As shown in the figure on the right, in a cube ABCD-EFGH with edge length 3, there are three points $\mathrm { P } , \mathrm { Q } , \mathrm { R }$ on the three edges AD, BC, FG such that $\overline { \mathrm { DP } } = \overline { \mathrm { BQ } } = \overline { \mathrm { GR } } = 1$. The angle between plane PQR and plane CGHD is $\theta$. What is the value of $\cos \theta$? (where $0 < \theta < \frac { \pi } { 2 }$) [3 points]
(1) $\frac { \sqrt { 10 } } { 5 }$
(2) $\frac { \sqrt { 10 } } { 10 }$
(3) $\frac { \sqrt { 11 } } { 11 }$
(4) $\frac { 2 \sqrt { 11 } } { 11 }$
(5) $\frac { 3 \sqrt { 11 } } { 11 }$

In coordinate space, there is a sphere $S : x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 50$ and a point $\mathrm { P } ( 0,5,5 )$. For all circles $C$ satisfying the following conditions, find the maximum area of the orthogonal projection of $C$ onto the $xy$-plane, expressed as $\frac { q } { p } \pi$. Find the value of $p + q$. (Here, $p$ and $q$ are coprime natural numbers.) [4 points] (가) Circle $C$ is formed by the intersection of a plane passing through point P and the sphere $S$. (나) The radius of circle $C$ is 1.

In coordinate space, there are a point $\mathrm { A } ( 2,2,1 )$ and a plane $\alpha : x + 2 y + 2 z - 14 = 0$. When point P on plane $\alpha$ satisfies $\overline { \mathrm { AP } } \leq 3$, what is the area of the projection of the figure traced by point P onto the $xy$-plane? [4 points]
(1) $\frac { 14 } { 3 } \pi$
(2) $\frac { 13 } { 3 } \pi$
(3) $4 \pi$
(4) $\frac { 11 } { 3 } \pi$
(5) $\frac { 10 } { 3 } \pi$

There is a plane $\alpha$ in coordinate space. Let $\mathrm{A}'$ and $\mathrm{B}'$ be the orthogonal projections of two distinct points $\mathrm{A}$ and $\mathrm{B}$ (not on plane $\alpha$) onto plane $\alpha$, respectively. $$\overline{\mathrm{AB}} = \overline{\mathrm{A'B'}} = 6$$ Let $\mathrm{M}'$ be the orthogonal projection of the midpoint M of segment AB onto plane $\alpha$. A point P is chosen on plane $\alpha$ such that $$\overline{\mathrm{PM'}} \perp \overline{\mathrm{A'B'}}, \quad \overline{\mathrm{PM'}} = 6$$ When the area of the orthogonal projection of triangle $\mathrm{A'B'P}$ onto plane ABP is $\frac{9}{2}$, what is the length of segment PM? [3 points]
(1) 12
(2) 15
(3) 18
(4) 21
(5) 24

17. In the geometric solid ABCDE shown in the figure, quadrilateral ABCD is a rectangle, $AB \perp$ plane $BEC$, $BE \perp EC$, $AB = BE = EC = 2$, and $G$ and $F$ are the midpoints of segments $BE$ and $DC$ respectively.
(1) Prove that $GF \parallel$ plane $ADE$.
(2) Find the cosine of the acute dihedral angle between plane $AEF$ and plane $BEC$.

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 $\mathrm { A } _ { 1 } \mathrm { CD }$.

Let the points on the plane $P$ be equidistant from the points $( - 4,2,1 )$ and $( 2 , - 2,3 )$. Then the acute angle between the plane $P$ and the plane $2 x + y + 3 z = 1$ is
(1) $\frac { \pi } { 6 }$
(2) $\frac { \pi } { 4 }$
(3) $\frac { \pi } { 3 }$
(4) $\frac { 5 \pi } { 12 }$

Let $\theta$ be the angle between the planes $P_1: \vec{r} \cdot (\hat{i} + \hat{j} + 2\hat{k}) = 9$ and $P_2: \vec{r} \cdot (2\hat{i} - \hat{j} + \hat{k}) = 15$. Let $L$ be the line that meets $P_2$ at the point $(4,-2,5)$ and makes an angle $\theta$ with the normal of $P_2$. If $\alpha$ is the angle between $L$ and $P_2$, then $\tan^2\theta \cdot \cot^2\alpha$ is equal to $\underline{\hspace{1cm}}$.