21. (Multiple Choice) This problem includes four sub-problems A, B, C, and D. Please select two of them and answer in the corresponding areas. If more are done, the first two sub-problems will be graded. When solving, you should write out text explanations, proofs, or calculation steps.
A. [Elective 4-1: Geometric Proof Selection] (This problem is worth 10 points) In $\triangle A B C$, $A B = A C$, the chord $A E$ of the circumcircle O of $\triangle A B C$ intersects $B C$ at point D. Prove: $\triangle A B D \sim \triangle A E B$ [Figure]
B. [Elective 4-2: Matrices and Transformations] (This problem is worth 10 points) Given $x , y \in R$, the vector $\alpha = \left[ \begin{array} { c } 1 \\ - 1 \end{array} \right]$ is an eigenvector of the matrix $A = \left[ \begin{array} { c c } x & 1 \\ y & 0 \end{array} \right]$ corresponding to the eigenvalue $- 2$. Find the matrix A and its other eigenvalue.

C. [Elective 4-4: Coordinate Systems and Parametric Equations]

The polar equation of circle C is $\rho ^ { 2 } + 2 \sqrt { 2 } \rho \sin \left( \theta - \frac { \pi } { 4 } \right) - 4 = 0$. Find the radius of circle C.
D. [Elective 4-5: Inequalities Selection] Solve the inequality $x + | 2 x + 3 | \geq 3$

22. (10 points) Elective 4-1: Geometric Proof As shown in the figure, AB is tangent to circle O at point B. Line AD intersects circle O at points D and E. $\mathrm { BC } \perp \mathrm { DE }$, with C as the foot of the perpendicular. (I) Prove that $\angle \mathrm { CBD } = \angle \mathrm { DBA }$; (II) If $\mathrm { AD } = 3 \mathrm { DC } , ~ \mathrm { BC }

23. Given the set $X = \{ 1,2,3 \} , Y _ { n } = \{ 1,2,3 , m , n \} \left( n \in N ^ { * } \right)$, let $S _ { n } = \left\{ ( a , b ) \mid a \text{ divides } b \text{ or } b \text{ divides } a , a \in X , b \in Y _ { n } \right\}$, and let $f ( n )$ denote the number of elements in set $S _ { n }$.
(1) Write out the value of $f ( 6 )$;
(2) When $n \geq 6$, write out the expression for $f ( n )$ and prove it using mathematical induction.

As shown in the figure, the three views of a certain geometric solid are three circles with equal radii, each with two mutually perpendicular radii. If the volume of this geometric solid is $\frac { 28 \pi } { 3 }$, then its surface area is
(A) $17 \pi$
(B) $18 \pi$
(C) $20 \pi$

Executing the program flowchart on the right, if the input is $x = 0 , y = 1$, $n = 1$, then the output values of $x , y$ satisfy
(A) $y = 2 x$
(B) $y = 3 x$
(C) $y = 4 x$
(D) $y = 5 x$

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

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 scratch paper or the exam paper are invalid. [Figure]

4. As shown in the figure, on grid paper with unit squares, the three-view of a geometric solid obtained by cutting off part of a cylinder with a plane is shown. The volume of this geometric solid is
A. $90 \pi$
B. $63 \pi$
C. $42 \pi$
D. $36 \pi$ [Figure] [Figure]
5. Let $x , y$ satisfy the constraints $\left\{ \begin{array} { l } 2 x + 3 y - 3 \leqslant 0 , \\ 2 x - 3 y + 3 \geqslant 0 , \\ y + 3 \geqslant 0 , \end{array} \right.$ then the minimum value of $z = 2 x + y$ is
A. $-3$
B. $- 9$
C. $1$
D. $9$

5. Keep the card surface clean, do not fold it, do not get it wet, and do not use correction fluid, correction tape, or scrapers.
I. Multiple Choice Questions: This section has 12 questions, each worth 5 points, totaling 60 points. For each question, only one of the four options is correct. $\frac { ( 3 + i ) ( 1 - i ) } { 2 }$ $3 - 3 i + i + 1 \quad 4 - 2 i$

As shown in the figure, in the following four cubes, $A$ and $B$ are two vertices of the cube, and $M$, $N$, $Q$ are midpoints of the respective edges. Among these four cubes, the line $AB$ is not parallel to plane $MNQ$ in which figure?

A polyhedron is formed by cutting off a corner of a triangular pyramid. Among the faces of this polyhedron, several are trapezoids. The sum of the areas of these trapezoids is
A. 10
B. 12
C. 14
D. 16

As shown in the figure, the side length of the small squares on the grid paper is 1. The three views shown are of a certain geometric solid, which is obtained by removing a part from a rectangular solid and a cylinder. The volume of this geometric solid is
A. $90\pi$
B. $63\pi$
C. $42\pi$
D. $36\pi$

8. Executing the flowchart on the right, if the input is $a = - 1$, then the output $S =$
A. $2$
B. $3$
C. $4$
D. $5$ [Figure]

As shown in the figure, in the quadrangular pyramid $P-ABCD$, $AB \parallel CD$, and $\angle BAP = \angle CDP = 90°$.
(1) Prove: plane $PAB \perp$ plane $PAD$;
(2) If $PA = PD = AB = DC$, $\angle APD = 90°$, and the volume of quadrangular pyramid $P-ABCD$ is $\frac{8}{3}$, find the lateral surface area of this quadrangular pyramid.

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. $1$

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 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