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QA 6 marks Vectors 3D & Lines Section Division and Coordinate Computation View

In coordinate space, let $O$ be the origin, and let point $P$ be the intersection of three planes $x - 3y - 5z = 0$, $x - 3y + 2z = 0$, $x + y = t$, where $t > 0$. If $\overline{OP} = 10$, then $t =$ （9）（10）（11）. (Express as a simplified radical)

QB 6 marks Circles Circle Equation Derivation View

Consider three distinct points $A$, $B$, $C$ in the coordinate plane, where point $A$ is $(1, 1)$. Circles are drawn with line segments $\overline{AB}$ and $\overline{AC}$ as diameters. These two circles intersect at point $A$ and point $P(4, 2)$. Given that $\overline{PB} = 3\sqrt{10}$ and point $B$ is in the fourth quadrant, the coordinates of point $B$ are (（12），（13）（14）).

QC 6 marks Sine and Cosine Rules Heights and distances / angle of elevation problem View

There is a triangular park with vertices at $O$, $A$, $B$. At vertex $O$ there is an observation tower 150 meters high. A person standing on the observation tower observes the other two vertices $A$, $B$ on the ground and the midpoint $C$ of $\overline{AB}$, measuring angles of depression of $30^{\circ}$, $60^{\circ}$, $45^{\circ}$ respectively. The area of this triangular park is （15）（16）（17）（18）$\sqrt{(19)}$ square meters. (Express as a simplified radical)

QI 12 marks Applied differentiation Tangent line computation and geometric consequences View

In the coordinate plane, a ``Bézier curve'' determined by four points $A$, $B$, $C$, $D$ refers to a polynomial function of degree at most 3 whose graph passes through points $A$ and $D$, and whose tangent line at point $A$ passes through point $B$, and whose tangent line at point $D$ passes through point $C$. Let $y = f(x)$ be the ``Bézier curve'' determined by the four points $A(0, 0)$, $B(1, 4)$, $C(3, 2)$, $D(4, 0)$. Answer the following questions.
(1) Let the equation of the tangent line to the graph of $y = f(x)$ at point $D$ be $y = ax + b$, where $a$ and $b$ are real numbers. Find the values of $a$ and $b$. (2 points)
(2) Prove that the polynomial $f(x)$ is divisible by $x^{2} - 4x$. (2 points)
(3) Find $f(x)$. (4 points)
(4) Find the value of the definite integral $\int_{2}^{6} |8f(x)|\, dx$. (4 points)

QII 12 marks Vectors 3D & Lines Multi-Part 3D Geometry Problem View

A unit cube $ABCD-EFGH$ with edge length 1. Point $P$ is the midpoint of edge $\overline{CG}$. Points $Q$ and $R$ are on edges $\overline{BF}$ and $\overline{DH}$ respectively, and $A$, $Q$, $P$, $R$ are the four vertices of a parallelogram, as shown in the figure below.
A coordinate system is established such that the coordinates of $D$, $A$, $C$, $H$ are $(0, 0, 0)$, $(1, 0, 0)$, $(0, 1, 0)$, $(0, 0, 1)$ respectively, and $\overline{BQ} = t$. Answer the following questions.
(1) Find the coordinates of point $P$. (2 points)
(2) Find the vector $\overrightarrow{AR}$ (express in terms of $t$). (2 points)
(3) Prove that the volume of the pyramid $G-AQPR$ is a constant (independent of $t$), and find this constant value. (4 points)
(4) When $t = \frac{1}{4}$, find the distance from point $G$ to the plane containing parallelogram $AQPR$. (4 points)

Q1 6 marks Trig Proofs Ordering or Comparing Trigonometric Expressions View

Given $45^{\circ} < \theta < 50^{\circ}$, and let $a = 1 - \cos^{2}\theta$, $b = \frac{1}{\cos\theta} - \cos\theta$, $c = \frac{\tan\theta}{\tan^{2}\theta + 1}$. Regarding the relative sizes of the three values $a$, $b$, $c$, select the correct option.
(1) $a < b < c$
(2) $a < c < b$
(3) $b < a < c$
(4) $b < c < a$
(5) $c < a < b$

Q2 6 marks Discrete Probability Distributions Expectation and Variance from Context-Based Random Variables View

There are two boxes $A$ and $B$. Box $A$ contains 6 white balls and 4 red balls, and box $B$ contains 8 white balls and 2 blue balls. There are three lottery methods (each ball in each box has equal probability of being drawn): (I) First draw one ball from box $A$; if a red ball is drawn, stop; if a white ball is drawn, then draw one ball from box $B$; (II) First draw one ball from box $B$; if a blue ball is drawn, stop; if a white ball is drawn, then draw one ball from box $A$; (III) Simultaneously draw one ball from each of boxes $A$ and $B$. The prize rules are: Among red and blue balls, if only a red ball is drawn, win 50 yuan; if only a blue ball is drawn, win 100 yuan; if both colors are drawn, still win only 100 yuan; if neither color is drawn, win nothing. Let $E_{1}$, $E_{2}$, $E_{3}$ denote the expected values of winnings for methods (I), (II), (III) respectively. Select the correct option.
(1) $E_{1} > E_{2} > E_{3}$
(2) $E_{1} = E_{2} > E_{3}$
(3) $E_{2} = E_{3} > E_{1}$
(4) $E_{1} = E_{3} > E_{2}$
(5) $E_{3} > E_{2} > E_{1}$

Q3 6 marks Exponential Equations & Modelling Threshold or Tipping-Point Calculation in Applied Exponential Models View

According to experimental statistics, a certain type of bacteria reproduces such that its quantity increases by a factor of 2.4 on average every 3.5 hours. Suppose a test tube in the laboratory initially contains 1000 of this type of bacteria. According to an exponential function model, approximately how many hours later will the quantity of this bacteria reach about $4 \times 10^{10}$? (Note: $\log 2 \approx 0.3010$, $\log 3 \approx 0.4771$)
(1) 63 hours
(2) 70 hours
(3) 77 hours
(4) 84 hours
(5) 91 hours

Q4 8 marks Vectors Introduction & 2D True/False or Multiple-Statement Verification View

In the coordinate plane, let $O$ be the origin, and let $A$ and $B$ be two distinct points different from $O$. Let $C_{1}$, $C_{2}$, $C_{3}$ be three points in the plane satisfying $\overrightarrow{OC}_{n} = \overrightarrow{OA} + n\overrightarrow{OB}$, $n = 1, 2, 3$. Select the correct options.
(1) $\overrightarrow{OC}_{1} \neq \overrightarrow{0}$
(2) $\overline{OC_{1}} < \overline{OC_{2}} < \overline{OC_{3}}$
(3) $\overrightarrow{OC}_{1} \cdot \overrightarrow{OA} < \overrightarrow{OC}_{2} \cdot \overrightarrow{OA} < \overrightarrow{OC}_{3} \cdot \overrightarrow{OA}$
(4) $\overrightarrow{OC_{1}} \cdot \overrightarrow{OB} < \overrightarrow{OC_{2}} \cdot \overrightarrow{OB} < \overrightarrow{OC_{3}} \cdot \overrightarrow{OB}$
(5) $C_{1}$, $C_{2}$, $C_{3}$ are collinear

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

For a real number $a$, let $[a]$ denote the greatest integer not exceeding $a$. For example: $[1.2] = [\sqrt{2}] = 1$, $[-1.2] = -2$. Consider the irrational number $\theta = \sqrt{10001}$. Select the correct options.
(1) $a - 1 < [a] \leq a$ holds for all real numbers $a$
(2) The sequence $b_{n} = \frac{[n\theta]}{n}$ diverges, where $n$ is a positive integer
(3) The sequence $c_{n} = \frac{[-n\theta]}{n}$ diverges, where $n$ is a positive integer
(4) The sequence $d_{n} = n\left[\frac{\theta}{n}\right]$ diverges, where $n$ is a positive integer
(5) The sequence $e_{n} = n\left[\frac{-\theta}{n}\right]$ diverges, where $n$ is a positive integer

Q6 8 marks Indefinite & Definite Integrals Antiderivative Verification and Construction View

Let $F(x)$ and $f(x)$ both be polynomial functions with real coefficients. Given that $F'(x) = f(x)$, select the correct options.
(1) If $a \geq 0$, then $F(a) - F(0) = \int_{0}^{a} f(t)\, dt$
(2) If $F(x)$ divided by $x$ has quotient $Q(x)$, then $Q(0) = f(0)$
(3) If $f(x)$ is divisible by $x + 1$, then $F(x) - F(0)$ is divisible by $(x+1)^{2}$
(4) If $F(x) \geq \frac{x^{2}}{2}$ holds for all real numbers $x$, then $f(x) \geq x$ also holds for all real numbers $x$
(5) If $f(x) \geq x$ holds for all $x > 0$, then $F(x) \geq \frac{x^{2}}{2}$ also holds for all $x > 0$

Q7 8 marks Complex Numbers Argand & Loci Geometric Properties of Triangles/Polygons from Affixes View

In the complex plane, let $O$ be the origin, and let $A$ and $B$ represent points with coordinates corresponding to complex numbers $z$ and $z + 1$ respectively. Given that both points $A$ and $B$ lie on the unit circle centered at $O$, select the correct options.
(1) Line $AB$ is parallel to the real axis
(2) $\triangle OAB$ is a right triangle
(3) Point $A$ is in the second quadrant
(4) $z^{3} = 1$
(5) The point with coordinate $1 + \frac{1}{z}$ also lies on the same unit circle

Q8 8 marks Linear transformations View

Let a $2 \times 2$ real matrix $A$ represent a reflection transformation of the coordinate plane and satisfy $A^{3} = \left[\begin{array}{cc} 0 & -1 \\ -1 & 0 \end{array}\right]$; let a $2 \times 2$ real matrix $B$ represent a rotation transformation (centered at the origin) of the coordinate plane and satisfy $B^{3} = \left[\begin{array}{cc} -1 & 0 \\ 0 & -1 \end{array}\right]$. Select the correct options.
(1) There are exactly three possible matrices for $A$
(2) There are exactly three possible matrices for $B$
(3) $AB = BA$
(4) The $2 \times 2$ matrix $AB$ represents a rotation transformation of the coordinate plane
(5) $BABA = \left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right]$

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