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 gsat__math-b

20 maths questions

Q1 5 marks Measures of Location and Spread View
A game has a total of 210 players, each holding gems. Among them, 1 player holds 1 gem, 2 players hold 2 gems, and so on, with 20 players holding 20 gems. What is the 90th percentile of the number of gems held by each player?
(1) 16
(2) 17
(3) 18
(4) 19
(5) 20
Q2 5 marks Laws of Logarithms Compare or Order Logarithmic Values View
Given that $a , b , c$ are real numbers satisfying $1 < a < 10$, $b = \log a$, $c = \log b$, select the correct option.
(1) $c < 0 < b < 1$
(2) $0 < c < 1 < b$
(3) $0 < c < b < 1$
(4) $1 < c < b$
(5) $c < b < 0$
Q3 5 marks Straight Lines & Coordinate Geometry Perspective, Projection, and Applied Geometry View
In a shooting game, a player must avoid obstacles to shoot a target. A rectangular coordinate system is set up on the game screen with the lower left corner $O$ of the screen as the origin, the lower edge of the screen as the $x$-axis, and the left edge of the screen as the $y$-axis. The target is placed at point $P ( 12,10 )$. There are two walls in the screen (wall thickness is negligible), one wall extends horizontally from point $A ( 10,5 )$ to point $B ( 15,5 )$, and another wall extends horizontally from point $C ( 0,6 )$ to point $D ( 9,6 )$, as shown in the schematic diagram on the right. If a player at point $Q$ can shoot the target at point $P$ in a straight line without being blocked by the two walls, which of the following options could be the coordinates of point $Q$?
(1) $( 6,3 )$
(2) $( 7,3 )$
(3) $( 8,5 )$
(4) $( 9,1 )$
(5) $( 9,2 )$
Q4 5 marks Completing the square and sketching Optimization on a constrained domain via completing the square View
Given a vector $\vec { v } = ( - 2,3 )$ and two points $A$ and $B$ on the coordinate plane, where the $x$-coordinate and $y$-coordinate of point $A$, and the $x$-coordinate and $y$-coordinate of point $B$ all lie in the interval $[ 0,1 ]$, what is the maximum value of $| \vec { v } + \overrightarrow { A B } |$?
(1) $\sqrt { 13 }$
(2) $\sqrt { 17 }$
(3) $3 \sqrt { 2 }$
(4) 5
(5) $\sqrt { 2 } + \sqrt { 13 }$
Q5 5 marks Completing the square and sketching Max/min of a quadratic function on a closed interval with parameter View
Let the quadratic function $f ( x ) = x ^ { 2 } + b x + c$, where $b , c$ are real numbers. Given that $f ( x - 2 ) = f ( - x - 2 )$ holds for all real numbers $x$, and when $- 3 \leq x \leq 1$, the maximum value of $f ( x )$ is 4 times its minimum value, what is the minimum value of $f ( x )$?
(1) 0
(2) $\frac { 5 } { 3 }$
(3) 3
(4) 4
(5) 6
Q6 5 marks Straight Lines & Coordinate Geometry Geometric Figure on Coordinate Plane View
A resident of a building displays a Christmas tree-shaped light decoration on the building's exterior wall, as shown in the figure. From a certain point $P$ on the fifth floor exterior wall, small light bulbs are pulled to the two ends $A , B$ of the fourth floor to form an isosceles triangle $P A B$, where $\overline { P A } = \overline { P B }$; light bulbs are pulled to the two ends $C , D$ of the third floor to form an isosceles triangle $P C D$; light bulbs are pulled to the two ends $E , F$ of the second floor to form an isosceles triangle $PEF$. Assume each floor has equal height and each floor has equal length. If the length of the line segment cut out by the fifth floor inside triangle $P A B$ is $\frac { 1 } { 3 }$ of the floor length, what fraction of the floor length is the length of the line segment cut out by the fifth floor inside triangle $PEF$? (Light decoration thickness is negligible)
(1) $\frac { 1 } { 7 }$
(2) $\frac { 1 } { 6 }$
(3) $\frac { 1 } { 5 }$
(4) $\frac { 2 } { 9 }$
(5) $\frac { 1 } { 4 }$
Q7 5 marks Measures of Location and Spread View
A city is divided into east and west regions. Each region has a temperature monitoring station. The city's daily maximum temperature (in degrees Celsius) is recorded as the maximum of the daily temperatures from the two regions. The table below shows the distribution of daily maximum temperatures for the east and west regions over a certain month (30 days).
Temperature $t$$18 \leq t < 24$$24 \leq t < 30$$30 \leq t < 36$$36 \leq t$
East Region (days)011145
West Region (days)312150

Based on the above table, the distribution of the city's daily maximum temperature for the month is shown in the following table.
Temperature $t$$18 \leq t < 24$$24 \leq t < 30$$30 \leq t < 36$$36 \leq t$
Days$A$$B$$C$$D$

Select the option that could be the tuple $(A , B , C , D)$.
(1) $( 0,15,15,0 )$
(2) $( 3,12,15,5 )$
(3) $( 0,9,16,5 )$
(4) $( 3,7,15,5 )$
(5) $( 0,12,13,5 )$
Q8 5 marks Geometric Sequences and Series True/False or Multiple-Statement Verification View
Given that positive real numbers $a , b , c , d , e$ form a geometric sequence with $a < b < c < d < e$, select the options that form a geometric sequence.
(1) $a , - b , c , - d , e$
(2) $e , d , c , b , a$
(3) $\log a , \log b , \log c , \log d , \log e$
(4) $3 ^ { a } , 3 ^ { b } , 3 ^ { c } , 3 ^ { d } , 3 ^ { e }$
(5) $a b c , b c d , c d e$
Q9 5 marks Factor & Remainder Theorem Euclidean Division: Quotient and Remainder Determination View
Given that when polynomial $f ( x )$ is divided by $x ^ { 2 } + 5 x + 1$, the quotient is $x ^ { 3 } + 7 x ^ { 2 } + x + 3$, select the options that could be $f ( x )$.
(1) $2 \left( x ^ { 3 } + 7 x ^ { 2 } + x + 3 \right) \left( x ^ { 2 } + 5 x + 1 \right)$
(2) $\left( x ^ { 3 } + 7 x ^ { 2 } + x + 3 \right) \left( x ^ { 2 } + 5 x + 1 \right) - x$
(3) $\left( x ^ { 3 } + 7 x ^ { 2 } + x + 3 \right) \left( x ^ { 2 } + 5 x + 1 \right) + x ^ { 2 }$
(4) $\left( x ^ { 3 } + 7 x ^ { 2 } + x + 4 \right) \left( x ^ { 2 } + 5 x + 1 \right) - x$
(5) $\left( x ^ { 3 } + 7 x ^ { 2 } + x + 4 \right) \left( x ^ { 2 } + 5 x + 1 \right) - x ^ { 2 }$
Q10 5 marks Travel graphs View
Two light spots move on a straight track of length 120 cm. When they reach an endpoint, they reverse direction and continue moving. Initially, the two spots are at the two ends of the track moving toward each other. Light spot A and light spot B move at speeds of 5 cm/s and 10 cm/s respectively. Select the correct options.
(1) The position of the first meeting of the two light spots is 40 cm away from one of the endpoints
(2) The position of light spot A exhibits periodic behavior with a period of 24 seconds
(3) When light spot A returns to its starting point, light spot B is also at its starting point
(4) The second meeting of the two light spots occurs at one of the endpoints
(5) There are 3 different meeting positions for the two light spots on the track
Q11 5 marks Exponential Functions Applied/Contextual Exponential Modeling View
Over the past five years, a country's total carbon emissions decreased from $X$ billion metric tons of CO2 equivalent (CO2e) in year 1 to $Y$ billion metric tons of CO2 equivalent (CO2e) in year 5, achieving an average annual carbon reduction of 5\%, that is, $Y = ( 1 - 0.05 ) ^ { 4 } X$. The five-year carbon emission totals and annual growth rates are recorded in the following table, where Year $n$ carbon emission growth rate $= \frac { \text {(Year } n \text { carbon emission total)} - \text {(Year } n - 1 \text { carbon emission total)} } { \text {Year } n - 1 \text { carbon emission total} }$, $n = 2,3,4,5$.
Year 1Year 2Year 3Year 4Year 5
\begin{tabular}{ c } Carbon Emission Total
$($ billion metric tons $\mathrm { CO } 2 \mathrm { e } )$
& $X$ & $A$ & $B$ & $C$ & $Y$ \hline Annual Carbon Emission Growth Rate & & - 0.07 & $p$ & $q$ & $r$ \hline \end{tabular}
Select the correct options.
(1) $A = 0.93 X$
(2) $Y \leq 0.8 X$
(3) $\frac { - 0.07 + p + q + r } { 4 } = - 0.05$
(4) $\sqrt [ 4 ] { \frac { Y } { X } } - 1 = - 0.05$
(5) $0.93 ( 1 + p ) ( 1 + q ) ( 1 + r ) = ( 0.95 ) ^ { 4 }$
Q12 5 marks Discrete Probability Distributions Markov Chain and Transition Matrix Analysis View
Xiaoming wrote a program to move a robot on a $2 \times 2$ chessboard, as shown in the figure. Each execution, the program selects one direction from ``up, down, left, right'' with equal probability for each direction, and instructs the robot to move one square in that direction. However, if the selected direction would move the robot off the board, the robot stays in place. Each execution starts from the robot's current position with a newly selected direction by the program. Assume the robot's initial position is $A$. Let $a _ { n } , b _ { n } , c _ { n }$, and $d _ { n }$ be the probabilities that the robot is at positions $A , B , C , D$ respectively after executing the program $n$ times. Select the correct options.
(1) $b _ { 1 } = \frac { 1 } { 4 }$
(2) $b _ { 2 } = \frac { 1 } { 8 }$
(3) $a _ { 2 } + d _ { 2 } = \frac { 3 } { 4 }$
$A$$B$
$C$$D$

(4) $b _ { 99 } = c _ { 99 }$
(5) $a _ { 100 } + d _ { 100 } > \frac { 1 } { 2 }$
Q13 5 marks Matrices Linear System and Inverse Existence View
Given that $a , b , c , d$ are real numbers, and $\left[ \begin{array} { l l } 1 & - 1 \\ 3 & - 2 \end{array} \right] \left[ \begin{array} { l } a \\ b \end{array} \right] = \left[ \begin{array} { l } 1 \\ 0 \end{array} \right]$. If $\left[ \begin{array} { l l } 1 & - 1 \\ 3 & - 2 \end{array} \right] \left[ \begin{array} { l } 2 a + 1 \\ 2 b + 1 \end{array} \right] = \left[ \begin{array} { l } c \\ d \end{array} \right]$, then the value of $c - 3 d$ is (13-1)(13-2).
Q14 5 marks Conditional Probability Total Probability via Tree Diagram (Two-Stage Partition) View
All senior high school students at a certain school have taken either Mathematics A or Mathematics B on the scholastic aptitude test. Among these students, those taking only Mathematics A account for $\frac { 3 } { 10 }$ of all senior high school students. Among students taking Mathematics A, $\frac { 5 } { 8 }$ also took Mathematics B. What is the proportion of students taking only Mathematics B among all students at the school taking Mathematics B? (Express as a fraction in lowest terms)
Q15 5 marks Vectors Introduction & 2D Expressing a Vector as a Linear Combination View
Given that $P _ { 1 } , P _ { 2 } , Q _ { 1 } , Q _ { 2 } , R$ are five distinct points on a plane, where $P _ { 1 } , P _ { 2 } , R$ are not collinear, and satisfy $\overrightarrow { P _ { 1 } R } = 4 \overrightarrow { P _ { 1 } Q _ { 1 } }$ and $\overrightarrow { P _ { 2 } R } = 7 \overrightarrow { P _ { 2 } Q _ { 2 } }$, then $\overrightarrow { Q _ { 1 } Q _ { 2 } } =$ (15-1) $\overrightarrow { P _ { 1 } Q _ { 1 } } +$ (15-2)(15-3) $\overrightarrow { P _ { 2 } Q _ { 2 } }$.
Q16 5 marks Circles Sphere and 3D Circle Problems View
In a spatial coordinate system, there is a globe with center at $O ( 0,0,0 )$ and north pole at $N ( 0,0,2 )$. A point $A$ on the sphere has coordinates $\left( \frac { \sqrt { 3 } } { 2 } , \frac { 1 } { 2 } , \sqrt { 3 } \right)$. The point on the equator farthest from point $A$ is point $P$. On the great circle passing through points $A$ and $P$, the length of the minor arc between these two points is (blank). (Express as a fraction in lowest terms)
Q17 5 marks Permutations & Arrangements Circular Arrangement View
On a circle, 12 equally spaced points are marked and numbered 1 to 12 in clockwise order. Among all triangles formed by choosing any 3 of these 12 points as vertices, the number of triangles whose three interior angles, arranged from smallest to largest, form an arithmetic sequence is (17-1)(17-2).
Q18 3 marks Proof Computation of a Limit, Value, or Explicit Formula View
As shown in the figure, consider a rectangular stone block with a vertex $A$ and a face containing point $A$. Let the midpoints of the edges of this face be $B , E , F , D$ respectively. Another face of the rectangular block containing point $B$ has its edge midpoints as $B , C , H , G$ respectively. Given that $\overline { B C } = 8$ and $\overline { B D } = \overline { D C } = 9$. The stone block is now cut to remove eight corners, such that the cutting plane for each corner passes through the midpoints of the three adjacent edges of that corner.
How many faces does the stone block have after cutting the corners? (Single choice question, 3 points)
(1) Octahedron
(2) Decahedron
(3) Dodecahedron
(4) Tetradecahedron
(5) Hexadecahedron
Q19 4 marks Sine and Cosine Rules Compute area of a triangle or related figure View
As shown in the figure, consider a rectangular stone block with a vertex $A$ and a face containing point $A$. Let the midpoints of the edges of this face be $B , E , F , D$ respectively. Another face of the rectangular block containing point $B$ has its edge midpoints as $B , C , H , G$ respectively. Given that $\overline { B C } = 8$ and $\overline { B D } = \overline { D C } = 9$. The stone block is now cut to remove eight corners, such that the cutting plane for each corner passes through the midpoints of the three adjacent edges of that corner.
Find the area of $\triangle B C D$. (Non-multiple choice question, 4 points)
Q20 8 marks Volumes of Revolution Volume of a 3D Geometric Solid (Pyramid/Tetrahedron) View
As shown in the figure, consider a rectangular stone block with a vertex $A$ and a face containing point $A$. Let the midpoints of the edges of this face be $B , E , F , D$ respectively. Another face of the rectangular block containing point $B$ has its edge midpoints as $B , C , H , G$ respectively. Given that $\overline { B C } = 8$ and $\overline { B D } = \overline { D C } = 9$. The stone block is now cut to remove eight corners, such that the cutting plane for each corner passes through the midpoints of the three adjacent edges of that corner.
Find the length of $\overline { A D }$ and the volume of tetrahedron $A B C D$, and find the height from vertex $A$ to the base plane when $\triangle B C D$ is the base of the tetrahedron. (Volume of pyramid $= \frac { \text{Base area} \times \text{Height} } { 3 }$) (Non-multiple choice question, 8 points)