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

\section*{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\\
\includegraphics[max width=\textwidth, alt={}, center]{85948435-6f44-4fbb-8256-7e2243e8de3a-2_542_639_868_310}\\
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) $\left( 2 \mathrm { k } - \frac { 1 } { 4 } , 2 \mathrm { k } + \frac { 3 } { 4 } \right) , \mathrm { k } \in \mathbf { z }$\\
\includegraphics[max width=\textwidth, alt={}, center]{85948435-6f44-4fbb-8256-7e2243e8de3a-3_531_661_301_388}\\
(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\\
\includegraphics[max width=\textwidth, alt={}, center]{85948435-6f44-4fbb-8256-7e2243e8de3a-3_851_384_990_501}\\
(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 $\mathrm { r } =$\\
\includegraphics[max width=\textwidth, alt={}, center]{85948435-6f44-4fbb-8256-7e2243e8de3a-4_604_447_287_310}\\
(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*{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.\\
\includegraphics[max width=\textwidth, alt={}, center]{85948435-6f44-4fbb-8256-7e2243e8de3a-5_417_673_280_303}\\
(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