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
2021 ast__math-b

10 maths questions

QA 8 marks Inequalities Solving inequalities involving modulus View
On a number line, there is the origin $O$ and three points $A ( - 2 ) , B ( 10 ) , C ( x )$, where $x$ is a real number. Given that the lengths of segments $\overline { B C } , \overline { A C } , \overline { O B }$ satisfy $\overline { B C } < \overline { A C } < \overline { O B }$, then the maximum range of $x$ is (8) $< x <$ (9).
QB 8 marks Matrices Matrix Power Computation and Application View
Let matrix $A = \left[ \begin{array} { c c } 1 & - 2 \\ 0 & 1 \end{array} \right] \left[ \begin{array} { l l } 1 & 0 \\ 0 & 6 \end{array} \right] \left[ \begin{array} { c c } 1 & - 2 \\ 0 & 1 \end{array} \right] ^ { - 1 } , B = \left[ \begin{array} { c c } 1 & - 2 \\ 0 & 1 \end{array} \right] \left[ \begin{array} { l l } 6 & 0 \\ 0 & 1 \end{array} \right] \left[ \begin{array} { c c } 1 & - 2 \\ 0 & 1 \end{array} \right] ^ { - 1 }$, where $\left[ \begin{array} { c c } 1 & - 2 \\ 0 & 1 \end{array} \right] ^ { - 1 }$ is the inverse matrix of $\left[ \begin{array} { c c } 1 & - 2 \\ 0 & 1 \end{array} \right]$. If $A + B = \left[ \begin{array} { l l } a & b \\ c & d \end{array} \right]$, then $a + b + c + d =$ (10)(11).
QC 8 marks Vectors Introduction & 2D Contextual Probability Requiring Binomial Modeling Setup View
A biased coin has probability $\frac { 1 } { 3 }$ of showing heads and probability $\frac { 2 } { 3 }$ of showing tails. On a coordinate plane, a game piece moves to the next position based on the result of flipping this coin, according to the following rules: (I) If heads appears, the piece moves from its current position in the direction and distance of vector $( - 1,2 )$ to the next position; (II) If tails appears, the piece moves from its current position in the direction and distance of vector $( 1,0 )$ to the next position. For example: If the game piece is currently at coordinates $( 2,4 )$ and tails appears, the piece moves to coordinates $( 3,4 )$. Suppose the game piece starts at the origin $( 0,0 )$ and, according to the above rules, flips the coin 6 times consecutively, with each flip being independent. After 6 moves, the game piece is most likely to stop at coordinates (12), (13)).
QI 12 marks Straight Lines & Coordinate Geometry Point-to-Line Distance Computation View
On a coordinate plane, there are two points $A ( - 3,4 ) , B ( 3,2 )$ and a line $L$. Points $A$ and $B$ are on opposite sides of line $L$, and $\vec { n } = ( 4 , - 3 )$ is a normal vector to line $L$. The distance from point $A$ to line $L$ is 5 times the distance from point $B$ to line $L$. Based on the above, answer the following questions.
(1) Find the dot product of vector $\overrightarrow { A B }$ and vector $\vec { n }$. (4 points)
(2) Find the equation of line $L$. (4 points)
(3) Point $P$ is on line $L$ and $\overline { P A } = \overline { P B }$. Find the coordinates of point $P$. (4 points)
Q2 6 marks Stationary points and optimisation Vertex location identification from evaluated conditions View
Given that a real-coefficient quadratic polynomial function $f ( x )$ satisfies $f ( - 1 ) = k , f ( 1 ) = 9 k , f ( 3 ) = - 15 k$, where $k > 0$. Let the $x$-coordinate of the vertex of the graph of $y = f ( x )$ be $a$. Select the correct option.
(1) $a \leq - 1$
(2) $- 1 < a < 1$
(3) $a = 1$
(4) $1 < a < 3$
(5) $3 \leq a$
Q3 6 marks Probability Definitions Expectation and Variance from Context-Based Random Variables View
A company holds a year-end lottery. Each person randomly draws two cards from six cards numbered 1 to 6. Assume each card has an equal chance of being drawn, and the rules are as follows: (I) If the sum of the numbers on the two cards is odd, the person wins 100 yuan and the lottery ends; (II) If the sum is even, the two cards are discarded, and two cards are randomly drawn from the remaining four cards. If the sum of their numbers is odd, the person wins 50 yuan; otherwise, there is no prize and the lottery ends. According to the above rules, what is the expected value of the prize money for each person participating in this lottery?
(1) 50
(2) 70
(3) 72
(4) 80
(5) 100
Q4 8 marks Laws of Logarithms Compare or Order Logarithmic Values View
Let $a = \log _ { 2 } 8 , ~ b = \log _ { 3 } 1 , ~ c = \log _ { 0.5 } 8$. Select the correct options.
(1) $b = 0$
(2) $a + b + c > 0$
(3) $a > b > c$
(4) $a ^ { 2 } > b ^ { 2 } > c ^ { 2 }$
(5) $2 ^ { a } > 3 ^ { b } > \left( \frac { 1 } { 2 } \right) ^ { c }$
Q5 8 marks Combinations & Selection Selection with Group/Category Constraints View
A convenience store packages three building block models (A, B, C) and five character figurines ($a, b, c, d, e$), totaling eight toys, into two bags for sale. Each bag contains four toys, and the packaging follows these principles: (I) A and $a$ must be in the same bag. (II) Each bag must contain at least one building block model. (III) $d$ and $e$ must be in different bags. Based on the above, select the correct options.
(1) Each bag must contain at least two character figurines
(2) B and C must be in different bags
(3) If B and $d$ are in the same bag, then C and $e$ must be in the same bag
(4) If B and $d$ are in different bags, then $b$ and $c$ must be in different bags
(5) If $b$ and $c$ are in different bags, then B and C must be in the same bag
Q6 8 marks Sequences and series, recurrence and convergence Multiple-choice on sequence properties View
Given a real number sequence $\left\langle a _ { n } \right\rangle$ satisfying $a _ { 1 } = 1 , a _ { n + 1 } = \frac { 2 n + 1 } { 2 n - 1 } a _ { n } , n$ is a positive integer. Select the correct options.
(1) $a _ { 2 } = 3$
(2) $a _ { 4 } = 9$
(3) $\left\langle a _ { n } \right\rangle$ is a geometric sequence
(4) $\sum _ { n = 1 } ^ { 20 } a _ { n } = 400$
(5) $\lim _ { n \rightarrow \infty } \frac { a _ { n } } { n } = 2$
Q7 8 marks Discrete Probability Distributions Verifying Statements About Probability Properties View
A person's probability of hitting a dart each time is $\frac { 1 } { 2 }$, and the results of each dart throw are independent. From the following options, select the events with probability $\frac { 1 } { 2 }$.
(1) Throwing darts 2 times consecutively, hitting exactly 1 time
(2) Throwing darts 4 times consecutively, hitting exactly 2 times
(3) Throwing darts 4 times consecutively, the total number of hits is odd
(4) Throwing darts 6 times consecutively, given that the first throw misses, the second throw hits
(5) Throwing darts 6 times consecutively, given that exactly 1 hit in the first 2 throws, exactly 2 hits in the last 4 throws