2. For multiple choice questions: After selecting the answer for each question, use a 2B pencil to blacken the corresponding answer choice on the answer sheet. Answers written on the test paper, scratch paper, and non-answer areas on the answer sheet are invalid.

2. For multiple choice questions, after selecting the answer to each question, use a pencil to blacken the corresponding answer letter on the answer sheet. If changes are needed, erase completely with an eraser before selecting another answer. For non-multiple choice questions, write your answers on the answer sheet. Answers written on this test paper are invalid.

2. Multiple choice questions must be filled in with 2B pencil; non-multiple choice questions must be written with a 0.5 mm black signature pen, with neat handwriting and clear strokes.

2. Multiple choice questions must be filled in with 2B pencil; non-multiple choice questions must be written with a 0.5mm black pen, with neat handwriting and clear strokes.

2. For multiple choice questions, after selecting the answer to each question, use a pencil to blacken the corresponding answer code on the answer sheet. If changes are needed, erase cleanly with an eraser before selecting another answer code. For non-multiple choice questions, write the answer on the answer sheet. Answers written on this test paper are invalid.

2. When answering multiple choice questions, after selecting the answer for each question, use a pencil to blacken the corresponding answer letter on the answer sheet. If changes are needed, erase completely with an eraser before selecting another answer. When answering non-multiple choice questions, write your answers on the answer sheet. Answers written on this test paper are invalid.

Journey to the West, Romance of the Three Kingdoms, Water Margin, and Dream of the Red Chamber are treasures of classical Chinese literature, collectively known as the Four Great Classical Novels of China. To understand the reading situation of these four classics among students in a school, a random survey was conducted of 100 students. Among them, 90 students had read either Journey to the West or Dream of the Red Chamber, 80 students had read Dream of the Red Chamber, and 60 students had read both Journey to the West and Dream of the Red Chamber. The estimated value of the ratio of the number of students who have read Journey to the West to the total number of students in the school is
A. 0.5
B. 0.6
C. 0.7
D. 0.8

3. For non-multiple choice questions: Use a purple pen to answer directly in the corresponding answer area on the answer sheet. Answers written on the test paper, scratch paper, and non-answer areas on the answer sheet are invalid.

3. Please answer according to the question numbers in the answer areas for each question on the answer sheet. Answers written outside the answer area are invalid; answers written on draft paper or the test paper are invalid.

3. Please answer according to the question numbers in the designated answer areas on the answer sheet. Answers written outside the answer area are invalid; answers written on draft paper or the test paper are invalid.

3. After the examination, submit this test paper and the answer sheet together. I. Multiple Choice Questions: This section has 12 questions, each worth 5 points, for a total of 60 points. For each question, only one of the four options is correct.
1. Given sets $A = \{ - 1,0,1,2 \} , B = \left\{ x \mid x ^ { 2 } \leq 1 \right\}$ , then $A \cap B =$
A. $\{ - 1,0,1 \}$
B. $\{ 0,1 \}$
C. $\{ - 1,1 \}$
D. $\{ 0,1,2 \}$
2. If $z ( 1 + \mathrm { i } ) = 2 \mathrm { i }$ , then $z =$
A. $-1-i$
B. $- 1 + \mathrm { i }$
C. $1 - \mathrm { i }$
D. $1 + \mathrm { i }$
3. Two male students and two female students are randomly arranged in a line. The probability that the two female students are adjacent is
A. $\frac { 1 } { 6 }$
B. $\frac { 1 } { 4 }$
C. $\frac { 1 } { 3 }$
D. $\frac { 1 } { 2 }$
4. ``Journey to the West,'' ``Romance of the Three Kingdoms,'' ``Water Margin,'' and ``Dream of the Red Chamber'' are treasures of classical Chinese literature, collectively known as the Four Great Classical Novels of China. A school conducted a random survey of 100 students to understand their reading of the Four Great Novels. Among them, 90 students had read either ``Journey to the West'' or ``Dream of the Red Chamber,'' 80 students had read ``Dream of the Red Chamber,'' and 60 students had read both ``Journey to the West'' and ``Dream of the Red Chamber.'' The estimated ratio of the number of students in the school who have read ``Journey to the West'' to the total number of students in the school is
A. 0.5
B. 0.6
C. 0.7
D. 0.8

3. After the examination ends, submit both this test paper and the answer sheet together. I. Multiple Choice Questions: This section has 12 questions, each worth 5 points, for a total of 60 points. For each question, only one of the four options is correct.
1. Given sets $A = \{ - 1,0,1,2 \} , B = \left\{ x \mid x ^ { 2 } \leq 1 \right\}$ , then $A \cap B =$
A. $\{ - 1,0,1 \}$
B. $\{ 0,1 \}$
C. $\{ - 1,1 \}$
D. $\{ 0,1,2 \}$
2. If $z ( 1 + \mathrm { i } ) = 2 \mathrm { i }$ , then $z =$
A. $-1-i$
B. $- 1 + \mathrm { i }$
C. $1-i$
D. $1 + \mathrm { i }$
3. Journey to the West, Romance of the Three Kingdoms, Water Margin, and Dream of the Red Chamber are treasures of classical Chinese literature, collectively known as the Four Great Classical Novels of China. To understand the reading situation of these four classics among students at a school, a random survey was conducted of 100 students. Among them, 90 students had read Journey to the West or Dream of the Red Chamber, 80 students had read Dream of the Red Chamber, and 60 students had read both Journey to the West and Dream of the Red Chamber. The estimated ratio of the number of students who have read Journey to the West to the total number of students in the school is
A. 0.5
B. 0.6
C. 0.7
D. 0.8

On January 3, 2019, the Chang'e-4 probe successfully achieved humanity's first soft landing on the far side of the moon. A key technical challenge in achieving soft landing on the far side of the moon is maintaining communication between the ground and the probe. To solve this problem, the Chang'e-4 relay satellite ``Queqiao'' was launched, which orbits around the Earth-Moon Lagrange point $L _ { 2 }$. The $L _ { 2 }$ point is an equilibrium point located on the extension of the Earth-Moon line. Let the mass of Earth be $M _ { 1 }$, the mass of the Moon be $M _ { 2 }$, the Earth-Moon distance be $R$, and the distance from the $L _ { 2 }$ point to the Moon be $r$. According to Newton's laws of motion and the law of universal gravitation, $r$ satisfies the equation: $\frac { M _ { 1 } } { ( R + r ) ^ { 2 } } + \frac { M _ { 2 } } { r ^ { 2 } } = ( R + r ) \frac { M _ { 1 } } { R ^ { 3 } }$.
Let $\alpha = \frac { r } { R }$. Since $\alpha$ is very small, in approximate calculations $\frac { 3 \alpha ^ { 3 } + 3 \alpha ^ { 4 } + \alpha ^ { 5 } } { ( 1 + \alpha ) ^ { 2 } } \approx 3 \alpha ^ { 3 }$. Then the approximate value of $r$ is
A．$\sqrt { \frac { M _ { 2 } } { M _ { 1 } } } R$
B．$\sqrt { \frac { M _ { 2 } } { 2 M _ { 1 } } R }$
C．$\sqrt [ 3 ] { \frac { 3 M _ { 2 } } { M _ { 1 } } R }$
D．$\sqrt [ 3 ] { \frac { M _ { 2 } } { 3 M _ { 1 } } } R$

4. According to historical records, the ``Hundred Family Names'' was written in the early Northern Song Dynasty. Table 1 records the top 24 surnames from the beginning of the ``Hundred Family Names'':
\begin{table}[h]
Table 1

赵	钱	孙	李	周	吴	郑	王	冯	陈	褚	卫
蒋	沈	韩	杨	朱	秦	尤	许	何	吕	施	张

\end{table}
Table 2 records the top 25 most populous surnames in China in 2018:
\begin{table}[h]
Table 2

1: Li	2: Wang	3: Zhang	4: Liu	5: Chen
6: Yang	7: Zhao	8: Huang	9: Zhou	10: Wu
11: Xu	12: Sun	13: Hu	14: Zhu	15: Gao
16: Lin	17: He	18: Guo	19: Ma	20: Luo

\end{table}

21: Liang

22: Song

23: Zheng

24: Xie

25: Han

If one surname is randomly selected from the top 24 surnames in the ``Hundred Family Names'', the probability that this surname is among the top 24 most populous surnames in China in 2018 is
A. $\frac { 5 } { 12 }$ B. $\frac { 11 } { 24 }$ C. $\frac { 13 } { 24 }$ D. $\frac { 1 } { 2 }$

4. For optional questions: First blacken the question number of your chosen question at the designated location on the answer sheet using a 2B pencil. Write your answer in the corresponding answer area on the answer sheet. Answers written on the test paper, scratch paper, and non-answer areas on the answer sheet are invalid.

4. In ancient Greece, people believed that the most beautiful human body has the ratio of the length from the top of the head to the navel to the length from the navel to the bottom of the feet equal to $\frac { \sqrt { 5 } - 1 } { 2 } \left( \frac { \sqrt { 5 } - 1 } { 2 } \approx 0.618 \right.$ , called the golden ratio), as exemplified by the famous ``Armless Venus''. Furthermore, the ratio of the length from the top of the head to the base of the neck to the length from the base of the neck to the navel is also $\frac { \sqrt { 5 } - 1 } { 2 }$ . If a person satisfies both golden ratio proportions above, with leg length 105 cm and head to base of neck length 26 cm, then their height is likely to be [Figure]
A. 165 cm
B. 175 cm
C. 185 cm
D. 190 cm

4. Drawings may first be made with a pencil, and after confirmation must be traced with a black signature pen.

4. For drawings, you may first use pencil, and after confirmation, you must trace over with a black pen.

5. Keep the card surface clean, do not fold, do not tear or wrinkle, and do not use correction fluid, correction tape, or scraping knives. I. Multiple Choice Questions: This section has 12 questions, 5 points each, 60 points total. For each question, only one of the four options is correct.
1. Given sets $A = \{ x \mid x > - 1 \} , B = \{ x \mid x < 2 \}$, then $A \cap B =$
A. $( - 1 , + \infty )$
B. $( - \infty , 2 )$
C. $( - 1,2 )$
D. $\varnothing$
2. Let $z = \mathrm { i } ( 2 + \mathrm { i } )$, then $\bar { z } =$
A. $1 + 2 \mathrm { i }$
B. $- 1 + 2 \mathrm { i }$
C. $1 - 2 \mathrm { i }$
D. $- 1 - 2 \mathrm { i }$
3. Given vectors $\boldsymbol { a } = ( 2,3 ) , \boldsymbol { b } = ( 3,2 )$, then $| \boldsymbol { a } - \boldsymbol { b } | =$
A. $\sqrt { 2 }$
B. 2
C. $5 \sqrt { 2 }$
D. 50
4. A biology laboratory has 5 rabbits, of which only 3 have been measured for a certain indicator. If 3 rabbits are randomly selected from these 5 rabbits, the probability that exactly 2 have been measured for this indicator is
A. $\frac { 2 } { 3 }$
B. $\frac { 3 } { 5 }$
C. $\frac { 2 } { 5 }$
D. $\frac { 1 } { 5 }$
5. After a knowledge test on the ``Belt and Road'', three people A, B, and C made predictions about their scores. A: My score is higher than B's.
B: C's score is higher than both mine and A's. C: My score is higher than B's. After the scores were announced, the three people's scores are all different and only one person's prediction is correct. Then the three people in order from highest to lowest score are
A. A, B, C
B. B, A, C
C. C, B, A
D. A, C, B

7. The three-view drawing of a certain solid is shown in the figure. The volume of this solid is
A. $24 \pi - 6$
B. $8 \pi - 6$
C. $24 \pi + 6$
D. $8 \pi + 6$

7. Let $\alpha , \beta$ be two planes. Then $\alpha / / \beta$ is a necessary and sufficient condition for
A. There are infinitely many lines in $\alpha$ parallel to $\beta$
B. There are two intersecting lines in $\alpha$ parallel to $\beta$
C. $\alpha , \beta$ are both parallel to the same line
D. $\alpha , \beta$ are both perpendicular to the same plane

7. Let $\alpha , \beta$ be two planes. Then $\alpha / / \beta$ is a necessary and sufficient condition for
A. $\alpha$ contains infinitely many lines parallel to $\beta$
B. $\alpha$ contains two intersecting lines parallel to $\beta$
C. $\alpha , \beta$ are both parallel to the same line
D. $\alpha , \beta$ are both perpendicular to the same plane

9. Executing the flowchart below, if the input $\varepsilon$ is 0.01 , then the output value of $s$ equals [Figure]
A. $2 - \frac { 1 } { 2 ^ { 4 } }$
B. $2 - \frac { 1 } { 2 ^ { 5 } }$
C. $2 - \frac { 1 } { 2 ^ { 6 } }$
D. $2 - \frac { 1 } { 2 ^ { 7 } }$

Li Ming started his own business and operates a fruit shop online, selling strawberries, Chinese pears, watermelons, and peaches at prices of 60 yuan/box, 65 yuan/box, 80 yuan/box, and 90 yuan/box respectively. To increase sales, Li Ming offers a promotion: when the total price of a purchase reaches 120 yuan, the customer pays $x$ yuan less. After a customer successfully pays online, Li Ming receives 80\% of the payment. (1) When $x = 10$, if a customer purchases 1 box each of strawberries and watermelons, the amount to be paid is $\_\_\_\_$ yuan; (2) In the promotion activity, to ensure that Li Ming receives at least 70\% of the pre-promotion total price for each order, the maximum value of $x$ is $\_\_\_\_$.

16. China has a long history of stone and seal culture, and seals are representatives of this culture. Seals are usually shaped as rectangular prisms, cubes, or cylinders, but the seal of the official Dugu Xin from the Northern and Southern Dynasties is shaped as a ``semi-regular polyhedron'' (Figure 1). A semi-regular polyhedron is a polyhedron formed by two or more types of regular polygons. Semi-regular polyhedra embody the symmetry beauty of mathematics. Figure 2 shows a semi-regular polyhedron with 48 edges, with all vertices on the surface of a cube with edge length 1. This semi-regular polyhedron has $\_\_\_\_$ faces, and its edge length is $\_\_\_\_$. (The first blank is worth 2 points, the second blank is worth 3 points.)
[Figure]
Figure 1
[Figure]
Figure 2
III. Solution Questions: 70 points total. Solutions should include written explanations, proofs, or calculation steps. Questions 17--21 are required questions that all candidates must answer. Questions 22 and 23 are optional questions; candidates should answer according to requirements. (I) Required Questions: 60 points total.