A cube has edge length 1. Each line containing an edge makes equal angles with plane $\alpha$. Then the maximum area of the cross-section obtained when plane $\alpha$ intersects this cube is
A. $\frac { 3 \sqrt { 3 } } { 4 }$
B. $\frac { 2 \sqrt { 3 } } { 3 }$
C. $\frac { 3 \sqrt { 2 } } { 4 }$
D. $\frac { \sqrt { 3 } } { 2 }$

A cone has apex $S$, and edges $S A , S B$ are mutually perpendicular. The angle between $S A$ and the base plane is $30 ^ { \circ }$. If the area of $\triangle S A B$ is 8, then the volume of the cone is \_\_\_\_ .

The apex of a cone is $S$. The cosine of the angle between generatrices $S A$ and $S B$ is $\frac { 7 } { 8 }$. The angle between $S A$ and the base of the cone is $45 ^ { \circ }$. The area of $\triangle S A B$ is $5 \sqrt { 15 }$. Then the lateral surface area of the cone is $\_\_\_\_$.

As shown in the figure, in parallelogram $A B C M$, $A B = A C = 3$ and $\angle A C M = 90 ^ { \circ }$. Fold $\triangle A C M$ along $AC$ so that point $M$ moves to position $D$, with $A B \perp A D$.
(1) Prove: Plane $A C D \perp$ Plane $A B C$
(2) Let $Q$ be a point on segment $AD$ and $P$ be a point on segment $BC$ such that $B P = D Q = \frac { 2 } { 3 } D A$. Find the volume of the tetrahedron $Q - A B P$.

As shown in the figure, quadrilateral $A B C D$ is a square, $E$ and $F$ are the midpoints of $A D$ and $B C$ respectively. Using $D F$ as the fold line, fold $\triangle D F C$ so that point $C$ reaches position $P$, with $P F \perp B F$.
(1) Prove: plane $P E F \perp$ plane $A B F D$;
(2) Find the sine of the angle between $D P$ and plane $A B F D$.

1. This test paper consists of Section I (Multiple Choice) and Section II (Non-Multiple Choice), totaling 150 points. Examination time: 120 minutes.

1. Before answering, write your name and admission ticket number on the test paper and answer sheet, and paste the barcode of your admission ticket number at the designated location on the answer sheet.

1. Before answering, candidates must fill in their name, candidate number, and other information in the designated positions on the answer sheet and test paper.

1. Before answering, candidates should first fill in their name and admission ticket number clearly, and accurately paste the barcode in the candidate information barcode area.

1. Before answering, candidates should first fill in their name and admission ticket number clearly, and accurately paste the barcode in the candidate information barcode area.

1. Before answering, candidates must fill in their name and admission ticket number on the answer sheet.

1. Before answering, candidates must fill in their name and admission ticket number on the answer sheet.

Execute the flowchart shown in the figure. The output value of $s$ is (A) 1 (B) 2 (C) 3 (D) 4

2. Please fill in the answers to each question on the answer sheet provided after the test paper.

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.

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