Read the flowchart on the right and run the corresponding program. The output value of S is
(A) $-10$
(B) 6
(C) 14
(D) 18

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

Reference Formulas:

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) $(4\vec{a} - \vec{b}) \perp \overrightarrow{BC}$ [Figure]
(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? [Figure] [Figure]
(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 II

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 \_\_\_\_ [Figure]
(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) $a = 1, b = 2$.

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 $E-A_1D-B_1$. [Figure]
(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 $D \leq 1$.

4. Let $\alpha , \beta$ be two different planes, and $m$ be a line with $m \subset \alpha$. ``$m / / \beta$'' is ``$\alpha / / \beta$'' a
A. sufficient but not necessary condition
B. necessary but not sufficient condition
C. sufficient and necessary condition
D. neither sufficient nor necessary condition

4. For optional questions: First fill in the question number of your choice at the designated position on the answer sheet with a 2B pencil, then answer in the corresponding answer area on the answer sheet. Answers written on the examination paper, scratch paper, and non-answer areas on the answer sheet are invalid.

4. According to the pseudocode shown in the figure, the output result S is $\_\_\_\_$ . [Figure]

4. After the examination, return this test paper and the answer sheet together.

Section I

I. Multiple Choice Questions: This section contains 12 questions, each worth 5 points. For each question, only one of the four options is correct.
(1) Let the complex number $z$ satisfy $\frac { 1 + z } { 1 - z } = i$. Then $| z | =$
(A) 1
(B) $\sqrt { 2 }$
(C) $\sqrt { 3 }$
(D) 2
(2) $\sin 20 ^ { \circ } \cos 10 ^ { \circ } - \cos 160 ^ { \circ } \sin 10 ^ { \circ } =$
(A) $- \frac { \sqrt { 3 } } { 2 }$
(B) $\frac { \sqrt { 3 } } { 2 }$
(C) $- \frac { 1 } { 2 }$
(D) $\frac { 1 } { 2 }$
(3) Let proposition P: $\exists \mathrm { n } \in \mathrm { N } , n ^ { 2 } > 2 ^ { n }$. Then $\neg P$ is
(A) $\forall \mathrm { n } \in \mathrm { N } , n ^ { 2 } > 2 ^ { n }$
(B) $\exists n \in N , n ^ { 2 } \leqslant 2 ^ { n }$
(C) $\forall \mathrm { n } \in \mathrm { N } , n ^ { 2 } \leqslant 2 ^ { n }$
(D) $\exists n \in N , n ^ { 2 } = 2 ^ { n }$
(4) In a basketball shooting test, each person shoots 3 times, and must make at least 2 shots to pass. It is known that a certain student has a probability of 0.6 of making each shot, and the results of each shot are independent. The probability that this student passes the test is
(A) 0.648
(B) 0.432
(C) 0.36
(D) 0.312
(5) Let $M \left( x _ { 0 } , y _ { 0 } \right)$ be a point on the hyperbola $C : \frac { x ^ 2 } { 2 } - y ^ 2 = 1$. Let $F _ { 1 }$ and $F _ { 2 }$ be the two foci of $C$. If $\overrightarrow { M F 1 } \cdot \overrightarrow { M F 2 } < 0$, then the range of $y _ { 0 }$ is
(A) $\left( - \frac { \sqrt { 3 } } { 3 } , \frac { \sqrt { 3 } } { 3 } \right)$
(B) $\left( - \frac { \sqrt { 3 } } { 6 } , \frac { \sqrt { 3 } } { 6 } \right)$
(C) $\left( - \frac { 2 \sqrt { 2 } } { 3 } , \frac { 2 \sqrt { 2 } } { 3 } \right)$
(D) $\left( - \frac { 2 \sqrt { 3 } } { 3 } , \frac { 2 \sqrt { 3 } } { 3 } \right)$ (6) The ``Nine Chapters on the Mathematical Art'' is a famous ancient Chinese mathematical work with extremely rich content. It contains the following problem: ``There is rice piled in the corner of a room inside a wall (as shown in the figure, the rice pile is one-quarter of a cone). The arc at the base of the rice pile is 8 chi, and the height of the rice pile is 5 chi. What are the volume of the rice pile and the amount of rice stored?'' It is known that the volume of 1 hu of rice is approximately 1.62 cubic chi, and the circumference ratio is approximately 3. Estimate the amount of rice stored in hu as approximately [Figure]
A. 14 hu
B. 22 hu
C. 36 hu
D. 66 hu (7) Let D be a point in the plane of $\triangle \mathrm { ABC }$ such that $\overrightarrow { B C } = 3 \overrightarrow { C D }$. Then
(A) $\overrightarrow { A D } = - \frac { 1 } { 3 } \overrightarrow { A B } + \frac { 4 } { 3 } \overrightarrow { A C }$
(B) $\overrightarrow { A D } = \frac { 1 } { 3 } \overrightarrow { A B } - \frac { 4 } { 3 } \overrightarrow { A C }$
(C) $\overrightarrow { A D } = \frac { 4 } { 3 } \overrightarrow { A B } + \frac { 1 } { 3 } \overrightarrow { A C }$
(D) $\overrightarrow { A D } = \frac { 4 } { 3 } \overrightarrow { A B } - \frac { 1 } { 3 } \overrightarrow { A C }$ (8) The function $f ( x ) = \cos ( \omega x + \varphi )$ has a partial graph as shown. The monotone decreasing interval of $f ( x )$ is
(A) $\left( \mathrm { k } \pi - \frac { 1 } { 4 } , \mathrm { k } \pi + \frac { 3 } { 4 } \right) , \mathrm { k } \in \mathbf { z }$
(B) $\left( 2 \mathrm { k } \pi - \frac { 1 } { 4 } , 2 \mathrm { k } \pi + \frac { 3 } { 4 } \right) , \mathrm { k } \in \mathbf { z }$
(C) $\left( \mathrm { k } - \frac { 1 } { 4 } , \mathrm { k } + \frac { 3 } { 4 } \right) , \mathrm { k } \in \mathbf { z }$
(D) $\left( 2 \mathrm { k } - \frac { 1 } { 4 } , 2 \mathrm { k } + \frac { 3 } { 4 } \right) , \mathrm { k } \in \mathbf { z }$ [Figure] (9) Executing the flowchart on the right, if the input is $\mathrm { t } = 0.01$, then the output is $\mathrm { n } =$
(A) 5
(B) 6
(C) 7
(D) 8 [Figure] (10) In the expansion of $\left( \mathrm { x } ^ { 2 } + \mathrm { x } + \mathrm { y } \right) ^ { 5 }$, the coefficient of $\mathrm { x } ^ { 5 } \mathrm { y } ^ { 2 }$ is
(A) 10
(B) 20
(C) 30
(D) 60 (11) A cylinder with part cut off by a plane, combined with a hemisphere (radius $r$), forms a geometric solid. The front view and top view of this solid in the three-view diagram are shown in the figure. If the surface area of this geometric solid is $16 + 20 \pi$, then $\mathrm { r } =$ [Figure]
(A) 1
(B) 2
(C) 4
(D) 8 (12) Let the function $f ( x ) = e ^ { x } ( 2 x - 1 ) - a x + a$, where $a < 1$. If there exists a unique integer $x _ { 0 }$ such that $f \left( x _ { 0 } \right) < 0$, then the range of $a$ is
A. $\left[ - \frac { 3 } { 2 e } , 1 \right)$
B. $\left[ - \frac { 3 } { 2 e } , \frac { 3 } { 4 } \right)$
C. $\left[ \frac { 3 } { 2 e } , \frac { 3 } { 4 } \right)$
D. $\left[ \frac { 3 } { 2 e } , 1 \right)$

Section II

This section includes both required questions and optional questions. Questions (13) through (21) are required; all candidates must answer them. Questions (22) through (24) are optional; candidates should answer according to requirements. II. Fill-in-the-Blank Questions: This section contains 3 questions, each worth 5 points. (13) If the function $\mathrm { f } ( \mathrm { x } ) = \mathrm { x } \ln ( \mathrm { x } + \sqrt { a + x ^ 2 } )$ is an even function, then $\mathrm { a } = $ \_\_\_\_ (14) A circle passes through three vertices of the ellipse $\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 4 } = 1$, and its center is on the $x$-axis. The standard equation of this circle is \_\_\_\_. (15) If $\mathrm { x } , \mathrm { y }$ satisfy the constraints $\left\{ \begin{array} { c } \mathrm { x } - 1 \geq 0 , \\ x - y \leq 0 , \\ x + y - 4 \leq 0 , \end{array} \right.$ then the maximum value of $\frac { x } { y }$ is \_\_\_\_. (16) In the planar quadrilateral ABCD, $\angle \mathrm { A } = \angle \mathrm { B } = \angle \mathrm { C } = 75 ^ { \circ }$, $\mathrm { BC } = 2$. The range of AB is \_\_\_\_. III. Solution Questions: Write out explanations, proofs, or calculation steps. (17) (This question is worth 12 points) Let $S_n$ be the sum of the first $n$ terms of the sequence $\{ a_n \}$. Given that $a_n > 0$ and $a _ { n } ^ { 2 } + 2 a _ { n } = 4 S _ { n } + 3$. (I) Find the general term formula for $\{ a_n \}$; (II) Let $b _ { n } = \frac { 1 } { a _ { n } a _ { n + 1 } }$. Find the sum of the first $n$ terms of the sequence $\left\{ b _ { n } \right\}$. (18) As shown in the figure, quadrilateral ABCD is a rhombus with $\angle \mathrm { ABC } = 120 ^ { \circ }$. E and F are two points on the same side of plane ABCD. $\mathrm { BE } \perp$ plane $\mathrm { ABCD }$, $\mathrm { DF } \perp$ plane $\mathrm { ABCD }$, $\mathrm { BE } = 2 \mathrm { DF }$, and $\mathrm { AE } \perp \mathrm { EC }$.
(1) Prove: plane $\mathrm { AEC } \perp$ plane AFC.
(2) Find the cosine of the angle between line AE and line CF. [Figure] (19) A company wants to determine the advertising expenditure for the next year for a certain product. It needs to understand how annual advertising expenditure $x$ (in units of thousand yuan) affects annual sales volume $y$ (in units of tons) and annual profit $z$ (in units of thousand yuan). Data on annual advertising expenditure $x_i$ and annual sales volume $y_i$ ($\mathrm { i } = 1,2 , \cdots , 8$) from the past 8 years were preliminarily processed to obtain the following scatter plot and some statistical values. \includegraph

4. For fill-in-the-blank questions, write the answer directly. For solution questions, write out explanations, proofs, or calculation steps.

Section I (50 points total)

I. Multiple Choice Questions: This section has 10 questions, each worth 5 points, totaling 50 points. For each question, only one of the four options is correct.
(1) Given sets $A = \{ x \mid 2 < x < 4 \}$ and $B = \{ x \mid (x - 1)(x - 3) < 0 \}$, then $A \cap B =$
(A) $(1,3)$
(B) $(1,4)$
(C) $(2,3)$
(D) $(2,4)$
(2) If the complex number $z$ satisfies $\frac{\bar{z}}{1-i} = i$, where $i$ is the imaginary unit, then $z =$
(A) $1 - i$
(B) $1 + i$
(C) $-1$
(D) $-1 + i$
(3) Let $a = 0.6^{0.6}$, $b = 0.6^{1.5}$, $c = 1.5^{0.6}$. The size relationship among $a$, $b$, $c$ is
(A) $a < b < c$
(B) $a < c < b$
(C) $b < a < c$
(D) $b < c < a$
(4) To obtain the graph of the function $y = \sin(4x - \frac{\pi}{3})$, we only need to shift the graph of the function $y = \sin 4x$ ()
(A) to the left by $\frac{\pi}{12}$ units
(B) to the right by $\frac{\pi}{12}$ units
(C) to the left by $\frac{\pi}{3}$ units
(D) to the right by $\frac{\pi}{3}$ units
(5) If $m \in \mathbb{N}$ and the proposition is ``if $m > 0$, then the equation $x^2 + x - m = 0$ has real roots'', then its contrapositive is
(A) If the equation $x^2 + x - m = 0$ has real roots, then $m > 0$
(B) If the equation $x^2 + x - m = 0$ has real roots, then $m \leq 0$
(C) If the equation $x^2 + x - m = 0$ has no real roots, then $m > 0$
(D) If the equation $x^2 + x - m = 0$ has no real roots, then $m \leq 0$ (6) To compare the 2 PM temperature situation in two locations A and B in a certain month, 5 days are randomly selected from that month. The temperature data at 2 PM on these 5 days (in ${}^{\circ}C$) are presented in the stem-and-leaf plot shown below. Consider the following conclusions:

\multicolumn{2}{c|}{Location A}		\multicolumn{2}{|c}{Location B}
9	9	6	2	8	9
	1	1	3	0	1	2

(1) The average temperature at 2 PM in Location A is lower than that in Location B;
(2) The average temperature at 2 PM in Location A is higher than that in Location B;
(3) The standard deviation of the temperature at 2 PM in Location A is smaller than that in Location B;
(4) The standard deviation of the temperature at 2 PM in Location A is larger than that in Location B. The numbers of the statistical conclusions that can be obtained from the stem-and-leaf plot are
(A) (1)(3)
(B) (1)(4)
(C) (2)(3)
(D) (2)(4) (7) A number $x$ is randomly selected from the interval $[0,2]$. The probability that the event ``$-1 \leq \log_{\frac{1}{2}}(x + \frac{1}{2}) \leq 1$'' occurs is
(A) $\frac{3}{4}$
(B) $\frac{2}{3}$
(C) $\frac{1}{3}$
(D) $\frac{1}{4}$ (8) If the function $f(x) = \frac{2^x + 1}{2^x - a}$ is an odd function, then the range of $x$ for which $f(x) > 3$ holds is
(A) $(-\infty, -1)$
(B) $(-1, 0)$
(C) $(0, 1)$
(D) $(1, +\infty)$ (9) An isosceles right triangle has legs of length 2. When the triangle is rotated one full revolution around the line containing its hypotenuse, the volume of the solid formed by the resulting surface is
(A) $\frac{2\sqrt{2}\pi}{3}$
(B) $\frac{4\sqrt{2}\pi}{3}$
(C) $2\sqrt{2}\pi$
(D) $4\sqrt{2}\pi$ (10) Let the function $f(x) = \begin{cases} 3x - b, & x < 1 \\ 2^x, & x \geq 1 \end{cases}$. If $f(f(\frac{5}{6})) = 4$, then $b =$
(A) $1$
(B) $\frac{7}{8}$
(C) $\frac{3}{4}$
(D) $\frac{1}{2}$

Section II (100 points total)

II. Fill-in-the-Blank Questions: This section has 5 questions, each worth 5 points, totaling 25 points

(11) Execute the flowchart on the right. If the input value of $x$ is 1, then the output value of $y$ is [Figure] (12) If $x, y$ satisfy the constraints $\begin{cases} y - x < 1 \\ x + y \leq 3 \\ y > 1 \end{cases}$, then the maximum value of $z = x + 3y$ is \_\_\_\_. (13) Two tangent lines are drawn from point $P(1, \sqrt{3})$ to the circle $x^2 + y^2 = 1$, with tangent points $A$ and $B$ respectively. Then $\overrightarrow{PA} \cdot \overrightarrow{PB} =$ (14) Define the operation ``$\otimes$'': $x \otimes y = \frac{x^2 - y^2}{xy}$ $(x, y \in \mathbb{R}, xy \neq 0)$. When $x > 0, y > 0$, the minimum value of $x \otimes y + 2y \otimes x$ is \_\_\_\_. (15) A line through the right focus of the hyperbola $C: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ $(a > 0, b > 0)$ parallel to one of its asymptotes intersects $C$ at point $P$. If the x-coordinate of point $P$ is $2a$, then the eccentricity of $C$ is \_\_\_\_.

III. Solution Questions: This section has 6 questions, totaling 75 points

(16) (This question is worth 12 points)
A school surveyed all 45 students in a certain class regarding their participation in calligraphy club and speech club. The data is shown in the table below (in persons):

	Participate in Calligraphy Club	Do Not Participate in Calligraphy Club
Participate in Speech Club	8	5
Do Not Participate in Speech Club	30

(1) If one student is randomly selected from the class, find the probability that the student participates in at least one of the two clubs;
(2) Among the 8 students who participate in both calligraphy club and speech club, there are 5 male students $A_1, A_2, A_3, A_4, A_5$ and 3 female students $B_1, B_2, B_3$. One person is randomly selected from the 5 male students and one person from the 3 female students. Find the probability that $A_1$ is selected and $B_1$ is not selected. (17) (This question is worth 12 points) In $\triangle ABC$, the sides opposite to angles $A, B, C$ are $a, b, c$ respectively. Given that $\cos B = \frac{\sqrt{3}}{3}$, $\sin(A + B) = \frac{\sqrt{6}}{9}$, and $ac = 2\sqrt{3}$, find $\sin A$ and the value of $c$. (18) As shown in the figure, in the triangular frustum $DEF - ABC$, $AB = 2DE$, and $G, H$ are the midpoints of $AC$ and $BC$ respectively. (I) Prove that $BD \parallel$ plane $FGH$; (II) If $CF \perp BC$ and $AB \perp BC$, prove that plane $BCD \perp$ plane $FGH$. [Figure] (19) (This question is worth 12 points) The sequence $\{a_n\}$ is an arithmetic sequence with a positive first term. The sum of the first $n$ terms of the sequence $\{\frac{1}{a_n \cdot a_{n+1}}\}$ is $\frac{n}{2n+1}$.
(1) Find the general term formula of the sequence $\{a_n\}$; (II) Let $b_n = (a_n + 1) \cdot 2^{a_n}$. Find the sum of the first $n$ terms $T_n$ of the sequence $\{b_n\}$. (20) (This question is worth 13 points) Let $f(x) = (x + a)\ln x$ and $g(x) = \frac{x^2}{e^x}$. The tangent line to the curve $y = f(x)$ at the point $(1, f(1))$ is parallel to the line $2x - y = 0$. (I) Find the value of $a$; (II) Does there exist a natural number $k$ such that the equation $f(x) = g(x)$ has a unique root in the interval $(k, k+1)$? If such a $k$ exists, find it; if not, explain why; (III) Let the function $m(x) = \min\{f(x), g(x)\}$ (where $\min\{p, q\}$ denotes the smaller of $p$ and $q$). Find the maximum value of $m(x)$.

(21) (This question is worth 14 points)

In the rectangular coordinate system $xOy$, the ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ $(a > b > 0)$ has eccentricity $\frac{\sqrt{3}}{2}$, and the point $(\sqrt{3}, \frac{1}{2})$ lies on the ellipse $C$. (I) Find the equation of ellipse $C$; (II) Let the ellipse $E: \frac{x^2}{4a^2} + \frac{y^2}{4b^2} = 1$. Let $P$ be an arbitrary point on ellipse $C$. A line $y = kx + m$ through point $P$ intersects ellipse $E$ at points $A$ and $B$. The ray $PO$ intersects ellipse $E$ at point $Q$.
(i) Find the value of $\frac{|OQ|}{|OP|}$;
(ii) Find the maximum area of $\triangle ABQ$.

4. For fill-in-the-blank questions, write the answer directly. For solution questions, write out explanations, proofs, or calculation steps. Reference Formulas: If events $A$ and $B$ are mutually exclusive, then $P(A + B) = P(A) + P(B)$.

Section I (50 points)

I. Multiple Choice Questions: This section has 10 questions, each worth 5 points, totaling 50 points. For each question, only one of the four options is correct.
(1) Given sets $A = \{x \mid x^2 - 4x + 3 < 0\}$ and $B = \{x \mid 2 < x < 4\}$, then $A \cap B =$
(A) $(1,3)$
(B) $(1,4)$
(C) $(2,3)$
(D) $(2,4)$
(2) If complex number $z$ satisfies $\frac{\bar{z}}{1-i} = i$, where $i$ is the imaginary unit, then $z =$
(A) $1 - i$
(B) $1 + i$
(C) $-1 - i$
(D) $-1 + i$
(3) To obtain the graph of function $y = \sin\left(4x - \frac{\pi}{3}\right)$, we only need to shift the graph of function $y = \sin 4x$ by ()
(A) $\frac{\pi}{12}$ units to the left
(B) $\frac{\pi}{12}$ units to the right
(C) $\frac{\pi}{3}$ units to the left
(D) $\frac{\pi}{3}$ units to the right
(4) Given that $ABCD$ is a rhombus with side length $a$ and $\angle ABC = 60°$, then $\overrightarrow{BD} \cdot \overrightarrow{CD} =$
(A) $-\frac{3}{2}a^2$
(B) $-\frac{3}{4}a^2$
(C) $\frac{3}{4}a^2$
(D) $\frac{3}{2}a^2$
(5) The solution set of the inequality $|x - 1| - |x - 5| < 2$ is
(A) $(-\infty, 4)$
(B) $(-\infty, 1)$
(C) $(1,4)$
(D) $(1,5)$ (6) Given that $x, y$ satisfy the constraints $\begin{cases} x - y \geq 0 \\ x + y \leq 2 \\ y \geq 0 \end{cases}$, if the maximum value of $z = ax + y$ is 4, then $a =$
(A) 3
(B) 2
(C) $-2$
(D) $-3$ (7) In trapezoid $ABCD$, $\angle ABC = \frac{\pi}{2}$, $AD \parallel BC$, $BC = 2AD = 2AB = 2$. When trapezoid $ABCD$ is rotated around the line containing $AD$ to form a surface, the volume of the solid enclosed by this surface is
(A) $\frac{2\pi}{3}$
(B) $\frac{4\pi}{3}$
(C) $\frac{5\pi}{3}$
(D) $2\pi$ (8) The length error (in millimeters) of a batch of parts follows a normal distribution $N(0,3)$. A part is randomly selected. The probability that its length error falls in the interval $(3,6)$ is (Note: If random variable $\xi$ follows normal distribution $N(\mu, \sigma^2)$, then $P(\mu - \sigma < \xi < \mu + \sigma) = 68.26\%$, $P(\mu - 2\sigma < \xi < \mu + 2\sigma) = 95.44\%$.)
(A) $4.56\%$
(B) $13.59\%$
(C) $27.18\%$
(D) $31.74\%$ (9) A light ray is emitted from point $(-2, -3)$, reflects off the $y$-axis, and is tangent to the circle $(x+3)^2 + (y-2)^2 = 1$. The slope of the reflected ray is ()
(A) $-\frac{5}{3}$ or $-\frac{3}{5}$
(B) $-\frac{3}{2}$ or $-\frac{2}{3}$
(C) $-\frac{5}{4}$ or $-\frac{4}{5}$
(D) $-\frac{4}{3}$ or $-\frac{3}{4}$ (10) Let function $f(x) = \begin{cases} 3x - 1, & x < 1 \\ 2^x, & x \geq 1 \end{cases}$. The range of values of $a$ satisfying $f(f(a)) = 2^{f(a)}$ is ()
(A) $\left[\frac{2}{3}, 1\right]$
(B) $[0,1]$
(C) $\left[\frac{2}{3}, +\infty\right)$
(D) $[1, +\infty)$

Section II (100 points)

II. Fill-in-the-Blank Questions: This section has 5 questions, each worth 5 points, totaling 25 points.

(11) Observe the following equations:
$$\begin{aligned} C_1^0 &= 4^0 \\ C_3^0 + C_3^1 &= 4^1 \\ C_5^0 + C_5^1 + C_5^2 &= 4^2 \\ C_7^0 + C_7^1 + C_7^2 + C_7^3 &= 4^3 \\ &\ldots \end{aligned}$$
Following this pattern, when $n \in \mathbb{N}$, $C_{2n-1}^0 + C_{2n-1}^1 + C_{2n-1}^2 + \cdots + C_{2n-1}^{n-1} = $ \_\_\_\_ (12) If ``$\forall x \in \left[0, \frac{\pi}{4}\right], \tan x \leq m$'' is a true statement, then the minimum value of real number $m$ is \_\_\_\_. (13) Execute the flowchart on the right. The output value of $T$ is \_\_\_\_. (14) Given that function $f(x) = a^x + b$ ($a > 0, a \neq 1$) has both domain and range equal to $[-1, 0]$, then $a + b = $ \_\_\_\_ (15) In the rectangular coordinate system $xOy$, the asymptotes of hyperbola $C_1: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ($a > 0, b > 0$) and parabola $C_2: x^2 = 2py$ ($p > 0$) intersect at points $O, A, B$. If the orthocenter of $\triangle OAB$ is the focus of $C_2$, then [Figure] [Figure] the eccentricity of $C_1$ is \_\_\_\_ — III. Solution Questions: This section has 6 questions, totaling 75 points. (16) (This question is worth 12 points) Let $f(x) = \sin x \cos x - \cos^2\left(x + \frac{\pi}{4}\right)$. (I) Find the monotonic intervals of $f(x)$; (II) In acute triangle $ABC$, let the sides opposite to angles $A, B, C$ be $a, b, c$ respectively. If $f\left(\frac{A}{2}\right) = 0$ and $a = 1$, find the maximum area of $\triangle ABC$. (17) (This question is worth 12 points)
As shown in the figure, in triangular frustum $DEF$-$ABC$, $AB = 2DE$, and $G, H$ are the midpoints of $AC, BC$ respectively. (I) Prove that $BC \parallel$ plane $FGH$; (II) If $CF \perp$ plane $ABC$, $AB \perp BC$, $CF = DE$, $\angle BAC = 45°$, [Figure] find the acute angle between plane $FGH$ and plane $ACFD$. (18) (This question is worth 12 points) Let the sum of the first $n$ terms of sequence $\{a_n\}$ be $S_n$. Given that $2S_n = 3^n + 3$. (I) Find the general term formula of $\{a_n\}$; (II) If sequence $\{b_n\}$ satisfies $a_n b_n = \log_3 n$, find the sum of the first $n$ terms $T_n$ of $\{b_n\}$. (19) (This question is worth 12 points) If $n$ is a three-digit positive integer where the units digit is greater than the tens digit, and the tens digit is greater than the hundreds digit, then $n$ is called a ``three-digit increasing number'' (such as 137, 359, 567, etc.). In a mathematics recreational activity, each participant needs to randomly draw 1 ``three-digit increasing number'' from all such numbers, and can only draw once. The scoring rule is as follows: if the product of the three digits of the drawn ``three-digit increasing number'' is not divisible by 5, the participant scores 0 points; if it is divisible by 5 but not by 10, the participant scores $-1$ point; if it is divisible by 10, the participant scores 1 point. (I) Write out all ``three-digit increasing numbers'' whose units digit is 5; (II) If participant A participates in the activity, find the probability distribution of A's score $X$ and the mathematical expectation $E(X)$. (20) (This question is worth 13 points) In the rectangular coordinate system $xOy$, the ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$) has eccentricity $\frac{\sqrt{3}}{2}$, with left and right foci $F_1$ and $F_2$ respectively. The circle centered at $F_1$ with radius 3 and the circle centered at $F_2$ with radius 1 intersect, and the intersection points lie on ellipse $C$. (I) Find the equation of ellipse $C$; (II) Let ellipse $E: \frac{x^2}{4a^2} + \frac{y^2}{4b^2} = 1$. Let $P$ be any point on ellipse $C$. A line $y = kx + m$ through point $P$ intersects ellipse $E$ at points $A, B$. Ray $PO$ intersects ellipse $E$ at point $Q$.
(i) Find the value of $\frac{|OQ|}{|OP|}$;
(ii) Find the maximum area of $\triangle ABQ$. (21) (This question is worth 14 points) Let function $f(x) = \ln(x+1) + \alpha(x^2 - x)$, where $\alpha \in \mathbb{R}$. (I) Discuss the number of extreme points of function $f(x)$ and explain the reasoning; (II) If $\forall x > 0, f(x) \geq 0$ holds, find the range of values for $\alpha$.

4. The negation of the proposition ``$\forall n \in \mathbb{N} ^ { * } , f ( n ) \in \mathbb{N} ^ { * }$ and $f ( n ) \leq n$'' is
A. $\forall n \in \mathbb{N} ^ { * } , f ( n ) \in \mathbb{N} ^ { * }$ and $f ( n ) > n$
B. $\forall n \in \mathbb{N} ^ { * } , f ( n ) \in \mathbb{N} ^ { * }$ or $f ( n ) > n$
C. $\exists n _ { 0 } \in \mathbb{N} ^ { * } , f \left( n _ { 0 } \right) \in \mathbb{N} ^ { * }$ and $f \left( n _ { 0 } \right) > n _ { 0 }$
D. $\exists n _ { 0 } \in \mathbb{N} ^ { * } , f \left( n _ { 0 } \right) \notin \mathbb{N} ^ { * }$ or $f \left( n _ { 0 } \right) > n _ { 0 }$

Executing the program flowchart shown, the output value of $k$ is

5. The three views of a certain triangular pyramid are shown in the figure. The surface area of this triangular pyramid is
[Figure]
Front (Main) View
[Figure]
Top View
[Figure]
Side (Left) View
A. $2 + \sqrt { 5 }$
B. $4 + \sqrt { 5 }$
C. $2 + 2 \sqrt { 5 }$
D. $5$

5. Let $l _ { 1 } , l _ { 2 }$ denote two lines in space. If $\mathrm { p } : l _ { 1 } , l _ { 2 }$ are skew lines, $\mathrm { q } : l _ { 1 } , l _ { 2 }$ do not intersect, then
A. p is a sufficient condition for q, but not a necessary condition
B. p is a necessary condition for q, but not a sufficient condition
C. p is a sufficient and necessary condition for q
D. p is neither a sufficient condition nor a necessary condition for q

5. The three views of a solid are shown in the figure. The surface area of this solid is
A. $3 \pi$
B. $4 \pi$
C. $2 \pi + 4$
D. $3 \pi + 4$ [Figure]
6. ``$\sin \alpha = \cos \alpha$'' is ``$\cos 2 \alpha = 0$'' a
A. sufficient but not necessary condition
B. necessary but not sufficient condition
C. necessary and sufficient condition
D. neither sufficient nor necessary condition

5. Among the following functions, the odd function with minimum positive period $\pi$ is
(A) $y = \cos \left( 2 x + \frac { \pi } { 2 } \right)$
(B) $y = \sin \left( x 2 + \frac { \pi } { 3 } \right)$
(C) $y = \sin z + \cos$ $y = \sin x +$

6. By reading the flowchart shown in the figure and running the corresponding program, the output result is
A. 2
B. 1
C. 0
D. $- 1$

6. A cube is cut by a plane, and the three-view drawing of the remaining part is shown on the right. The ratio of the volume of the cut-off part to the volume of the remaining part is [Figure]
A. $\frac { 1 } { 8 }$
B. $\frac { 1 } { 7 }$
C. $\frac { 1 } { 6 }$
D. $\frac { 1 } { 5 }$

A cube is cut by a plane, and the orthogonal projections of the remaining part are shown in the figure on the right. The ratio of the volume of the cut-off part to the volume of the remaining part is
(A) $\frac { 1 } { 8 }$
(B) $\frac { 1 } { 7 }$
(C) $\frac { 1 } { 6 }$
(D) $\frac { 1 } { 5 }$

6. Executing the flowchart shown in the figure, the output value of $S$ is
(A) $- \frac { \sqrt { 3 } } { 2 }$
(B) $\frac { \sqrt { 3 } } { 2 }$
(C) $- \frac { 1 } { 2 }$
(D) $\frac { 1 } { 2 }$

6. As shown in the figure, in circle $O$, $M, N$ are trisection points of chord $AB$. Chords $CD, CE$ pass through points $M, N$ respectively. If $CM = 3$, then the length of segment $NE$ is [Figure]
(A) $\frac { 8 } { 3 }$
(B) 3
(C) $\frac { 10 } { 3 }$
(D) $\frac { 5 } { 2 }$

6. Let $A , B$ be finite sets, and define $d ( A , B ) = \operatorname { card } ( A \cup B ) - \operatorname { card } ( A \cap B )$ , where $\operatorname { card } ( A )$ denotes the number of elements in the finite set $A$.
Proposition (1): For any finite sets $A , B$, ``$A \neq B$'' is a [Figure]
necessary and sufficient condition for ``$d ( A , B ) > 0$'';
Proposition (2): For any finite sets $A , B , C$, $d ( A , C ) \leq d ( A , B ) + d ( B , C )$.
A. Both Proposition (1) and Proposition (2) are true
B. Both Proposition (1) and Proposition (2) are false
C. Proposition (1) is true, Proposition (2) is false
D. Proposition (1) is false, Proposition (2) is true

7. Execute the program flowchart shown in the figure. The output value of $n$ is [Figure]
(A) $3$
(B) $4$
(C) $5$
(D) $6$

The three-view drawing of a certain quadrangular pyramid is shown in the figure. The length of the longest edge of this quadrangular pyramid is

7. If $l$ and $m$ are two different lines, and $m$ is perpendicular to plane $\alpha$, then ``$l \perp m$'' is ``$l \parallel \alpha$'' a [Figure]
A. sufficient but not necessary condition
B. necessary but not sufficient condition
C. sufficient and necessary condition
D. neither sufficient nor necessary condition

8. The ``fuel efficiency'' of a car refers to the distance a car can travel per liter of gasoline consumed. The figure below describes the fuel efficiency of three cars, A, B, and C, at different speeds. The correct statement is [Figure]
A. Consuming 1 liter of gasoline, car B can travel at most 5 kilometers
B. Traveling at the same speed for the same distance, among the three cars, car A consumes the most gasoline
C. Car A travels at 80 kilometers per hour for 1 hour, consuming 10 liters of gasoline
D. A certain city has a maximum speed limit of 80 kilometers per hour. Under the same conditions, using car C in this city is more fuel-efficient than using car B

Part Two (Non-Multiple Choice Questions, 110 points)

II. Fill-in-the-Blank Questions: There are 6 questions in total, each worth 5 points, for a total of 30 points.

8. The algorithm idea of the flowchart on the right comes from the ``Mutual Subtraction Method'' in the ancient Chinese mathematical classic ``The Nine Chapters on the Mathematical Art''. When executing this flowchart, if the input values of $a$ and $b$ are 14 and 18 respectively, then the output value of $a$ is ( )
A. $0$
B. $2$
C. $4$
D. $14$