14. In the triangular prism $A B C - A _ { 1 } B _ { 1 } C _ { 1 }$, $\angle B A C = 90 ^ { \circ }$. Both the front view and side view are squares with side length 1, and the top view is an isosceles right triangle with legs of length 1. Let $M , N , P$ be the midpoints of edges $A B , B C , B _ { 1 } C _ { 1 }$ respectively. Then the volume of the triangular pyramid $P - A _ { 1 } M N$ is \_\_\_\_.

14. If real numbers $x , y$ satisfy $x ^ { 2 } + y ^ { 2 } \leq 1$ , then the minimum value of $| 2 x + y - 2 | + | 6 - x - 3 y |$ is $\_\_\_\_$ .
[Figure]
(Figure for Question 13)

15. A binary code is a string of digits $x _ { 1 } x _ { 2 } \cdots x _ { n }$ ($n \in \mathbb{N} ^ { * }$) composed of 0 and 1, where $x _ { k }$ ($k = 1,2, \cdots, n$) is called the $k$-th bit code element. Binary codes are commonly used in communication, but during the communication process [Figure]code element errors sometimes occur (that is, a code element changes from 0 to 1, or from 1 to 0).
A certain type of binary code $x _ { 1 } x _ { 2 } \cdots x _ { 7 }$ satisfies the following system of check equations: $\left\{ \begin{array} { l } x _ { 4 } \oplus x _ { 5 } \oplus x _ { 6 } \oplus x _ { 7 } = 0 , \\ x _ { 2 } \oplus x _ { 3 } \oplus x _ { 6 } \oplus x _ { 7 } = 0 , \\ x _ { 1 } \oplus x _ { 3 } \oplus x _ { 5 } \oplus x _ { 7 } = 0 , \end{array} \right.$ where the operation $\oplus$ is defined as: $0 \oplus 0 = 0$, $0 \oplus 1 = 1$, $1 \oplus 0 = 1$, $1 \oplus 1 = 0$. It is now known that such a binary code became 1101101 after a code element error occurred at the $k$-th position during transmission. Using the above system of check equations, we can determine that $k$ equals $\_\_\_\_$.

18. (This question is worth 12 points)

A net of a cube and a schematic diagram of the cube are shown in the figure. (1) Mark the letters $F , G , H$ at the corresponding vertices of the cube (no explanation needed); (2) Determine the positional relationship between plane $B E G$ and plane $A C H$, and prove your conclusion; (3) Prove: line $D F \perp$ plane $B E G$. [Figure] [Figure]

19. (This question is worth 13 points) As shown in Figure 6, a frustum of a pyramid $A B C D - A _ { 1 } B _ { 1 } C _ { 1 } D _ {

20. In ``The Nine Chapters on the Mathematical Art,'' a quadrangular pyramid with a rectangular base and one lateral edge perpendicular to the base is called a ``yang ma'', and a tetrahedron with all four faces being right triangles is called a ``bie nao''. In the yang ma $\mathrm { P } - \mathrm { ABCD }$ shown in the figure, the lateral edge $\mathrm { PD } \perp$ base ABCD, and $\mathrm { PD } = \mathrm { CD }$. Point E is the midpoint of PC. Connect $\mathrm { DE } , \mathrm { BD } , \mathrm { BE }$.
[Figure]
Figure for Question 20
(I) Prove that $\mathrm { DE } \perp$ plane PBC. Determine whether the tetrahedron EBCD is a ``bie nao''. If yes, write out the right angle of each face (only conclusions are needed); if no, please explain the reason; (II) Let the volume of the yang ma $\mathr

As shown in the figure for question (21), [content incomplete in source document]

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) $y = 5 x$

Let the set $A = \{1,2,3\}$, $B = \{2,3,4\}$, then $A \cup B =$
A. $\{1,2,3,4\}$
B. $\{1,2,3\}$
C. $\{2,3,4\}$
D. $\{1,1,4\}$

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]

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. $36\pi$

Let $x, y$ satisfy the constraint conditions $\left\{ \begin{array}{l} x - y \geq 1 \\ y \geq 0 \end{array} \right.$. Then the maximum value of $z = x + y$ is
A. 0
B. 1
C. 2
D. 3

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.