taiwan-gsat

Papers (23)
2025
ast__math-a 16 ast__math-b 18 gsat__math-a 20 gsat__math-b 18
2024
ast__math-a 17 gsat__math-a 20 gsat__math-b 15
2023
ast__math-a 17 gsat__math-a 20 gsat__math-b 15
2022
ast__math-a 16 gsat__math-a 20 gsat__math-b 19
2021
ast__math-a 13 ast__math-b 10 gsat__math 20
2020
ast__math-a 13 ast__math-b 11
2010
gsat__math 12
2009
gsat__math 10
2008
gsat__math 9
2007
gsat__math 10
2006
gsat__math 9
2010 gsat__math

12 maths questions

Q1 Permutations & Arrangements Counting or Combinatorial Problems on APs View
1. If each term in the sequence $a_{1}, a_{2}, \ldots, a_{k}, \ldots, a_{10}$ is either 1 or $-1$, how many possible values can $a_{1} + a_{2} + \cdots + a_{k} + \cdots + a_{10}$ take?
(1) 10
(2) 11
(3) $P_{2}^{10}$
(4) $C_{2}^{10}$
(5) $2^{10}$
Q2 Matrices Determinant and Rank Computation View
2. Given that $a, b$ are integers and the determinant $\left| \begin{array} { c c } 5 & a \\ b & 7 \end{array} \right| = 4$, what is the absolute value $|a + b|$?
(1) 16
(2) 31
(3) 32
(4) 39
(5) Insufficient information to determine
Q3 Discrete Probability Distributions Expectation and Variance from Context-Based Random Variables View
3. A box contains three red balls and three white balls. A lottery game involves randomly drawing two balls simultaneously from the box. If the two balls are different colors, the player wins 100 dollars; if the two balls are the same color, there is no prize. What is the expected value of the prize for this game?
(1) 20 dollars
(2) 30 dollars
(3) 40 dollars
(4) 50 dollars
(5) 60 dollars
Q4 Straight Lines & Coordinate Geometry Area Computation in Coordinate Geometry View
4. On a coordinate plane, two points $A(1, 0)$ and $B(0, 1)$ are given. Consider three additional points $P(\pi, 1)$, $Q(-\sqrt{3}, 6)$, and $R\left(2, \log_{4} 32\right)$. Let the area of $\triangle PAB$ be $p$, the area of $\triangle QAB$ be $q$, and the area of $\triangle RAB$ be $r$. Which of the following options is correct?
(1) $p < q < r$
(2) $p < r < q$
(3) $q < p < r$
(4) $q < r < p$
(5) $r < q < p$
Q5 Exponential Functions Applied/Contextual Exponential Modeling View
5. In a sealed laboratory, initially there are 1000 bacteria of a certain species, and they reproduce at a rate of 8\% per hour. If reproduction continues at this rate, which of the following options best approximates the number of bacteria after 100 hours?
(1) 9 thousand
(2) 108 thousand
(3) 2200 thousand
(4) 3200 thousand
(5) 32000 thousand
Q6 Circles Sphere and 3D Circle Problems View
6. In coordinate space, $O$ is the origin and point $A$ has coordinates $(1, 2, 1)$. Let $S$ be the sphere with center $O$ and radius 4. What is the figure formed by all points $P$ on $S$ that satisfy the dot product $\overrightarrow{OA} \cdot \overrightarrow{OP} = 6$?
(1) Empty set
(2) A single point
(3) Two points
(4) A circle
(5) Two circles
Q7 5 marks Conic sections Conic Identification and Conceptual Properties View
7. Let the ellipses $\Gamma_{1}: \frac{x^{2}}{5^{2}} + \frac{y^{2}}{3^{2}} = 1$, $\Gamma_{2}: \frac{x^{2}}{5^{2}} + \frac{y^{2}}{3^{2}} = 2$, $\Gamma_{3}: \frac{x^{2}}{5^{2}} + \frac{y^{2}}{3^{2}} = \frac{2x}{5}$ have major axis lengths $l_{1}$, $l_{2}$, $l_{3}$ respectively. Which of the following options is correct?
(1) $l_{1} = l_{2} = l_{3}$
(2) $l_{1} = l_{2} < l_{3}$
(3) $l_{1} < l_{2} < l_{3}$
(4) $l_{1} = l_{3} < l_{2}$
(5) $l_{1} < l_{3} < l_{2}$
II. Multiple-Choice Questions (25 points)
Instructions: For questions 8 through 12, each of the five options is independent, and at least one option is correct. Select all correct options and mark them on the ``Answer Sheet''. No points are deducted for incorrect answers. Full marks (5 points) are awarded for all five options correct; 2.5 points are awarded if only one option is incorrect; no points are awarded if two or more options are incorrect.
Q8 Trig Graphs & Exact Values View
8. Let $\theta_{1}, \theta_{2}, \theta_{3}, \theta_{4}$ be angles in the first, second, third, and fourth quadrants respectively, all between 0 and $2\pi$. Given that $|\cos \theta_{1}| = |\cos \theta_{2}| = |\cos \theta_{3}| = |\cos \theta_{4}| = \frac{1}{3}$, which of the following options are correct?
(1) $\theta_{1} < \frac{\pi}{4}$
(2) $\theta_{1} + \theta_{2} = \pi$
(3) $\cos \theta_{3} = -\frac{1}{3}$
(4) $\sin \theta_{4} = \frac{2\sqrt{2}}{3}$
(5) $\theta_{4} = \theta_{3} + \frac{\pi}{2}$
Q9 Standard trigonometric equations Solve trigonometric equation for solutions in an interval View
9. Which of the following equations have real solutions?
(1) $x^{3} + x - 1 = 0$
(2) $2^{x} + 2^{-x} = 0$
(3) $\log_{2} x + \log_{x} 2 = 1$
(4) $\sin x + \cos 2x = 3$
(5) $4 \sin x + 3 \cos x = \frac{9}{2}$
Q10 Sequences and series, recurrence and convergence True/false or conceptual reasoning about sequences View
10. Let $a_{1}, a_{2}, \ldots, a_{n}, \ldots$ be a sequence of real numbers satisfying $a_{n+1} = \frac{n(n+1)}{2} - a_{n}$ for all positive integers $n$. Which of the following options are correct?
(1) If $a_{1} = 1$, then $a_{2} = 1$
(2) If $a_{1}$ is an integer, then every term of the sequence is an integer
(3) If $a_{1}$ is irrational, then every term of the sequence is irrational
(4) $a_{2} \leq a_{4} \leq \cdots \leq a_{2n} \leq \cdots$ (where $n$ is a positive integer)
(5) If $a_{k}$ is odd, then $a_{k+2}, a_{k+4}, \ldots, a_{k+2n}, \ldots$ are all odd (where $n$ is a positive integer)
Q11 Vectors 3D & Lines MCQ: Distance or Length Optimization on a Line View
11. In coordinate space, the point on line $L$ closest to point $Q$ is called the projection of $Q$ onto $L$. Given that $L$ is a line on the plane $2x - y = 2$ passing through the point $(2, 2, 2)$. Which of the following points could be the projection of the origin $O$ onto $L$?
(1) $(2, 2, 2)$
(2) $(2, 0, 2)$
(3) $\left(\frac{4}{5}, -\frac{2}{5}, 0\right)$
(4) $\left(\frac{4}{5}, -\frac{2}{5}, -2\right)$
(5) $\left(\frac{8}{9}, -\frac{2}{9}, -\frac{2}{9}\right)$
Q12 40 marks Confidence intervals Compute confidence interval for a proportion (estimation) View
12. A sampling survey was conducted to understand the level of support among Taiwan's citizens for a certain issue. The results, classified by gender, are shown in the table below:
Female CitizensMale Citizens
Proportion supporting the issue $\hat{p}$0.520.59
Standard deviation of $\hat{p}$: $\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$0.020.04

Which of the following conclusions can be drawn from this sampling result?
(1) The proportion of male citizens in Taiwan supporting this issue is greater than the proportion of female citizens supporting this issue
(2) At a 95\% confidence level, the confidence interval for the proportion of female citizens in Taiwan supporting this issue is $[0.48, 0.56]$ (rounded to the second decimal place)
(3) The number of female citizens in this sample is less than the number of male citizens
(4) If gender is not distinguished, the proportion of people in this sample supporting the issue $\hat{p}$ is between 0.52 and 0.59
(5) If gender is not distinguished, the standard deviation of $\hat{p}$ in this sample $\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$ is between 0.02 and 0.04
Part II: Fill-in-the-Blank Questions (40 points)
Instructions