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).

In space with an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k})$ we are given the points: $$\mathrm{A}(1;2;3),\ \mathrm{B}(3;0;1),\ \mathrm{C}(-1;0;1),\ \mathrm{D}(2;1;-1),\ \mathrm{E}(-1;-2;3)\ \text{and}\ \mathrm{F}(-2;-3;4).$$
For each statement, say whether it is true or false by justifying your answer. An unjustified answer will not be taken into account.
Statement 1: The three points $\mathrm{A}$, $\mathrm{B}$, and C are collinear. Statement 2: The vector $\vec{n}(0;1;-1)$ is a normal vector to the plane (ABC). Statement 3: The line $(\mathrm{EF})$ and the plane $(\mathrm{ABC})$ are secant and their point of intersection is the midpoint of segment [BC]. Statement 4: The lines (AB) and (CD) are secant.

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).

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.

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.

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.

For each of the four statements below, indicate whether it is true or false, by justifying the answer. One point is awarded for each correct answer with proper justification. An answer without justification is not taken into account. An absence of answer is not penalized.

1. We have two dice, identical in appearance, one of which is biased so that 6 appears with probability $\frac{1}{2}$. We take one of the two dice at random, roll it, and obtain 6. Statement 1: the probability that the die rolled is the biased die is equal to $\frac{2}{3}$.
   
2. In the complex plane, consider the points M and N with affixes respectively $z_{\mathrm{M}} = 2 \mathrm{e}^{-\mathrm{i} \frac{\pi}{3}}$ and $z_{\mathrm{N}} = \frac{3 - \mathrm{i}}{2 + \mathrm{i}}$. Statement 2: the line $(MN)$ is parallel to the imaginary axis.
   
3. In an orthonormal frame $(\mathrm{O}, \vec{\imath}, \vec{\jmath}, \vec{k})$ of space, consider the line $d$ with parametric representation: $\left\{ \begin{array}{l} x = 1 + t \\ y = 2 \\ z = 3 + 2t \end{array} \quad t \in \mathbf{R} \right.$. Consider the points $\mathrm{A}$, $\mathrm{B}$ and $\mathrm{C}$ with $\mathrm{A}(-2; 2; 3)$, $\mathrm{B}(0; 1; 2)$ and $\mathrm{C}(4; 2; 0)$. We admit that the points $\mathrm{A}$, $\mathrm{B}$ and $\mathrm{C}$ are not collinear. Statement 3: the line $d$ is orthogonal to the plane (ABC).
   
4. In an orthonormal frame $(\mathrm{O}, \vec{\imath}, \vec{\jmath}, \vec{k})$ of space, consider the line $d$ with parametric representation: $\left\{ \begin{array}{l} x = 1 + t \\ y = 2 \\ z = 3 + 2t \end{array} \quad t \in \mathbf{R} \right.$. Consider the line $\Delta$ passing through the point $\mathrm{D}(1; 4; 1)$ and with direction vector $\vec{v}(2; 1; 3)$. Statement 4: the line $d$ and the line $\Delta$ are not coplanar.

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?

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 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.

Exercise 3 — Part A
In a plane P, consider a triangle ABC right-angled at A. Let $d$ be the line orthogonal to plane P and passing through point B. Consider a point D on this line distinct from point B.
1. Show that the line (AC) is orthogonal to the plane (BAD).
A bicoin is called a tetrahedron whose four faces are right triangles.
2. Show that the tetrahedron ABCD is a bicoin.
3. a. Justify that the edge $[CD]$ is the longest edge of the bicoin ABCD.
b. Let I be the midpoint of edge $[CD]$. Show that point I is equidistant from the 4 vertices of the bicoin ABCD.