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.

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 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.

The objective is to determine a measure of the angle between two carbon-hydrogen bonds.
A regular tetrahedron is a polyhedron whose four faces are equilateral triangles.
Electrical interactions lead to modeling the methane molecule $\mathrm{CH}_4$ as follows:

• The nuclei of hydrogen atoms occupy the positions of the four vertices of a regular tetrahedron.
• The carbon nucleus at the center of the molecule is equidistant from the four hydrogen atoms.

1. Justify that we can inscribe this tetrahedron in a cube ABCDEFGH by positioning two hydrogen atoms at vertices A and C of the cube and the two other hydrogen atoms at two other vertices of the cube. Represent the molecule in the cube given in the appendix on page 6. In the rest of the exercise, we can work in the coordinate system $(A; \overrightarrow{AB}; \overrightarrow{AD}; \overrightarrow{AE})$.
2. Prove that the carbon atom is at the center $\Omega$ of the cube.
3. Determine the approximation to the nearest tenth of a degree of the measure of the angle formed between the carbon-hydrogen bonds, that is, the angle $\widehat{A\Omega C}$.

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 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 $V = \frac{1}{3} \times \text{base area} \times \text{height}$.

(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.

The figure below represents a cube ABCDEFGH. The three points I, J, K are defined by the following conditions:

• I is the midpoint of segment [AD];
• J is such that $\overrightarrow{\mathrm{AJ}} = \frac{3}{4}\overrightarrow{\mathrm{AE}}$;
• K is the midpoint of segment [FG].

1. On the figure provided in the appendix, construct without justification the point of intersection P of the plane (IJK) and the line (EH). Leave the construction lines on the figure.
2. Deduce from this, by justifying, the intersection of the plane (IJK) and the plane (EFG).

The figure below represents a cube ABCDEFGH. The three points I, J, K are defined by the following conditions:

• I is the midpoint of segment [AD];
• J is such that $\overrightarrow{\mathrm{AJ}} = \frac{3}{4}\overrightarrow{\mathrm{AE}}$;
• K is the midpoint of segment [FG].

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

1. a. Give without justification the coordinates of points I, J and K. b. Determine the real numbers $a$ and $b$ such that the vector $\vec{n}(4; a; b)$ is orthogonal to the vectors $\overrightarrow{\mathrm{IJ}}$ and $\overrightarrow{\mathrm{IK}}$. c. Deduce that a Cartesian equation of the plane (IJK) is: $4x - 6y - 4z + 3 = 0$.
2. a. Give a parametric representation of the line (CG). b. Calculate the coordinates of point N, the intersection of the plane (IJK) and the line (CG). c. Place point N on the figure and construct in colour the cross-section of the cube by the plane (IJK).

The figure below represents a cube ABCDEFGH with the plane (IJK) having Cartesian equation $4x - 6y - 4z + 3 = 0$ in the orthonormal coordinate system $(\mathrm{A}; \overrightarrow{\mathrm{AB}}, \overrightarrow{\mathrm{AD}}, \overrightarrow{\mathrm{AE}})$. We denote by R the orthogonal projection of point F onto the plane (IJK). Point R is therefore the unique point of the plane (IJK) such that the line (FR) is orthogonal to the plane (IJK). We define the interior of the cube as the set of points $M(x; y; z)$ such that $\left\{\begin{array}{l} 0 < x < 1 \\ 0 < y < 1 \\ 0 < z < 1 \end{array}\right.$ Is point R inside the cube?

An artist wishes to create a sculpture composed of a tetrahedron placed on a cube with 6-metre edges. These two solids are represented by the cube $ABCDEFGH$ and by the tetrahedron $SELM$.
The space is equipped with an orthonormal coordinate system $(A; \overrightarrow{AI}, \overrightarrow{AJ}, \overrightarrow{AK})$ such that: $I \in [AB]$, $J \in [AD]$, $K \in [AE]$ and $AI = AJ = AK = 1$, the graphical unit representing 1 metre.
The points $L$, $M$ and $S$ are defined as follows:

• $L$ is the point such that $\overrightarrow{FL} = \frac{2}{3}\overrightarrow{FE}$;
• $M$ is the point of intersection of the plane $(BDL)$ and the line $(EH)$;
• $S$ is the point of intersection of the lines $(BL)$ and $(AK)$.

1. Prove, without calculating coordinates, that the lines $(LM)$ and $(BD)$ are parallel.
2. Prove that the coordinates of point $L$ are $(2; 0; 6)$.
3. a. Give a parametric representation of the line $(BL)$. b. Verify that the coordinates of point $S$ are $(0; 0; 9)$.
4. Let $\vec{n}$ be the vector with coordinates $(3; 3; 2)$. a. Verify that $\vec{n}$ is normal to the plane $(BDL)$. b. Prove that a Cartesian equation of the plane $(BDL)$ is: $$3x + 3y + 2z - 18 = 0$$ c. It is admitted that the line $(EH)$ has the parametric representation: $$\left\{\begin{array}{l} x = 0 \\ y = s \\ z = 6 \end{array} \quad (s \in \mathbb{R})\right.$$ Calculate the coordinates of point $M$.
5. Calculate the volume of the tetrahedron $SELM$. Recall that the volume $V$ of a tetrahedron is given by the following formula: $$V = \frac{1}{3} \times \text{Area of base} \times \text{Height}$$
6. The artist wishes the measure of angle $\widehat{SLE}$ to be between $55^\circ$ and $60^\circ$. Is this angle constraint satisfied?

Exercise 3 (5 points)

We place ourselves in space equipped with an orthonormal coordinate system whose origin is point A. We consider the points $\mathrm{B}(10; -8; 2)$, $\mathrm{C}(-1; -8; 5)$ and $\mathrm{D}(14; 4; 8)$.

1. a. Determine a system of parametric equations for each of the lines (AB) and (CD). b. Verify that the lines (AB) and (CD) are not coplanar.
2. We consider the point I on the line (AB) with abscissa 5 and the point J on the line (CD) with abscissa 4. a. Determine the coordinates of points I and J and deduce the distance IJ. b. Demonstrate that the line (IJ) is perpendicular to the lines (AB) and (CD). The line (IJ) is called the common perpendicular to the lines (AB) and (CD).
3. The purpose of this question is to verify that the distance IJ is the minimum distance between the lines (AB) and (CD). We consider a point $M$ on the line (AB) distinct from point I. We consider a point $M'$ on the line (CD) distinct from point J. a. Justify that the parallel to the line (IJ) passing through point $M'$ intersects the line $\Delta$ (the line parallel to (CD) passing through I) at a point that we will denote $P$. b. Demonstrate that the triangle $MPM'$ is right-angled at $P$. c. Justify that $MM' > IJ$ and conclude.

We place ourselves in an orthonormal coordinate system with origin O and axes $( \mathrm { O } x )$, $( \mathrm { O } y )$ and $( \mathrm { O } z )$. In this coordinate system, we are given the points $\mathrm { A } ( - 3 ; 0 ; 0 ) , \mathrm { B } ( 3 ; 0 ; 0 ) , \mathrm { C } ( 0 ; 3 \sqrt { 3 } ; 0 )$ and $\mathrm { D } ( 0 ; \sqrt { 3 } ; 2 \sqrt { 6 } )$. We denote H as the midpoint of segment [CD] and I as the midpoint of segment [BC].

1. Calculate the lengths AB and AD.

We admit for the rest that all edges of the solid ABCD have the same length, that is, the tetrahedron ABCD is a regular tetrahedron. We call $\mathscr { P }$ the plane with normal vector $\overrightarrow { \mathrm { OH } }$ and passing through point I.
2. Study of the cross-section of tetrahedron ABCD by plane $\mathscr { P }$ a. Show that a Cartesian equation of plane $\mathscr { P }$ is : $2 y \sqrt { 3 } + z \sqrt { 6 } - 9 = 0$. b. Prove that the midpoint J of $[ \mathrm { BD } ]$ is the intersection point of line (BD) and plane $\mathscr { P }$. c. Give a parametric representation of line (AD), then prove that plane $\mathscr { P }$ and line (AD) intersect at a point K whose coordinates you will determine. d. Prove that lines (IJ) and (JK) are perpendicular. e. Determine precisely the nature of the cross-section of tetrahedron ABCD by plane $\mathscr { P }$.
3. Can we place a point M on edge $[ \mathrm { BD } ]$ such that triangle OIM is right-angled at M?

Let ABCDEFGH be a cube. We consider:

• I and J the midpoints respectively of segments [AD] and [BC];
• P the center of face ABFE, that is, the intersection of diagonals (AF) and (BE);
• Q the midpoint of segment [FG].

We place ourselves in the orthonormal coordinate system $( \mathrm { A } ; \frac { 1 } { 2 } \overrightarrow { \mathrm { AB } } , \frac { 1 } { 2 } \overrightarrow { \mathrm { AD } } , \frac { 1 } { 2 } \overrightarrow { \mathrm { AE } } )$. Throughout the exercise, we may use the coordinates of the points in the figure without justifying them. We admit that a parametric representation of the line (IJ) is
$$\left\{ \begin{array} { l } x = r \\ y = 1 , \quad r \in \mathbf { R } \\ z = 0 \end{array} \right.$$

1. Verify that a parametric representation of the line (PQ) is

$$\left\{ \begin{array} { l } x = 1 + t \\ y = t , \quad t \in \mathbf { R } \\ z = 1 + t \end{array} \right.$$
Let $t$ be a real number and $\mathrm { M } ( 1 + t ; t ; 1 + t )$ be the point on the line (PQ) with parameter $t$.
2. a. We admit that there exists a unique point K belonging to the line (IJ) such that (MK) is orthogonal to (IJ). Prove that the coordinates of this point $\mathrm { K }$ are $( 1 + t ; 1 ; 0 )$. b. Deduce that $\mathrm { MK } = \sqrt { 2 + 2 t ^ { 2 } }$.
3. a. Verify that $y - z = 0$ is a Cartesian equation of the plane (HGB). b. We admit that there exists a unique point L belonging to the plane (HGB) such that (ML) is orthogonal to (HGB). Verify that the coordinates of this point L are $\left( 1 + t ; \frac { 1 } { 2 } + t ; \frac { 1 } { 2 } + t \right)$. c. Deduce that the distance ML is independent of $t$.
4. Does there exist a value of $t$ for which the distance MK is equal to the distance ML?

The objective of this exercise is to study the trajectories of two submarines in the diving phase. We consider that these submarines move in a straight line, each at constant speed. At each instant $t$, expressed in minutes, the first submarine is located by the point $S_{1}(t)$ and the second submarine is located by the point $S_{2}(t)$ in an orthonormal reference frame $(\mathrm{O}, \vec{\imath}, \vec{\jmath}, \vec{k})$ with unit metre. The plane defined by $(\mathrm{O}, \vec{\imath}, \vec{\jmath})$ represents the sea surface. The $z$ coordinate is zero at sea level, negative underwater.

1. We admit that, for every real $t \geqslant 0$, the point $S_{1}(t)$ has coordinates: $$\left\{ \begin{array}{l} x(t) = 140 - 60t \\ y(t) = 105 - 90t \\ z(t) = -170 - 30t \end{array} \right.$$ a. Give the coordinates of the submarine at the beginning of the observation. b. What is the speed of the submarine? c. We place ourselves in the vertical plane containing the trajectory of the first submarine.
   Determine the angle $\alpha$ that the submarine's trajectory makes with the horizontal plane. Give the value of $\alpha$ rounded to 0.1 degree.
2. At the beginning of the observation, the second submarine is located at the point $S_{2}(0)$ with coordinates $(68; 135; -68)$ and reaches after three minutes the point $S_{2}(3)$ with coordinates $(-202; -405; -248)$ at constant speed. At what instant $t$, expressed in minutes, are the two submarines at the same depth?

This exercise is a multiple choice questionnaire. For each question, four answers are proposed and only one of them is correct. No justification is required. 1.5 points are awarded for each correct answer. No points are deducted for no answer or an incorrect answer.
Question 1 In space with an orthonormal coordinate system ($\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k}$), we consider the line ($D$) with parametric representation $\left\{\begin{array}{l} x = 2 + t \\ y = 1 - 3t \\ z = 2t \end{array} \quad (t \in \mathbb{R})\right.$, and the plane $(P)$ with Cartesian equation $x + y + z - 3 = 0$.
We can assert that: Answer A: the line ($D$) and the plane ($P$) are strictly parallel. Answer B: the line ($D$) is contained in the plane ($P$). Answer C: the line ($D$) and the plane ($P$) intersect at the point with coordinates $(4; -5; 4)$. Answer D: the line ($D$) and the plane ($P$) are orthogonal.
Question 2 In the computer department of a large store, only one salesperson is present and there are many customers. We assume that the random variable $T$, which associates to each customer the waiting time in minutes for the salesperson to be available, follows an exponential distribution with parameter $\lambda$. The average waiting time is 20 minutes. Given that a customer has already waited 20 minutes, the probability that their total waiting time exceeds half an hour is: Answer A: $\mathrm{e}^{-\frac{1}{2}}$ Answer B: $\mathrm{e}^{-\frac{3}{2}}$ Answer C: $1 - \mathrm{e}^{-\frac{1}{2}}$ Answer D: $1 - \mathrm{e}^{-10\lambda}$
Question 3
A factory manufactures tennis balls in large quantities. To comply with international competition regulations, the diameter of a ball must be between $63.5\text{ mm}$ and $66.7\text{ mm}$. We denote by $D$ the random variable which associates to each ball produced its diameter measured in millimetres. We assume that $D$ follows a normal distribution with mean 65.1 and standard deviation $\sigma$. We call $P$ the probability that a ball chosen at random from the total production is compliant. The factory decides to adjust the machines so that $P$ equals 0.99. The value of $\sigma$, rounded to the nearest hundredth, allowing this objective to be achieved is: Answer A: 0.69 Answer B: 2.58 Answer C: 0.62 Answer D: 0.80
Question 4 The curve below is the graph, in an orthonormal coordinate system, of the function $f$ defined by: $$f(x) = \frac{4x}{x^{2} + 1}$$ The exact value of the positive real number $a$ such that the line with equation $x = a$ divides the shaded region into two regions of equal area is: Answer A: $\sqrt{\sqrt{\frac{3}{2}}}$ Answer B: $\sqrt{\sqrt{5} - 1}$ Answer C: $\ln 5 - 0.5$ Answer D: $\frac{10}{9}$

Exercise 3 (5 points)
The purpose of this exercise is to examine, in different cases, whether the altitudes of a tetrahedron are concurrent, that is, to study the existence of an intersection point of its four altitudes. We recall that in a tetrahedron MNPQ, the altitude from M is the line passing through M perpendicular to the plane (NPQ).
Part A: Study of particular cases
We consider a cube ABCDEFGH. We admit that the lines (AG), (BH), (CE) and (DF), called ``main diagonals'' of the cube, are concurrent.

1. We consider the tetrahedron ABCE. a. Specify the altitude from E and the altitude from C in this tetrahedron. b. Are the four altitudes of the tetrahedron ABCE concurrent?
   
2. We consider the tetrahedron ACHF and work in the coordinate system $(\mathrm{A} ; \overrightarrow{\mathrm{AB}}, \overrightarrow{\mathrm{AD}}, \overrightarrow{\mathrm{AE}})$. a. Verify that a Cartesian equation of the plane (ACH) is: $x - y + z = 0$. b. Deduce that (FD) is the altitude from F of the tetrahedron ACHF. c. By analogy with the previous result, specify the altitudes of the tetrahedron ACHF from the vertices $\mathrm{A}$, $\mathrm{C}$ and H respectively. Are the four altitudes of the tetrahedron ACHF concurrent?

In the rest of this exercise, a tetrahedron whose four altitudes are concurrent will be called an orthocentric tetrahedron.
Part B: A property of orthocentric tetrahedra
In this part, we consider a tetrahedron MNPQ whose altitudes from vertices M and N intersect at a point K. The lines (MK) and (NK) are therefore perpendicular to the planes (NPQ) and (MPQ) respectively.

1. a. Justify that the line (PQ) is perpendicular to the line (MK); we admit likewise that the lines (PQ) and (NK) are perpendicular. b. What can we deduce from the previous question regarding the line (PQ) and the plane (MNK)? Justify the answer.
2. Show that the edges [MN] and [PQ] are perpendicular.

Thus, we obtain the following property: If a tetrahedron is orthocentric, then its opposite edges are perpendicular in pairs.
Part C: Application
In an orthonormal coordinate system, we consider the points: $$\mathrm { R } ( - 3 ; 5 ; 2 ) , \mathrm { S } ( 1 ; 4 ; - 2 ) , \mathrm { T } ( 4 ; - 1 ; 5 ) \quad \text { and } \mathrm { U } ( 4 ; 7 ; 3 ) .$$ Is the tetrahedron RSTU orthocentric? Justify.

In space equipped with the orthonormal coordinate system ($\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k}$) with unit 1 cm, we consider the points $\mathrm{A}$, $\mathrm{B}$, C and D with coordinates respectively $(2; 1; 4)$, $(4; -1; 0)$, $(0; 3; 2)$ and $(4; 3; -2)$.

1. Determine a parametric representation of the line (CD).
2. Let $M$ be a point on the line (CD). a. Determine the coordinates of the point $M$ such that the distance $BM$ is minimal. b. We denote H the point on the line $(\mathrm{CD})$ with coordinates $(3; 3; -1)$. Verify that the lines $(\mathrm{BH})$ and $(\mathrm{CD})$ are perpendicular. c. Show that the area of triangle BCD is equal to $12\,\mathrm{cm}^2$.
3. a. Prove that the vector $\vec{n}\begin{pmatrix}2\\1\\2\end{pmatrix}$ is a normal vector to the plane (BCD). b. Determine a Cartesian equation of the plane (BCD).

For each of the following questions, only one of the four statements is correct. Indicate on your answer sheet the question number and copy the letter corresponding to the correct statement. One point is awarded if the letter corresponds to the correct statement, 0 otherwise.
Throughout the exercise, we work in an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k})$ in space. The four questions are independent. No justification is required.

1. Consider the plane $P$ with Cartesian equation $3x + 2y + 9z - 5 = 0$ and the line $d$ with parametric representation: $\left\{\begin{array}{l} x = 4t + 3 \\ y = -t + 2 \\ z = -t + 9 \end{array}, t \in \mathbb{R}\right.$. Statement A: the intersection of plane $P$ and line $d$ is reduced to the point with coordinates $(3;2;9)$. Statement B: plane $P$ and line $d$ are orthogonal. Statement C: plane $P$ and line $d$ are parallel. Statement D: the intersection of plane $P$ and line $d$ is reduced to the point with coordinates $(-353; 91; 98)$.
   
2. Consider the cube ABCDEFGH and the points I, J and K defined by the vector equalities: $$\overrightarrow{\mathrm{AI}} = \frac{3}{4}\overrightarrow{\mathrm{AB}}, \quad \overrightarrow{\mathrm{DJ}} = \frac{1}{4}\overrightarrow{\mathrm{DC}}, \quad \overrightarrow{\mathrm{HK}} = \frac{3}{4}\overrightarrow{\mathrm{HG}}$$ Statement A: the cross-section of cube ABCDEFGH by plane (IJK) is a triangle. Statement B: the cross-section of cube ABCDEFGH by plane (IJK) is a quadrilateral. Statement C: the cross-section of cube ABCDEFGH by plane (IJK) is a pentagon. Statement D: the cross-section of cube ABCDEFGH by plane (IJK) is a hexagon.
   
3. Consider the line $d$ with parametric representation $\left\{\begin{aligned} x &= t + 2 \\ y &= 2 \\ z &= 5t - 6 \end{aligned}\right.$, with $t \in \mathbb{R}$, and the point $\mathrm{A}(-2; 1; 0)$. Let $M$ be a variable point on line $d$. Statement A: the smallest length $AM$ is equal to $\sqrt{53}$. Statement B: the smallest length $AM$ is equal to $\sqrt{27}$. Statement C: the smallest length $AM$ is attained when point $M$ has coordinates $(-2; 1; 0)$. Statement D: the smallest length $AM$ is attained when point $M$ has coordinates $(2; 2; -6)$.
   
4. Consider the plane $P$ with Cartesian equation $x + 2y - 3z + 1 = 0$ and the plane $P'$ with Cartesian equation $2x - y + 2 = 0$. Statement A: planes $P$ and $P'$ are parallel. Statement B: the intersection of planes $P$ and $P'$ is a line passing through points $\mathrm{A}(5; 12; 10)$ and $\mathrm{B}(3; 1; 2)$. Statement C: the intersection of planes $P$ and $P'$ is a line passing through point $\mathrm{C}(2; 6; 5)$ and having a direction vector $\vec{u}(1; 2; 2)$. Statement D: the intersection of planes $P$ and $P'$ is a line passing through point $\mathrm{D}(-1; 0; 0)$ and having a direction vector $\vec{v}(3; 6; 5)$.

Let ABCDEFGH be a cube and I the center of the square ADHE, that is, the midpoint of segment [AH] and segment [ED]. Let J be a point on segment [CG]. The cross-section of the cube ABCDEFGH by the plane (FIJ) is the quadrilateral FKLJ.
We place ourselves in the orthonormal coordinate system $(\mathrm{A}; \overrightarrow{\mathrm{AB}}, \overrightarrow{\mathrm{AD}}, \overrightarrow{\mathrm{AE}})$. We have therefore $\mathrm{A}(0;0;0)$, $\mathrm{B}(1;0;0)$, $\mathrm{D}(0;1;0)$ and $\mathrm{E}(0;0;1)$. Parts A and B can be treated independently.

Part A

In this part, the point J has coordinates $\left(1; 1; \frac{2}{5}\right)$.

1. Prove that the coordinates of point I are $\left(0; \frac{1}{2}; \frac{1}{2}\right)$.
2. a. Prove that the vector $\vec{n}\begin{pmatrix} -1 \\ 3 \\ 5 \end{pmatrix}$ is a normal vector to the plane (FIJ). b. Prove that a Cartesian equation of the plane (FIJ) is $$-x + 3y + 5z - 4 = 0.$$
3. Let $d$ be the line perpendicular to the plane (FIJ) and passing through B. a. Determine a parametric representation of the line $d$. b. We denote by M the point of intersection of the line $d$ and the plane (FIJ). Prove that $\mathrm{M}\left(\frac{6}{7}; \frac{3}{7}; \frac{5}{7}\right)$.
4. a. Calculate $\overrightarrow{\mathrm{BM}} \cdot \overrightarrow{\mathrm{BF}}$. b. Deduce an approximate value to the nearest degree of the angle $\widehat{\mathrm{MBF}}$.

Part B

In this part, J is an arbitrary point on segment [CG]. Its coordinates are therefore $(1; 1; a)$, where $a$ is a real number in the interval $[0; 1]$.

1. Show that the cross-section of the cube by the plane (FIJ) is a parallelogram.
2. We admit that L has coordinates $\left(0; 1; \frac{a}{2}\right)$. For which value(s) of $a$ is the quadrilateral FKLJ a rhombus?

(For candidates who have not followed the specialization course)
We connect the centres of each face of a cube ABCDEFGH to form a solid IJKLMN. More precisely, the points I, J, K, L, M and N are the centres respectively of the square faces ABCD, BCGF, CDHG, ADHE, ABFE and EFGH (thus the midpoints of the diagonals of these squares).

1. Without using a coordinate system (and thus coordinates) in the reasoning, justify that the lines (IN) and (ML) are orthogonal.

In what follows, we consider the orthonormal coordinate system $(\mathrm{A}; \overrightarrow{\mathrm{AB}}; \overrightarrow{\mathrm{AD}}; \overrightarrow{\mathrm{AE}})$ in which, for example, the point N has coordinates $\left(\frac{1}{2}; \frac{1}{2}; 1\right)$.

1. a. Give the coordinates of the vectors $\overrightarrow{\mathrm{NC}}$ and $\overrightarrow{\mathrm{ML}}$. b. Deduce that the lines (NC) and (ML) are orthogonal. c. From the previous questions, deduce a Cartesian equation of the plane (NCI).
2. a. Show that a Cartesian equation of the plane (NJM) is: $x - y + z = 1$. b. Is the line (DF) perpendicular to the plane (NJM)? Justify. c. Show that the intersection of the planes (NJM) and (NCI) is a line for which you will give a point and a direction vector. Name the line thus obtained using two points from the figure.

Exercise 4 (For candidates who have not followed the specialty course)

We consider a cube $ABCDEFGH$ with edge length 1. We denote $I$ the midpoint of segment $[EF]$, $J$ the midpoint of segment $[EH]$ and $K$ the point of segment $[AD]$ such that $\overrightarrow{AK} = \frac{1}{4}\overrightarrow{AD}$. We denote $\mathscr{P}$ the plane passing through $I$ and parallel to the plane $(FHK)$.

Part A

In this part, the constructions requested will be performed without justification on the figure given in the appendix.

1. The plane $(FHK)$ intersects the line $(AE)$ at a point which we denote $M$. Construct the point $M$.
2. Construct the cross-section of the cube by the plane $\mathscr{P}$.

Part B

In this part, we equip the space with the orthonormal coordinate system $(A; \overrightarrow{AB}, \overrightarrow{AD}, \overrightarrow{AE})$. We recall that $\mathscr{P}$ is the plane passing through $I$ and parallel to the plane $(FHK)$.

1. a. Show that the vector $\vec{n}\left(\begin{array}{c} 4 \\ 4 \\ -3 \end{array}\right)$ is a normal vector to the plane $(FHK)$. b. Deduce that a Cartesian equation of the plane $(FHK)$ is: $4x + 4y - 3z - 1 = 0$. c. Determine an equation of the plane $\mathscr{P}$. d. Calculate the coordinates of the point $M'$, the point of intersection of the plane $\mathscr{P}$ and the line $(AE)$.
2. We denote $\Delta$ the line passing through point $E$ and perpendicular to the plane $\mathscr{P}$. a. Determine a parametric representation of the line $\Delta$. b. Calculate the coordinates of point $L$, the intersection of line $\Delta$ and plane $(ABC)$. c. Draw the line $\Delta$ on the figure provided in the appendix. d. Are the lines $\Delta$ and $(BF)$ intersecting? What about the lines $\Delta$ and $(CG)$? Justify.

Exercise 4 — For candidates who have not followed the speciality
5 points
On the figure given in appendix 2 to be returned with the copy:

• ABCDEFGH is a rectangular parallelepiped such that $\mathrm { AB } = 12 , \mathrm { AD } = 18$ and $\mathrm { AE } = 6$
• EBDG is a tetrahedron.

Space is referred to an orthonormal coordinate system with origin A in which the points $\mathrm { B } , \mathrm { D }$ and E have respective coordinates $\mathrm { B } ( 12 ; 0 ; 0 ) , \mathrm { D } ( 0 ; 18 ; 0 )$ and $\mathrm { E } ( 0 ; 0 ; 6 )$.

1. Prove that the plane (EBD) has the Cartesian equation $3 x + 2 y + 6 z - 36 = 0$.
2. a. Determine a parametric representation of the line (AG). b. Deduce that the line (AG) intersects the plane (EBD) at a point K with coordinates (4;6;2).
3. Is the line (AG) orthogonal to the plane (EBD)? Justify.
4. a. Let M be the midpoint of the segment $[ \mathrm { ED } ]$. Prove that the points B, K and M are collinear. b. Then construct the point K on the figure given in appendix 2 to be returned with the copy.
5. We denote by P the plane parallel to the plane (ADE) passing through the point K. a. Prove that the plane P intersects the plane (EBD) along a line parallel to the line (ED). b. Then construct on appendix 2 to be returned with the copy the intersection of the plane P and the face EBD of the tetrahedron EBDG.