Space is referred to the orthonormal frame $(\mathrm{O}, \vec{\imath}, \vec{\jmath}, \vec{k})$. We consider the plane $\mathscr{P}$ with equation $2x + y - 2z + 4 = 0$ and the points A with coordinates $(3; 2; 6)$, B with coordinates $(1; 2; 4)$, and C with coordinates $(4; -2; 5)$.

1. a. Verify that the points A, B and C define a plane. b. Verify that this plane is the plane $\mathscr{P}$.
2. a. Show that the triangle ABC is right-angled. b. Write a system of parametric equations for the line $\Delta$ passing through O and perpendicular to the plane $\mathscr{P}$. c. Let K be the orthogonal projection of O onto $\mathscr{P}$. Calculate the distance OK. d. Calculate the volume of the tetrahedron OABC.
3. We consider, in this question, the system of weighted points $$S = \{(\mathrm{O}, 3), (\mathrm{A}, 1), (\mathrm{B}, 1), (\mathrm{C}, 1)\}$$ a. Verify that this system admits a centroid, which we denote G. b. Let I denote the centroid of the triangle ABC. Show that G belongs to (OI). c. Determine the distance from G to the plane $\mathscr{P}$.
4. Let $\Gamma$ be the set of points $M$ in space satisfying: $$\|3\overrightarrow{M\mathrm{O}} + \overrightarrow{M\mathrm{A}} + \overrightarrow{M\mathrm{B}} + \overrightarrow{M\mathrm{C}}\| = 5.$$ Determine $\Gamma$. What is the nature of the set of points common to $\mathscr{P}$ and $\Gamma$?

We work in space with an orthonormal coordinate system. We consider the points $\mathrm { A } ( 0 ; 4 ; 1 ) , \mathrm { B } ( 1 ; 3 ; 0 ) , \mathrm { C } ( 2 ; - 1 ; - 2 )$ and $\mathrm { D } ( 7 ; - 1 ; 4 )$.

1. Prove that the points $\mathrm { A } , \mathrm { B }$ and C are not collinear.
2. Let $\Delta$ be the line passing through point D with direction vector $\vec { u } ( 2 ; - 1 ; 3 )$. a. Prove that the line $\Delta$ is orthogonal to the plane ( ABC ). b. Deduce a Cartesian equation of the plane (ABC). c. Determine a parametric representation of the line $\Delta$. d. Determine the coordinates of point H, the intersection of the line $\Delta$ and the plane (ABC).
3. Let $\mathscr { P } _ { 1 }$ be the plane with equation $x + y + z = 0$ and $\mathscr { P } _ { 2 }$ be the plane with equation $x + 4 y + 2 = 0$. a. Prove that the planes $\mathscr { P } _ { 1 }$ and $\mathscr { P } _ { 2 }$ are secant. b. Verify that the line $d$, the intersection of the planes $\mathscr { P } _ { 1 }$ and $\mathscr { P } _ { 2 }$, has the parametric representation $$\left\{ \begin{array} { l } x = - 4 t - 2 \\ y = t \\ z = 3 t + 2 \end{array} , t \in \mathbb { R } . \right.$$ c. Are the line $d$ and the plane ( ABC ) secant or parallel?

Exercise 2 -- Common to all candidates

We consider the cube ABCDEFGH, with edge length 1, represented below, and we equip space with the orthonormal coordinate system $(\mathrm{A}; \overrightarrow{\mathrm{AB}}, \overrightarrow{\mathrm{AD}}, \overrightarrow{\mathrm{AE}})$.

1. Determine a parametric representation of the line (FD).
2. Prove that the vector $\vec{n}\begin{pmatrix}1\\-1\\1\end{pmatrix}$ is a normal vector to the plane (BGE) and determine an equation of the plane (BGE).
3. Show that the line (FD) is perpendicular to the plane (BGE) at a point K with coordinates $\mathrm{K}\left(\frac{2}{3}; \frac{1}{3}; \frac{2}{3}\right)$.
4. What is the nature of triangle BEG? Determine its area.
5. Deduce the volume of the tetrahedron BEGD.

Exercise 3 (4 points)

We consider a cube ABCDEFCH given in Appendix 2 (to be returned with your work). We denote M the midpoint of segment $[\mathrm{EH}]$, N that of $[\mathrm{FC}]$ and P the point such that $\overrightarrow{\mathrm{HP}} = \frac{1}{4} \overrightarrow{\mathrm{HG}}$.

Part A: Section of the cube by the plane (MNP)

1. Justify that the lines (MP) and (FG) are secant at a point L.

Construct the point L.
2. We admit that the lines (LN) and (CG) are secant and we denote T their point of intersection.
We admit that the lines (LN) and (BF) are secant and we denote Q their point of intersection. a. Construct the points T and Q leaving the construction lines visible. b. Construct the intersection of the planes (MNP) and (ABF).
3. Deduce a construction of the section of the cube by the plane (MNP).

Part B

The space is referred to the coordinate system $(\mathrm{A}; \overrightarrow{\mathrm{AB}}, \overrightarrow{\mathrm{AD}}, \overrightarrow{\mathrm{AE}})$.

1. Give the coordinates of points $\mathrm{M}, \mathrm{N}$ and P in this coordinate system.
2. Determine the coordinates of point L.
3. We admit that point T has coordinates $\left(1 ; 1 ; \frac{5}{8}\right)$. Is the triangle TPN right-angled at T?

Space is referred to an orthonormal coordinate system $(\mathrm{O}, \vec{\imath}, \vec{\jmath}, \vec{k})$. We are given the points $\mathrm{A}(1;0;-1)$, $\mathrm{B}(1;2;3)$, $\mathrm{C}(-5;5;0)$ and $\mathrm{D}(11;1;-2)$. The points I and J are the midpoints of the segments $[\mathrm{AB}]$ and $[\mathrm{CD}]$ respectively. The point K is defined by $\overrightarrow{\mathrm{BK}} = \frac{1}{3}\overrightarrow{\mathrm{BC}}$.

1. a. Determine the coordinates of points I, J and K. b. Prove that the points I, J and K define a plane. c. Show that the vector $\vec{n}$ with coordinates $(3;1;4)$ is a normal vector to the plane (IJK). Deduce a Cartesian equation of this plane.
2. Let $\mathscr{P}$ be the plane with equation $3x + y + 4z - 8 = 0$. a. Determine a parametric representation of the line (BD). b. Prove that the plane $\mathscr{P}$ and the line $(\mathrm{BD})$ are secant and give the coordinates of L, the point of intersection of the plane $\mathscr{P}$ and the line (BD). c. Is the point L the symmetric of point D with respect to point B?

In space equipped with an orthonormal coordinate system, we consider the points:
$$\mathrm{A}(1; 2; 7), \quad \mathrm{B}(2; 0; 2), \quad \mathrm{C}(3; 1; 3), \quad \mathrm{D}(3; -6; 1) \text{ and } \mathrm{E}(4; -8; -4).$$

1. Show that the points $\mathrm{A}, \mathrm{B}$ and C are not collinear.
2. Let $\vec{u}(1; b; c)$ be a vector in space, where $b$ and $c$ denote two real numbers. a) Determine the values of $b$ and $c$ such that $\vec{u}$ is a normal vector to the plane (ABC). b) Deduce that a Cartesian equation of the plane (ABC) is: $$x - 2y + z - 4 = 0$$ c) Does the point D belong to the plane (ABC)?
3. We consider the line $\mathscr{D}$ in space whose parametric representation is: $$\left\{\begin{aligned} x & = 2t + 3 \\ y & = -4t + 5 \end{aligned}\right.$$

In an orthonormal coordinate system in space, we consider the points $$\mathrm { A } ( 5 ; - 5 ; 2 ) , \mathrm { B } ( - 1 ; 1 ; 0 ) , \mathrm { C } ( 0 ; 1 ; 2 ) \text { and } \mathrm { D } ( 6 ; 6 ; - 1 ) .$$

1. Determine the nature of triangle BCD and calculate its area.
2. a. Show that the vector $\vec { n } \left( \begin{array} { c } - 2 \\ 3 \\ 1 \end{array} \right)$ is a normal vector to the plane (BCD). b. Determine a Cartesian equation of the plane (BCD).
3. Determine a parametric representation of the line $\mathfrak { D }$ perpendicular to the plane (BCD) and passing through point A.
4. Determine the coordinates of point H, the intersection of line $\mathcal { D }$ and plane (BCD).
5. Determine the volume of tetrahedron ABCD.

Recall that the volume of a tetrahedron is given by the formula $\mathcal { V } = \frac { 1 } { 3 } \mathcal { B } \times h$, where $\mathcal { B }$ is the area of a base of the tetrahedron and h is the corresponding height. 6. We admit that $\mathrm { AB } = \sqrt { 76 }$ and $\mathrm { AC } = \sqrt { 61 }$.
Determine an approximate value to the nearest tenth of a degree of the angle $\widehat { \mathrm { BAC } }$.

ABCDEFGH is a cube.
I is the midpoint of segment $[\mathrm{AB}]$, J is the midpoint of segment $[\mathrm{EH}]$, K is the midpoint of segment [BC] and L is the midpoint of segment [CG]. We equip space with the orthonormal coordinate system (A ; $\overrightarrow{\mathrm{AB}}, \overrightarrow{\mathrm{AD}}, \overrightarrow{\mathrm{AE}}$).

1. a) Prove that the line (FD) is orthogonal to the plane (IJK). b) Deduce a Cartesian equation of the plane (IJK).
2. Determine a parametric representation of the line (FD).
3. Let $M$ be the point of intersection of the line (FD) and the plane (IJK). Determine the coordinates of point $M$.
4. Determine the nature of triangle IJK and calculate its area.
5. Calculate the volume of the tetrahedron FIJK.
6. Are the lines (IJ) and (KL) intersecting?

In an orthonormal reference frame ( $\mathrm { O } , \mathrm { I } , \mathrm { J } , \mathrm { K }$ ) with unit 1 cm, we consider the points $\mathrm { A } ( 0 ; - 1 ; 5 )$, $\mathrm { B } ( 2 ; - 1 ; 5 ) , \mathrm { C } ( 11 ; 0 ; 1 ) , \mathrm { D } ( 11 ; 4 ; 4 )$.
A point $M$ moves on the line ( AB ) in the direction from A to B at a speed of 1 cm per second.
A point $N$ moves on the line (CD) in the direction from C to D at a speed of 1 cm per second. At time $t = 0$ the point $M$ is at A and the point $N$ is at C. We denote $M _ { t }$ and $N _ { t }$ the positions of points $M$ and $N$ after $t$ seconds, $t$ denoting a positive real number. We admit that $M _ { t }$ and $N _ { t }$ have coordinates: $M _ { t } ( t ; - 1 ; 5 )$ and $N _ { t } ( 11 ; 0,8 t ; 1 + 0,6 t )$. Questions 1 and 2 are independent.
1. a. The line $( \mathrm { AB } )$ is parallel to one of the axes $( \mathrm { OI } )$, (OJ) or (OK). Which one? b. The line $( \mathrm { CD } )$ lies in a plane $\mathscr { P }$ parallel to one of the planes $( \mathrm { OIJ } )$, (OIK) or (OJK). Which one? An equation of this plane $\mathscr { P }$ will be given. c. Verify that the line $( \mathrm { AB } )$, orthogonal to the plane $\mathscr { P }$, intersects this plane at the point $\mathrm { E } ( 11 ; - 1 ; 5 )$. d. Are the lines ( AB ) and ( CD ) secant?
2. a. Show that $M _ { t } N _ { t } ^ { 2 } = 2 t ^ { 2 } - 25,2 t + 138$. b. At what time $t$ is the length $M _ { t } N _ { t }$ minimal?

Consider the rectangular prism ABCDEFGH below, for which $\mathrm { AB } = 6 , \mathrm { AD } = 4$ and $\mathrm { AE } = 2$. I, J and K are points such that $\overrightarrow { A I } = \frac { 1 } { 6 } \overrightarrow { A B } , \overrightarrow { A J } = \frac { 1 } { 4 } \overrightarrow { A D } , \overrightarrow { A K } = \frac { 1 } { 2 } \overrightarrow { A E }$. We use the orthonormal coordinate system ( $A$; $\overrightarrow { A I } , \overrightarrow { A J } , \overrightarrow { A K }$ ).

1. Verify that the vector $\vec { n }$ with coordinates $\left( \begin{array} { c } 2 \\ 2 \\ - 9 \end{array} \right)$ is normal to the plane (IJG).
2. Determine an equation of the plane (IJG).
3. Determine the coordinates of the intersection point L of the plane (IJG) and the line (BF).
4. Draw the cross-section of the rectangular prism ABCDEFGH by the plane (IJG). This drawing should be done on the figure provided in the appendix to be returned with your work). No justification is required.

Exercise 4 (5 points) -- Candidate who has NOT followed the specialization course

Let a cube ABCDEFGH with edge length 1. In the coordinate system $(A;\,\overrightarrow{AB},\,\overrightarrow{AD},\,\overrightarrow{AE})$, we consider the points $M$, $N$ and $P$ with respective coordinates $\mathrm{M}\!\left(1\,;\,1\,;\,\tfrac{3}{4}\right)$, $\mathrm{N}\!\left(0\,;\,\tfrac{1}{2}\,;\,1\right)$, $\mathrm{P}\!\left(1\,;\,0\,;\,-\tfrac{5}{4}\right)$.

1. Plot $\mathrm{M}$, $\mathrm{N}$ and $\mathrm{P}$ on the figure provided in the appendix.
2. Determine the coordinates of the vectors $\overrightarrow{\mathrm{MN}}$ and $\overrightarrow{\mathrm{MP}}$.
   Deduce that the points $\mathrm{M}$, $\mathrm{N}$ and $\mathrm{P}$ are not collinear.
3. We consider algorithm 1 given in the appendix. a. Execute this algorithm by hand with the coordinates of the points $\mathrm{M}$, $\mathrm{N}$ and $\mathrm{P}$ given above. b. What does the result displayed by the algorithm correspond to? What can we deduce about triangle MNP?
4. We consider algorithm 2 given in the appendix. Complete it so that it tests and displays whether a triangle MNP is right-angled and isosceles at M.
5. We consider the vector $\vec{n}(5\,;\,-8\,;\,4)$ normal to the plane (MNP). a. Determine a Cartesian equation of the plane (MNP). b. We consider the line $\Delta$ passing through F and with direction vector $\vec{n}$.
   Determine a parametric representation of the line $\Delta$.
6. Let K be the point of intersection of the plane (MNP) and the line $\Delta$. a. Prove that the coordinates of point K are $\left(\dfrac{4}{7}\,;\,\dfrac{24}{35}\,;\,\dfrac{23}{35}\right)$. b. We are given $FK = \sqrt{\dfrac{27}{35}}$.
   Calculate the volume of the tetrahedron MNPF.

Exercise 4 — Candidates who have not followed the speciality course
We consider the regular pyramid $SABCD$ with apex $S$ consisting of the square base $ABCD$ and equilateral triangles.
The point O is the centre of the base ABCD with $\mathrm{OB} = 1$. We recall that the segment $[\mathrm{SO}]$ is the height of the pyramid and that all edges have the same length.

1. Justify that the coordinate system $(\mathrm{O}; \overrightarrow{\mathrm{OB}}, \overrightarrow{\mathrm{OC}}, \overrightarrow{\mathrm{OS}})$ is orthonormal.
2. We define the point K by the relation $\overrightarrow{\mathrm{SK}} = \frac{1}{3}\overrightarrow{\mathrm{SD}}$ and we denote by I the midpoint of segment $[\mathrm{SO}]$. a. Determine the coordinates of point K. b. Deduce that the points B, I and K are collinear. c. We denote by L the point of intersection of the edge $[\mathrm{SA}]$ with the plane (BCI). Justify that the lines (AD) and (KL) are parallel. d. Determine the coordinates of point L.
3. We consider the vector $\vec{n}\begin{pmatrix}1\\1\\2\end{pmatrix}$ in the coordinate system $(\mathrm{O}; \overrightarrow{\mathrm{OB}}, \overrightarrow{\mathrm{OC}}, \overrightarrow{\mathrm{OS}})$. a. Show that $\vec{n}$ is a normal vector to the plane (BCI). b. Show that the vectors $\vec{n}$, $\overrightarrow{\mathrm{AS}}$ and $\overrightarrow{\mathrm{DS}}$ are coplanar. c. What is the relative position of the planes (BCI) and (SAD)?

Part A: a volume calculation without a coordinate system We consider an equilateral pyramid SABCD (pyramid with a square base whose lateral faces are all equilateral triangles). The diagonals of the square ABCD measure 24 cm. We denote O the center of the square ABCD. We will admit that $\mathrm { OS } = \mathrm { OA }$.

1. Without using a coordinate system, prove that the line (SO) is orthogonal to the plane (ABC).
2. Deduce the volume, in $\mathrm { cm } ^ { 3 }$, of the pyramid SABCD.

Part B: in a coordinate system We consider the orthonormal coordinate system ( $\mathrm { O } ; \overrightarrow { \mathrm { OA } } , \overrightarrow { \mathrm { OB } } , \overrightarrow { \mathrm { OS } }$ ).

1. We denote P and Q the midpoints of the segments [AS] and [BS] respectively. a. Justify that $\vec { n } ( 1 ; 1 ; - 3 )$ is a normal vector to the plane (PQC). b. Deduce a Cartesian equation of the plane (PQC).
2. Let H be the point of the plane (PQC) such that the line (SH) is orthogonal to the plane (PQC). a. Give a parametric representation of the line (SH). b. Calculate the coordinates of the point H. c. Show then that the length SH, in unit of length, is $\frac { 2 \sqrt { 11 } } { 11 }$.
3. We will admit that the area of the quadrilateral PQCD, in unit of area, is equal to $\frac { 3 \sqrt { 11 } } { 8 }$. Calculate the volume of the pyramid SPQCD, in unit of volume.

Part C: fair sharing For the birthday of her twin daughters Anne and Fanny, Mrs. Nova has made a beautiful cake in the shape of an equilateral pyramid whose diagonals of the square base measure 24 cm. She is about to share it equally by placing her knife on the apex. That is when Anne stops her and proposes a more original cut: ``Place the blade on the midpoint of an edge, parallel to a side of the base, then cut towards the opposite side''. Is this the case? Justify the answer.

$ABCDEFGH$ is a cube with edge length equal to 1. The space is equipped with the orthonormal coordinate system ($D; \overrightarrow{DC}, \overrightarrow{DA}, \overrightarrow{DH}$). In this coordinate system, we have: $D(0;0;0)$, $C(1;0;0)$, $A(0;1;0)$, $H(0;0;1)$ and $E(0;1;1)$. Let $I$ be the midpoint of $[AB]$. Let $\mathscr{P}$ be the plane parallel to the plane $(BGE)$ and passing through the point $I$. It is admitted that the section of the cube by the plane $\mathscr{P}$ is a hexagon whose vertices $I, J, K, L, M$, and $N$ belong respectively to the edges $[AB], [BC], [CG], [GH], [HE]$ and $[AE]$.

1. a. Show that the vector $\overrightarrow{DF}$ is normal to the plane $(BGE)$. b. Deduce a Cartesian equation of the plane $\mathscr{P}$.
2. Show that the point $N$ is the midpoint of the segment $[AE]$.
3. a. Determine a parametric representation of the line $(HB)$. b. Deduce that the line $(HB)$ and the plane $\mathscr{P}$ intersect at a point $T$ whose coordinates you will specify.
4. Calculate, in units of volume, the volume of the tetrahedron $FBGE$.

The points O, A, B and C are vertices of a cube, such that the coordinate system $(\mathrm{O} ; \overrightarrow{\mathrm{OA}}, \overrightarrow{\mathrm{OB}}, \overrightarrow{\mathrm{OC}})$ is an orthonormal coordinate system. This coordinate system will be used throughout the exercise. The three mirrors of the retroreflector are represented by the planes (OAB), (OBC) and (OAC). Light rays are modeled by lines.
Rules for reflection of a light ray (admitted):

• when a light ray with direction vector $\vec{v}(a ; b ; c)$ is reflected by the plane (OAB), a direction vector of the reflected ray is $\vec{v}(a ; b ; -c)$;
• when a light ray with direction vector $\vec{v}(a ; b ; c)$ is reflected by the plane (OBC), a direction vector of the reflected ray is $\vec{v}(-a ; b ; c)$;
• when a light ray with direction vector $\vec{v}(a ; b ; c)$ is reflected by the plane (OAC), a direction vector of the reflected ray is $\vec{v}(a ; -b ; c)$.

1. Property of retroreflectors
Using the above rules, prove that if a light ray with direction vector $\vec{v}(a ; b ; c)$ is reflected successively by the planes (OAB), (OBC) and (OAC), the final ray is parallel to the initial ray.
For the rest, we consider a light ray modeled by a line $d _ { 1 }$ with direction vector $\overrightarrow{v _ { 1 }}(-2 ; -1 ; -1)$ which strikes the plane (OAB) at the point $\mathrm{I} _ { 1 }(2 ; 3 ; 0)$. The reflected ray is modeled by the line $d _ { 2 }$ with direction vector $\overrightarrow{v _ { 2 }}(-2 ; -1 ; 1)$ and passing through the point $\mathrm{I} _ { 1 }$.
2. Reflection of $d_2$ on the plane (OBC)
a. Give a parametric representation of the line $d _ { 2 }$. b. Give, without justification, a normal vector to the plane (OBC) and a Cartesian equation of this plane. c. Let $\mathrm{I} _ { 2 }$ be the point with coordinates $(0 ; 2 ; 1)$. Verify that the plane (OBC) and the line $d _ { 2 }$ intersect at $\mathrm{I} _ { 2 }$.
We denote by $d _ { 3 }$ the line representing the light ray after reflection on the plane (OBC). $d _ { 3 }$ is therefore the line with direction vector $\overrightarrow{v _ { 3 }}(2 ; -1 ; 1)$ passing through the point $\mathrm{I} _ { 2 }(0 ; 2 ; 1)$.
3. Reflection of $d_3$ on the plane (OAC)
Calculate the coordinates of the intersection point $\mathrm{I} _ { 3 }$ of the line $d _ { 3 }$ with the plane (OAC).
We denote by $d _ { 4 }$ the line representing the light ray after reflection on the plane (OAC). It is therefore parallel to the line $d _ { 1 }$.
4. Study of the light path
We are given the vector $\vec{u}(1 ; -2 ; 0)$, and we denote by $\mathscr{P}$ the plane defined by the lines $d _ { 1 }$ and $d _ { 2 }$. a. Prove that the vector $\vec{u}$ is a normal vector to the plane $\mathscr{P}$. b. Are the lines $d _ { 1 }$, $d _ { 2 }$ and $d_3$ coplanar?

Consider the cube ABCDEFGH represented below. We define the points I and J respectively by $\overrightarrow { \mathrm { HI } } = \frac { 3 } { 4 } \overrightarrow { \mathrm { HG } }$ and $\overrightarrow { \mathrm { JG } } = \frac { 1 } { 4 } \overrightarrow { \mathrm { CG } }$.

1. On the answer sheet provided in the appendix, to be returned with your work, draw, without justification, the cross-section of the cube by the plane (IJK) where K is a point of the segment [BF].
2. On the answer sheet provided in the appendix, to be returned with your work, draw, without justification, the cross-section of the cube by the plane (IJL) where L is a point of the line (BF).
3. Does there exist a point P on the line (BF) such that the cross-section of the cube by the plane (IJP) is an equilateral triangle? Justify your answer.

Consider a solid ADECBF consisting of two identical pyramids with the square ABCD as common base with centre I. A perspective representation of this solid is given in the appendix (to be returned with the answer sheet). All edges have length 1. The space is referred to the orthonormal coordinate system ( $\mathrm { A } ; \overrightarrow { \mathrm { AB } } , \overrightarrow { \mathrm { AD } } , \overrightarrow { \mathrm { AK } }$ ).

1. a) Show that $\mathrm { IE } = \frac { \sqrt { 2 } } { 2 }$. Deduce the coordinates of points I, E and F. b) Show that the vector $\vec { n } \left( \begin{array} { c } 0 \\ - 2 \\ \sqrt { 2 } \end{array} \right)$ is normal to the plane (ABE). c) Determine a Cartesian equation of the plane (ABE).
2. Let M be the midpoint of segment [DF] and N the midpoint of segment [AB]. a) Prove that the planes $( \mathrm { FDC } )$ and $( \mathrm { ABE } )$ are parallel. b) Determine the intersection of planes (EMN) and (FDC). c) Construct on the appendix (to be returned with the answer sheet) the cross-section of solid ADECBF by plane (EMN).

ABCDEFGH designates a cube with side length 1. Point I is the midpoint of segment [BF]. Point J is the midpoint of segment [BC]. Point K is the midpoint of segment [CD].

Part A

In this part, no justification is required. We admit that the lines (IJ) and (CG) intersect at a point L. Construct, on the figure provided in the appendix and leaving the construction lines visible:

• the point L;
• the intersection $\mathscr { D }$ of the planes (IJK) and (CDH);
• the cross-section of the cube by the plane (IJK).

Part B

Space is referred to the coordinate system ( $\mathrm { A } ; \overrightarrow { \mathrm { AB } } , \overrightarrow { \mathrm { AD } } , \overrightarrow { \mathrm { AE } }$ ).

1. Give the coordinates of $\mathrm { A } , \mathrm { G } , \mathrm { I } , \mathrm { J }$ and K in this coordinate system.
2. a. Show that the vector $\overrightarrow { \mathrm { AG } }$ is normal to the plane (IJK). b. Deduce a Cartesian equation of the plane (IJK).
3. We denote by $M$ a point of the segment [AG] and $t$ the real number in the interval $[ 0 ; 1 ]$ such that $\overrightarrow { \mathrm { AM } } = t \overrightarrow { \mathrm { AG } }$. a. Prove that $M \mathrm { I } ^ { 2 } = 3 t ^ { 2 } - 3 t + \frac { 5 } { 4 }$. b. Prove that the distance $M I$ is minimal for the point $\mathrm { N } \left( \frac { 1 } { 2 } ; \frac { 1 } { 2 } ; \frac { 1 } { 2 } \right)$.
4. Prove that for this point $\mathrm { N } \left( \frac { 1 } { 2 } ; \frac { 1 } { 2 } ; \frac { 1 } { 2 } \right)$: a. N belongs to the plane (IJK). b. The line (IN) is perpendicular to the lines (AG) and (BF).

A homeowner is interested in the shadow cast on his future veranda by the roof of his house when the sun is at its zenith. This veranda is schematized in cavalier perspective in an orthonormal coordinate system $(\mathrm{O}, \vec{\imath}, \vec{\jmath}, \vec{k})$. The roof of the veranda consists of two triangular faces SEF and SFG.

• The planes (SOA) and (SOC) are perpendicular.
• The planes (SOC) and (EAB) are parallel, as are the planes (SOA) and (GCB).
• The edges [UV) and [EF] of the roofs are parallel.

The point K belongs to the segment [SE], the plane (UVK) separates the veranda into two zones, one illuminated and the other shaded. The plane (UVK) cuts the veranda along the polygonal line KMNP which is the shadow-sun boundary.

1. Without calculation, justify that: a. the segment $[\mathrm{KM}]$ is parallel to the segment $[\mathrm{UV}]$; b. the segment [NP] is parallel to the segment [UK].
2. In the rest of the exercise, we place ourselves in the orthonormal coordinate system $(\mathrm{O}, \vec{\imath}, \vec{\jmath}, \vec{k})$. The coordinates of the different points are as follows: $\mathrm{A}(4 ; 0 ; 0)$, \ldots

Exercise 2 (4 points)
We consider a cube ABCDEFGH.

1. a. Simplify the vector $\overrightarrow{\mathrm{AC}} + \overrightarrow{\mathrm{AE}}$. b. Deduce that $\overrightarrow{\mathrm{AG}} \cdot \overrightarrow{\mathrm{BD}} = 0$. c. It is admitted that $\overrightarrow{\mathrm{AG}} \cdot \overrightarrow{\mathrm{BE}} = 0$. Prove that the line (AG) is orthogonal to the plane (BDE).
   
2. Space is equipped with the orthonormal coordinate system $(\mathrm{A}; \overrightarrow{\mathrm{AB}}, \overrightarrow{\mathrm{AD}}, \overrightarrow{\mathrm{AE}})$. a. Prove that a Cartesian equation of the plane (BDE) is $x + y + z - 1 = 0$. b. Determine the coordinates of the intersection point K of the line (AG) and the plane (BDE). c. It is admitted that the area, in square units, of triangle BDE is equal to $\dfrac{\sqrt{3}}{2}$. Calculate the volume of the pyramid BDEG.

(Candidates who did not follow the specialization course)
We denote by $\mathbb { R }$ the set of real numbers. The space is equipped with an orthonormal coordinate system ( $\mathrm { O } ; \vec { \imath } , \vec { \jmath } , \vec { k }$ ). Consider the points $\mathrm { A } ( - 1 ; 2 ; 0 ) , \mathrm { B } ( 1 ; 2 ; 4 )$ and $\mathrm { C } ( - 1 ; 1 ; 1 )$.

1. a. Prove that points $\mathrm { A } , \mathrm { B }$ and C are not collinear. b. Calculate the dot product $\overrightarrow { \mathrm { AB } } \cdot \overrightarrow { \mathrm { AC } }$. c. Deduce the measure of angle $\widehat { \mathrm { BAC } }$, rounded to the nearest degree.
2. Let $\vec { n }$ be the vector with coordinates $\left( \begin{array} { c } 2 \\ - 1 \\ - 1 \end{array} \right)$. a. Prove that $\vec { n }$ is a normal vector to plane (ABC). b. Determine a Cartesian equation of plane ( ABC ).
3. Let $\mathscr { P } _ { 1 }$ be the plane with equation $3 x + y - 2 z + 3 = 0$ and $\mathscr { P } _ { 2 }$ the plane passing through O and parallel to the plane with equation $x - 2 z + 6 = 0$. a. Prove that plane $\mathscr { P } _ { 2 }$ has equation $x = 2z$. b. Prove that planes $\mathscr{P}_1$, $\mathscr{P}_2$ and (ABC) have a common point, and determine its coordinates.

Space is equipped with an orthonormal coordinate system $( O ; \vec { i } ; \vec { j } ; \vec { k } )$. We consider two lines $d _ { 1 }$ and $d _ { 2 }$ defined by the parametric representations:
$$d _ { 1 } : \left\{ \begin{array} { l } { x = 2 + t } \\ { y = 3 - t } \\ { z = t } \end{array} , t \in \mathbb { R } \text { and } \left\{ \begin{array} { l } x = - 5 + 2 t ^ { \prime } \\ y = - 1 + t ^ { \prime } \\ z = 5 \end{array} , t ^ { \prime } \in \mathbb { R } . \right. \right.$$
We admit that the lines $d _ { 1 }$ and $d _ { 2 }$ are non-coplanar. The purpose of this exercise is to determine, if it exists, a third line $\Delta$ that is simultaneously secant to both lines $d _ { 1 }$ and $d _ { 2 }$ and orthogonal to these two lines.

1. Verify that the point $\mathrm { A } ( 2 ; 3 ; 0 )$ belongs to the line $d _ { 1 }$.
2. Give a direction vector $\overrightarrow { u _ { 1 } }$ of the line $d _ { 1 }$ and a direction vector $\overrightarrow { u _ { 2 } }$ of the line $d _ { 2 }$. Are the lines $d _ { 1 }$ and $d _ { 2 }$ parallel?
3. Verify that the vector $\vec { v } ( 1 ; - 2 ; - 3 )$ is orthogonal to the vectors $\overrightarrow { u _ { 1 } }$ and $\overrightarrow { u _ { 2 } }$.
4. Let $P$ be the plane passing through point A, and directed by the vectors $\overrightarrow { u _ { 1 } }$ and $\vec { v }$. In this question we study the intersection of the line $d _ { 2 }$ and the plane $P$. a. Show that a Cartesian equation of the plane $P$ is: $5 x + 4 y - z - 22 = 0$. b. Show that the line $d _ { 2 }$ intersects the plane $P$ at the point $\mathrm { B } ( 3 ; 3 ; 5 )$.
5. We now consider the line $\Delta$ directed by the vector $\vec { v} \left( \begin{array} { c } 1 \\ - 2 \\ - 3 \end{array} \right)$, and passing through the point $\mathrm { B } ( 3 ; 3 ; 5 )$. a. Give a parametric representation of this line $\Delta$. b. Are the lines $d _ { 1 }$ and $\Delta$ secant? Justify your answer. c. Explain why the line $\Delta$ answers the problem posed.

Consider a cube ABCDEFGH whose graphical representation in cavalier perspective is given below. The edges have length 1. Space is referred to the orthonormal coordinate system $( \mathrm { D } ; \overrightarrow { \mathrm { DA } } , \overrightarrow { \mathrm { DC } } , \overrightarrow { \mathrm { DH } } )$.

Part A

1. Show that the vector $\overrightarrow { \mathrm { DF } }$ is normal to the plane (EBG).
2. Determine a Cartesian equation of the plane (EBG).
3. Deduce the coordinates of point I, the intersection of line (DF) and plane (EBG).

One would show in the same way that point J, the intersection of line (DF) and plane (AHC), has coordinates $\left( \frac { 1 } { 3 } ; \frac { 1 } { 3 } ; \frac { 1 } { 3 } \right)$.

Part B

For any real number $x$ in the interval $[ 0 ; 1 ]$, we associate the point $M$ of segment $[ \mathrm{DF} ]$ such that $\overrightarrow { \mathrm { DM } } = x \overrightarrow { \mathrm { DF } }$. We are interested in the evolution of the measure $\theta$ in radians of angle $\widehat { \mathrm { EMB } }$ as point $M$ moves along segment [DF]. We have $0 \leqslant \theta \leqslant \pi$.

1. What is the value of $\theta$ if point $M$ coincides with point D? with point F?
2. a. Justify that the coordinates of point $M$ are $( x ; x ; x )$. b. Show that $\cos ( \theta ) = \frac { 3 x ^ { 2 } - 4 x + 1 } { 3 x ^ { 2 } - 4 x + 2 }$. For this, one may consider the dot product of vectors $\overrightarrow { M \mathrm { E } }$ and $\overrightarrow { M \mathrm {~B} }$.
3. The table of variations of the function below has been constructed $$f : x \longmapsto \frac { 3 x ^ { 2 } - 4 x + 1 } { 3 x ^ { 2 } - 4 x + 2 }$$
   

   $x$	0	$\frac { 1 } { 3 }$	$\frac { 2 } { 3 }$	1
   \begin{tabular}{ c } Variations
   of $f$

   & $\frac { 1 } { 2 }$ & & & & & & 0 & \hline \end{tabular}
   For which positions of point $M$ on segment [DF]: a. is triangle $MEB$ right-angled at $M$? b. is angle $\theta$ maximal?

Exercise 4 -- For candidates who have not followed the specialized course

In space, consider the cube ABCDEFGH. We denote I and J the midpoints of segments [EH] and [FB] respectively. We equip space with the orthonormal coordinate system $(\mathrm{A}; \overrightarrow{\mathrm{AB}}, \overrightarrow{\mathrm{AD}}, \overrightarrow{\mathrm{AE}})$.

1. Give the coordinates of points I and J.
   
2. a. Show that the vector $\vec{n}\begin{pmatrix} 1 \\ -2 \\ 2 \end{pmatrix}$ is a normal vector to the plane (BGI). b. Deduce a Cartesian equation of the plane (BGI). c. We denote K the midpoint of segment [HJ]. Does point K belong to the plane (BGI)?
   
3. The purpose of this question is to calculate the area of triangle BGI. a. Using for example triangle FIG as a base, prove that the volume of tetrahedron FBIG equals $\frac{1}{6}$. We recall that the volume $V$ of a tetrahedron is given by the formula $V = \frac{1}{3} \times \text{base area} \times \text{height}$.

Exercise 2 -- 3 points -- Common to all candidates

Space is equipped with a coordinate system $(\mathrm{O}, \vec{\imath}, \vec{\jmath}, \vec{k})$. Let $\mathcal{P}$ be the plane with Cartesian equation: $2x - z - 3 = 0$. We denote $A$ the point with coordinates $(1 ; a ; a^{2})$ where $a$ is a real number.

1. Justify that, regardless of the value of $a$, the point $A$ does not belong to the plane $\mathcal{P}$.
2. a. Determine a parametric representation of the line $\mathcal{D}$ (with parameter $t$) passing through the point $A$ and orthogonal to the plane $\mathcal{P}$. b. Let $M$ be a point belonging to the line $\mathcal{D}$, associated with the value $t$ of the parameter in the previous parametric representation. Express the distance $AM$ as a function of the real number $t$.
3. We denote $H$ the point of intersection of the plane $\mathcal{P}$ and the line $\mathcal{D}$ orthogonal to $\mathcal{P}$ and passing through the point $A$. The point $H$ is called the orthogonal projection of the point $A$ onto the plane $\mathcal{P}$ and the distance $AH$ is called the distance from the point $A$ to the plane $\mathcal{P}$. Is there a value of $a$ for which the distance $AH$ from the point $A$ with coordinates $(1 ; a ; a^{2})$ to the plane $\mathcal{P}$ is minimal? Justify the answer.