Let $L$ be the line of intersection of the planes $x + y = 0$ and $y + z = 0$.
(a) Write the vector equation of $L$, i.e., find $(a, b, c)$ and $(p, q, r)$ such that $$L = \{(a, b, c) + \lambda(p, q, r) \mid \lambda \text{ is a real number.}\}$$ (b) Find the equation of a plane obtained by rotating $x + y = 0$ about $L$ by $45^\circ$.

For a regular hexahedron (cube) $\mathrm { ABCD } - \mathrm { EFGH }$, let $\theta$ be the angle between plane AFG and plane AGH. What is the value of $\cos ^ { 2 } \theta$? [3 points]
(1) $\frac { 1 } { 6 }$
(2) $\frac { 1 } { 5 }$
(3) $\frac { 1 } { 4 }$
(4) $\frac { 1 } { 3 }$
(5) $\frac { 1 } { 2 }$

In coordinate space, let $\theta$ be the acute angle between the plane $2 x + 2 y - z + 5 = 0$ and the $xy$-plane. What is the value of $\cos \theta$? [3 points]
(1) $\frac { 1 } { 12 }$
(2) $\frac { 1 } { 6 }$
(3) $\frac { 1 } { 4 }$
(4) $\frac { 1 } { 3 }$
(5) $\frac { 5 } { 12 }$

In the coordinate plane, consider two lines $$\frac { x + 1 } { 2 } = y - 3 , \quad x - 2 = \frac { y - 5 } { 3 }$$ If $\theta$ is the acute angle between these lines, what is the value of $\cos \theta$? [3 points]
(1) $\frac { 1 } { 2 }$
(2) $\frac { \sqrt { 5 } } { 4 }$
(3) $\frac { \sqrt { 6 } } { 4 }$
(4) $\frac { \sqrt { 7 } } { 4 }$
(5) $\frac { \sqrt { 2 } } { 2 }$

(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 $P A M$.

A square $ABCD$ with side length 1 is rotated around $BC$ to form a cylinder.
(1) Find the surface area of the cylinder;
(2) The square $ABCD$ is rotated counterclockwise by $\frac { \pi } { 2 }$ around $BC$ to position $A _ { 1 } B C D _ { 1 }$. Find the angle between $A D _ { 1 }$ and plane $ABCD$.

(12 points) As shown in the figure, in tetrahedron $ABCD$, $AD \perp CD, AD = CD, \angle ADB = \angle BDC$, and $E$ is the midpoint of $AC$.
(1) Prove: Plane $BED \perp$ plane $ACD$;
(2) Given $AB = BD = 2, \angle ACB = 60°$, point $F$ is on $BD$. When the area of $\triangle AFC$ is minimum, find the sine of the angle between $CF$ and plane $ABD$.

Given a quadrangular pyramid $P - ABCD$, where $AD \parallel BC$, $AB = BC = 1$, $AD = 3$, $DE = PE = 2$, $E$ is a point on $AD$, and $PE \perp AD$.
(1) If $F$ is the midpoint of $PE$, prove that $BF \parallel$ plane $PCD$.
(2) If $AB \perp$ plane $PED$, find the cosine of the dihedral angle between plane $PAB$ and plane $PCD$.

As shown in the figure, in planar quadrilateral $A B C D$, $A B = 8$, $C D = 3$, $A D = 5 \sqrt { 3 }$, $\angle A D C = 90 ^ { \circ }$, $\angle B A D = 30 ^ { \circ }$. Points $E$ and $F$ satisfy $\overrightarrow { A E } = \frac { 2 } { 5 } \overrightarrow { A D }$ and $\overrightarrow { A F } = \frac { 1 } { 2 } \overrightarrow { A B }$. Fold $\triangle A E F$ along $E F$ to $\triangle P E F$ such that $P C = 4 \sqrt { 3 }$.
(1) Prove: $E F \perp P D$;
(2) Find the sine of the dihedral angle between plane $P C D$ and plane $P B F$.

(15 points) As shown in the figure, in the quadrangular pyramid $P - A B C D$ , $P A \perp$ base $A B C D , P A = A C = 2$ , $B C = 1 , A B = \sqrt { 3 }$ .
(1) If $A D \perp P B$ , prove that $A D \|$ plane $P B C$ ;
(2) If $A D \perp D C$ , and the sine of the dihedral angle $A - C P - D$ is $\frac { \sqrt { 42 } } { 7 }$ , find $A D$ .

In the quadrangular pyramid $P - ABCD$ shown in the figure, $PA \perp$ plane $ABCD$, $BC \parallel AD$, $AB \perp AD$.
(1) Prove that plane $PAB \perp$ plane $PAD$.
(2) If $PA = AB = \sqrt{2}$, $AD = \sqrt{3} + 1$, $BC = 2$, and $P, B, C, D$ lie on the same sphere with center $O$.
(i) Prove that $O$ lies on plane $ABCD$.
(ii) Find the cosine of the angle between line $AC$ and line $PO$.

The angle between the lines whose direction cosines satisfy the equations $l + m + n = 0$ and $l ^ { 2 } = m ^ { 2 } + n ^ { 2 }$ is
(1) $\frac { \pi } { 6 }$
(2) $\frac { \pi } { 2 }$
(3) $\frac { \pi } { 3 }$
(4) $\frac { \pi } { 4 }$

A line in the 3-dimensional space makes an angle $\theta \left( 0 < \theta \leq \frac { \pi } { 2 } \right)$ with both the $X$ and $Y$-axes. Then, the set of all values of $\theta$ is in the interval:
(1) $\left( \frac { \pi } { 3 } , \frac { \pi } { 2 } \right]$
(2) $\left( 0 , \frac { \pi } { 4 } \right]$
(3) $\left[ \frac { \pi } { 4 } , \frac { \pi } { 2 } \right]$
(4) $\left[ \frac { \pi } { 6 } , \frac { \pi } { 3 } \right]$

If the angle between the line $2 ( x + 1 ) = y = z + 4$ and the plane $2 x - y + \sqrt { \lambda } z + 4 = 0$ is $\frac { \pi } { 6 }$, then the value of $\lambda$ is
(1) $\frac { 45 } { 7 }$
(2) $\frac { 135 } { 11 }$
(3) $\frac { 135 } { 7 }$
(4) $\frac { 45 } { 11 }$

An angle between the plane $x + y + z = 5$ and the line of intersection of the planes, $3 x + 4 y + z - 1 = 0$ and $5 x + 8 y + 2 z + 14 = 0$ is
(1) $\cos ^ { - 1 } \left( \sqrt { \frac { 3 } { 17 } } \right)$
(2) $\cos ^ { - 1 } \left( \frac { 3 } { \sqrt { 17 } } \right)$
(3) $\sin ^ { - 1 } \left( \frac { 3 } { \sqrt { 17 } } \right)$
(4) $\sin ^ { - 1 } \left( \sqrt { \frac { 3 } { 17 } } \right)$

An angle between the plane, $x + y + z = 5$ and the line of intersection of the planes, $3 x + 4 y + z - 1 = 0$ and $5 x + 8 y + 2 z + 14 = 0$, is
(1) $\cos ^ { - 1 } \left( \frac { 3 } { \sqrt { 17 } } \right)$
(2) $\cos ^ { - 1 } \left( \sqrt { \frac { 3 } { 17 } } \right)$
(3) $\sin ^ { - 1 } \left( \frac { 3 } { \sqrt { 17 } } \right)$
(4) $\sin ^ { - 1 } \left( \sqrt { \frac { 3 } { 17 } } \right)$

Let $\alpha$ be the angle between the lines whose direction cosines satisfy the equations $l + m - n = 0$ and $l ^ { 2 } + m ^ { 2 } - n ^ { 2 } = 0$. Then the value of $\sin ^ { 4 } \alpha + \cos ^ { 4 } \alpha$ is :
(1) $\frac { 5 } { 8 }$
(2) $\frac { 1 } { 2 }$
(3) $\frac { 3 } { 8 }$
(4) $\frac { 3 } { 4 }$

The line of shortest distance between the lines $\frac { x - 2 } { 0 } = \frac { y - 1 } { 1 } = \frac { z } { 1 }$ and $\frac { x - 3 } { 2 } = \frac { y - 5 } { 2 } = \frac { z - 1 } { 1 }$ makes an angle of $\sin ^ { - 1 } \sqrt { \frac { 2 } { 27 } }$ with the plane $P : ax - y - z = 0$, $a > 0$. If the image of the point $(1,1,-5)$ in the plane $P$ is $(\alpha, \beta, \gamma)$, then $\alpha + \beta - \gamma$ is equal to $\_\_\_\_$.

If the line of intersection of the planes $a x + b y = 3$ and $a x + b y + c z = 0 , a > 0$ makes an angle $30 ^ { \circ }$ with the plane $y - z + 2 = 0$, then the direction cosines of the line are
(1) $\frac { 1 } { \sqrt { 2 } } , \frac { 1 } { \sqrt { 2 } } , 0$
(2) $\frac { 1 } { \sqrt { 2 } } , \frac { - 1 } { \sqrt { 2 } } , 0$
(3) $\frac { 1 } { \sqrt { 5 } } , - \frac { 2 } { \sqrt { 5 } } , 0$
(4) $\frac { 1 } { 2 } , - \frac { \sqrt { 3 } } { 2 } , 0$

In coordinate space, consider three planes $E_{1}: x + y + z = 7$, $E_{2}: x - y + z = 3$, $E_{3}: x - y - z = -5$. Let $L_{3}$ be the line of intersection of $E_{1}$ and $E_{2}$; $L_{1}$ be the line of intersection of $E_{2}$ and $E_{3}$; $L_{2}$ be the line of intersection of $E_{3}$ and $E_{1}$. Explain that among $L_{1}, L_{2}, L_{3}$, the acute angle between any two lines is $60^{\circ}$. (Note: Let the acute angle between $L_{1}$ and $L_{2}$ be $\alpha$, the acute angle between $L_{2}$ and $L_{3}$ be $\beta$, and the acute angle between $L_{3}$ and $L_{1}$ be $\gamma$)

In coordinate space, let $O$ be the origin and $E$ be the plane $x - z = 4$.
It is known that there is a point $P(a, b, c)$ in space such that the angle $\theta$ between vector $\overrightarrow{OP}$ and vector $(1, 0, 0)$ satisfies $\theta \leq \frac{\pi}{6}$. Show that the real numbers $a, b, c$ satisfy the inequality $a^{2} \geq 3\left(b^{2} + c^{2}\right)$. (Non-multiple choice question, 4 points)