taiwan-gsat

Q1 5 marks Modulus function Counting solutions satisfying modulus conditions View

A point $P$ on a number line satisfies the condition that the distance from $P$ to 1 plus the distance from $P$ to 4 equals 4. How many such points $P$ are there?
(1) 0
(2) 1
(3) 2
(4) 3
(5) Infinitely many

Q2 5 marks Matrices Linear System and Inverse Existence View

Let $A$ be a $3 \times 2$ matrix such that $A \left[ \begin{array} { c c } 1 & 0 \\ - 1 & 1 \end{array} \right] = \left[ \begin{array} { c c } 4 & - 6 \\ - 2 & 1 \\ 3 & 5 \end{array} \right]$ . If $A \left[ \begin{array} { l } 1 \\ 0 \end{array} \right] = \left[ \begin{array} { l } a \\ b \\ c \end{array} \right]$ , what is the value of $a + b + c$?
(1) 0
(2) 2
(3) 4
(4) 5
(5) 8

Q3 5 marks Laws of Logarithms Compare or Order Logarithmic Values View

Given that real numbers $a , b$ satisfy $\frac { 1 } { 2 } < a < 1$ and $1 < b < 2$ . Which of the following options has the smallest value?
(1) 0
(2) $\log a$
(3) $\log \left( a ^ { 2 } \right)$
(4) $\log b$
(5) $\frac { 1 } { \log b }$

Q4 5 marks Binomial Distribution Contextual Probability Requiring Binomial Modeling Setup View

A store launches a lottery activity offering four different styles of fruit figurines as prizes. Each lottery draw yields 1 figurine, and each style has an equal probability of being drawn. A person decides to draw 4 times. What is the probability that he draws exactly 3 different styles of figurines?
(1) $\frac { 5 } { 16 }$
(2) $\frac { 3 } { 8 }$
(3) $\frac { 1 } { 2 }$
(4) $\frac { 9 } { 16 }$
(5) $\frac { 5 } { 8 }$

Q5 5 marks Conic sections Conic Identification and Conceptual Properties View

In space, two intersecting lines $L , M$ form an angle of $24 ^ { \circ }$ . Rotating $M$ around $L$ one complete revolution generates a right circular cone surface. A plane $E$ is parallel to line $L$. What is the cross-section formed by plane $E$ and this cone surface?
(1) Hyperbola
(2) Parabola
(3) Ellipse (with unequal major and minor axes)
(4) Circle
(5) Two intersecting lines

Q6 5 marks Polynomial Division & Manipulation View

Let $a , b , c$ be real numbers, and the polynomial $f ( x ) = a ( x - 1 ) ( x - 3 ) + b ( x - 1 ) ( x - 4 ) + c ( x - 3 ) ( x - 4 )$ simplifies to $f ( x ) = x ^ { 2 }$ . Regarding the magnitude relationship of $a , b , c$, select the correct option.
(1) $a > b > c$
(2) $a > c > b$
(3) $b > c > a$
(4) $c > a > b$
(5) $c > b > a$

Q7 5 marks Curve Sketching Lattice Points and Counting via Graph Geometry View

A person uses single-point perspective with a point on the horizon as the vanishing point to draw six vertical pillars $A , B , C , D , E , F$ on a coordinate plane. The coordinates of the top and base of each pillar are shown in the table below, with point $V ( 4,9 )$ representing the vanishing point, as shown in the figure. Since the base line and top line of pillars $A$ and $F$ in the figure are both parallel to the horizon, the actual heights of pillars $A$ and $F$ are equal. Based on the above, select the pillar with the maximum actual height.

Pillar	$A$	$B$	$C$	$D$	$E$	$F$
Top coordinate	$( 0,8 )$	$( 2,3 )$	$( 4,6 )$	$( 6,8 )$	$( 8,5 )$	$( 10,8 )$
Base coordinate	$( 0,6 )$	$( 2,0 )$	$( 4,3 )$	$( 6,5 )$	$( 8,1 )$	$( 10,6 )$

(1) $A$
(2) $B$
(3) $C$
(4) $D$
(5) $E$

Q8 5 marks Curve Sketching Multi-Statement Verification (Remarks/Options) View

Let $\Gamma$ be the graph of the function $y = x ^ { 3 } - x$ on the coordinate plane. Select the correct options.
(1) The center of symmetry of $\Gamma$ is the origin
(2) $\Gamma$ approximates the line $y = x$ near $x = 0$
(3) $\Gamma$ can coincide with the graph of the function $y = x ^ { 3 } + x + 3$ after appropriate translation
(4) $\Gamma$ and the graph of the function $y = x ^ { 3 } + x$ are symmetric about the $x$-axis
(5) $\Gamma$ and the graph of the function $y = - x ^ { 3 } + x$ are symmetric about the $y$-axis

Q9 5 marks Vectors Introduction & 2D Expressing a Vector as a Linear Combination View

On the coordinate plane, let $O$ be the origin and point $P$ have coordinates $( 2,2 )$ . Given that $\overrightarrow { O P } = \alpha \overrightarrow { O A } + \beta \overrightarrow { O B }$ , where real numbers $\alpha , \beta$ satisfy $0 \leq \alpha \leq 1,0 \leq \beta \leq 1$ . From the following options, select the possible coordinates of points $A$ and $B$.
(1) $A ( 2 , - 3 ) , B ( - 4,3 )$
(2) $A ( 3,2 ) , B ( 3,4 )$
(3) $A ( 3,4 ) , B ( 4 , - 1 )$
(4) $A ( 1,2 ) , B ( 2,1 )$
(5) $A ( 1 , - 1 ) , B ( 1,1 )$

Q10 5 marks Measures of Location and Spread View

A badminton player competes against four opponents: A, B, C, and D, one match each. After the competition, data from these four matches were collected, recording the total number of smashes by each opponent and the average and standard deviation of the time used per smash. The results are shown in the table below. For example, opponent A made 25 smashes in that match, with an average time of 1.2 seconds per smash and a standard deviation of 0.5 seconds.

Opponent	Number of smashes in that match	Average time per smash (seconds)	Standard deviation of time per smash (seconds)
A	25	1.2	0.5
B	14	1.5	0.3
C	20	1.7	0.2
D	30	1.2	0.4

Based on the above, regarding the performance of opponents A, B, C, and D, select the correct options.
(1) C had the highest average time per smash among the four in that match
(2) D spent the most total time on smashing among the four in that match
(3) A's time per smash in that match was the same as D's for every smash
(4) The range of A's smash times in that match is greater than the range of D's smash times in that match
(5) It is impossible for all of B's smash times in that match to be between 1.4 and 1.6 seconds

Q11 5 marks Geometric Probability View

The Earth is a sphere. Five points $A , B , C , D , E$ on the Earth's surface have the following latitude and longitude coordinates, for example, point $A$ is located at longitude 0 degrees, north latitude 60 degrees.

Location	Longitude 0 degrees	Longitude 180 degrees
North latitude 60 degrees	$A$	$B$
North latitude 30 degrees	$C$	$D$
Latitude 0 degrees	$E$

A great circle is the circle formed by the intersection of a plane passing through the center of the sphere with the sphere's surface. The shorter arc formed by two distinct points on the sphere on a great circle is the shortest path. Based on the above, select the correct options.
(1) ``The shortest path length from the North Pole to $A$'' equals ``the shortest path length from the North Pole to $B$''
(2) ``The shortest path length from $A$ to $B$'' equals ``the shortest path length from $C$ to $D$''
(3) The shortest path from $A$ to $E$ must pass through $C$
(4) The shortest path from $C$ to $D$ must pass through the North Pole
(5) The ratio of ``the shortest path length from $E$ to the North Pole'' to ``the shortest path length from $C$ to $D$'' is $2 : 3$

Q12 5 marks Arithmetic Sequences and Series Properties of AP Terms under Transformation View

An arithmetic sequence has a first term of 1, a last term of 81, and 9 is also in the sequence. Let the number of terms in this sequence be $n$, where $n \leq 100$ . Select the correct options.
(1) $n$ is odd
(2) 41 must be in this arithmetic sequence
(3) The common difference of all arithmetic sequences satisfying the conditions is an integer
(4) There are 10 arithmetic sequences satisfying the conditions
(5) If $n$ is a multiple of 7, then $n = 21$

Q13 5 marks Probability Definitions Probability Using Set/Event Algebra View

There are two parking lots next to a scenic spot. Assume that on a certain day, the probability that either parking lot has no available spaces is 0.7, and whether the two parking lots have available spaces is independent. If a car arrives at these two parking lots on that day, the probability that at least one parking lot has available spaces is 0.(13--1)(13--2).

Q14 5 marks Straight Lines & Coordinate Geometry Area Computation in Coordinate Geometry View

On the coordinate plane, given three points $A ( 0,2 )$ , $B ( - 1,0 )$ , $C ( 4,0 )$ . If the line $y = m x$ divides triangle $A B C$ into two equal areas, then $m = \frac { \text{(14--1)} } { \text{(14--2)} }$ . (Reduce to lowest terms)

Q15 5 marks Permutations & Arrangements Distribution of Objects into Bins/Groups View

A company hires 8 new employees, including 2 translators, 3 engineers, and 3 assistants. These 8 people are assigned to research and testing departments, with 4 people assigned to each department. Each department must include 1 translator and at least 1 engineer. There are (15--1)(15--2) ways to make such assignments.

Q16 5 marks Vectors Introduction & 2D Area Computation Using Vectors View

A corner of a classroom is formed by two walls and the floor, which are mutually perpendicular. Let the corner be point $O$. There is a triangular baffle $A B C$ with vertices $A , B , C$ located on the intersection lines between walls or between walls and the floor, at distances of 20, 20, and 10 centimeters from corner $O$ respectively. The three sides $\overline { A B }$ , $\overline { B C }$ , $\overline { C A }$ are flush with the walls or floor, as shown in the figure. Find the area of the triangular baffle $ABC$.

Q18 3 marks Exponential Functions Applied/Contextual Exponential Modeling View

It is known that UVI values have an exponential relationship with altitude: for every 300-meter increase in altitude, the UVI value increases by 4\% of the value before the increase. At ground level, the ultraviolet radiation received from the sun is 400 joules per square meter. At a mountain 4500 meters above ground level, the UVI value of the ultraviolet radiation received is which of the following options? (Single choice question, 3 points)
(1) $4 ( 1 + 0.04 \times 15 )$
(2) $4 \left( 1 + 0.04 ^ { 15 } \right)$
(3) $4 ( 1 + 0.04 ) ^ { 15 }$
(4) $4 \times 100 ( 1 + 0.04 ) ^ { 15 }$
(5) $4 \times 100 \left( 1 + 0.04 ^ { 45 } \right)$

Q19 6 marks Trig Graphs & Exact Values View

On a certain day at a certain location, the duration of daylight (from sunrise to sunset) is exactly 12 hours. The UVI value at that location $x$ hours after sunrise ($0 \leq x \leq 12$) can be expressed by the function $f ( x ) = a \sin ( b x )$ , where $a , b > 0$ . Assume that the UVI value is positive during daylight and 0 during non-daylight hours (i.e., $f ( 0 ) = f ( 12 ) = 0$), and the UVI value 2 hours after sunrise on that day is 4. Find the values of $a$ and $b$.

Q20 6 marks Standard trigonometric equations Applied trigonometric modeling View

Continuing from question 19, where $f ( x ) = a \sin ( b x )$ with $f(0) = f(12) = 0$ and $f(2) = 4$, a person wants to sunbathe when the UVI value is between $4 \sqrt { 2 }$ and $4 \sqrt { 3 }$ (inclusive). The time during which he can sunbathe is set as $t$ hours after sunrise. Find the maximum possible range of $t$.

taiwan-gsat

Papers (18)

2025 gsat__math-b