Following the flowchart below, if the input is $a = -1$, then the output is $S =$
A. $2$
B. $3$
C. $4$
D. $5$

11.
A. $- 1$
B. $0$
C. $\frac{1}{2}$
D. $1$

As shown in the figure, a circular piece of paper has center $O$ and radius $5$ cm. An equilateral triangle $ABC$ on this paper has center at $O$. Points $D, E, F$ are on circle $O$. Triangles $DBC, ECA, FAB$ are isosceles triangles with $BC, CA, AB$ as their bases respectively. After cutting along the dashed lines and folding triangles $DBC, ECA, FAB$ along $BC, CA, AB$ respectively so that $D, E, F$ coincide, a triangular pyramid is formed. As the side length of $\triangle ABC$ varies, the maximum volume (in $\mathrm { cm } ^ { 3 }$) of the resulting triangular pyramid is \_\_\_\_

17. (12 points) [Figure] [Figure]

In the pyramid $P$-$ABCD$, $AB \parallel CD$ and $\angle BAP = \angle CDP = 90 ^ { \circ }$.
(1) Prove that plane $PAB \perp$ plane $PAD$;
(2) If $PA = PD = AB = DC$ and $\angle APD = 90 ^ { \circ }$, find the cosine of the dihedral angle along edge $PD$ between plane $PAD$ and plane $PCD$.

(12 points)
As shown in the figure, in the pyramid $P$-$ABCD$, the base face $PAD$ is an equilateral triangle, $\angle BAD = \angle ADC = 90°$.
(1) Prove that the plane $PAD$ is perpendicular to the base $ABCD$ (or a related perpendicularity result).
(2) If the area of $\triangle PCD$ is $2\sqrt{7}$, find the volume of pyramid $P$-$ABCD$.

18. (12 points) [Figure] [Figure] [Figure]

Definition	[Figure]	Weight $\geqslant 50 \mathrm {~kg}$
Traditional Method
New Method

\includegraphics[max width=\textwidth, alt={}]{cd808fdb-5cf6-4c

Given sets $A = \{ 1,3,5,7 \} , B = \{ 2,3,4,5 \}$, then $A \cap B =$
A. $\{ 3 \}$
B. $\{ 5 \}$
C. $\{ 3,5 \}$
D. $\{ 1,2,3,4,5,7 \}$

Given a cylinder with centers of the upper and lower bases at $O _ { 1 }$ and $O _ { 2 }$ respectively, a plane passing through the line $O _ { 1 } O _ { 2 }$ intersects the cylinder in a rectangle with area 8. Then the surface area of the cylinder is
A. $- 12 \sqrt { 2 } \pi$
B. $12 \pi$
C. $8 \sqrt { 2 } \pi$
D. $10 \pi$

A certain cylinder has height 2 and base circumference 16. Its three-view diagram is shown on the right. Point $M$ on the cylinder surface corresponds to point $A$ in the front view, and point $N$ on the cylinder surface corresponds to point $B$ in the left view. Then on the lateral surface of this cylinder, the length of the shortest path from $M$ to $N$ is
A. $2 \sqrt { 17 }$
B. $2 \sqrt { 5 }$
C. 3
D. 2

To calculate $S = 1 - \frac { 1 } { 2 } + \frac { 1 } { 3 } - \frac { 1 } { 4 } + \cdots + \frac { 1 } { 99 } - \frac { 1 } { 100 }$, a flowchart was designed. The statement that should be filled in the blank is
A. $i = i + 1$
B. $i = i + 2$
C. $i = i + 3$
D. $i = i + 4$

To calculate $S = 1 - \frac { 1 } { 2 } + \frac { 1 } { 3 } - \frac { 1 } { 4 } + \cdots + \frac { 1 } { 99 } - \frac { 1 } { 100 }$, a flowchart was designed. What should be filled in the blank box?
A. $i = i + 1$
B. $i = i + 2$
C. $i = i + 3$
D. $i = i + 4$

A certain cylinder has height 2 and base circumference 16. Its three-view diagram is shown on the right. Point $M$ on the cylinder surface corresponds to point $A$ in the front view, and point $N$ on the cylinder surface corresponds to point $B$ in the left view. Then on the lateral surface of the cylinder, the length of the shortest path from $M$ to $N$ is
A. $2 \sqrt { 17 }$
B. $2 \sqrt { 5 }$
C. 3
D. 2

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. $\frac { \sqrt { 3 } } { 2 }$

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.