4. At the end of the examination, you must submit both the test paper and the answer sheet.

\section*{Reference Formulas:}
\section*{Section I (Multiple Choice, Total 50 Points)}
I. Multiple Choice Questions: This section has 10 questions; each question is worth 5 points, for a total of 50 points. For each question, there are four options, and exactly one is correct.\\
(1) Let $i$ be the imaginary unit. The complex number $\frac{2i}{1-i}$ corresponds to a point in the complex plane located in\\
(A) the first quadrant\\
(B) the second quadrant\\
(C) the third quadrant\\
(D) the fourth quadrant\\
(2) Among the following functions, which one is both an even function and has a zero?\\
(A) $y = \cos x$\\
(B) $y = \sin x$\\
(C) $y = \ln x$\\
(D) $y = x^2 + 1$\\
(3) Let $p$ and $q$ be two propositions. Then $p$ is a \_\_\_\_ condition for $q$ to hold.\\
(A) sufficient but not necessary condition\\
(B) necessary but not sufficient condition\\
(C) sufficient and necessary condition\\
(D) neither sufficient nor necessary condition

(4) Among the following hyperbolas, which one has its foci on the $y$-axis and asymptote equations $y = \pm 2x$?\\
(A) $x^2 - \frac{y^2}{4} = 1$\\
(B) $\frac{x^2}{4} - y^2 = 1$\\
(C) $\frac{y^2}{4} - x^2 = 1$\\
(D) $y^2 - \frac{x^2}{4} = 1$\\

(5) Let $m, n$ be two different lines, and $\alpha, \beta$ be two different planes. Which of the following propositions is correct?\\
(A) If $\alpha$ and $\beta$ are both perpendicular to the same plane, then $\alpha$ is parallel to $\beta$\\
(B) If $m$ and $n$ are both parallel to the same plane, then $m$ is parallel to $n$\\
(C) If $\alpha$ and $\beta$ are not parallel, then there does not exist a line in $\alpha$ that is parallel to $\beta$\\
(D) If $m$ and $n$ are not parallel, then $m$ and $n$ cannot both be perpendicular to the same plane\\

(6) If the sample data $x_1, x_2, \cdots, x_{10}$ has a standard deviation of 8, then the standard deviation of the data $2x_1 - 1, 2x_2 - 1, \cdots, 2x_{10} - 1$ is ( )\\
(A) 8\\
(B) 15\\
(C) 16\\
(D) 32

(7) The three-view drawing of a tetrahedron is shown in the figure. The surface area of this tetrahedron is\\
(A) $1 + \sqrt{3}$\\
(B) $2 + \sqrt{3}$\\
(C) $1 + 2\sqrt{2}$\\
(D) $2\sqrt{2}$

(8) $\triangle ABC$ is an equilateral triangle with side length 2. Given that vectors $\vec{a}, \vec{b}$ satisfy $\overrightarrow{AB} = 2\vec{a}$ and $\overrightarrow{AC} = 2\vec{a} + \vec{b}$, which of the following conclusions is correct?\\
(A) $|\vec{b}| = 1$\\
(B) $\vec{a} \perp \vec{b}$\\
(C) $\vec{a} \cdot \vec{b} = 1$\\
(D) $(4\vec{a} - \vec{b}) \perp \overrightarrow{BC}$\\
\includegraphics[max width=\textwidth, alt={}, center]{89d037be-b1bf-4f0d-a99c-dd7d08586dfe-2_161_403_721_1452}

(9) The graph of the function $f(x) = \frac{ax + b}{(x + c)^2}$ is shown in the figure. Which of the following conclusions is correct?\\
(A) $a > 0, b > 0, c < 0$\\
(B) $a < 0, b > 0, c > 0$\\
(C) $a < 0, b > 0, c < 0$\\
(D) $a < 0, b < 0, c < 0$

(10) Given the function $f(x) = A\sin(\omega x + \varphi)$ (where $A, \omega, \varphi$ are all positive constants) with minimum positive period $\pi$. When $x = \frac{2\pi}{3}$, the function $f(x)$ attains its minimum value. Which of the following conclusions is correct?\\
\includegraphics[max width=\textwidth, alt={}, center]{89d037be-b1bf-4f0d-a99c-dd7d08586dfe-2_268_391_1297_1480}\\
\includegraphics[max width=\textwidth, alt={}, center]{89d037be-b1bf-4f0d-a99c-dd7d08586dfe-2_250_422_877_1452}\\
(A) $f(2) < f(-2) < f(0)$\\
(B) $f(0) < f(2) < f(-2)$\\
(C) $f(-2) < f(0) < f(2)$\\
(D) $f(2) < f(0) < f(-2)$

\section*{Section II}
\section*{II. Fill-in-the-Blank Questions}
(11) The coefficient of $x^3$ in the expansion of $\left(x^3 + \frac{1}{x}\right)^7$ is \_\_\_\_ (answer with a number)

(12) In polar coordinates, the maximum distance from a point on the circle $\rho = 8\sin\theta$ to the line $\theta = \frac{\pi}{3}$ ($\rho \in \mathbb{R}$) is \_\_\_\_

(13) Execute the program flowchart shown in the figure. The output value of $a$ is \_\_\_\_\\
\includegraphics[max width=\textwidth, alt={}, center]{89d037be-b1bf-4f0d-a99c-dd7d08586dfe-3_615_479_251_230}

(14) Given that the sequence $\{a_n\}$ is an increasing geometric sequence with $a_2 + a_4 = 9$ and $a_2a_3 = 8$, the sum of the first $n$ terms of the sequence $\{a_n\}$ equals \_\_\_\_

(15) Consider the cubic equation $x^3 + ax + b = 0$, where $a, b$ are real numbers. Which of the following conditions ensure that this cubic equation has only one real root? (Write out all correct condition numbers)\\
(1) $a = -3, b = -3$;\\
(2) $a = -3, b = 2$;\\
(3) $a = -3, b > 2$;\\
(4) $a = 0, b = 2$;\\
(5) $a = 1, b = 2$.

\section*{III. Solution Questions}
(16) In $\triangle ABC$, $A = \frac{\pi}{4}$, $AB = 6$, $AC = 3\sqrt{2}$. Point $D$ is on side $BC$ with $AD = BD$. Find the length of $AD$.

(17) There are 2 defective items and 3 good items mixed together. We need to distinguish them through inspection. Each time we randomly inspect one item, and after inspection it is not returned. The inspection stops when either 2 defective items or 3 good items have been detected.\\
(1) Find the probability that the first item inspected is defective and the second item inspected is good.\\
(2) Each inspection of one item costs 100 yuan. Let $X$ denote the total inspection cost (in yuan) until either 2 defective items or 3 good items have been detected. Find the probability distribution and expected value of $X$.

(18) (This question is worth 12 points)\\
Let $n \in \mathbb{N}^*$. Let $x_n$ be the $x$-coordinate of the intersection point of the tangent line to the curve $y = x^{2n+3} + 1$ at the point $(1, 2)$ with the $x$-axis.\\
(1) Find the general term formula of the sequence $\{x_n\}$;\\
(2) Let $T_n = x_1^2 x_2^2 \cdots x_{2n-1}^2$. Prove that $T_n \geq \frac{1}{4n}$.

(19) As shown in the figure, in the polyhedron $A_1B_1D_1DCBA$, the quadrilaterals $AA_1B_1B$, $ADD_1A_1$, and $ABCD$ are all squares. Let $E$ be the midpoint of $B_1D_1$. A plane through $A_1$, $D$, and $E$ intersects $CD_1$ at $F$.\\
(1) Prove that $EF \parallel B_1C_1$.\\
(2) Find the cosine of the dihedral angle $E-A_1D-B_1$.\\
\includegraphics[max width=\textwidth, alt={}, center]{89d037be-b1bf-4f0d-a99c-dd7d08586dfe-3_330_335_2179_230}

(20) (This question is worth 13 points)\\
Let the equation of ellipse $E$ be $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$). Let $O$ be the origin, point $A$ have coordinates $(a, 0)$, and point $B$ have coordinates $(0, b)$. Point $M$ is on segment $AB$ and satisfies $|BM| = 2|MA|$. The slope of line $OM$ is $\frac{\sqrt{5}}{10}$.\\
(I) Find the eccentricity $e$ of $E$;\\
(II) Let point $C$ have coordinates $(0, -b)$, and let $N$ be the midpoint of segment $AC$. The $y$-coordinate of the symmetric point of $N$ with respect to line $AB$ is $\frac{7}{2}$. Find the equation of $E$.

(21) Let the function $f(x) = x^2 - ax + b$.\\
(1) Discuss the monotonicity of $f(\sin x)$ on $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ and determine whether it has extrema. If it has extrema, find them.\\
(2) Let $f_0(x) = x^2 - a_0x + b_0$. Find the maximum value $D$ of the function $|f(\sin x) - f_0(\sin x)|$ on $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$.\\
(3) In part (2), take $a_0 = b_0 = 0$. Find the maximum value of $z = b - \frac{a^2}{4}$ subject to the constraint $D \leq 1$.