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) $\frac { \sqrt { 2 } } { 2 }$

As shown in the figure, there is a cube $\mathrm { ABCD } - \mathrm { EFGH }$ with edge length 4. Let M be the midpoint of segment AD. What is the area of triangle MEG? [3 points]
(1) $\frac { 21 } { 2 }$
(2) 11
(3) $\frac { 23 } { 2 }$
(4) 12
(5) $\frac { 25 } { 2 }$

In the coordinate plane, for a parallelogram OACB with $\overline { \mathrm { OA } } = \sqrt { 2 } , \overline { \mathrm { OB } } = 2 \sqrt { 2 }$ and $\cos ( \angle \mathrm { AOB } ) = \frac { 1 } { 4 }$, point P satisfies the following conditions. (가) $\overrightarrow { \mathrm { OP } } = s \overrightarrow { \mathrm { OA } } + t \overrightarrow { \mathrm { OB } } ( 0 \leq s \leq 1, 0 \leq t \leq 1 )$ (나) $\overrightarrow { \mathrm { OP } } \cdot \overrightarrow { \mathrm { OB } } + \overrightarrow { \mathrm { BP } } \cdot \overrightarrow { \mathrm { BC } } = 2$
For a point X moving on a circle centered at O and passing through point A, let $M$ and $m$ be the maximum and minimum values of $| 3 \overrightarrow { \mathrm { OP } } - \overrightarrow { \mathrm { OX } } |$ respectively. When $M \times m = a \sqrt { 6 } + b$, find the value of $a ^ { 2 } + b ^ { 2 }$. (Given that $a$ and $b$ are rational numbers.) [4 points]

As shown in the figure, for a tetrahedron ABCD with $\overline{\mathrm{AB}} = 6$, $\overline{\mathrm{BC}} = 4\sqrt{5}$, let M be the midpoint of segment BC. Triangle AMD is equilateral and line BC is perpendicular to plane AMD. Find the area of the orthogonal projection of the circle inscribed in triangle ACD onto plane BCD. [3 points]
(1) $\frac{\sqrt{10}}{4}\pi$
(2) $\frac{\sqrt{10}}{6}\pi$
(3) $\frac{\sqrt{10}}{8}\pi$
(4) $\frac{\sqrt{10}}{10}\pi$
(5) $\frac{\sqrt{10}}{12}\pi$

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

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 $ABCD$.

As shown in the figure, $D$ is the apex of the cone, $O$ is the center of the base of the cone, $\triangle A B C$ is an equilateral triangle inscribed in the base, and $P$ is a point on $D O$ with $\angle A P C = 90 ^ { \circ }$ .
(1) Prove that plane $P A B \perp$ plane $P A C$ ;
(2) Given $D O = \sqrt { 2 }$ and the lateral surface area of the cone is $\sqrt { 3 } \pi$ , find the volume of the triangular pyramid $P - A B C$ .

As shown in the figure, in the rectangular prism $A B C D - A _ { 1 } B _ { 1 } C _ { 1 } D _ { 1 }$, points $E , F$ are on edges $D D _ { 1 } , B B _ { 1 }$ respectively, with $2 D E = E D _ { 1 } , B F = 2 F B _ { 1 }$. Prove:
(1) When $A B = B C$, $E F \perp A C$;
(2) Point $C _ { 1 }$ lies in plane $A E F$.

In the cube $ABCD-A_1B_1C_1D_1$, $E, F$ are the midpoints of $AB, BC$ respectively. Then
A. Plane $B_1EF \perp$ plane $BDD_1$
B. Plane $B_1EF \perp$ plane $A_1BD$
C. Plane $B_1EF \parallel$ plane $A_1AC$
D. Plane $B_1EF \parallel$ plane $A_1C_1D$

In the cube $A B C D - A _ { 1 } B _ { 1 } C _ { 1 } D _ { 1 }$ , $E , F$ are the midpoints of $A B , B C$ respectively, then
A. Plane $B _ { 1 } E F \perp$ plane $B D D _ { 1 }$
B. Plane $B _ { 1 } E F \perp$ plane $A _ { 1 } B D$
C. Plane $B _ { 1 } E F \parallel$ plane $A _ { 1 } A C$
D. Plane $B _ { 1 } E F \parallel$ plane $A _ { 1 } C _ { 1 } D$

(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 $ABD$.

Xiaoming designed a closed packaging box as shown in the figure: the bottom face $ABCD$ is a square with side length 2. Triangles $\triangle E A B , \triangle F B C , \triangle G C D , \triangle H D A$ are all equilateral triangles, and the planes containing them are perpendicular to the bottom face.
(1) Prove that $E F \parallel$ plane $A B C D$ ;
(2) Find the volume of the packaging box (disregarding the thickness of the material).

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 right triangular prism $ABC - A_1B_1C_1$, let $D$ be the midpoint of $BC$. Then
A. $AD \perp A_1C$
B. $BC \perp$ plane $AA_1D$
C. $AD \parallel A_1B_1$
D. $CC_1 \parallel$ plane $AA_1D$

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 $PO$.

As shown in the figure, in quadrilateral $ABCD$, $AB \parallel CD$, $\angle DAB = 90°$, $F$ is the midpoint of $CD$, point $E$ is on $AB$, $EF \parallel AD$, $AB = 3AD$, $CD = 2AD$. Fold quadrilateral $EFDA$ along $EF$ to quadrilateral $EFD'A'$ such that the dihedral angle between plane $EFD'A'$ and plane $EFCB$ is $60°$.
(1) Prove: $A'B \parallel$ plane $CD'F$;
(2) Find the sine of the dihedral angle between plane $BCD'$ and plane $EFD'A'$.

(15 points) In the quadrangular pyramid $P - ABCD$, $PA \perp$ 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 $PO$.

Consider three planes $$\begin{aligned} & P _ { 1 } : x - y + z = 1 \\ & P _ { 2 } : x + y - z = - 1 \\ & P _ { 3 } : x - 3 y + 3 z = 2 . \end{aligned}$$ Let $L _ { 1 } , L _ { 2 } , L _ { 3 }$ be the lines of intersection of the planes $P _ { 2 }$ and $P _ { 3 } , P _ { 3 }$ and $P _ { 1 }$, and $P _ { 1 }$ and $P _ { 2 }$, respectively.
STATEMENT-1: At least two of the lines $L _ { 1 } , L _ { 2 }$ and $L _ { 3 }$ are non-parallel. and STATEMENT-2 : The three planes do not have a common point.
(A) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1
(B) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for STATEMENT-1
(C) STATEMENT-1 is True, STATEMENT-2 is False
(D) STATEMENT-1 is False, STATEMENT-2 is True

Consider the lines
$$\begin{aligned} & L _ { 1 } : \frac { x + 1 } { 3 } = \frac { y + 2 } { 1 } = \frac { z + 1 } { 2 } \\ & L _ { 2 } : \frac { x - 2 } { 1 } = \frac { y + 2 } { 2 } = \frac { z - 3 } { 3 } \end{aligned}$$
The distance of the point $( 1,1,1 )$ from the plane passing through the point $( - 1 , - 2 , - 1 )$ and whose normal is perpendicular to both the lines $L _ { 1 }$ and $L _ { 2 }$ is
(A) $\frac { 2 } { \sqrt { 75 } }$
(B) $\frac { 7 } { \sqrt { 75 } }$
(C) $\frac { 13 } { \sqrt { 75 } }$
(D) $\frac { 23 } { \sqrt { 75 } }$

Let $P ( 3,2,6 )$ be a point in space and $Q$ be a point on the line
$$\vec { r } = ( \hat { i } - \hat { j } + 2 \hat { k } ) + \mu ( - 3 \hat { i } + \hat { j } + 5 \hat { k } )$$
Then the value of $\mu$ for which the vector $\overrightarrow { P Q }$ is parallel to the plane $x - 4 y + 3 z = 1$ is
(A) $\frac { 1 } { 4 }$
(B) $- \frac { 1 } { 4 }$
(C) $\frac { 1 } { 8 }$
(D) $- \frac { 1 } { 8 }$

If the distance of the point $\mathrm { P } ( 1 , - 2,1 )$ from the plane $\mathrm { x } + 2 \mathrm { y } - 2 z = \alpha$, where $\alpha > 0$, is 5 , then the foot of the perpendicular from $P$ to the plane is
A) $\left( \frac { 8 } { 3 } , \frac { 4 } { 3 } , - \frac { 7 } { 3 } \right)$
B) $\left( \frac { 4 } { 3 } , - \frac { 4 } { 3 } , \frac { 1 } { 3 } \right)$
C) $\left( \frac { 1 } { 3 } , \frac { 2 } { 3 } , \frac { 10 } { 3 } \right)$
D) $\left( \frac { 2 } { 3 } , - \frac { 1 } { 3 } , \frac { 5 } { 2 } \right)$