For two points $\mathrm { A } ( 1,6,4 ) , \mathrm { B } ( a , 2 , - 4 )$ in coordinate space, the point that divides segment AB internally in the ratio $1 : 3$ has coordinates $( 2,5,2 )$. What is the value of $a$? [2 points]
(1) 1
(2) 3
(3) 5
(4) 7
(5) 9

For two points $\mathrm { A } ( 2 , a , - 2 ) , \mathrm { B } ( 5 , - 2,1 )$ in coordinate space, when the point that divides segment AB internally in the ratio $2 : 1$ lies on the $x$-axis, what is the value of $a$? [2 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

For two points $\mathrm { A } ( 2,0,1 ) , \mathrm { B } ( 3,2,0 )$ in coordinate space, if the coordinates of a point on the $y$-axis that is equidistant from both points is $( 0 , a , 0 )$, what is the value of $a$? [2 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

In coordinate space, let P be the point obtained by reflecting point $\mathrm { A } ( 2,1,3 )$ across the xy-plane, and let Q be the point obtained by reflecting point A across the yz-plane. What is the length of segment PQ? [2 points]
(1) $5 \sqrt { 2 }$
(2) $2 \sqrt { 13 }$
(3) $3 \sqrt { 6 }$
(4) $2 \sqrt { 14 }$
(5) $2 \sqrt { 15 }$

In coordinate space, there is a sphere $$S : ( x - 2 ) ^ { 2 } + ( y - \sqrt { 5 } ) ^ { 2 } + ( z - 5 ) ^ { 2 } = 25$$ with center $\mathrm { C } ( 2 , \sqrt { 5 } , 5 )$ passing through point $\mathrm { P } ( 0,0,1 )$. For a point Q moving on the circle formed by the intersection of sphere $S$ and plane OPC, and a point R moving on sphere $S$, let $\mathrm { Q } _ { 1 }$ and $\mathrm { R } _ { 1 }$ be the orthogonal projections of points $\mathrm { Q }$ and $\mathrm { R }$ onto the xy-plane respectively.
For two points $\mathrm { Q } , \mathrm { R }$ that maximize the area of triangle $\mathrm { OQ } _ { 1 } \mathrm { R } _ { 1 }$, the area of the orthogonal projection of triangle $\mathrm { OQ } _ { 1 } \mathrm { R } _ { 1 }$ onto plane PQR is $\frac { q } { p } \sqrt { 6 }$. Find the value of $p + q$. [4 points]

For two points $\mathrm{A}(a, -2, 6)$ and $\mathrm{B}(9, 2, b)$ in coordinate space, the midpoint of segment AB has coordinates $(4, 0, 7)$. What is the value of $a + b$? [2 points]
(1) 1
(2) 3
(3) 5
(4) 7
(5) 9

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

In coordinate space, there is a right triangle ABC with $\overline{\mathrm{AB}} = 8$, $\overline{\mathrm{BC}} = 6$, $\angle\mathrm{ABC} = \frac{\pi}{2}$ and a sphere $S$ with diameter AC. Let $O$ be the circle formed by the intersection of sphere $S$ with the plane that contains line AB and is perpendicular to plane ABC. Let P and Q be two distinct points on circle $O$ such that the distance from each to line AC is 4. Find the length of segment PQ. [4 points]
(1) $\sqrt{43}$
(2) $\sqrt{47}$
(3) $\sqrt{51}$
(4) $\sqrt{55}$
(5) $\sqrt{59}$

13. As shown in the figure, in the triangular pyramid $A - B C D$, $AB = AC = BD = CD = 3$ , $AD = BC = 2$ , and $M , N$ are the midpoints of $AD , BC$ respectively. Then the cosine of the angle between the skew lines $AN$ and $CM$ is $\_\_\_\_$ .

14. As shown in the figure, quadrilaterals $ABCD$ and $ADPQ$ are both squares, and the planes they lie in are mutually perpendicular. A moving point $M$ is on segment $PQ$. $\mathrm { E }$ and $\mathrm { F }$ are the midpoints of $\mathrm { AB }$ and $\mathrm { BC }$ respectively. Let the angle between skew lines $EM$ and $AF$ be $\theta$, then the maximum value of $\cos \theta$ is $\_\_\_\_$. [Figure]

16. As shown in the figure, in the right triangular prism $A B C - A _ { 1 } B _ { 1 } C _ { 1 }$, given $A C \perp B C$. Let D be the midpoint of $A B _ { 1 }$, $B _ { 1 } C \cap B C _ { 1 } = E$ . Prove: (1) $D E \parallel$ plane $A A _ { 1 } C C _ { 1 }$
(2) $B C _ { 1 } \perp A B _ { 1 }$ [Figure]

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 $BEC$.

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]

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. (This question is worth 12 points)

A net of a cube and a schematic diagram of the cube are shown in the figure. (1) Mark the letters $F , G , H$ at the corresponding vertices of the cube (no explanation needed); (2) Determine the positional relationship between plane $B E G$ and plane $A C H$, and prove your conclusion; (3) Prove: line $D F \perp$ plane $B E G$. [Figure] [Figure]

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]

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 $BC$.

20. In ``The Nine Chapters on the Mathematical Art,'' a quadrangular pyramid with a rectangular base and one lateral edge perpendicular to the base is called a ``yang ma'', and a tetrahedron with all four faces being right triangles is called a ``bie nao''. In the yang ma $\mathrm { P } - \mathrm { ABCD }$ shown in the figure, the lateral edge $\mathrm { PD } \perp$ base ABCD, and $\mathrm { PD } = \mathrm { CD }$. Point E is the midpoint of PC. Connect $\mathrm { DE } , \mathrm { BD } , \mathrm { BE }$.
[Figure]
Figure for Question 20
(I) Prove that $\mathrm { DE } \perp$ plane PBC. Determine whether the tetrahedron EBCD is a ``bie nao''. If yes, write out the right angle of each face (only conclusions are needed); if no, please explain the reason; (II) Let the volume of the yang ma $\mathr

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. $\frac { \sqrt { 2 } } { 2 }$