.3$
(3) $\langle p_{n} \rangle$ is an arithmetic sequence
(4) Playing this game two or more times, the probability of not winning on the first play and winning on the second play equals $p_{2} - p_{1}$
(5) When playing this game $n$ times with $n \geq 2$, the probability of winning at least 2 times equals $2p_{n}$
Q9 5 marks Sequences and series, recurrence and convergence Summation of sequence terms View
Let $a_{1}, a_{2}, a_{3}, \ldots, a_{n}$ be a geometric sequence with first term 3 and common ratio $3\sqrt{3}$. Select the number of terms $n$ that satisfy the inequality
$$\log_{3} a_{1} - \log_{3} a_{2} + \log_{3} a_{3} - \log_{3} a_{4} + \ldots + (-1)^{n+1} \log_{3} a_{n} > 18$$
among the possible options.
(1) 23
(2) 24
(3) 25
(4) 26
(5) 27
Q10 5 marks Straight Lines & Coordinate Geometry Line Equation and Parametric Representation View
Consider the line $L: 5y + (2k-4)x - 10k = 0$ on the coordinate plane (where $k$ is a real number), and the rectangle $OABC$ with vertices at $O(0,0)$, $A(10,0)$, $B(10,6)$, $C(0,6)$. Let $L$ intersect the line $OC$ and the line $AB$ at points $D$ and $E$ respectively. Select the correct options.
(1) When $k = 4$, the line $L$ passes through point $A$
(2) If the line $L$ passes through point $C$, then the slope of $L$ is $-\frac{5}{2}$
(3) If point $D$ is on the line segment $\overline{OC}$, then $0 \leq k \leq 3$
(4) If $k = \frac{1}{2}$, then the line segment $\overline{DE}$ is inside the rectangle $OABC$ (including the boundary)
(5) If the line segment $\overline{DE}$ is inside the rectangle $OABC$ (including the boundary), then the slope of $L$ could be $\frac{3}{10}$
Q11 5 marks Matrices True/False or Multiple-Select Conceptual Reasoning View
On the coordinate plane, let $A$ and $B$ denote the rotation matrices for clockwise and counterclockwise rotation by $90^{\circ}$ about the origin respectively. Let $C$ and $D$ denote the reflection matrices with reflection axes $x = y$ and $x = -y$ respectively. Select the correct options.
(1) $A$ and $C$ map the point $(1,0)$ to the same point
(2) $A = -B$
(3) $C = D^{-1}$
(4) $AB = CD$
(5) $AC = BD$
Q12 5 marks Trig Graphs & Exact Values View
Let $f(x) = \sin x + \sqrt{3} \cos x$. Select the correct options.
(1) The vertical line $x = \frac{\pi}{6}$ is an axis of symmetry of the graph of $y = f(x)$
(2) If the vertical lines $x = a$ and $x = b$ are both axes of symmetry of the graph of $y = f(x)$, then $f(a) = f(b)$
(3) In the interval $[0, 2\pi)$, there is only one real number $x$ satisfying $f(x) = \sqrt{3}$
(4) In the interval $[0, 2\pi)$, the sum of all real numbers $x$ satisfying $f(x) = \frac{1}{2}$ does not exceed $2\pi$
(5) The graph of $y = f(x)$ can be obtained from the graph of $y = 4\sin^{2}\frac{x}{2}$ by appropriate (left-right, up-down) translation
Q13 5 marks Simultaneous equations View
A newly opened juice shop offers three types of beverages: juice, milk tea, and coffee. The sales volume (in cups) and total revenue (in yuan) for each type of beverage over the first 3 days are shown in the table below. For example, on the first day, the sales of juice, milk tea, and coffee were 60 cups, 80 cups, and 50 cups respectively, with total revenue of 12,900 yuan.
It is known that the price of each type of beverage is the same every day. Then the price per cup of coffee is
Juice (cups)Milk Tea (cups)Coffee (cups)Total Revenue (yuan)
Day 160805012900
Day 23040306850
Day 350704010800
Q14 5 marks Factor & Remainder Theorem Remainder by Quadratic or Higher Divisor View
Let $a$ and $b$ be real numbers (where $a > 0$). If the polynomial $ax^{2} + (2a+b)x - 12$ divided by $x^{2} + (2-a)x - 2a$ gives a remainder of 6, then the ordered pair $(a, b) = $ (14--1), 14--2).
Q15 5 marks Vectors Introduction & 2D Perpendicularity or Parallel Condition View
Let $O$, $A$, $B$ be three non-collinear points on the coordinate plane, where the vector $\overrightarrow{OA}$ is perpendicular to $\overrightarrow{OB}$. If points $C$ and $D$ are on the line $AB$ satisfying $\overrightarrow{OC} = \frac{3}{5}\overrightarrow{OA} + \frac{2}{5}\overrightarrow{OB}$, $3\overline{AD} = 8\overline{BD}$, and $\overrightarrow{OC}$ is perpendicular to $\overrightarrow{OD}$, then $\frac{\overline{OB}}{\overline{OA}} = $ (Express as a fraction in lowest terms)
Q16 5 marks Vectors: Lines & Planes Perpendicular/Orthogonal Projection onto a Plane View
Let $E: x + z = 2$ be the plane in coordinate space passing through the three points $A(2,-1,0)$, $B(0,1,2)$, $C(-2,1,4)$. There is another point $P$ on the plane $z = 1$ whose projection onto $E$ is equidistant from points $A$, $B$, and $C$. Then the distance from point $P$ to plane $E$ is (16--1)$\sqrt{16\text{-}2}$. (Express as a simplified radical)
Q17 5 marks Vectors: Lines & Planes Distance Computation (Point-to-Plane or Line-to-Line) View
In coordinate space, there are two non-intersecting lines $L_{1}: \left\{\begin{array}{l} x = 1+t \\ y = 1-t \\ z = 2+t \end{array}\right.$, $t$ is a real number, $L_{2}: \left\{\begin{array}{l} x = 2+2s \\ y = 5+s \\ z = 6-s \end{array}\right.$, $s$ is a real number. Another line $L_{3}$ intersects both $L_{1}$ and $L_{2}$ and is perpendicular to both. If points $P$ and $Q$ are on $L_{1}$ and $L_{2}$ respectively and both are at distance 3 from $L_{3}$, then the distance between points $P$ and $Q$ is (17--1)$\sqrt{17\text{-}2}$. (Express as a simplified radical)
Q18 3 marks Standard trigonometric equations Evaluate trigonometric expression given a constraint View
On the coordinate plane, $O$ is the origin, and points $A(1,0)$ and $B(-2,0)$ are given. There are also two points $P$ and $Q$ in the upper half-plane satisfying $\overline{AP} = \overline{OA}$, $\overline{BQ} = \overline{OB}$, $\angle POQ$ is a right angle. Let $\angle AOP = \theta$.
The length of line segment $\overline{OP}$ is which of the following options? (Single choice question, 3 points)
(1) $\sin\theta$
(2) $\cos\theta$
(3) $2\sin\theta$
(4) $2\cos\theta$
(5) $\cos 2\theta$
Q19 6 marks Sine and Cosine Rules Multi-step composite figure problem View
On the coordinate plane, $O$ is the origin, and points $A(1,0)$ and $B(-2,0)$ are given. There are also two points $P$ and $Q$ in the upper half-plane satisfying $\overline{AP} = \overline{OA}$, $\overline{BQ} = \overline{OB}$, $\angle POQ$ is a right angle. Let $\angle AOP = \theta$.
If $\sin\theta = \frac{3}{5}$, find the coordinates of point $Q$ and explain that $\overrightarrow{BQ} = 2\overrightarrow{AP}$. (Non-multiple choice question, 6 points)
Q20 6 marks Straight Lines & Coordinate Geometry Area Computation in Coordinate Geometry View
On the coordinate plane, $O$ is the origin, and points $A(1,0)$ and $B(-2,0)$ are given. There are also two points $P$ and $Q$ in the upper half-plane satisfying $\overline{AP} = \overline{OA}$, $\overline{BQ} = \overline{OB}$, $\angle POQ$ is a right angle. Let $\angle AOP = \theta$.
(Continuing from question 19, where $\sin\theta = \frac{3}{5}$) Find the distance from point $A$ to line $BQ$, and find the area of quadrilateral $PABQ$. (Non-multiple choice question, 6 points)