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]

As shown in the figure, in the quadrangular pyramid $P - A B C D$, $P A \perp$ plane $A B C D$, $A D \perp C D$, $A D \| B C$, $P A = A D = C D = 2$, $B C =$ 3. $E$ is the midpoint of $P D$, and point $F$ is on $P C$ such that $\frac { P F } { P C } = \frac { 1 } { 3 }$. (I) Prove that: $C D \perp$ plane $P A D$; (II) Find the cosine of the dihedral angle $F - A E - P$; (III) Let point $G$ be on $P B$ such that $\frac { P G } { P B } = \frac { 2 } { 3 }$. Determine whether line $A G$ lies in plane $A E F$, and explain the reason.

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.

17. (12 points) As shown in the figure, the rectangular prism $A B C D - A _ { 1 } B _ { 1 } C _ { 1 } D _ { 1 }$ has a square base $A B C D$. Point $E$ is on edge $A A _ { 1 }$, and $B E \perp E C _ { 1 }$. [Figure]
(1) Prove: $B E \perp$ plane $E B _ { 1 } C _ { 1 }$;
(2) If $A E = A _ { 1 } E , A B = 3$, find the volume of the quadrangular pyramid $E - B B _ { 1 } C _ { 1 } C$.

17. (12 points) As shown in the figure, the rectangular prism $A B C D - A _ { 1 } B _ { 1 } C _ { 1 } D _ { 1 }$ has a square base $A B C D$. Point $E$ is on edge $A A _ { 1 }$, and $B E \perp E C _ { 1 }$. [Figure]
(1) Prove that $B E \perp$ plane $E B _ { 1 } C _ { 1 }$;
(2) If $A E = A _ { 1 } E$, find the sine of the dihedral angle $B - E C - C _ { 1 }$.

19. (12 points) Figure 1 is a planar figure composed of rectangle $A D E B$ , right triangle $A B C$ , and rhombus $B F G C$ , where $A B = 1 , B E = B F = 2$ , $\angle F B C = 60 ^ { \circ }$ . Fold it along $A B$ and $B C$ so that $B E$ and $B F$ coincide, and connect $D G$ , as shown in Figure 2.
(1) Prove: In Figure 2, points $A , C , G , D$ are coplanar, and plane $A B C \perp$ plane $B C G E$ .
(2) Therefore, from the known condition we have $( x - 2 ) ^ { 2 } + ( y - 1 ) ^ { 2 } + ( z - a ) ^ { 2 } \geq \frac { ( 2 + a ) ^ { 2 } } { 3 }$ , equality holds if and only if $x = \frac { 4 - a } { 3 } , y = \frac { 1 - a } { 3 } , z = \frac { 2 a - 2 } { 3 }$ . Thus the minimum value of $( x - 2 ) ^ { 2 } + ( y - 1 ) ^ { 2 } + ( z - a ) ^ { 2 }$ is $\frac { ( 2 + a ) ^ { 2 } } { 3 }$ .
From the given condition we have $\frac { ( 2 + a ) ^ { 2 } } { 3 } \geq \frac { 1 } { 3 }$ , solving gives $a \leq - 3$ or $a \geq - 1$ .

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

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 $BCC_{1}B_{1}$ .

9. Given in three-dimensional space the points $A(3, 1, 0)$, $B(3, -1, 2)$, $C(1, 1, 2)$. After verifying that $ABC$ is an equilateral triangle and that it is contained in the plane $\alpha$ with equation $x + y + z - 4 = 0$, establish which are the points $P$ such that $ABCP$ is a regular tetrahedron.

3. Verify that the points $O ( 0,0,0 ) , A ( 1,4,8 ) , B ( - 6,0,12 )$ and $C ( - 7 , - 4,4 )$ are coplanar. Calculate the area and perimeter of the quadrilateral $O A B C$ and classify it.

Consider a pyramid $OPQRS$ located in the first octant $(x \geq 0, y \geq 0, z \geq 0)$ with $O$ as origin, and $OP$ and $OR$ along the $x$-axis and the $y$-axis, respectively. The base $OPQR$ of the pyramid is a square with $OP = 3$. The point $S$ is directly above the mid-point $T$ of diagonal $OQ$ such that $TS = 3$. Then
(A) the acute angle between $OQ$ and $OS$ is $\frac{\pi}{3}$
(B) the equation of the plane containing the triangle $OQS$ is $x - y = 0$
(C) the length of the perpendicular from $P$ to the plane containing the triangle $OQS$ is $\frac{3}{\sqrt{2}}$
(D) the perpendicular distance from $O$ to the straight line containing $RS$ is $\sqrt{\frac{15}{2}}$

Let $P _ { 1 } : 2 x + y - z = 3$ and $P _ { 2 } : x + 2 y + z = 2$ be two planes. Then, which of the following statement(s) is (are) TRUE?
(A) The line of intersection of $P _ { 1 }$ and $P _ { 2 }$ has direction ratios $1,2 , - 1$
(B) The line $$\frac { 3 x - 4 } { 9 } = \frac { 1 - 3 y } { 9 } = \frac { z } { 3 }$$ is perpendicular to the line of intersection of $P _ { 1 }$ and $P _ { 2 }$
(C) The acute angle between $P _ { 1 }$ and $P _ { 2 }$ is $60 ^ { \circ }$
(D) If $P _ { 3 }$ is the plane passing through the point $( 4,2 , - 2 )$ and perpendicular to the line of intersection of $P _ { 1 }$ and $P _ { 2 }$, then the distance of the point $( 2,1,1 )$ from the plane $P _ { 3 }$ is $\frac { 2 } { \sqrt { 3 } }$

Let a line passing through the point $(-1, 2, 3)$ intersect the lines $L_1: \frac{x-1}{3} = \frac{y-2}{2} = \frac{z+1}{-2}$ at $M(\alpha, \beta, \gamma)$ and $L_2: \frac{x+2}{-3} = \frac{y-2}{-2} = \frac{z-1}{4}$ at $N(a, b, c)$. Then the value of $\frac{(\alpha + \beta + \gamma)^2}{(a + b + c)^2}$ equals $\underline{\hspace{1cm}}$.

Given the lines $r _ { 1 } \equiv \left\{ \begin{array} { l } 6 x - y - z = 1 , \\ 2 x - y + z = 1 \end{array} \quad \right.$ and $r _ { 2 } \equiv \left\{ \begin{array} { l } 3 x - 5 y - 2 z = 3 , \\ 3 x + y + 4 z = 3 \end{array} \right.$ it is requested:\ a) (1 point) Study the relative position of $r _ { 1 }$ and $r _ { 2 }$.\ b) (1 point) Calculate the distance between the two lines.\ c) (1 point) Find the equation of the plane that contains $r _ { 1 }$ and the point $\mathrm { P } ( 1,2,3 )$.

Given the points $P ( 1 , - 2,1 ) , Q ( - 4,0,1 ) , R ( - 3,1,2 ) , S ( 0 , - 3,0 )$, it is requested:
a) (1 point) Find the equation of the plane containing $\mathrm { P } , \mathrm { Q }$ and R.
b) (1 point) Study the relative position of line $r$, which passes through points P and Q, and line $s$, which passes through R and S.
c) (1 point) Find the area of the triangle formed by points $\mathrm { P } , \mathrm { Q }$ and R.

Given the planes $\pi _ { 1 } \equiv 4 x + 6 y - 12 z + 1 = 0 , \pi _ { 2 } \equiv - 2 x - 3 y + 6 z - 5 = 0$, it is requested:
a) (1 point) Calculate the volume of a cube that has two of its faces in these planes.
b) (1.5 points) For the square with consecutive vertices ABCD , with $\mathrm { A } ( 2,1,3 )$ and $\mathrm { B } ( 1,2,3 )$, calculate the vertices C and D , knowing that C belongs to the planes $\pi _ { 2 } \mathrm { and } \pi _ { 3 } \equiv x - y + z = 2$.

Given the points $\mathrm { A } ( 1,1,1 ) , \mathrm { B } ( 1,3 , - 3 )$ and $\mathrm { C } ( - 3 , - 1,1 )$, it is requested:
a) (1 point) Determine the equation of the plane containing the three points.
b) ( 0.5 points) Obtain a point D (different from $\mathrm { A } , \mathrm { B }$ and C ) such that the vectors $\overrightarrow { A B } , \overrightarrow { A C } , \overrightarrow { A D }$ are linearly dependent.
c) (1 point) Find a point P on the OX axis, such that the volume of the tetrahedron with vertices A, B, C and P equals 1.

Given the point $P(3,3,0)$ and the line $r \equiv \frac{x-2}{-1} = \frac{y}{1} = \frac{z+1}{0}$, find:\ a) (0.75 points) Write the equation of the plane that contains point $P$ and line $r$.\ b) (1 point) Calculate the point symmetric to $P$ with respect to $r$.\ c) (0.75 points) Find two points $A$ and $B$ on $r$ such that triangle $ABP$ is right-angled, has area $\frac{3}{\sqrt{2}}$ and the right angle is at $A$.

For the parallelogram $ABCD$, the consecutive vertices $A(1,0,-1)$, $B(2,1,0)$ and $C(4,3,-2)$ are known. Find:\ a) (1 point) Calculate an equation of the line that passes through the midpoint of segment $AC$ and is perpendicular to segments $AC$ and $BC$.\ b) (1 point) Find the coordinates of vertex $D$ and the area of the resulting parallelogram.\ c) (0.5 points) Calculate the cosine of the angle formed by vectors $\overrightarrow{AB}$ and $\overrightarrow{AC}$.

Given the lines $r \equiv \left\{ \begin{array} { l } x - y = 2 \\ 3 x - z = - 1 \end{array} \right. , s \equiv \left\{ \begin{array} { l } x = - 1 + 2 \lambda \\ y = - 4 - \lambda \\ z = \lambda \end{array} \right.$ it is requested:\ a) (1 point) Calculate the relative position of lines $r$ and $s$.\ b) (0.5 points) Find the equation of the plane perpendicular to line $r$ and passing through point $P ( 2 , - 1,5 )$.\ c) (1 point) Find the equation of the plane parallel to line $r$ that contains line s.

With a laser device located at point $\mathrm { P } ( 1,1,1 )$ it has been possible to follow the trajectory of a particle that moves along the line with equations $r \equiv \left\{ \begin{array} { l } 2 x - y = 10 \\ x - z = - 90 \end{array} \right.$. a) ( 0.5 points) Calculate a direction vector of $r$ and the position of the particle when its trajectory intersects the plane $z = 0$. b) (1.25 points) Calculate the closest position of the particle to the laser device. c) ( 0.75 points) Determine the angle between the plane with equation $x + y = 2$ and the line $r$.

Let the plane $\pi \equiv x + y + z = 1$, the line $r _ { 1 } \equiv \left\{ \begin{array} { l } x = 1 + \lambda \\ y = 1 - \lambda \\ z = - 1 \end{array} , \lambda \in \mathbb { R } \right.$ and the point $P ( 0,1,0 )$. a) ( 0.5 points) Verify that the line $r _ { 1 }$ is contained in the plane $\pi$ and that the point P belongs to the same plane. b) ( 0.75 points) Find an equation of the line contained in the plane $\pi$ that passes through P and is perpendicular to $r _ { 1 }$. c) (1.25 points) Calculate an equation of the line, $r _ { 2 }$, that passes through P and is parallel to $r _ { 1 }$. Find the area of a square that has two of its sides on the lines $r _ { 1 }$ and $r _ { 2 }$.

Let the plane $\pi \equiv z = x$ and the points $\mathrm { A } ( 0 , - 1,0 )$ and $\mathrm { B } ( 0,1,0 )$ belonging to the plane $\pi$. a) ( 1.25 points) If points A and B are adjacent vertices of a square with vertices $\{ \mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D } \}$ that lies in the plane $\pi$, find the possible points C and D. b) ( 1.25 points) If points A and B are opposite vertices of a square that lies in the plane $\pi$, determine the other two vertices of it.