We consider the cube ABCDEFGH which is represented in APPENDIX. In the orthonormal coordinate system ( $A$; $\overrightarrow { A B }$; $\overrightarrow { A D }$; $\overrightarrow { A E }$ ), we consider the points $M , N$ and $P$ with coordinates:
$$\mathrm { M } \left( 1 ; 1 ; \frac { 3 } { 4 } \right) , \quad \mathrm { N } \left( 0 ; \frac { 1 } { 2 } ; 1 \right) , \quad \mathrm { P } \left( 1 ; 0 ; - \frac { 5 } { 4 } \right)$$
In this exercise, we propose to calculate the volume of the tetrahedron FMNP.

1. Give the coordinates of the vectors $\overrightarrow { \mathrm { MN } }$ and $\overrightarrow { \mathrm { MP } }$.
2. Place the points $\mathrm { M} , \mathrm { N }$ and P on the figure given in APPENDIX which must be returned with your work.
3. Justify that the points $\mathrm { M } , \mathrm { N }$ and P are not collinear.

From then on, the three points define the plane (MNP).
4. a. Calculate the dot product $\overrightarrow { \mathrm { MN } } \cdot \overrightarrow { \mathrm { MP } }$, then deduce the nature of the triangle MNP. b. Calculate the area of the triangle MNP.
5. a. Show that the vector $\vec { n } ( 5 ; - 8 ; 4 )$ is a normal vector to the plane (MNP). b. Deduce that a Cartesian equation of the plane (MNP) is $5 x - 8 y + 4 z = 0$. 6. We recall that the point F has coordinates $\mathrm { F } ( 1 ; 0 ; 1 )$.
Determine a parametric representation of the line $d$ orthogonal to the plane (MNP) and passing through the point F. 7. We denote L the orthogonal projection of the point F onto the plane (MNP).
Show that the coordinates of the point L are: $\mathrm { L } \left( \frac { 4 } { 7 } ; \frac { 24 } { 35 } ; \frac { 23 } { 35 } \right)$. 8. Show that $\mathrm { FL } = \frac { 3 \sqrt { 105 } } { 35 }$ then calculate the volume of the tetrahedron FMNP.
We recall that the volume V of a tetrahedron is given by the formula:
$$V = \frac { 1 } { 3 } \times \text { area of a base } \times \text{ height associated with this base. }$$

We consider the cube ABCDEFGH represented below. The points I and J are the midpoints of segments $[\mathrm{AB}]$ and $[\mathrm{CG}]$ respectively. The point N is the midpoint of segment [IJ]. The objective of this exercise is to calculate the volume of the tetrahedron HFIJ. We place ourselves in the orthonormal coordinate system ($A$; $\overrightarrow{AB}, \overrightarrow{AD}, \overrightarrow{AE}$).

1. a. Give the coordinates of points I and J.
   Deduce the coordinates of N. b. Justify that the vectors $\overrightarrow{\mathrm{IJ}}$ and $\overrightarrow{\mathrm{NF}}$ have the respective coordinates: $$\overrightarrow{\mathrm{IJ}} \left(\begin{array}{c} 0.5 \\ 1 \\ 0.5 \end{array}\right) \text{ and } \overrightarrow{\mathrm{NF}} \left(\begin{array}{c} 0.25 \\ -0.5 \\ 0.75 \end{array}\right)$$ c. Prove that the vectors $\overrightarrow{\mathrm{IJ}}$ and $\overrightarrow{\mathrm{NF}}$ are orthogonal.
   We admit that $\mathrm{NF} = \frac{\sqrt{14}}{4}$. d. Deduce that the area of triangle FIJ is equal to $\frac{\sqrt{21}}{8}$.
   
2. We consider the vector $\vec{u}\left(\begin{array}{c} 4 \\ -1 \\ -2 \end{array}\right)$. a. Prove that the vector $\vec{u}$ is normal to the plane (FIJ). b. Deduce that a Cartesian equation of the plane (FIJ) is: $4x - y - 2z - 2 = 0$. c. We denote by $d$ the line perpendicular to the plane (FIJ) passing through point H. Determine a parametric representation of the line $d$. d. Show that the distance from point H to the plane (FIJ) is equal to $\frac{5\sqrt{21}}{21}$. e. We recall that the volume of a pyramid is given by the formula $V = \frac{1}{3} \times \mathscr{B} \times h$ where $\mathscr{B}$ is the area of a base and $h$ is the length of the height relative to this base. Calculate the volume of the tetrahedron HFIJ. Give the answer in the form of an irreducible fraction.

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]

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) $\frac { 25 } { 2 }$

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).

Let $P$ be the plane $\sqrt { 3 } x + 2 y + 3 z = 16$ and let $S = \left\{ \alpha \hat { i } + \beta \hat { j } + \gamma \hat { k } : \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = 1 \right.$ and the distance of $( \alpha , \beta , \gamma )$ from the plane $P$ is $\left. \frac { 7 } { 2 } \right\}$. Let $\vec { u } , \vec { v }$ and $\vec { w }$ be three distinct vectors in $S$ such that $| \vec { u } - \vec { v } | = | \vec { v } - \vec { w } | = | \vec { w } - \vec { u } |$. Let $V$ be the volume of the parallelepiped determined by vectors $\vec { u } , \vec { v }$ and $\vec { w }$. Then the value of $\frac { 80 } { \sqrt { 3 } } V$ is

A plane $P$ is parallel to two lines whose direction ratios are $- 2,1 , - 3$ and $- 1,2 , - 2$ and it contains the point $( 2,2 , - 2 )$. Let $P$ intersect the co-ordinate axes at the points $A , B , C$ making the intercepts $\alpha , \beta , \gamma$. If $V$ is the volume of the tetrahedron $O A B C$, where $O$ is the origin and $p = \alpha + \beta + \gamma$, then the ordered pair $( V , p )$ is equal to
(1) $( 48 , - 13 )$
(2) $( 24 , - 13 )$
(3) $( 48,11 )$
(4) $( 24 , - 5 )$