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

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

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

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

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.

Exercise 2 -- Common to all candidates

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

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

In space with an orthonormal coordinate system, we consider

• the points $\mathrm{A}(12;0;0), \mathrm{B}(0;-15;0), \mathrm{C}(0;0;20), \mathrm{D}(2;7;-6), \mathrm{E}(7;3;-3)$;
• the plane $\mathscr{P}$ with Cartesian equation: $2x + y - 2z - 5 = 0$

For each of the following statements, indicate whether it is true or false by justifying your answer. An unjustified answer will not be taken into account.
Statement 1
A Cartesian equation of the plane parallel to $\mathscr{P}$ and passing through point A is: $$2x + y + 2z - 24 = 0$$
Statement 2
A parametric representation of line (AC) is: $\left\{ \begin{array}{rl} x &= 9 - 3t \\ y &= 0 \\ z &= 5 + 5t \end{array}, t \in \mathbb{R} \right.$.
Statement 3 Line (DE) and plane $\mathscr{P}$ have at least one point in common.
Statement 4 Line (DE) is orthogonal to plane (ABC).

For each question, four answer options are given, of which only one is correct. For each question, indicate, without justification, the correct answer on your paper. A correct answer is worth 1 point. An incorrect answer or the absence of an answer gives neither points nor deducts any points.
Space is referred to an orthonormal coordinate system. $t$ and $t ^ { \prime }$ denote real parameters. The plane (P) has equation $x - 2 y + 3 z + 5 = 0$. The plane (S) has parametric representation $\left\{ \begin{aligned} x & = - 2 + t + 2 t ^ { \prime } \\ y & = - t - 2 t ^ { \prime } \\ z & = - 1 - t + 3 t ^ { \prime } \end{aligned} \right.$ The line (D) has parametric representation $\left\{ \begin{aligned} x & = - 2 + t \\ y & = - t \\ z & = - 1 - t \end{aligned} \right.$ We are given the points in space $\mathrm { M } ( - 1 ; 2 ; 3 )$ and $\mathrm { N } ( 1 ; - 2 ; 9 )$.

1. A parametric representation of the plane (P) is: a. $\left\{ \begin{array} { r l r } x & = & t \\ y & = & 1 - 2 t \\ z & = & - 1 + 3 t \end{array} \right.$ b. $\left\{ \begin{array} { r l r } x & = t + 2 t ^ { \prime } \\ y & = 1 - t + t ^ { \prime } \\ z & = - 1 - t \end{array} \right.$ c. $\left\{ \begin{aligned} x & = t + t ^ { \prime } \\ y & = 1 - t - 2 t ^ { \prime } \\ z & = 1 - t - 3 t ^ { \prime } \end{aligned} \right.$ d. $\left\{ \begin{array} { l } x = 1 + 2 t + t ^ { \prime } \\ y = 1 - 2 t + 2 t ^ { \prime } \\ z = - 1 - t ^ { \prime } \end{array} \right.$
2. a. The line (D) and the plane (P) are secant at point A(-8;3;2). b. The line (D) and the plane (P) are perpendicular. c. The line (D) is a line of the plane (P). d. The line (D) and the plane (P) are strictly parallel.
3. a. The line (MN) and the line (D) are orthogonal. b. The line (MN) and the line (D) are parallel. c. The line (MN) and the line (D) are secant. d. The line (MN) and the line (D) are coincident.
4. a. The planes $( \mathrm { P } )$ and $( \mathrm { S } )$ are parallel. b. The line $( \Delta )$ with parametric representation $\left\{ \begin{aligned} x & = t \\ y & = - 2 - t \\ z & = - 3 - t \end{aligned} \right.$ is the line of intersection of the planes (P) and (S). c. The point M belongs to the intersection of the planes (P) and (S). d. The planes $( \mathrm { P } )$ and $( \mathrm { S } )$ are perpendicular.

In the orthonormal frame $(A ; \overrightarrow{AB} ; \overrightarrow{AD} ; \overrightarrow{AE})$, $ABCDEFGH$ denotes a cube with side length 1. $\mathscr{P}$ denotes the plane $(AFH)$.
In the orthonormal frame $(A ; \overrightarrow{AB} ; \overrightarrow{AD} ; \overrightarrow{AE})$: a. Plane $\mathscr{P}$ has Cartesian equation: $x + y + z - 1 = 0$. b. Plane $\mathscr{P}$ has Cartesian equation: $x - y + z = 0$. c. Plane $\mathscr{P}$ has Cartesian equation: $-x + y + z = 0$. d. Plane $\mathscr{P}$ has Cartesian equation: $x + y - z = 0$.

In the orthonormal frame $(A ; \overrightarrow{AB} ; \overrightarrow{AD} ; \overrightarrow{AE})$, $ABCDEFGH$ denotes a cube with side length 1. $\mathscr{P}$ denotes the plane $(AFH)$. $I$ is the midpoint of $[AE]$, $J$ is the midpoint of $[BC]$, $L$ is the intersection point of line $(EC)$ and plane $\mathscr{P}$.
a. $\overrightarrow{EG}$ is a normal vector to plane $\mathscr{P}$. b. $\overrightarrow{EL}$ is a normal vector to plane $\mathscr{P}$. c. $\overrightarrow{IJ}$ is a normal vector to plane $\mathscr{P}$. d. $\overrightarrow{DI}$ is a normal vector to plane $\mathscr{P}$.

In the orthonormal frame $(A ; \overrightarrow{AB} ; \overrightarrow{AD} ; \overrightarrow{AE})$, $ABCDEFGH$ denotes a cube with side length 1. $\mathscr{P}$ denotes the plane $(AFH)$. $I$ is the midpoint of $[AE]$, $J$ is the midpoint of $[BC]$, $K$ is the midpoint of $[HF]$, $L$ is the intersection point of line $(EC)$ and plane $\mathscr{P}$.
a. $\overrightarrow{AL} = \frac{1}{2}\overrightarrow{AH} + \frac{1}{2}\overrightarrow{AF}$. b. $\overrightarrow{AL} = \frac{1}{3}\overrightarrow{AK}$. c. $\overrightarrow{ID} = \frac{1}{2}\overrightarrow{IJ}$. d. $\overrightarrow{AL} = \frac{1}{3}\overrightarrow{AB} + \frac{1}{3}\overrightarrow{AD} + \frac{2}{3}\overrightarrow{AE}$.

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

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

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

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

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

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?

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

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

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 $\Delta$.

For each of the four following statements, indicate whether it is true or false, and justify the answer. An unjustified answer is not taken into account. An absence of response is not penalized.
In questions 1 and 2, the space is equipped with an orthonormal coordinate system, and we consider the planes $\mathscr{P}_{1}$ and $\mathscr{P}_{2}$ with equations $x + y + z - 5 = 0$ and $7x - 2y + z - 2 = 0$ respectively.

1. Statement 1: the planes $\mathscr{P}_{1}$ and $\mathscr{P}_{2}$ are perpendicular.
2. Statement 2: the planes $\mathscr{P}_{1}$ and $\mathscr{P}_{2}$ intersect along the line with parametric representation: $$\left\{ \begin{aligned} x & = t \\ y & = 2t + 1, \quad t \in \mathbb{R} \\ z & = -3t + 4 \end{aligned} \right.$$
3. A video game player always adopts the same strategy. Out of the first 312 games played, he wins 223. The games played are treated as a random sample of size 312 from the set of all games. It is desired to estimate the proportion of games that the player will win in the next games he plays, while maintaining the same strategy. Statement 3: at the 95\% confidence level, the proportion of games won should belong to the interval $[0.658; 0.771]$.
4. Consider the following algorithm:

VARIABLES	\begin{tabular}{l} $a, b$ are two real numbers such that $a < b$
$x$ is a real number
$f$ is a function defined on the interval $[a; b]$

\hline PROCESSING &

Read $a$ and $b$

While $b - a > 0.3$

$x$ takes the value $\frac{a + b}{2}$

If $f(x)f(a) > 0$, then $a$ takes the value $x$ otherwise $b$ takes the value $x$

End If

End While

Display $\frac{a + b}{2}$

\hline \end{tabular}
Statement 4: if we enter $a = 1, b = 2$ and $f(x) = x^{2} - 3$, then the algorithm displays as output the number 1.6875.

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.

In space referred to an orthonormal coordinate system, consider the line $\mathscr{D}_1$ with parametric representation: $$\left\{ \begin{array}{l} x = 1 + 2t \\ y = 2 - 3t \\ z = 4t \end{array} \right., t \in \mathbb{R}$$
Statement 4: The line $\mathscr{D}_1$ is parallel to the plane with equation $x + 2y + z - 3 = 0$.
Indicate whether this statement is true or false, justifying your answer.