18. (12 points) As shown in the figure, the right prism $A B C D - A _ { 1 } B _ { 1 } C _ { 1 } D _ { 1 }$ has a rhombus base, with $A A _ { 1 } = 4 , A B = 2 , \angle B A D = 60 ^ { \circ }$ . Let $E , M , N$ be the midpoints of $B C$ , $B B _ { 1 Therefore $f(x)$ has a unique zero point on $\left[\frac{\pi}{2}, \pi\right]$.
(iv) When $x \in (\pi, +\infty)$, $\ln(x+1) > 1$, so $f(x) < 0$, thus $f(x)$ has no zero points on $(\pi, +\infty)$.
In conclusion, $f(x)$ has exactly 2 zero points.

In a cube $A B C D - A _ { 1 } B _ { 1 } C _ { 1 } D _ { 1 }$ with edge length 10, $P$ is a point on the left face $A D D _ { 1 } A _ { 1 }$. Given that the distance from point $P$ to $A _ { 1 } D _ { 1 }$ is 3 and the distance from point $P$ to $A A _ { 1 }$ is 2, a line through point $P$ parallel to $A _ { 1 } C$ intersects the cube at points $P$ and $Q$. On which face of the cube is point $Q$ located? ( )
A. $A A _ { 1 } B _ { 1 } B$
B. $B B _ { 1 } C _ { 1 } C$
C. $C C _ { 1 } D _ { 1 } D$
D. $A B C D$

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 $A E F$.

19. In a right triangular prism $A B C - A _ { 1 } B _ { 1 } C _ { 1 }$, the lateral face $A A _ { 1 } B _ { 1 } B$ is a square. $A B = B C = 2$. Let $E , F$ be the midpoints of $A C$ and $C C _ { 1 }$ respectively, and $B F \perp A _ { 1 } B _ { 1 }$.
(1) Find the volume of the triangular pyramid $F - E B C$;
(2) Let $D$ be a point on edge $A _ { 1 } B _ { 1 }$. Prove that $B F \perp D E$. [Figure]

19. In the quadrangular pyramid $Q - A B C D$ , the base $A B C D$ is a square with $A D = 2$ , $Q D = Q A = \sqrt { 5 }$ , $Q C = 3$ . [Figure]
(1) Prove: plane $Q A D \perp$ plane $A B C D$ ;
(2) Find the cosine of the dihedral angle $B - Q D - A$ . Answer: (1) See proof below; (2) $\frac { 2 } { 3 }$ .

[Solution]

[Analysis] (1) Let $O$ be the midpoint of $A D$ , and connect $Q O$ and $C O$ . We can prove that $Q O \perp$ plane $A B C D$ , thus obtaining plane $Q A D \perp$ plane $A B C D$ .
(2) In plane $A B C D$ , through $O$ draw $O T \parallel C D$ , intersecting $B C$ at $T$ . Then $O T \perp A D$ . Establish a coordinate system as shown in the figure. After finding the normal vectors of planes $Q A D$ and $B Q D$ , we can find the cosine of the dihedral angle.

[Detailed Solution]

[Figure]
(1) Let $O$ be the midpoint of $A D$ , and connect $Q O$ and $C O$ . Since $Q A = Q D$ and $O A = O D$ , we have $Q O \perp A D$ . Since $A D = 2$ and $Q A = \sqrt { 5 }$ , we have $Q O = \sqrt { 5 - 1 } = 2$ . In square $A B C D$ , since $A D = 2$ , we have $D O = 1$ , thus $C O = \sqrt { 1 + 4 } = \sqrt { 5 }$ . Since $Q C = 3$ , we have $Q C ^ { 2 } = 9 = 4 + 5 = Q O ^ { 2 } + O C ^ { 2 }$ , so $\triangle Q O C$ is a right triangle with $Q O \perp O C$ . Since $O C \cap A D = O$ , we have $Q O \perp$ plane $A B C D$ . Since $Q O \subset$ plane $Q A D$ , we have plane $Q A D \perp$ plane $A B C D$ .
(2) In plane $A B C D$ , through $O$ draw $O T \parallel C D$ , intersecting $B C$ at $T$ . Then $O T \perp A D$ . Combined with $Q O \perp$ plane $A B C D$ from part (1), we can establish a coordinate system as shown in the figure. [Figure]
Then $D ( 0,1,0 ) , Q ( 0,0,2 ) , B ( 2 , -1,0 )$ , so $\overrightarrow { B Q } = ( -2,1,2 ) , \overrightarrow { B D } = ( -2,2,0 )$ . Let the normal vector of plane $Q B D$ be $\vec { n } = ( x , y , z )$ . Then $\left\{ \begin{array} { l } \vec { n } \cdot \overrightarrow { B Q } = 0 \\ \vec { n } \cdot \overrightarrow { B D } = 0 \end{array} \right.$ , i.e., $\left\{ \begin{array} { l } - 2 x + y + 2 z = 0 \\ - 2 x + 2 y = 0 \end{array} \right.$ . Taking $x = 1$ , we get $y = 1 , z = \frac { 1 } { 2 }$ , Thus $\vec { n } = \left( 1,1 , \frac { 1 } { 2 } \right)$ . The normal vector of plane $Q A D$ is $\vec { m } = ( 1,0,0 )$ . Therefore $\cos \langle \vec { m } , \vec { n } \rangle = \frac { |\vec{m} \cdot \vec{n}| } { |\vec{m}| \cdot |\vec{n}| } = \frac

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 $A _ { 1 } C _ { 1 } D$

9. Given a cube $A B C D - A _ { 1 } B _ { 1 } C _ { 1 } D _ { 1 }$, then
A. The angle between lines $B C _ { 1 }$ and $D A _ { 1 }$ is $90 ^ { \circ }$
B. The angle between lines $B C _ { 1 }$ and $C A _ { 1 }$ is $90 ^ { \circ }$
C. The angle between line $B C _ { 1 }$ and plane $B B _ { 1 } D _ { 1 } D$ is $45 ^ { \circ }$
D. The angle between line $B C _ { 1 }$ and plane $A B C D$ is $45 ^ { \circ }$

In the pyramid $P - ABCD$, $PD \perp$ base $ABCD$, $CD \parallel AB$, $AD = DC = CB = 1$, $AB = 2$.
(1) Prove that $BD \perp PA$;
(2) Find the sine of the angle between $PD$ and plane $PAB$.

(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 $ABD$.

19. (12 points) As shown in the figure, a right triangular prism $A B C - A _ { 1 } B _ { 1 } C _ { 1 }$ has volume 4, and the area of $\triangle A _ { 1 } B C$ is $2 \sqrt { 2 }$.
(1) Find the distance from $A$ to plane $A _ { 1 } B C$;
(2) Let $D$ be the midpoint of $A _ { 1 } C$, with $A A _ { 1 } = A B$ and plane $A _ { 1 } B C \perp$ plane [Figure] $A B B _ { 1 } A _ { 1 }$. Find the sine of the dihedral angle $A - B D - C$.

In the triangular prism $ABC - A_{1}B_{1}C_{1}$ , $AA_{1} = 2$ , $A_{1}C \perp$ base $ABC$ , $\angle ACB = 90^{\circ}$ , the distance from $A_{1}$ to plane $BCC_{1}B_{1}$ is 1 .
(1) Prove: $AC = A_{1}C$ ;
(2) If the distance between lines $AA_{1}$ and $BB_{1}$ is 2 , find the sine of the angle between $AB_{1}$ and plane $BCC_{1}B_{1}$ .

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 $PO$.

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 $PO$.

On the basis of the model, calculate the total length of the borehole rounded to the nearest metre.

At the transition between the two sections of the borehole, the drilling direction must be changed by the angle that is described in the model by the angle of intersection of the two lines $A P$ and $P Q$. Determine the size of this angle.

Determine an equation of the plane $E$ in normal form. (for verification: $E : 4 x _ { 1 } + 4 x _ { 2 } - 10 x _ { 3 } - 43 = 0$ )

The borehole is extended in a straight line and leaves the water-bearing rock layer at a depth of 3600 m below the Earth's surface. The exit point is described in the model as a point $R$ on the line $P Q$. Determine the coordinates of $R$ and calculate the thickness of the water-bearing rock layer rounded to the nearest metre. (for verification: $x _ { 1 }$ and $x _ { 2 }$ coordinate of $R : 1.04$ )

Show by calculation that the second borehole reaches the water-bearing rock layer in the model at the point $T ( t | - t | - 4,3 )$, and explain how the length of the second borehole to the water-bearing rock layer is influenced by the location of the associated drilling site.

The points $A _ { 1 } ( 0 | 0 | 0 ) , A _ { 2 } ( 20 | 0 | 0 ) , A _ { 3 }$ and $A _ { 4 } ( 0 | 10 | 0 )$ represent the vertices of the base of the multipurpose hall in the model, and the points $B _ { 1 } , B _ { 2 } , B _ { 3 }$ and $B _ { 4 }$ represent the vertices of the roof surface. The side wall that lies in the $x _ { 1 } x _ { 3 }$-plane in the model is 6 m high, and the opposite wall is only 4 m high.
Give the coordinates of the points $B _ { 2 } , B _ { 3 }$ and $B _ { 4 }$ and confirm that these points lie in the plane $E : x _ { 2 } + 5 x _ { 3 } - 30 = 0$.

Calculate the angle of inclination of the roof surface with respect to the horizontal.