taiwan-gsat

Q1 6 marks Standard trigonometric equations Solve trigonometric equation for solutions in an interval View

On the coordinate plane, the graph of the function $y = \sin x$ is symmetric about $x = \frac{\pi}{2}$, as shown in the figure. Find the value of $\theta$ in the range $0 < \theta \leq \pi$ that satisfies $\sin \theta = \sin\left(\theta + \frac{\pi}{5}\right)$.
(1) $\frac{\pi}{5}$
(2) $\frac{2\pi}{5}$
(3) $\frac{3\pi}{5}$
(4) $\frac{4\pi}{5}$
(5) $\pi$

Q2 6 marks Vectors: Lines & Planes Prove Perpendicularity/Orthogonality of Line and Plane View

In space, there is a regular cube $ABCD-EFGH$, where vertices $A, B, C, D$ lie on the same plane, and $\overline{AE}$ is one of its edges, as shown in the figure. Among the following options, select the plane that is perpendicular to both plane $BGH$ and plane $CFE$.
(1) Plane $ADH$
(2) Plane $BCD$
(3) Plane $CDG$
(4) Plane $DFG$
(5) Plane $DFH$

Q3 6 marks Combinations & Selection Geometric Combinatorics View

In the Elements of Geometry, it is stated: "Two distinct points determine a line." In general, three distinct points determine $C_{2}^{3} = 3$ lines; however, if these three points are collinear, only one line is determined. On the coordinate plane, circle $\Gamma_{1}: x^{2} + y^{2} = 4$ intersects the two coordinate axes at 4 points, circle $\Gamma_{2}: x^{2} + y^{2} = 2$ intersects the line $x - y = 0$ at 2 points, and circle $\Gamma_{2}$ intersects the line $x + y = 0$ at 2 points. How many different lines can these 8 points determine?
(1) 12
(2) 16
(3) 20
(4) 24
(5) 28

Q4 8 marks Conic sections Conic Identification and Conceptual Properties View

From the following conic sections on the coordinate plane, select the options that intersect all vertical lines.
(1) $\frac{x^{2}}{9} + \frac{y^{2}}{4} = 1$
(2) $\frac{x^{2}}{9} - \frac{y^{2}}{4} = 1$
(3) $-\frac{x^{2}}{9} + \frac{y^{2}}{4} = 1$
(4) $y = \frac{4}{9}x^{2}$
(5) $x = \frac{4}{9}y^{2}$

Q5 8 marks Sequences and series, recurrence and convergence True/false or conceptual reasoning about sequences View

There is a real number sequence $\left\langle a_{n} \right\rangle$, where $a_{n} = \cos\left(n\pi - \frac{\pi}{6}\right)$, and $n$ is a positive integer. Select the correct options.
(1) $a_{1} = -\frac{1}{2}$
(2) $a_{2} = a_{3}$
(3) $a_{4} = a_{24}$
(4) $\left\langle a_{n} \right\rangle$ is a convergent sequence, and $\lim_{n \rightarrow \infty} a_{n} < 1$
(5) $\sum_{n=1}^{\infty} \left(a_{n}\right)^{n} = 3 - 2\sqrt{3}$

Q6 8 marks Exponential Functions True/False or Multiple-Statement Verification View

Let the exponential function $f(x) = 1.2^{x}$. Select the correct options.
(1) $f(0) > 0$
(2) $f(10) > 10$
(3) On the coordinate plane, the graph of $y = 1.2^{x}$ intersects the line $y = x$
(4) On the coordinate plane, the graphs of $y = 1.2^{x}$ and $y = \log(1.2^{x})$ are symmetric about the line $y = x$
(5) For any positive real number $b$, $\log_{1.2} b \neq 1.2^{b}$

Q7 8 marks Stationary points and optimisation Analyze function behavior from graph or table of derivative View

A real coefficient polynomial $f(x)$ has degree greater than 5, and its leading coefficient is positive. Moreover, $f(x)$ has local minima at $x = 1, 2, 4$ and local maxima at $x = 3, 5$. Based on the above, select the correct options.
(1) $f(1) < f(3)$
(2) There exist real numbers $a, b$ satisfying $1 < a < b < 2$ such that $f'(a) > 0$ and $f'(b) < 0$
(3) $f''(3) > 0$
(4) There exists a real number $c > 5$ such that $f'(c) > 0$
(5) The degree of $f(x)$ is greater than 7

Q8 8 marks Complex Numbers Arithmetic True/False or Property Verification Statements View

Let the complex number $z$ have a nonzero imaginary part and $|z| = 2$. It is known that on the complex plane, $1, z, z^{3}$ are collinear. Select the correct options.
(1) $z \cdot \bar{z} = 2$
(2) The imaginary part of $\frac{z^{3} - z}{z - 1}$ is 0
(3) The real part of $z$ is $-\frac{1}{2}$
(4) $z$ satisfies $z^{2} - z + 4 = 0$
(5) On the complex plane, $-2, z, z^{2}$ are collinear

Q9 6 marks Linear transformations View

Let $A$ be the rotation matrix that rotates counterclockwise by angle $\theta$ about the origin, and let $B$ be the reflection matrix with the $x$-axis as the axis of reflection (axis of symmetry). Let $A = \left[\begin{array}{ll} a_{1} & a_{2} \\ a_{3} & a_{4} \end{array}\right]$ and $BA = \left[\begin{array}{ll} c_{1} & c_{2} \\ c_{3} & c_{4} \end{array}\right]$.
Given that $a_{1} + a_{2} + a_{3} + a_{4} = 2(c_{1} + c_{2} + c_{3} + c_{4})$, then $\tan\theta =$ \hspace{2cm}. (Express as a fraction in lowest terms)

Q10 6 marks Vectors 3D & Lines Shortest Distance Between Two Lines View

In coordinate space, a plane intersects the plane $x = 0$ and the plane $z = 0$ at lines $L_{1}$ and $L_{2}$, respectively.
Given that $L_{1}$ and $L_{2}$ are parallel, $L_{1}$ passes through the point $(0, 2, -11)$, and $L_{2}$ passes through the point $(8, 21, 0)$,
the distance between $L_{1}$ and $L_{2}$ is $\sqrt{(10-1)(10-2)(10-3)}$. (Express as a simplified radical)

Q11 6 marks Circles Area and Geometric Measurement Involving Circles View

On the coordinate plane, there is a parallelogram $\Gamma$, where two sides lie on lines parallel to $5x - y = 0$, and the other two sides lie on lines perpendicular to $3x - 2y = 0$. Let $Q$ be the intersection point of the two diagonals of $\Gamma$. It is known that $\Gamma$ has a vertex $P$ satisfying $\overrightarrow{PQ} = (10, -1)$. The area of $\Gamma$ is (11--1)(11--2)(11--3).

Q12 2 marks Discrete Probability Distributions Multiple Choice: Direct Probability or Distribution Calculation View

A store sells a popular action figure through a lottery. Each lottery draw is independent with a probability of winning of $\frac{2}{5}$. Participants can participate in the lottery using one of the following two methods.
Method 1: Pay 225 yuan to get two lottery chances. Stop drawing as soon as you win and receive one action figure. If you fail to win in both draws, you must pay an additional 75 yuan to receive one action figure.
Method 2: Unlimited number of lottery draws, paying 100 yuan per draw.
If using Method 1 to participate in the lottery, what is the probability of paying a total of 300 yuan to obtain one action figure? (Single choice question, 2 points)
(1) $\left(\frac{2}{5}\right)^{2}$
(2) $\left(\frac{2}{5}\right)^{3}$
(3) $\left(\frac{3}{5}\right)^{2}$
(4) $\left(\frac{3}{5}\right)^{3}$
(5) $\left(\frac{2}{5}\right) \times \left(\frac{3}{5}\right)^{2}$

Q13 4 marks Discrete Probability Distributions Properties of Named Discrete Distributions (Non-Binomial) View

A store sells a popular action figure through a lottery. Each lottery draw is independent with a probability of winning of $\frac{2}{5}$. Participants can participate in the lottery using one of the following two methods.
Method 1: Pay 225 yuan to get two lottery chances. Stop drawing as soon as you win and receive one action figure. If you fail to win in both draws, you must pay an additional 75 yuan to receive one action figure.
Method 2: Unlimited number of lottery draws, paying 100 yuan per draw.
If using Method 2 to participate in the lottery until winning one action figure, express the expected value of the number of lottery draws needed using the definition of expected value and the $\sum$ notation, and find its value. (Non-multiple choice question, 4 points)

Q14 6 marks Discrete Probability Distributions Expectation and Variance from Context-Based Random Variables View

A store sells a popular action figure through a lottery. Each lottery draw is independent with a probability of winning of $\frac{2}{5}$. Participants can participate in the lottery using one of the following two methods.
Method 1: Pay 225 yuan to get two lottery chances. Stop drawing as soon as you win and receive one action figure. If you fail to win in both draws, you must pay an additional 75 yuan to receive one action figure.
Method 2: Unlimited number of lottery draws, paying 100 yuan per draw.
Assuming there is no limit on spending until obtaining one action figure, find the expected value of the amount paid to obtain one action figure using each of the two lottery methods, and explain the relationship between these two expected values. (Non-multiple choice question, 6 points)

Q15 4 marks Completing the square and sketching Vertex and parameter conditions for a quadratic graph View

Let $f(x) = 3ax^{2} + (1 - a)$ be a real coefficient polynomial function, where $-\frac{1}{2} \leq a \leq 1$. On the coordinate plane, let $\Gamma$ be the region enclosed by $y = f(x)$ and the $x$-axis for $-1 \leq x \leq 1$.
Prove that when $-1 \leq x \leq 1$, $f(x) \geq 0$ always holds. (Non-multiple choice question, 4 points)

Q16 2 marks Areas by integration View

Let $f(x) = 3ax^{2} + (1 - a)$ be a real coefficient polynomial function, where $-\frac{1}{2} \leq a \leq 1$. On the coordinate plane, let $\Gamma$ be the region enclosed by $y = f(x)$ and the $x$-axis for $-1 \leq x \leq 1$.
Prove that for all $a \in \left[-\frac{1}{2}, 1\right]$, the area of $\Gamma$ is always 2. (Non-multiple choice question, 2 points)

Q17 6 marks Volumes of Revolution Volume of Revolution with Parameter Determination View

Let $f(x) = 3ax^{2} + (1 - a)$ be a real coefficient polynomial function, where $-\frac{1}{2} \leq a \leq 1$. On the coordinate plane, let $\Gamma$ be the region enclosed by $y = f(x)$ and the $x$-axis for $-1 \leq x \leq 1$.
Let $V$ be the volume of the solid of revolution obtained by rotating $\Gamma$ about the $x$-axis. For all $a \in \left[-\frac{1}{2}, 1\right]$, is $V$ always equal? If equal, find its value; if not equal, find the value of $a$ for which $V$ has a maximum value, and find this maximum value. (Non-multiple choice question, 6 points)

taiwan-gsat

Papers (18)

2025 ast__math-a