17. (13 points) As shown in the figure, $AA _ { 1 } \perp$ plane $ABC$, $BB _ { 1 } \parallel AA _ { 1 }$, $AB = AC = 3$, $BC = 2 \sqrt { 5 }$, $AA _ { 1 } = \sqrt { 7 }$, $BB _ { 1 } = 2 \sqrt { 7 }$. Points $E$ and $F$ are the midpoints of $BC$ and $A _ { 1 } C$ respectively. (I) Prove that $EF \parallel$ plane $A _ { 1 } B _ { 1 } BA$; (II) Prove that plane $AEA _ { 1 } \perp$ plane $BCB _ { 1 }$. (III) Find the angle between line $A _ { 1 } B _ { 1 }$ and plane $BCB _ { 1 }$. [Figure]

As shown in the figure, in the quadrangular prism $\mathrm{ABCD} - A_1B_1C_1D_1$, the lateral edge $AA_1 \perp$ base $\mathrm{ABCD}$, $\mathrm{AB} \perp \mathrm{AC}$, $\mathrm{AB} = 1$, $\mathrm{AC} = AA_1 = 2$, $AD = CD = \sqrt{5}$, and points M and N are the midpoints of $B_1C$ and $D_1D$ respectively.
(I) Prove: $\mathrm{MN} \parallel$ plane ABCD
(II) Find the sine value of the dihedral angle $D_1 - AC - B_1$;
(III) Let E be a point on edge $A_1B_1$. If the sine value of the angle between line NE and plane ABCD is $\frac{1}{3}$, find the length of segment $A_1E$.

17. (This question is worth 15 points) As shown in the figure, in the triangular prism $ABC - A _ { 1 } B _ { 1 } C _ { 1 }$, $\angle BAC = 90 ^ { \circ }$ , $AB = AC = 2$ , $A _ { 1 } A = 4$ , the projection of $A _ { 1 }$ on the base plane $ABC$ is the midpoint of $BC$, and $D$ is the midpoint of $B _ { 1 } C _ { 1 }$ . (I) Prove that $A _ { 1 } D \perp$ plane $A _ { 1 } B C _ { 1 }$ ; (II) Find the cosine of the plane angle of the dihedral angle $A _ { 1 } - BD - B _ { 1 }$ . [Figure]

As shown in the figure, in the triangular pyramid $E - ABC$, plane $EAB \perp$ plane $ABC$, triangle $EAB$ is equilateral, $AC \perp BC$, and $AC = BC = \sqrt { 2 }$. $O$ and $M$ are the midpoints of $AB$ and $EA$ respectively.\n(1) Prove that $EB \parallel$ plane $MOC$.\n(2) Prove that plane $MOC \perp$ plane $EAB$.\n(3) Find the volume of the triangular pyramid $E - ABC$.

18. (This question is worth 12 points) As shown in Figure 4, the right triangular prism $\mathrm { ABC } - \mathrm { A } _ { 1 } \mathrm { B } _ { 1 } \mathrm { C } _ { 1 }$ has an equilateral triangle base with side length 2. $\mathrm { E }$ and $\mathrm { F }$ are the midpoints of $\mathrm { BC }$ and $\mathrm { CC } _ { 1 }$ respectively. (I) Prove that: plane $\mathrm { AEF } \perp$ plane $\mathrm { B } _ { 1 } \mathrm { BCC } _ { 1 }$ (II) If the angle between line $A _ { 1 } C$ and plane $A _ { 1 } A B B _ { 1 }$ is $45 ^ { \circ }$, find the volume of the triangular pyramid F-AEC. [Figure]

18. (12 points) As shown in Figure 1, in right trapezoid ABCD, $\mathrm { AD } / / \mathrm { BC } , ~ \angle \mathrm { BAD } = \frac { \pi } { 2 } , \mathrm { AB } = \mathrm { BC } = 1$, $\mathrm { AD } = 2$, E is the midpoint of AD, and O is the intersection of AC and BE. Fold $\triangle \mathrm { ABE }$ along BE to the position $\Delta \mathrm { A } _ { 1 } \mathrm { BE }$ as shown in Figure 2.
[Figure]
Figure 1
[Figure]
Figure 2
(I) Prove that $\mathrm { CD } \perp$ plane $\mathrm { A } _ { 1 } \mathrm { OC }$; (II) If plane $\mathrm { A } _ { 1 } \mathrm { BE } \perp$ plane BCDE, find the cosine of the dihedral angle between plane $\mathrm { A } _ { 1 } \mathrm { BC }$ and plane $\mathrm { A } _ { 1 } \mathrm { CD }$.

18. A net of a cube and a schematic diagram of the cube are shown in the figure. In the cube, let $M$ be the midpoint of $BC$ and $N$ be the midpoint of $GH$.
(1) Mark the letters $F$, $G$, $H$ at the corresponding vertices of the cube (no explanation needed).
(2) Prove: line $MN \parallel$ plane $BDH$.
(3) Find the cosine of the dihedral angle $A - E G - M$. [Figure]

18. (15 points) As shown in the figure, in the triangular prism $A B C - A _ { 1 } B _ { 1 } C _ { 1 }$ , $\angle \mathrm { ABC } = 90 ^ { \circ } , \mathrm { AB } = \mathrm { AC } = 2 , \mathrm { AA } _ { 1 } = 4$ , the projection of $A _ { 1 }$ on the base plane ABC is the midpoint of BC, and D is the midpoint of $B _ { 1 } C _ { 1 }$.
(1) Prove that $A _ { 1 } \mathrm { D } \perp$ plane $\mathrm { A } _ { 1 } \mathrm { BC }$ ;
(2) Find the sine of the angle between line $\mathrm { A } _ { 1 } \mathrm { B}$ and plane $\mathrm { BB } _ { 1 } \mathrm { C } C _ { 1 }$ . [Figure]

19. (This question is worth 12 points). In the rectangular prism $A B C D - A _ { 1 } B _ { 1 } C _ { 1 } D _ { 1 }$, $A B = 16 , B C = 10 , A A _ { 1 } = 8$. Points $E$ and $F$ are on $A _ { 1 } B _ { 1 }$ and $D _ { 1 } C _ { 1 }$ respectively, with $A _ { 1 } E = D _ { 1 } F = 4$. A plane $\alpha$ passes through points $E$ and $F$ and intersects the faces of the rectangular prism, with the intersection lines forming a square. [Figure] (I) Draw this square in the figure (no need to explain the method or reasoning); (II) Find the ratio of the volumes of the two parts that plane $\alpha$ divides the rectangular prism into.

As shown in the figure, in rectangular prism $\mathrm { ABCD } - \mathrm { A } _ { 1 } \mathrm { B } _ { 1 } \mathrm { C } _ { 1 } \mathrm { D } _ { 1 }$, we have $\mathrm { AB } = 16 , \mathrm { BC } = 10 , \mathrm { AA } _ { 1 } = 8$. Points $\mathrm { E }$ and $\mathrm { F }$ are on $\mathrm { A } _ { 1 } \mathrm { B } _ { 1 }$ and $\mathrm { D } _ { 1 } \mathrm { C } _ { 1 }$ respectively, with $\mathrm { A } _ { 1 } \mathrm { E } = \mathrm { D } _ { 1 } \mathrm { F }$. A plane $\alpha$ passes through points $E$ and $F$ and intersects the faces of the rectangular prism, with the intersection lines forming a square.
(I) Draw this square in the figure (no need to explain the method or reasoning)
(II) Find the sine of the angle between line

As shown in the figure for question (20), in the triangular pyramid $\mathrm { P } - \mathrm { ABC }$, plane $\mathrm { PAC } \perp$ plane $\mathrm { ABC }$, $\angle \mathrm { ABC } = \frac { \pi } { 2 }$. Points $D$ and $E$ lie on segment $AC$ with $\mathrm { AD } = \mathrm { DE } = \mathrm { EC } = 2$, $\mathrm { PD } = \mathrm { PC } = 4$. Point $F$ lies on segment $AB$ with $\mathrm { EF } \parallel \mathrm { BC }$ .
(I) Prove that $\mathrm { AB } \perp$ plane $PFE$ .
(II) If the volume of the quadrangular pyramid $\mathrm { P } - \mathrm { DFBC }$ is 7, find the length of segment $BC$.

21. (14 marks) (I) Let point $D ( t , 0 ) ( | t | \leq 2 ) , N \left( x _ { 0 } , y _ { 0 } \right) , M ( x , y )$. According to the problem,
$$\overrightarrow { M D } = 2 \overrightarrow { D N } \text {, and } | \overrightarrow { D N } | = | \overrightarrow { O N } | = 1,$$
[Figure]
so $( t - x , - y ) = 2 \left( x _ { 0 } - t , y _ { 0 } \right)$, and $\left\{ \begin{array} { l } \left( x _ { 0 } - t \right) ^ { 2 } + y _ { 0 } ^ { 2 } = 1 , \\ x _ { 0 } ^ { 2 } + y _ { 0 } ^ { 2 } = 1 . \end{array} \right.$ That is, $\left\{ \begin{array} { l } t - x = 2 x _ { 0 } - 2 t , \\ y = - 2 y _ { 0 } . \end{array} \right.$ and $t \left( t - 2 x _ { 0 } \right) = 0$. Since when point $D$ is fixed, point $N$ is also fixed, $t$ is not identically zero, thus $t = 2 x _ { 0 }$, so $x _ { 0 } = \frac { x } { 4 } , y _ { 0 } = - \frac { y } { 2 }$. Substituting into $x _ { 0 } ^ { 2 } + y _ { 0 } ^ { 2 } = 1$, we obtain $\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 4 } = 1$, that is, the equation of the required curve $C$ is $\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 4 } = 1$. (II) (1) When the slope of line $l$ does not exist, line $l$ is $x = 4$ or $x = - 4$, and we have $S _ { \triangle O P Q } = \frac { 1 } { 2 } \times 4 \times 4 = 8$.
(2) When the slope of line $l$ exists, let line $l : y = k x + m \left( k \neq \pm \frac { 1 } { 2 } \right)$. From $\left\{ \begin{array} { l } y = k x + m , \\ x ^ { 2 } + 4 y ^ { 2 } = 16 , \end{array} \right.$ eliminating $y$, we obtain $\left( 1 + 4 k ^ { 2 } \right) x ^ { 2 } + 8 k m x + 4 m ^ { 2 } - 16 = 0$. Since line $l$ always has exactly one common point with ellipse $C$, we have $\Delta = 64 k ^ { 2 } m ^ { 2 } - 4 \left( 1 + 4 k ^ { 2 } \right) \left( 4 m ^ { 2 } - 16 \right) = 0$, that is, $m ^ { 2 } = 16 k ^ { 2 } + 4$.
From $\left\{ \begin{array} { l } y = k x + m , \\ x - 2 y = 0 , \end{array} \right.$ we obtain $P \left( \frac { 2 m } { 1 - 2 k } , \frac { m } { 1 - 2 k } \right)$; similarly, we obtain $Q \left( \frac { - 2 m } { 1 + 2 k } , \frac { m } { 1 + 2 k } \right)$. From the distance from origin $O$ to line $P Q$ being $d = \frac { | m | } { \sqrt { 1 + k ^ { 2 } } }$ and $| P Q | = \sqrt { 1 + k ^ { 2 } } \left| x _ { P } - x _ { Q } \right|$, we obtain $S _ { \triangle O P Q } = \frac { 1 } { 2 } | P Q | \cdot d = \frac { 1 } { 2 } | m | \left| x _ { P } - x _ { Q } \right| = \frac { 1 } { 2 } \cdot | m | \left| \frac { 2 m } { 1 - 2 k } + \frac { 2 m } { 1 + 2 k } \right| = \left| \frac { 2 m ^ { 2 } } { 1 - 4 k ^ { 2 } } \right|$.
Substituting (1) into (2), we obtain $S _ { \triangle O P Q } = \left| \frac { 2 m ^ { 2 } } { 1 - 4 k ^ { 2 } } \right| = 8 \frac { \left| 4 k ^ { 2 } + 1 \right| } { \left| 4 k ^ { 2 } - 1 \right| }$. When $k ^ { 2 } > \frac { 1 } { 4 }$, $S _ { \triangle O P Q } = 8 \left( \frac { 4 k ^ { 2 } + 1 } { 4 k ^ { 2 } - 1 } \right) = 8 \left( 1 + \frac { 2 } { 4 k ^ { 2 } - 1 } \right) > 8$; When $0 \leq k ^ { 2 } < \frac { 1 } { 4 }$, $S _ { \triangle O P Q } = 8 \left( \frac { 4 k ^ { 2 } + 1 } { 1 - 4 k ^ { 2 } } \right) = 8 \left( - 1 + \frac

22. As shown in the figure, in the quadrangular pyramid $P - A B C D$, given that $P A \perp$ plane $A B C D$, and the quadrilateral $A B C D$ is a right trapezoid, $\angle A B C = \angle B A D = \frac { \pi } { 2 } , P A = A D = 2 , A B = B C = 1$
(1) Find the cosine of the dihedral angle between plane $P A B$ and plane $P C D$;
(2) Point Q is a moving point on segment BP. When the angle between line CQ and DP is minimized, find the length of segment BQ. [Figure]

10. In a right triangular prism $A B C - A _ { 1 } B _ { 1 } C _ { 1 }$, $\angle A B C = 120 ^ { \circ } , A B = 2 , B C = C C _ { 1 } = 1$. The cosine of the angle between skew lines $A B _ { 1 }$ and $B C _ { 1 }$ is
A. $\frac { \sqrt { 3 } } { 2 }$
B. $\frac { \sqrt { 15 } } { 5 }$
C. $\frac { \sqrt { 10 } } { 5 }$
D. $\frac { \sqrt { 3 } } { 3 }$ [Figure]

In the rectangular prism $A B C D - A _ { 1 } B _ { 1 } C _ { 1 } D _ { 1 }$, $E$ is the midpoint of edge $C C _ { 1 }$. The tangent of the angle between skew lines $A E$ and $C D$ is
A. $\frac { \sqrt { 2 } } { 2 }$
B. $\frac { \sqrt { 3 } } { 2 }$
C. $\frac { \sqrt { 5 } } { 2 }$
D. $\frac { \sqrt { 7 } } { 2 }$

In rectangular prism $A B C D - A _ { 1 } B C _ { 1 } D _ { 1 }$, $A B = B C = 1 , A A _ { 1 } = \sqrt { 3 }$, the cosine of the angle between skew lines $A D _ { 1 }$ and $D B _ { 1 }$ is
A. $\frac { 1 } { 5 }$
B. $\frac { \sqrt { 5 } } { 6 }$
C. $\frac { \sqrt { 5 } } { 5 }$
D. $\frac { \sqrt { 2 } } { 2 }$

In rectangular parallelepiped $A B C D - A _ { 1 } B _ { 1 } C _ { 1 } D _ { 1 }$, $A B = B C = 2$, and the angle between $A C _ { 1 }$ and plane $B B _ { 1 } C _ { 1 } C$ is $30 ^ { \circ }$. Then the volume of the rectangular parallelepiped is
A. 8
B. $6 \sqrt { 2 }$
C. $8 \sqrt { 2 }$
D. $8 \sqrt { 3 }$

As shown in the figure, in the triangular pyramid $P - A B C$, $A B = B C = 2 \sqrt { 2 }$, $P A = P B = P C = A C = 4$, and $O$ is the midpoint of $A C$.
(1) Prove: $P O \perp$ plane $A B C$;
(2) If point $M$ is on edge $B C$ such that $M C = 2 M B$, find the distance from point $C$ to plane $P O M$.

(12 points)
As shown in the figure, in triangular pyramid $P - A B C$, $A B = B C = 2 \sqrt { 2 }$, $P A = P B = P C = A C = 4$, $O$ is the midpoint of $A C$.
(1) Prove that $P O \perp$ plane $A B C$;
(2) Point $M$ is on edge $B C$ such that the dihedral angle $M - P A - C$ is $30 ^ { \circ }$. Find the sine of the angle between $P C$ and plane $P A M$.

As shown in the figure, point $N$ is the center of square $ABCD$, $\triangle ECD$ is an equilateral triangle, plane $ECD \perp$ plane $ABCD$, and $M$ is the midpoint of segment $ED$. Then
A. $BM = EN$, and lines $BM$ and $EN$ are intersecting lines
B. $BM \neq EN$, and lines $BM$ and $EN$ are intersecting lines
C. $BM = EN$, and lines $BM$ and $EN$ are skew lines
D. $BM \neq EN$, and lines $BM$ and $EN$ are skew lines

8. As shown in the figure, point $N$ is the center of square $A B C D$ , $\triangle E C D$ is an equilateral triangle, plane $E C D \perp$ plane $A B C D$ , and $M$ is the midpoint of segment $E D$ . Then [Figure]
A. $B M = E N$ , and lines $B M$ and $E N$ are intersecting lines
B. $B M \neq E N$ , and lines $B M$ and $E N$ are intersecting lines
C. $B M = E N$ , and lines $B M$ and $E N$ are skew lines
D. $B M \neq E N$ , and lines $B M$ and $E N$ are skew lines

8. As shown in the figure, point $N$ is the center of square $A B C D$ , $\triangle E C D$ is an equilateral triangle, plane $E C D \perp$ plane $A B C D$ , and $M$ is the midpoint of segment $E D$ . Then [Figure]
A. $B M = E N$ , and lines $B M$ and $E N$ are intersecting lines
B. $B M \neq E N$ , and lines $B M$ and $E N$ are intersecting lines
C. $B M = E N$ , and lines $B M$ and $E N$ are skew lines
D. $B M \neq E N$ , and lines $B M$ and $E N$ are skew lines

As shown in the figure, in the quadrangular pyramid $P - A B C D$, $P A \perp$ plane $A B C D$, $A D \perp C D$, $A D \| B C$, $P A = A D = C D = 2$, $B C =$ 3. $E$ is the midpoint of $P D$, and point $F$ is on $P C$ such that $\frac { P F } { P C } = \frac { 1 } { 3 }$. (I) Prove that: $C D \perp$ plane $P A D$; (II) Find the cosine of the dihedral angle $F - A E - P$; (III) Let point $G$ be on $P B$ such that $\frac { P G } { P B } = \frac { 2 } { 3 }$. Determine whether line $A G$ lies in plane $A E F$, and explain the reason.

17. (12 points) As shown in the figure, the rectangular prism $A B C D - A _ { 1 } B _ { 1 } C _ { 1 } D _ { 1 }$ has a square base $A B C D$. Point $E$ is on edge $A A _ { 1 }$, and $B E \perp E C _ { 1 }$. [Figure]
(1) Prove: $B E \perp$ plane $E B _ { 1 } C _ { 1 }$;
(2) If $A E = A _ { 1 } E , A B = 3$, find the volume of the quadrangular pyramid $E - B B _ { 1 } C _ { 1 } C$.

17. (12 points) As shown in the figure, the rectangular prism $A B C D - A _ { 1 } B _ { 1 } C _ { 1 } D _ { 1 }$ has a square base $A B C D$. Point $E$ is on edge $A A _ { 1 }$, and $B E \perp E C _ { 1 }$. [Figure]
(1) Prove that $B E \perp$ plane $E B _ { 1 } C _ { 1 }$;
(2) If $A E = A _ { 1 } E$, find the sine of the dihedral angle $B - E C - C _ { 1 }$.