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 $\Gamma$?

In the orthonormal frame $(A ; \overrightarrow{AB} ; \overrightarrow{AD} ; \overrightarrow{AE})$, $ABCDEFGH$ denotes a cube with side length 1. $I$ is the midpoint of $[AE]$, $J$ is the midpoint of $[BC]$.
a. Lines $(IJ)$ and $(EC)$ are strictly parallel. b. Lines $(IJ)$ and $(EC)$ are non-coplanar. c. Lines $(IJ)$ and $(EC)$ are intersecting. d. Lines $(IJ)$ and $(EC)$ are coincident.

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 space, with respect to an orthonormal coordinate system, we consider the points $\mathrm { A } ( 1 ; - 1 ; - 1 )$, $\mathrm { B } ( 1 ; 1 ; 1 ) , \mathrm { C } ( 0 ; 3 ; 1 )$ and the plane $\mathscr { P }$ with equation $2 x + y - z + 5 = 0$.
Let $\mathscr { D } _ { 1 }$ be the line with direction vector $\vec { u } ( 2 ; - 1 ; 1 )$ passing through A.
A parametric representation of the line $\mathscr { D } _ { 1 }$ is : a. $\left\{ \begin{array} { l } x = 2 + t \\ y = - 1 - t \\ z = 1 - t \end{array} \quad ( t \in \mathbb { R } ) \right.$ b. $\left\{ \begin{array} { l } x = - 1 + 2 t \\ y = 1 - t \\ z = 1 + t \end{array} \quad ( t \in \mathbb { R } ) \right.$ c. $\left\{ \begin{array} { l } x = 5 + 4 t \\ y = - 3 - 2 t \\ z = 1 + 2 t \end{array} \quad ( t \in \mathbb { R } ) \right.$ d. $\left\{ \begin{array} { l } x = 4 - 2 t \\ y = - 2 + t \\ z = 3 - 4 t \end{array} \quad ( t \in \mathbb { R } ) \right.$

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 $\widehat { \mathrm { BAC } }$.

This exercise is a multiple choice questionnaire. No justification is required. For each question, only one of the four propositions is correct. Each correct answer earns one point. An incorrect answer or no answer does not deduct any points.

1. In an orthonormal coordinate system in space, consider the points $\mathrm { A } ( 2 ; 5 ; - 1 ) , \mathrm { B } ( 3 ; 2 ; 1 )$ and $\mathrm { C } ( 1 ; 3 ; - 2 )$. Triangle ABC is: a. right-angled and not isosceles b. isosceles and not right-angled c. right-angled and isosceles d. equilateral
2. In an orthonormal coordinate system in space, consider the plane $P$ with equation $2 x - y + 3 z - 1 = 0$ and the point $\mathrm { A } ( 2 ; 5 ; - 1 )$. A parametric representation of the line $d$, perpendicular to plane $P$ and passing through A is: a. $\left\{ \begin{aligned} x & = 2 + 2 t \\ y & = 5 + t \\ z & = - 1 + 3 t \end{aligned} \right.$ b. $\left\{ \begin{aligned} x & = 2 + 2 t \\ y & = - 1 + 5 t \\ z & = 3 - t \end{aligned} \right.$ c. $\left\{ \begin{aligned} x & = 6 - 2 t \\ y & = 3 + t \\ z & = 5 - 3 t \end{aligned} \right.$ d. $\left\{ \begin{aligned} x & = 1 + 2 t \\ y & = 4 - t \\ z & = - 2 + 3 t \end{aligned} \right.$
3. Let A and B be two distinct points in the plane. The set of points $M$ in the plane such that $\overrightarrow { M A } \cdot \overrightarrow { M B } = 0$ is: a. the empty set b. the perpendicular bisector of segment [AB] c. the circle with diameter $[ \mathrm { AB } ]$ d. the line (AB)
4. The figure below represents a cube ABCDEFGH. Points I and J are the midpoints of edges $[ \mathrm { GH } ]$ and $[ \mathrm { FG } ]$ respectively. Points M and N are the centres of faces ABFE and BCGF respectively. Lines (IJ) and (MN) are: a. perpendicular b. intersecting, non-perpendicular c. orthogonal d. parallel

In space, with respect to an orthonormal coordinate system, we consider the points $\mathrm { A } ( 1 ; - 1 ; - 1 )$, $\mathrm { B } ( 1 ; 1 ; 1 ) , \mathrm { C } ( 0 ; 3 ; 1 )$ and the plane $\mathscr { P }$ with equation $2 x + y - z + 5 = 0$.
Let $\mathscr { D } _ { 2 }$ be the line with parametric representation $\left\{ \begin{aligned} x & = 1 + t \\ y & = - 3 - t \\ z & = 2 - 2 t \end{aligned} \quad ( t \in \mathbb { R } ) \right.$. a. The line $\mathscr { D } _ { 2 }$ and the plane $\mathscr { P }$ are not secant. b. The line $\mathscr { D } _ { 2 }$ is contained in the plane $\mathscr { P }$. c. The line $\mathscr { D } _ { 2 }$ and the plane $\mathscr { P }$ intersect at the point $\mathrm { E } \left( \frac { 1 } { 3 } ; - \frac { 7 } { 3 } ; \frac { 10 } { 3 } \right)$. d. The line $\mathscr { D } _ { 2 }$ and the plane $\mathscr { P }$ intersect at the point $\mathrm { F } \left( \frac { 4 } { 3 } ; - \frac { 1 } { 3 } ; \frac { 22 } { 3 } \right)$.

For each of the following propositions, indicate whether it is true or false and justify each answer. An unjustified answer will not be taken into account.
We are in space with an orthonormal coordinate system. We consider the plane $\mathscr{P}$ with equation $x - y + 3z + 1 = 0$ and the line $\mathscr{D}$ whose parametric representation is $\left\{\begin{array}{l} x = 2t \\ y = 1 + t \\ z = -5 + 3t \end{array}, \quad t \in \mathbb{R}\right.$ We are given the points $A(1; 1; 0)$, $B(3; 0; -1)$ and $C(7; 1; -2)$
Proposition 1:
A parametric representation of the line $(AB)$ is $\left\{\begin{array}{l} x = 5 - 2t \\ y = -1 + t \\ z = -2 + t \end{array}, t \in \mathbb{R}\right.$
Proposition 2: The lines $\mathscr{D}$ and $(AB)$ are orthogonal.
Proposition 3: The lines $\mathscr{D}$ and $(AB)$ are coplanar.
Proposition 4: The line $\mathscr{D}$ intersects the plane $\mathscr{P}$ at point $E$ with coordinates $(8; -3; -4)$.
Proposition 5: The planes $\mathscr{P}$ and $(ABC)$ are parallel.

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?

Exercise 3 (4 points)

For each of the four following propositions, indicate whether it is true or false by justifying the answer. One point is awarded for each correct answer with proper justification. An unjustified answer is not taken into account. An absence of answer is not penalized. Space is equipped with an orthonormal coordinate system ( $\mathrm { O } , \vec { \imath } , \vec { \jmath } , \vec { k }$ ). We consider the points $\mathrm { A } ( 1 ; 2 ; 5 ) , \mathrm { B } ( - 1 ; 6 ; 4 ) , \mathrm { C } ( 7 ; - 10 ; 8 )$ and $\mathrm { D } ( - 1 ; 3 ; 4 )$.

1. Proposition 1: The points $\mathrm { A } , \mathrm { B }$ and C define a plane.
2. We admit that the points $\mathrm { A } , \mathrm { B }$ and D define a plane. Proposition 2: A Cartesian equation of the plane (ABD) is $x - 2 z + 9 = 0$.
3. Proposition 3: A parametric representation of the line (AC) is $$\left\{ \begin{aligned} x & = \frac { 3 } { 2 } t - 5 \\ y & = - 3 t + 14 \quad t \in \mathbb { R } \\ z & = - \frac { 3 } { 2 } t + 2 \end{aligned} \right.$$
4. Let $\mathscr { P }$ be the plane with Cartesian equation $2 x - y + 5 z + 7 = 0$ and $\mathscr { P } ^ { \prime }$ the plane with Cartesian equation $- 3 x - y + z + 5 = 0$. Proposition 4: The planes $\mathscr { P }$ and $\mathscr { P } ^ { \prime }$ are parallel.

In space, with respect to an orthonormal coordinate system, we consider the points $\mathrm { A } ( 1 ; - 1 ; - 1 )$, $\mathrm { B } ( 1 ; 1 ; 1 ) , \mathrm { C } ( 0 ; 3 ; 1 )$ and the plane $\mathscr { P }$ with equation $2 x + y - z + 5 = 0$.
a. The intersection of the plane $\mathscr { P }$ and the plane $( \mathrm { ABC } )$ is reduced to a single point. b. The plane $\mathscr { P }$ and the plane ( ABC ) are identical. c. The plane $\mathscr { P }$ intersects the plane $( \mathrm { ABC } )$ along a line. d. The plane $\mathscr { P }$ and the plane ( ABC ) are strictly parallel.

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, with respect to an orthonormal coordinate system, we consider the points $\mathrm { A } ( 1 ; - 1 ; - 1 )$, $\mathrm { B } ( 1 ; 1 ; 1 ) , \mathrm { C } ( 0 ; 3 ; 1 )$ and the plane $\mathscr { P }$ with equation $2 x + y - z + 5 = 0$.
A measure of the angle $\widehat { \mathrm { BAC } }$ rounded to the nearest tenth of a degree is equal to : a. $22,2 ^ { \circ }$ b. $0,4 ^ { \circ }$ c. $67,8 ^ { \circ }$ d. $1,2 ^ { \circ }$

(For candidates who have not followed the specialization course)
In space equipped with an orthonormal coordinate system $(\mathrm{O}, \vec{\imath}, \vec{\jmath}, \vec{k})$, we consider the tetrahedron ABCD whose vertices have coordinates: $$\mathrm{A}(1;-\sqrt{3};0);\quad \mathrm{B}(1;\sqrt{3};0);\quad \mathrm{C}(-2;0;0);\quad \mathrm{D}(0;0;2\sqrt{2}).$$

1. Prove that the plane (ABD) has the Cartesian equation $4x + z\sqrt{2} = 4$.
2. We denote by $\mathscr{D}$ the line whose parametric representation is $$\left\{\begin{array}{l} x = t \\ y = 0 \\ z = t\sqrt{2} \end{array}, t \in \mathbb{R}\right.$$ a. Prove that $\mathscr{D}$ is the line that is parallel to $(\mathrm{CD})$ and passes through O. b. Determine the coordinates of point G, the intersection of the line $\mathscr{D}$ and the plane (ABD).
3. a. We denote by L the midpoint of segment $[\mathrm{AC}]$. Prove that the line (BL) passes through point O and is orthogonal to the line (AC). b. Prove that triangle ABC is equilateral and determine the centre of its circumscribed circle.
4. Prove that the tetrahedron ABCD is regular, that is, a tetrahedron whose six edges all have the same length.

Exercise 4 — Candidates who have not followed the specialization course
In space, we consider a tetrahedron ABCD whose faces ABC, ACD and ABD are right-angled and isosceles triangles at A. We denote by E, F and G the midpoints of sides $[\mathrm{AB}]$, $[\mathrm{BC}]$ and $[\mathrm{CA}]$ respectively. We choose AB as the unit of length and we place ourselves in the orthonormal coordinate system $(\mathrm{A}; \overrightarrow{\mathrm{AB}}, \overrightarrow{\mathrm{AC}}, \overrightarrow{\mathrm{AD}})$ of space.

1. We denote by $\mathscr { P }$ the plane that passes through A and is perpendicular to the line $(\mathrm{DF})$. We denote by H the point of intersection of plane $\mathscr { P }$ and line (DF). a. Give the coordinates of points D and F. b. Give a parametric representation of line (DF). c. Determine a Cartesian equation of plane $\mathscr { P }$. d. Calculate the coordinates of point H. e. Prove that the angle $\widehat{\mathrm{EHG}}$ is a right angle.
2. We denote by $M$ a point on line (DF) and by $t$ the real number such that $\overrightarrow{\mathrm{DM}} = t \overrightarrow{\mathrm{DF}}$. We denote by $\alpha$ the measure in radians of the geometric angle $\widehat{\mathrm{EMG}}$. The purpose of this question is to determine the position of point $M$ so that $\alpha$ is maximum. a. Prove that $ME^{2} = \frac{3}{2} t^{2} - \frac{5}{2} t + \frac{5}{4}$. b. Prove that triangle $M\mathrm{EG}$ is isosceles at $M$. Deduce that $ME \sin\left(\frac{\alpha}{2}\right) = \frac{1}{2\sqrt{2}}$. c. Justify that $\alpha$ is maximum if and only if $\sin\left(\frac{\alpha}{2}\right)$ is maximum. Deduce that $\alpha$ is maximum if and only if $ME^{2}$ is minimum. d. Conclude.

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

• the points $\mathrm { A } ( 0 ; 1 ; - 1 )$ and $\mathrm { B } ( - 2 ; 2 ; - 1 )$.
• the line $\mathscr { D }$ with parametric representation $\left\{ \begin{array} { r l } x & = - 2 + t \\ y & = 1 + t \\ z & = - 1 - t \end{array} , t \in \mathbb { R } \right.$.

1. Determine a parametric representation of the line ( AB ).
2. a. Show that the lines $( \mathrm { AB } )$ and $\mathscr { D }$ are not parallel. b. Show that the lines $( \mathrm { AB } )$ and $\mathscr { D }$ are not intersecting.

In the following, the letter $u$ denotes a real number. We consider the point $M$ of the line $\mathscr { D }$ with coordinates ( $- 2 + u ; 1 + u ; - 1 - u$ ).
3. Verify that the plane $\mathscr { P }$ with equation $x + y - z - 3 u = 0$ is orthogonal to the line $\mathscr { D }$ and passes through the point $M$.
4. Show that the plane $\mathscr { P }$ and the line (AB) intersect at a point $N$ with coordinates $( - 4 + 6 u ; 3 - 3 u ; - 1 )$.
5. a. Show that the line $( M N )$ is perpendicular to the line $\mathscr { D }$. b. Does there exist a value of the real number $u$ for which the line ( $M N$ ) is perpendicular to the line $( \mathrm { AB } )$ ? 6. a. Express $M N ^ { 2 }$ as a function of $u$. b. Deduce the value of the real number $u$ for which the distance $M N$ is minimal.

In space, we consider a pyramid SABCE with square base ABCE with centre O. Let D be the point in space such that ( O ; $\overrightarrow { \mathrm { OA } } , \overrightarrow { \mathrm { OB } } , \overrightarrow { \mathrm { OD } }$ ) is an orthonormal frame. The point S has coordinates $( 0 ; 0 ; 3 )$ in this frame.

Part A

1. Let U be the point on the line $( \mathrm { SB } )$ with height 1. Construct the point U on the attached figure in appendix 1.
2. Let V be the intersection point of the plane (AEU) and the line (SC). Show that the lines (UV) and (BC) are parallel. Construct the point V on the attached figure in appendix 1.
3. Let K be the point with coordinates $\left( \frac { 5 } { 6 } ; - \frac { 1 } { 6 } ; 0 \right)$. Show that K is the foot of the altitude from U in the trapezoid AUVE.

Part B

In this part, we admit that the area of the quadrilateral AUVE is $\frac { 5 \sqrt { 43 } } { 18 }$.

1. We admit that the point $U$ has coordinates $\left( 0 ; \frac { 2 } { 3 } ; 1 \right)$. Verify that the plane (EAU) has equation $3 x - 3 y + 5 z - 3 = 0$.
2. Give a parametric representation of the line (d) perpendicular to the plane (EAU) passing through the point $S$.
3. Determine the coordinates of H, the intersection point of the line (d) and the plane (EAU).
4. The plane (EAU) divides the pyramid (SABCE) into two solids. Do these two solids have the same volume?

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.

For each of the following statements, indicate whether it is true or false and justify the answer.
Space is equipped with an orthonormal coordinate system $(\mathrm{O}, \vec{\imath}, \vec{\jmath}, \vec{k})$. The points $\mathrm{A}$, $\mathrm{B}$, $\mathrm{C}$ are defined by their coordinates: $$\mathrm{A}(3; -1; 4), \quad \mathrm{B}(-1; 2; -3), \quad \mathrm{C}(4; -1; 2).$$ The plane $\mathscr{P}$ has the Cartesian equation: $2x - 3y + 2z - 7 = 0$. The line $\Delta$ has the parametric representation $\left\{\begin{array}{rl} x &= -1 + 4t \\ y &= 4 - t \\ z &= -8 + 2t \end{array}, t \in \mathbb{R}\right.$.
Statement 1: The lines $\Delta$ and $(AC)$ are orthogonal.
Statement 2: The points $\mathrm{A}$, $\mathrm{B}$ and $\mathrm{C}$ determine a plane and this plane has the Cartesian equation $2x + 5y + z - 5 = 0$.
Statement 3: All points whose coordinates $(x; y; z)$ are given by $$\left\{\begin{array}{rl} x &= 1 + s - 2s' \\ y &= 1 - 2s + s' \\ z &= 1 - 4s + 2s' \end{array}\right., \quad s, s' \in \mathbb{R}$$ lie in the plane $\mathscr{P}$.
Statement 4: There exists a plane parallel to the plane $\mathscr{P}$ which contains the line $\Delta$.

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?

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.

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).

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)?

$ABCDEFGH$ 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 $FBGE$.