As shown in the figure, there is a rectangular piece of paper ABCD with $\overline { \mathrm { AB } } = 9$ and $\overline { \mathrm { AD } } = 3$. Using the line connecting point E on segment AB and point F on segment DC as the fold line, the paper is folded so that the orthogonal projection of point B onto the plane AEFD is point D. When $\overline { \mathrm { AE } } = 3$, the angle between the two planes AEFD and EFCB is $\theta$. Find the value of $60 \cos \theta$. (Given that $0 < \theta < \frac { \pi } { 2 }$ and the thickness of the paper is negligible.) [4 points]
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 $\mathrm { A } _ { 1 } \mathrm { CD }$.
In a model for a coastal section by the sea, the $x _ { 1 } x _ { 2 }$-plane describes the horizontal water surface and the line $g$ describes the shoreline. The plane $E$ represents the sea floor in the considered section. A buoy floats on the water surface at the location corresponding to the coordinate origin $O$. One unit of length corresponds to one meter in reality. Determine the angle at which the sea floor slopes down relative to the water surface.
Determine the change in height of the structure caused by the change in construction plan, in meters. Justify that in the lower part of the structure, the angle of inclination of the lateral faces with respect to the ground is more than $9 ^ { \circ }$ greater than in the upper part of the structure.
Using the result from task $a$, give the width of the gate to the nearest meter. Justify using the statement from task $d$ that the straight line of travel of the skier is inclined at less than $30 ^ { \circ }$ to the horizontal.
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 $\underline{\hspace{1cm}}$.