Exercise 2
Consider the cube ABCDEFGH. We place the point M such that $\overrightarrow{\mathrm{BM}} = \overrightarrow{\mathrm{AB}}$.
Part A

1. Show that the lines (FG) and (FM) are perpendicular.
2. Show that the points A, M, G and H are coplanar.

Part B
We place ourselves in the orthonormal coordinate system $(A;\overrightarrow{AB},\overrightarrow{AD},\overrightarrow{AE})$.

1. Determine the coordinates of the vectors $\overrightarrow{\mathrm{GM}}$ and $\overrightarrow{\mathrm{AH}}$ and show that they are not collinear.
2. 1. [a.] Justify that a parametric representation of the line (GM) is: $$\left\{\begin{aligned} x &= 1+t \\ y &= 1-t \quad \text{with } t \in \mathbb{R}. \\ z &= 1-t \end{aligned}\right.$$
   2. [b.] We admit that a parametric representation of the line (AH) is: $$\left\{\begin{aligned} x &= 0 \\ y &= k \\ z &= k \end{aligned}\right. \text{ with } k \in \mathbb{R}.$$ Show that the intersection point of (GM) and (AH), which we will call N, has coordinates $(0;2;2)$.
3. 1. [a.] Show that the triangle AMN is a right-angled triangle at A.
   2. [b.] Calculate the area of this triangle.
4. Let J be the centre of the face BCGF.
   1. [a.] Determine the coordinates of point J.
   2. [b.] Show that the vector $\overrightarrow{\mathrm{FJ}}$ is a normal vector to the plane (AMN).
   3. [c.] Show that J belongs to the plane (AMN). Deduce that it is the orthogonal projection of point F onto the plane (AMN).
5. We recall that the volume $V$ of a tetrahedron or a pyramid is given by the formula: $$V = \frac{1}{3} \times \mathscr{B} \times h$$ $\mathscr{B}$ being the area of a base and $h$ the height relative to this base. Show that the volume of the tetrahedron AMNF is twice the volume of the pyramid BCGFM.

Two aircraft are approaching an airport. We equip space with an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k})$ whose origin O is the base of the control tower, and the ground is the plane $P_0$ with equation $z = 0$. The unit of the axes corresponds to 1 km. We model the aircraft as points.
Aircraft Alpha transmits to the tower its position at $\mathrm{A}(-7; 1; 7)$ and its trajectory is directed by the vector $\vec{u}\begin{pmatrix} 2 \\ -1 \\ -3 \end{pmatrix}$.
Aircraft Beta transmits a trajectory defined by the line $d_{\mathrm{B}}$ passing through point B with a parametric representation: $$\left\{\begin{aligned} x &= -11 + 5t \\ y &= -5 + t \\ z &= 11 - 4t \end{aligned}\right. \text{ where } t \text{ describes } \mathbb{R}.$$

1. If it does not deviate from its trajectory, determine the coordinates of point S where aircraft Beta will touch the ground.
2. a. Determine a parametric representation of the line $d_{\mathrm{A}}$ characterizing the trajectory of aircraft Alpha. b. Can the two aircraft collide?
3. a. Prove that aircraft Alpha passes through position $\mathrm{E}(-3; -1; 1)$. b. Justify that a Cartesian equation of the plane $P_{\mathrm{E}}$ passing through E and perpendicular to the line $d_{\mathrm{A}}$ is: $$2x - y - 3z + 8 = 0.$$ c. Verify that the point $\mathrm{F}(-1; -3; 3)$ is the intersection point of the plane $P_{\mathrm{E}}$ and the line $d_{\mathrm{B}}$. d. Calculate the exact value of the distance EF, then verify that this corresponds to a distance of 3464 m, to the nearest 1 m.
4. Air traffic regulations stipulate that two aircraft on approach must be at least 3 nautical miles apart at all times (1 nautical mile equals 1852 m). If aircraft Alpha and Beta are respectively at E and F at the same instant, is their safety distance respected?

For each of the following statements, indicate whether it is true or false. Each answer must be justified. An unjustified answer earns no points. Space is referred to an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k})$. We consider the line $(d)$ whose parametric representation is: $$\left\{\begin{array}{rl} x & = 3 - 2t \\ y & = -1 \\ z & = 2 - 6t \end{array}, \text{ where } t \in \mathbb{R}\right.$$ We also consider the following points:

• $\mathrm{A}(3; -3; -2)$
• $\mathrm{B}(5; -4; -1)$
• C the point on line $(d)$ with x-coordinate 2
• H the orthogonal projection of point B onto the plane $\mathscr{P}$ with equation $x + 3z - 7 = 0$

Statement 1: The line $(d)$ and the y-axis are two non-coplanar lines.
Statement 2: The plane passing through $A$ and perpendicular to line $(d)$ has the Cartesian equation: $$x + 3z + 3 = 0$$
Statement 3: A measure, expressed in radians, of the geometric angle $\widehat{\mathrm{BAC}}$ is $\frac{\pi}{6}$.
Statement 4: The distance BH is equal to $\frac{\sqrt{10}}{2}$.

For each of the following statements, indicate whether it is true or false. Each answer must be justified. An unjustified answer earns no points.

PART A

ABCDEFGH is a cube with edge length 1. The points I, J, K, L and M are the midpoints respectively of the edges [AB], [BF], [AE], [CD] and [DH].
Statement 1: $\ll \overrightarrow { \mathrm { JH } } = 2 \overrightarrow { \mathrm { BI } } + \overrightarrow { \mathrm { DM } } - \overrightarrow { \mathrm { CB } } \gg$ Statement 2: ``The triplet of vectors ( $\overrightarrow { \mathrm { AB } } , \overrightarrow { \mathrm { AH } } , \overrightarrow { \mathrm { AG } }$ ) is a basis of space.'' Statement 3: ``$\overrightarrow { \mathrm { IB } } \cdot \overrightarrow { \mathrm { LM } } = - \frac { 1 } { 4 }$.''

PART B

In space equipped with an orthonormal coordinate system, we consider:

• the plane $\mathscr { P }$ with Cartesian equation $2 x - y + 3 z + 6 = 0$
• the points $\mathrm { A } ( 2 ; 0 ; - 1 )$ and $\mathrm { B } ( 5 ; - 3 ; 7 )$

Statement 4: ``The plane $\mathscr { P }$ and the line ( AB ) are parallel.'' Statement 5: ``The plane $\mathscr { P } ^ { \prime }$ parallel to $\mathscr { P }$ passing through B has Cartesian equation $- 2 x + y - 3 z + 34 = 0$'' Statement 6: ``The distance from point A to plane $\mathscr { P }$ is equal to $\frac { \sqrt { 14 } } { 2 }$.'' We denote by (d) the line with parametric representation
$$\left\{ \begin{array} { r l } x & = - 12 + 2 k \\ y & = 6 \\ z & = 3 - 5 k \end{array} , \text { where } k \in \mathbb { R } \right.$$
Statement 7: ``The lines (AB) and (d) are not coplanar.''

Space is referred to an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k})$. We consider:

• the points $\mathrm{C}(3; 0; 0)$, $\mathrm{D}(0; 2; 0)$, $\mathrm{H}(-6; 2; 2)$ and $\mathrm{J}\left(\frac{-54}{13}; \frac{62}{13}; 0\right)$;
• the plane $P$ with Cartesian equation $2x + 3y + 6z - 6 = 0$;
• the plane $P'$ with Cartesian equation $x - 2y + 3z - 3 = 0$;
• the line $(d)$ with a parametric representation: $\left\{\begin{array}{l} x = -8 + \frac{1}{3}t \\ y = -1 + \frac{1}{2}t \\ z = -4 + t \end{array}, t \in \mathbb{R}\right.$

For each of the following statements, specify whether it is true or false, then justify the answer given. An answer without justification will not be taken into account.
Statement 1: The line $(d)$ is orthogonal to the plane $P$ and intersects this plane at $H$.
Statement 2: The measure in degrees of the angle $\widehat{\mathrm{DCH}}$, rounded to $10^{-1}$, is $17.3^{\circ}$.
Statement 3: The planes $P$ and $P'$ are secant and their intersection is the line $\Delta$ with a parametric representation: $\left\{\begin{array}{l} x = 3 - 3t \\ y = 0 \\ z = t \end{array}, t \in \mathbb{R}\right.$.
Statement 4: Point J is the orthogonal projection of point H onto the line (CD).

For each of the following statements, indicate whether it is true or false. Justify each answer. An unjustified answer earns no points.
We equip space with an orthonormal coordinate system ( $\mathrm { O } ; \vec { \imath } , \vec { \jmath } , \vec { k }$ ).

1. We consider the points $\mathrm { A } ( - 1 ; 0 ; 5 )$ and $\mathrm { B } ( 3 ; 2 ; - 1 )$.
   Statement 1: A parametric representation of the line (AB) is $$\left\{ \begin{aligned} x & = 3 - 2 t \\ y & = 2 - t \text { with } t \in \mathbb { R } \\ z & = - 1 + 3 t \end{aligned} \right.$$
   Statement 2: The vector $\vec { n } \left( \begin{array} { c } 5 \\ - 2 \\ 1 \end{array} \right)$ is normal to the plane (OAB).
   
2. We consider:
   • the line $d$ with parametric representation $\left\{ \begin{aligned} x & = 15 + k \\ y & = 8 - k \\ z & = - 6 + 2 k \end{aligned} \right.$ with $k \in \mathbb { R }$;
   • the line $d ^ { \prime }$ with parametric representation $\left\{ \begin{array} { l } x = 1 + 4 s \\ y = 2 + 4 s \\ z = 1 - 6 s \end{array} \right.$ with $s \in \mathbb { R }$.
   
   Statement 3: The lines $d$ and $d ^ { \prime }$ are not coplanar.
   
3. We consider the plane $\mathscr { P }$ with equation $x - y + z + 1 = 0$.
   Statement 4: The distance from point $\mathrm { C } ( 2 ; - 1 ; 2 )$ to the plane $\mathscr { P }$ is equal to $2 \sqrt { 3 }$.

Exercise 2 (5 points)
The space is equipped with an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k})$. We consider:

• the points $\mathrm{A}(-1; 2; 1)$, $\mathrm{B}(1; -1; 2)$ and $\mathrm{C}(1; 1; 1)$;
• the line $d$ whose parametric representation is given by: $$d : \left\{ \begin{aligned} x &= \frac{3}{2} + 2t \\ y &= 2 + t \\ z &= 3 - t \end{aligned} \quad \text{with } t \in \mathbb{R}; \right.$$
• the line $d'$ whose parametric representation is given by: $$d' : \left\{ \begin{aligned} x &= s \\ y &= \frac{3}{2} + s \\ z &= 3 - 2s \end{aligned} \quad \text{with } s \in \mathbb{R}. \right.$$

Part A

1. Show that the lines $d$ and $d'$ intersect at the point $\mathrm{S}\left(-\frac{1}{2}; 1; 4\right)$.
2. 1. [a.] Show that the vector $\vec{n}\begin{pmatrix}1\\2\\4\end{pmatrix}$ is a normal vector to the plane (ABC).
   2. [b.] Deduce that a Cartesian equation of the plane (ABC) is: $$x + 2y + 4z - 7 = 0$$
3. Prove that the points $\mathrm{A}$, $\mathrm{B}$, $\mathrm{C}$ and S are not coplanar.
4. 1. [a.] Prove that the point $\mathrm{H}(-1; 0; 2)$ is the orthogonal projection of S onto the plane (ABC).
   2. [b.] Deduce that there is no point $M$ in the plane (ABC) such that $\mathrm{S}M < \frac{\sqrt{21}}{2}$.

Part B
We consider a point $M$ belonging to the segment [CS]. We thus have $\overrightarrow{\mathrm{CM}} = k\overrightarrow{\mathrm{CS}}$ with $k$ a real number in the interval $[0; 1]$.

1. Determine the coordinates of point $M$ as a function of $k$.
2. Does there exist a point $M$ on the segment [CS] such that the triangle $(MAB)$ is right-angled at $M$?

Space is referred to an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k})$. We consider the following points: $$\mathrm{A}(1; 3; 0), \quad \mathrm{B}(-1; 4; 5), \quad \mathrm{C}(0; 1; 0) \quad \text{and} \quad \mathrm{D}(-2; 2; 1).$$

1. Show that the points $\mathrm{A}$, $\mathrm{B}$ and $\mathrm{C}$ determine a plane.
2. Show that the triangle ABC is right-angled at A.
3. Let $\Delta$ be the line passing through point D and with direction vector $\vec{u}\begin{pmatrix} 2 \\ -1 \\ 1 \end{pmatrix}$. a. Prove that the line $\Delta$ is orthogonal to the plane (ABC). b. Justify that the plane (ABC) admits the Cartesian equation: $$2x - y + z + 1 = 0$$ c. Determine a parametric representation of the line $\Delta$.
4. We call H the point with coordinates $\left(-\dfrac{2}{3}; \dfrac{4}{3}; \dfrac{5}{3}\right)$. Verify that $H$ is the orthogonal projection of point $D$ onto the plane (ABC).
5. We recall that the volume of a tetrahedron is given by $V = \dfrac{1}{3} B \times h$, where $B$ is the area of a base of the tetrahedron and $h$ is its height relative to this base. a. Show that $\mathrm{DH} = \dfrac{2\sqrt{6}}{3}$. b. Deduce the volume of the tetrahedron ABCD.
6. We consider the line $d$ with parametric representation: $$\left\{\begin{aligned} x &= 1 - 2k \\ y &= -3k \\ z &= 1 + k \end{aligned}\right. \text{ where } k \text{ describes } \mathbb{R}.$$ Are the line $d$ and the plane (ABC) secant or parallel?

Space is equipped with an orthonormal coordinate system ( $\mathrm { O } ; \vec { \imath } , \vec { \jmath } , \vec { k }$ ). We consider the points
$$\mathrm { A } ( 2 \sqrt { 3 } ; 0 ; 0 ) , \quad \mathrm { B } ( 0 ; 2 ; 0 ) , \quad \mathrm { C } ( 0 ; 0 ; 1 ) \quad \text { and } \quad \mathrm { K } \left( \frac { \sqrt { 3 } } { 2 } ; \frac { 3 } { 2 } ; 0 \right) .$$

1. Justify that a parametric representation of the line (CK) is:

$$\left\{ \begin{aligned} x & = \frac { \sqrt { 3 } } { 2 } t \\ y & = \frac { 3 } { 2 } t \quad ( t \in \mathbb { R } ) \\ z & = 1 - t \end{aligned} \right.$$

1. Let $\mathrm { M } ( t )$ be a point on the line (CK) parametrized by a real number $t$. Establish that $\mathrm { OM} ( t ) = \sqrt { 4 t ^ { 2 } - 2 t + 1 }$.
2. Let $f$ be the function defined and differentiable on $\mathbb { R }$ by $f ( t ) = \mathrm { OM } ( t )$. a. Study the variations of the function $f$ on $\mathbb { R }$. b. Deduce the value of $t$ for which $f$ reaches its minimum.
3. Deduce that the point $\mathrm { H } \left( \frac { \sqrt { 3 } } { 8 } ; \frac { 3 } { 8 } ; \frac { 3 } { 4 } \right)$ is the orthogonal projection of point O onto the line (CK).
4. Prove, using the dot product tool, that point H is the orthocenter (intersection of the altitudes of a triangle) of triangle ABC.
5. a. Prove that the line $( \mathrm { OH } )$ is orthogonal to the plane $( \mathrm { ABC } )$. b. Deduce an equation of the plane (ABC).
6. Calculate, in square units, the area of triangle ABC.

Exercise 4

We place ourselves in an orthonormal frame $(\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k})$ of space. We consider the points $\mathrm{A}(1; 0; 3)$, $\mathrm{B}(-2; 1; 2)$ and $\mathrm{C}(0; 3; 2)$.

1. a. Show that the points $\mathrm{A}$, $\mathrm{B}$ and $\mathrm{C}$ are not collinear. b. Let $\vec{n}$ be the vector with coordinates $\left(\begin{array}{c} -1 \\ 1 \\ 4 \end{array}\right)$. Verify that the vector $\vec{n}$ is orthogonal to the plane (ABC). c. Deduce that the plane $(\mathrm{ABC})$ has for Cartesian equation $-x + y + 4z - 11 = 0$.

We consider the plane $\mathscr{P}$ with Cartesian equation $3x - 3y + 2z - 9 = 0$ and the plane $\mathscr{P}'$ with Cartesian equation $x - y - z + 2 = 0$.

1. a. Prove that the planes $\mathscr{P}$ and $\mathscr{P}'$ are secant. We denote by (d) their line of intersection. b. Determine whether the planes $\mathscr{P}$ and $\mathscr{P}'$ are perpendicular.
2. Show that the line (d) is directed by the vector $\vec{u}\left(\begin{array}{l} 1 \\ 1 \\ 0 \end{array}\right)$.
3. Show that the point $\mathrm{M}(2; 1; 3)$ belongs to the planes $\mathscr{P}$ and $\mathscr{P}'$. Deduce a parametric representation of the line (d).
4. Show that the line (d) is also included in the plane (ABC). What can we say about the three planes (ABC), $\mathscr{P}$ and $\mathscr{P}'$?

Space is referred to an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k})$. We consider the points $$\mathrm{A}(4; -1; 3), \quad \mathrm{B}(-1; 1; -2), \quad \mathrm{C}(0; 4; 5) \text{ and } \mathrm{D}(-3; -4; 6).$$

1. a. Verify that the points $\mathrm{A}, \mathrm{B}, \mathrm{C}$ are not collinear.
   We admit that a Cartesian equation of the plane (ABC) is: $29x + 30y - 17z = 35$. b. Are the points A, B, C, D coplanar? Justify.
   
2. Let $P _ { 1 }$ be the perpendicular bisector plane of the segment $[\mathrm{AB}]$. a. Determine the coordinates of the midpoint of the segment $[\mathrm{AB}]$. b. Deduce that a Cartesian equation of $P _ { 1 }$ is: $5x - 2y + 5z = 10$.
   
3. We denote by $P _ { 2 }$ the perpendicular bisector plane of the segment $[\mathrm{CD}]$. a. Let M be a point of the plane $P _ { 2 }$ with coordinates $(x; y; z)$.
   Express $\mathrm{MC}^{2}$ and $\mathrm{MD}^{2}$ as functions of the coordinates of M. Deduce that a Cartesian equation of the plane $P _ { 2 }$ is: $-3x - 8y + z = 10$. b. Justify that the planes $P _ { 1 }$ and $P _ { 2 }$ are secant.
   
4. Let $\Delta$ be the line with a parametric representation: $$\left\{ \begin{array} { r l r l } x & = & -2 - 1.9t \\ y & = & t & \text{ where } t \in \mathbb{R} \\ z & = & 4 + 2.3t \end{array} \right.$$ Demonstrate that $\Delta$ is the line of intersection of $P _ { 1 }$ and $P _ { 2 }$.
   We denote by $P _ { 3 }$ the perpendicular bisector plane of the segment $[\mathrm{AC}]$. We admit that a Cartesian equation of the plane $P _ { 3 }$ is: $8x - 10y - 4z = -15$.
   
5. Demonstrate that the line $\Delta$ and the plane $P _ { 3 }$ are secant.
   
6. Justify that the point of intersection between $\Delta$ and $P _ { 3 }$ is the point H.

For each of the following statements, indicate whether it is true or false. Each answer must be justified. An unjustified answer earns no points.

1. We consider the function $f$ defined on $]0; +\infty[$ by: $f(x) = x\ln(x)$.
   Statement 1: $$\int_1^{\mathrm{e}} f(x)\,\mathrm{d}x = \frac{\mathrm{e}^2 + 1}{4}$$
   
2. Let $n$ and $k$ be two non-zero natural integers such that $k \leqslant n$.
   Statement 2: $$n \times \binom{n-1}{k-1} = k \times \binom{n}{k}$$
   
3. For the three following statements, we consider that space is equipped with an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k})$.
   Let $d$ be the line with parametric representation: $\left\{\begin{array}{rl} x &= t + 1 \\ y &= 2t + 1 \\ z &= -t \end{array}\right., t \in \mathbb{R}$. Let $d'$ be the line with parametric representation: $\left\{\begin{array}{rl} x &= 2t' - 1 \\ y &= -t' + 2 \\ z &= t' + 1 \end{array}\right., t' \in \mathbb{R}$. Let $P$ be the plane with Cartesian equation: $2x + y - 2z + 18 = 0$. Let A be the point with coordinates $(-1; -3; 2)$ and B be the point with coordinates $(-5; -5; 6)$. We call the perpendicular bisector plane of segment $[\mathrm{AB}]$ the plane passing through the midpoint of segment $[\mathrm{AB}]$ and perpendicular to the line $(\mathrm{AB})$.
   Statement 3: Point A belongs to line $d$. Statement 4: Lines $d$ and $d'$ are secant. Statement 5: Plane $P$ is the perpendicular bisector plane of segment $[\mathrm{AB}]$.

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 $45^\circ$.

Let P be the plane containing the vectors $(6,6,9)$ and $(7,8,10)$. Find a unit vector that is perpendicular to $(2,-3,4)$ and that lies in the plane P. (Note: all vectors are considered as line segments starting at the origin $(0,0,0)$. In particular the origin lies in the plane P.)

Let C be the circle formed by the intersection of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4$ and the plane $z = - 1$. When a plane $\alpha$ containing the $x$-axis intersects the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4$ to form a circle that meets C at exactly one point, and a normal vector to plane $\alpha$ is $\vec { n } = ( a , 3 , b )$, find the value of $a ^ { 2 } + b ^ { 2 }$. [4 points]

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, the equation of the plane that is perpendicular to the line $\frac { x - 2 } { 2 } = \frac { y - 2 } { 3 } = z - 1$ and passes through the point $( 1 , - 5,2 )$ is $2 x + a y + b z + c = 0$. Find the value of $a + b + c$. [3 points]

In coordinate space, there are two mutually perpendicular planes $\alpha$ and $\beta$. For two points $\mathrm { A }$ and $\mathrm { B }$ on plane $\alpha$, $\overline { \mathrm { AB } } = 3 \sqrt { 5 }$, and line AB is parallel to plane $\beta$. The distance between point A and plane $\beta$ is 2, and the distance between a point P on plane $\beta$ and plane $\alpha$ is 4. Find the area of triangle PAB. [4 points]

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 coordinate space, there are three points $\mathrm { A } , \mathrm { B } , \mathrm { C }$ not on the same line. For a plane $\alpha$ satisfying the following conditions, let $d ( \alpha )$ be the minimum distance among the distances from each point $\mathrm { A } , \mathrm { B } , \mathrm { C }$ to plane $\alpha$.
(a) Plane $\alpha$ intersects segment AC and also intersects segment BC.
(b) Plane $\alpha$ does not intersect segment AB.
Among planes $\alpha$ satisfying the above conditions, let $\beta$ be the plane where $d ( \alpha )$ is maximized. Which of the following statements in are correct? [4 points]
$\text{ㄱ}$. Plane $\beta$ is perpendicular to the plane passing through the three points $\mathrm { A } , \mathrm { B } , \mathrm { C }$. $\text{ㄴ}$. Plane $\beta$ passes through the midpoint of segment AC or the midpoint of segment BC. $\text{ㄷ}$. When the three points are $\mathrm { A } ( 2,3,0 ) , \mathrm { B } ( 0,1,0 ) , \mathrm { C } ( 2 , - 1,0 )$, $d ( \beta )$ equals the distance between point B and plane $\beta$.
(1) ㄱ
(2) ㄷ
(3) ㄱ, ㄴ
(4) ㄴ, ㄷ
(5) ㄱ, ㄴ, ㄷ

In coordinate space, there is a circle $C$ formed by the intersection of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 6$ and the plane $x + 2 z - 5 = 0$. Let P be the point on circle $C$ with the minimum $y$-coordinate, and let Q be the foot of the perpendicular from point P to the $xy$-plane. For a point X moving on circle $C$, the maximum value of $| \overrightarrow { \mathrm { PX } } + \overrightarrow { \mathrm { QX } } | ^ { 2 }$ is $a + b \sqrt { 30 }$.
Find the value of $10 ( a + b )$. (Here, $a$ and $b$ are rational numbers.) [4 points]

In coordinate space, the $x$-coordinate of the point where the plane passing through the point $( 2,0,5 )$ and containing the line $x - 1 = 2 - y = \frac { z + 1 } { 2 }$ meets the $x$-axis is? [3 points]
(1) $\frac { 9 } { 2 }$
(2) 4
(3) $\frac { 7 } { 2 }$
(4) 3
(5) $\frac { 5 } { 2 }$

For a tetrahedron ABCD with an equilateral triangle BCD of side length 12 as one face, let H be the foot of the perpendicular from vertex A to plane BCD. The point H lies inside triangle BCD. The area of triangle CDH is 3 times the area of triangle BCH, the area of triangle DBH is 2 times the area of triangle BCH, and $\overline { \mathrm { AH } } = 3$. Let M be the midpoint of segment BD, and let Q be the foot of the perpendicular from point A to segment CM. What is the length of segment AQ? [4 points]
(1) $\sqrt { 11 }$
(2) $2 \sqrt { 3 }$
(3) $\sqrt { 13 }$
(4) $\sqrt { 14 }$
(5) $\sqrt { 15 }$

As shown in the figure, there is a rhombus-shaped piece of paper ABCD with side length 4 and $\angle \mathrm { BAD } = \frac { \pi } { 3 }$. Let M and N be the midpoints of sides BC and CD respectively. The paper is folded along the three line segments $\mathrm { AM } , \mathrm { AN } , \mathrm { MN }$ to form a tetrahedron PAMN. The area of the orthogonal projection of triangle AMN onto the plane PAM is $\frac { q } { p } \sqrt { 3 }$. Find the value of $p + q$. (Here, the thickness of the paper is neglected, P is the point where the three points $\mathrm { B } , \mathrm { C } , \mathrm { D }$ coincide when the paper is folded, and $p$ and $q$ are coprime natural numbers.) [4 points]

In coordinate space, for two points $\mathrm { A } ( 3 , - 3,3 ) , \mathrm { B } ( - 2,7 , - 2 )$, let $\alpha , \beta$ be the two planes that contain segment AB and are tangent to the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 1$. Let C and D be the points of tangency of the two planes $\alpha , \beta$ with the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 1$ respectively. If the volume of tetrahedron ABCD is $\frac { q } { p } \sqrt { 3 }$, find the value of $p + q$. (Here, $p$ and $q$ are coprime natural numbers.) [4 points]