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

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

Exercise 2 -- Common to all candidates

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

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

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. $\overrightarrow{AL} = \frac{1}{3}\overrightarrow{AB} + \frac{1}{3}\overrightarrow{AD} + \frac{2}{3}\overrightarrow{AE}$.

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?

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 space referred to an orthonormal coordinate system, consider the lines $\mathscr{D}_1$ and $\mathscr{D}_2$ which have the following parametric representations respectively: $$\left\{ \begin{array}{l} x = 1 + 2t \\ y = 2 - 3t \\ z = 4t \end{array} \right., t \in \mathbb{R} \quad \text{and} \quad \left\{ \begin{array}{l} x = -5t' + 3 \\ y = 2t' \\ z = t' + 4 \end{array} \right., t' \in \mathbb{R}$$
Statement 3: The lines $\mathscr{D}_1$ and $\mathscr{D}_2$ are secant.
Indicate whether this statement is true or false, justifying your answer.

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.

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.

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.

Exercise 2 -- Common to all candidates
Alex and Élisa, two drone pilots, are training on a terrain consisting of a flat part bordered by an obstacle. We consider an orthonormal coordinate system ( $\mathrm { O } ; \vec { \imath } , \vec { \jmath } , \vec { k }$ ), with one unit corresponding to ten metres. Six points are defined by their coordinates: $$\mathrm { O } ( 0 ; 0 ; 0 ) , \mathrm { P } ( 0 ; 10 ; 0 ) , \mathrm { Q } ( 0 ; 11 ; 1 ) , \mathrm { T } ( 10 ; 11 ; 1 ) , \mathrm { U } ( 10 ; 10 ; 0 ) \text { and } \mathrm { V } ( 10 ; 0 ; 0 )$$ The flat part is delimited by the rectangle OPUV and the obstacle by the rectangle PQTU.
The two drones are assimilable to two points and follow rectilinear trajectories:

• Alex's drone follows the trajectory carried by the line $( \mathrm { AB } )$ with $\mathrm { A } ( 2 ; 4 ; 0.25 )$ and $\mathrm { B } ( 2 ; 6 ; 0.75 )$;
• Élisa's drone follows the trajectory carried by the line (CD) with C(4; 6; 0.25) and D(2; 6; 0.25).

Part A: Study of Alex's drone trajectory

1. Determine a parametric representation of the line ( AB ).
2. 1. [a.] Justify that the vector $\vec { n } ( 0 ; 1 ; - 1 )$ is a normal vector to the plane (PQU).
   2. [b.] Deduce a Cartesian equation of the plane (PQU).
3. Prove that the line (AB) and the plane (PQU) intersect at the point I with coordinates $\left( 2 ; \frac { 37 } { 3 } ; \frac { 7 } { 3 } \right)$.
4. Explain why, following this trajectory, Alex's drone does not encounter the obstacle.

Part B: Minimum distance between the two trajectories
To avoid a collision between their two devices, Alex and Élisa impose a minimum distance of 4 metres between the trajectories of their drones. For this, we consider a point $M$ on the line (AB) and a point $N$ on the line (CD). There then exist two real numbers $a$ and $b$ such that $\overrightarrow { \mathrm { A } M } = a \overrightarrow { \mathrm { AB } }$ and $\overrightarrow { \mathrm { C } N } = b \overrightarrow { \mathrm { CD } }$.

1. Prove that the coordinates of the vector $\overrightarrow { M N }$ are $( 2 - 2 b ; 2 - 2 a ; - 0.5 a )$.
2. It is admitted that the lines (AB) and (CD) are not coplanar. It is also admitted that the distance $MN$ is minimal when the line ( $MN$ ) is perpendicular to both the line ( AB ) and the line (CD). Prove then that the distance $MN$ is minimal when $a = \frac { 16 } { 17 }$ and $b = 1$.
3. Deduce the minimum value of the distance $MN$ and conclude.

Exercise 3 — Part B
In an orthonormal coordinate system of space, consider the point $\mathrm{A}(3; 1; -5)$ and the line $d$ with parametric representation $\left\{\begin{array}{rl} x &= 2t + 1 \\ y &= -2t + 9 \\ z &= t - 3 \end{array}\right.$ where $t \in \mathbb{R}$.
1. Determine a Cartesian equation of the plane $P$ orthogonal to the line $d$ and passing through point A.
2. Show that the intersection point of plane $P$ and line $d$ is point $\mathrm{B}(5; 5; -1)$.
3. Justify that point $\mathrm{C}(7; 3; -9)$ belongs to plane $P$ then show that triangle ABC is a right isosceles triangle at A.
4. Let $t$ be a real number different from 2 and $M$ the point with parameter $t$ belonging to line $d$.
a. Justify that triangle $\mathrm{AB}M$ is right-angled.
b. Show that triangle $\mathrm{AB}M$ is isosceles at B if and only if the real number $t$ satisfies the equation $t^2 - 4t = 0$.
c. Deduce the coordinates of points $M_1$ and $M_2$ on line $d$ such that the right triangles $\mathrm{AB}M_1$ and $\mathrm{AB}M_2$ are isosceles at B.

Let ABCDEFGH be a cube. The space is referred to the orthonormal coordinate system ( A ; $\overrightarrow { \mathrm { AB } } , \overrightarrow { \mathrm { AD } } , \overrightarrow { \mathrm { AE } }$ ).
For every real $t$, consider the point $M$ with coordinates ( $1 - t ; t ; t$ ).
It is admitted that the lines (BH) and (FC) have respectively the following parametric representations: $$\left\{ \begin{array} { l } { x = 1 - t } \\ { y = t } \\ { z = t } \end{array} \quad \text { where } t \in \mathbb { R } \quad \text { and } \left\{ \begin{array} { r l } x & = 1 \\ y & = t ^ { \prime } \\ z & = 1 - t ^ { \prime } \end{array} \quad \text { where } t ^ { \prime } \in \mathbb { R } . \right. \right.$$

1. Show that for every real $t$, the point $M$ belongs to the line (BH).
2. Show that the lines (BH) and (FC) are orthogonal and non-coplanar.
3. For every real $t ^ { \prime }$, consider the point $M ^ { \prime } \left( 1 ; t ^ { \prime } ; 1 - t ^ { \prime } \right)$. a. Show that for all real numbers $t$ and $t ^ { \prime } , M M ^ { \prime 2 } = 3 \left( t - \frac { 1 } { 3 } \right) ^ { 2 } + 2 \left( t ^ { \prime } - \frac { 1 } { 2 } \right) ^ { 2 } + \frac { 1 } { 6 }$. b. For which values of $t$ and $t ^ { \prime }$ is the distance $M M ^ { \prime }$ minimal? Justify. c. Let P be the point with coordinates $\left( \frac { 2 } { 3 } ; \frac { 1 } { 3 } ; \frac { 1 } { 3 } \right)$ and Q the point with coordinates $\left( 1 ; \frac { 1 } { 2 } ; \frac { 1 } { 2 } \right)$. Justify that the line (PQ) is perpendicular to both lines (BH) and (FC).

Space is referred to an orthonormal coordinate system $( \mathrm { O } ; \vec { \imath } , \vec { \jmath } , \vec { k } )$. We consider the points $\mathrm { A } ( 1 ; 1 ; 4 ) ; \mathrm { B } ( 4 ; 2 ; 5 ) ; \mathrm { C } ( 3 ; 0 ; - 2 )$ and $\mathrm { J } ( 1 ; 4 ; 2 )$. We denote:

• $\mathscr { P }$ the plane passing through points $\mathrm { A }$, $\mathrm { B }$ and $\mathrm { C }$;
• $\mathscr { D }$ the line passing through point $\mathrm { J }$ and with direction vector $\overrightarrow { \mathrm { u } } \left( \begin{array} { l } 1 \\ 1 \\ 3 \end{array} \right)$.

1. Relative position of $\mathscr { P }$ and $\mathscr { D }$ a. Show that the vector $\vec { n } \left( \begin{array} { c } 1 \\ - 4 \\ 1 \end{array} \right)$ is normal to $\mathscr { P }$. b. Determine a Cartesian equation of the plane $\mathscr { P }$. c. Show that $\mathscr { D }$ is parallel to $\mathscr { P }$.

We consider the point $\mathrm { I } ( 1 ; 9 ; 0 )$ and we call $\mathscr { S }$ the sphere with center $\mathrm { I }$ and radius 6.

1. Relative position of $\mathscr { P }$ and $\mathscr { S }$ a. Show that the line $\Delta$ passing through $\mathrm { I }$ and perpendicular to the plane $\mathscr { P }$ intersects this plane $\mathscr { P }$ at the point $\mathrm { H } ( 3 ; 1 ; 2 )$. b. Calculate the distance $\mathrm { IH }$. We admit that for every point $M$ of the plane $\mathscr { P }$ we have $\mathrm { I } M \geqslant \mathrm { IH }$. c. Does the plane $\mathscr { P }$ intersect the sphere $\mathscr { S }$? Justify your answer.
2. Relative position of $\mathscr { D }$ and $\mathscr { S }$ a. Determine a parametric representation of the line $\mathscr { D }$. b. Show that a point $M$ with coordinates $( x ; y ; z )$ belongs to the sphere $\mathscr { S }$ if and only if: $$( x - 1 ) ^ { 2 } + ( y - 9 ) ^ { 2 } + z ^ { 2 } = 36 .$$ c. Show that the line $\mathscr { D }$ intersects the sphere at two distinct points.

Exercise 4 Geometry in space
In the figure below, ABCDEFGH is a rectangular parallelepiped such that $\mathrm{AB} = 5$, $\mathrm{AD} = 3$ and $\mathrm{AE} = 2$. The space is equipped with an orthonormal coordinate system with origin A in which the points B, D and E have coordinates respectively $(5; 0; 0)$, $(0; 3; 0)$ and $(0; 0; 2)$.

1. a. Give, in the coordinate system considered, the coordinates of points H and G. b. Give a parametric representation of the line (GH).
2. Let M be a point of the segment $[\mathrm{GH}]$ such that $\overrightarrow{\mathrm{HM}} = k\overrightarrow{\mathrm{HG}}$ with $k$ a real number in the interval $[0; 1]$. a. Justify that the coordinates of M are $(5k; 3; 2)$. b. Deduce from this that $\overrightarrow{\mathrm{AM}} \cdot \overrightarrow{\mathrm{CM}} = 25k^{2} - 25k + 4$. c. Determine the values of $k$ for which AMC is a triangle right-angled at M.

For the rest of the exercise, we consider that point M has coordinates $(1; 3; 2)$. We admit that triangle AMC is right-angled at M. We recall that the volume of a tetrahedron is given by the formula $\frac{1}{3} \times$ Area of the base $\times h$ where $h$ is the height relative to the base.

1. We consider the point K with coordinates $(1; 3; 0)$. a. Determine a Cartesian equation of the plane (ACD). b. Justify that point K is the orthogonal projection of point M onto the plane (ACD). c. Deduce from this the volume of the tetrahedron MACD.
2. We denote P the orthogonal projection of point D onto the plane (AMC). Calculate the distance DP; give a value rounded to $10^{-1}$.

Exercise 4: Geometry in Space
In space equipped with an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k})$, we consider the points $$\mathrm{A}(0; 8; 6), \quad \mathrm{B}(6; 4; 4) \quad \text{and} \quad \mathrm{C}(2; 4; 0).$$

1. a. Justify that the points $\mathrm{A}$, $\mathrm{B}$ and $\mathrm{C}$ are not collinear. b. Show that the vector $\vec{n}(1; 2; -1)$ is a normal vector to the plane (ABC). c. Determine a Cartesian equation of the plane (ABC).
2. Let D and E be the points with respective coordinates $(0; 0; 6)$ and $(6; 6; 0)$. a. Determine a parametric representation of the line (DE). b. Show that the midpoint I of the segment [BC] belongs to the line (DE).
3. We consider the triangle ABC. a. Determine the nature of triangle ABC. b. Calculate the area of triangle ABC in square units. c. Calculate $\overrightarrow{\mathrm{AB}} \cdot \overrightarrow{\mathrm{AC}}$. d. Deduce a measure of the angle $\widehat{\mathrm{BAC}}$ rounded to 0.1 degree.
4. We consider the point H with coordinates $\left(\dfrac{5}{3}; \dfrac{10}{3}; -\dfrac{5}{3}\right)$. Show that $H$ is the orthogonal projection of the point $O$ onto the plane (ABC). Deduce the distance from point O to the plane (ABC).

In space with an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k})$, we consider:

• the plane $\mathscr{P}_1$ whose Cartesian equation is $2x + y - z + 2 = 0$,
• the plane $\mathscr{P}_2$ passing through point $\mathrm{B}(1; 1; 2)$ and whose normal vector is $\overrightarrow{n_2}\left(\begin{array}{c}1\\-1\\1\end{array}\right)$.

1. a. Give the coordinates of a vector $\overrightarrow{n_1}$ normal to the plane $\mathscr{P}_1$. b. We recall that two planes are perpendicular if a normal vector to one of the planes is orthogonal to a normal vector to the other plane. Show that the planes $\mathscr{P}_1$ and $\mathscr{P}_2$ are perpendicular.
2. a. Determine a Cartesian equation of the plane $\mathscr{P}_2$. b. We denote by $\Delta$ the line whose parametric representation is: $$\left\{\begin{array}{rl} x &= 0 \\ y &= -2 + t \\ z &= t \end{array},\quad t \in \mathbb{R}\right.$$ Show that the line $\Delta$ is the intersection of the planes $\mathscr{P}_1$ and $\mathscr{P}_2$.
3. We consider the point $\mathrm{A}(1; 1; 1)$ and we admit that point A belongs to neither $\mathscr{P}_1$ nor $\mathscr{P}_2$. We denote by H the orthogonal projection of point A onto the line $\Delta$. We recall that, from question 2.b, the line $\Delta$ is the set of points $M_t$ with coordinates $(0; -2+t; t)$, where $t$ denotes any real number. a. Show that, for every real $t$, $\mathrm{A}M_t = \sqrt{2t^2 - 8t + 11}$. b. Deduce that $\mathrm{AH} = \sqrt{3}$.
4. We denote by $\mathscr{D}_1$ the line perpendicular to the plane $\mathscr{P}_1$ passing through point A and $\mathrm{H}_1$ the orthogonal projection of point A onto the plane $\mathscr{P}_1$. a. Determine a parametric representation of the line $\mathscr{D}_1$. b. Deduce that the point $\mathrm{H}_1$ has coordinates $\left(-\frac{1}{3}; \frac{1}{3}; \frac{5}{3}\right)$.
5. Let $\mathrm{H}_2$ be the orthogonal projection of A onto the plane $\mathscr{P}_2$. We admit that $\mathrm{H}_2$ has coordinates $\left(\frac{4}{3}; \frac{2}{3}; \frac{4}{3}\right)$ and that H has coordinates $(0; 0; 2)$. Show that $\mathrm{AH}_1\mathrm{HH}_2$ is a rectangle.

We place ourselves in space with an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k})$.
We consider the point $\mathrm{A}(1; 1; 0)$ and the vector $\vec{u}\begin{pmatrix} 0 \\ 2 \\ -1 \end{pmatrix}$.
We consider the plane $\mathscr{P}$ with equation: $x + 4y + 2z + 1 = 0$.

1. We denote (d) the line passing through A and directed by the vector $\vec{u}$. Determine a parametric representation of (d).
2. Justify that the line (d) and the plane $\mathscr{P}$ intersect at a point B whose coordinates are $(1; -1; 1)$.
3. We consider the point $\mathrm{C}(1; -1; -1)$.
   a. Verify that the points $\mathrm{A}$, $\mathrm{B}$ and C do indeed define a plane.
   b. Show that the vector $\vec{n}\begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}$ is a normal vector to the plane (ABC).
   c. Determine a Cartesian equation of the plane (ABC).
   
4. a. Justify that the triangle ABC is isosceles at A.
   b. Let H be the midpoint of segment [BC]. Calculate the length AH then the area of triangle ABC.
   
5. Let D be the point with coordinates $(0; -1; 1)$.
   a. Show that the line (BD) is a height of the pyramid ABCD.
   b. Deduce from the previous questions the volume of the pyramid ABCD.

We recall that the volume $V$ of a pyramid is given by: $$V = \frac{1}{3}\mathscr{B} \times h,$$ where $\mathscr{B}$ is the area of a base and $h$ the corresponding height.

Exercise 4

In space equipped with an orthonormal coordinate system ( $\mathrm { O } ; \vec { \imath } , \vec { \jmath } , \vec { k }$ ), we consider the points
$$\mathrm { A } ( - 1 ; - 3 ; 2 ) , \quad \mathrm { B } ( 3 ; - 2 ; 6 ) \quad \text { and } \quad \mathrm { C } ( 1 ; 2 ; - 4 ) .$$

1. Prove that the points $\mathrm { A } , \mathrm { B }$ and C define a plane which we will denote $\mathscr { P }$.
2. a. Show that the vector $\vec { n } \left( \begin{array} { c } 13 \\ - 16 \\ - 9 \end{array} \right)$ is normal to the plane $\mathscr { P }$. b. Prove that a Cartesian equation of the plane $\mathscr{P}$ is $13 x - 16 y - 9 z - 17 = 0$.

We denote $\mathscr { D }$ the line passing through the point $\mathrm { F } ( 15 ; - 16 ; - 8 )$ and perpendicular to the plane $\mathscr { P }$.
3. Give a parametric representation of the line $\mathscr { D }$.
4. We call E the point of intersection of the line $\mathscr { D }$ and the plane $\mathscr { P }$. Prove that the point E has coordinates $( 2 ; 0 ; 1 )$.
5. Determine the exact value of the distance from point F to the plane $\mathscr { P }$. 6. Determine the coordinates of the point(s) on the line $\mathscr { D }$ whose distance to the plane $\mathscr { P }$ is equal to half the distance from point F to the plane $\mathscr { P }$.

For each of the following statements, indicate whether it is true or false. Each answer must be justified. An unjustified answer earns no points. In this exercise, the questions are independent of one another. The four statements are placed in the following situation: In space equipped with an orthonormal reference frame ( $\mathrm { O } ; \vec { \imath } , \vec { \jmath } , \vec { k }$ ), we consider the points:
$$\mathrm { A } ( 2 ; 1 ; - 1 ) , \quad \mathrm { B } ( - 1 ; 2 ; 1 ) \text { and } \quad \mathrm { C } ( 5 ; 0 ; - 3 ) .$$
We denote $\mathscr { P }$ the plane with Cartesian equation:
$$x + 5 y - 2 z + 3 = 0 .$$
We denote $\mathscr { D }$ the line with parametric representation:
$$\left\{ \begin{array} { r l } x & = - t + 3 \\ y & = t + 2 \\ z & = 2 t + 1 \end{array} , t \in \mathbb { R } . \right.$$
Statement 1: The vector $\vec { n } \left( \begin{array} { l } 1 \\ 0 \\ 2 \end{array} \right)$ is normal to the plane (OAC).
Statement 2: The lines $\mathscr { D }$ and ( AB ) intersect at point C .
Statement 3: The line $\mathscr { D }$ is parallel to the plane $\mathscr { P }$.
Statement 4: The perpendicular bisector plane of segment $[ \mathrm { BC } ]$, denoted $Q$, has Cartesian equation:
$$3 x - y - 2 z - 7 = 0 .$$
Recall that the perpendicular bisector plane of a segment is the plane perpendicular to this segment and passing through its midpoint.

This exercise is a multiple choice questionnaire. For each question, only one of the four proposed answers is correct. No justification is required. A wrong answer, multiple answers, or the absence of an answer to a question earns neither points nor deducts points. The four questions are independent.
Space is referred to an orthonormal coordinate system $(O; \vec{\imath}, \vec{\jmath}, \vec{k})$.

1. Consider the points $A(1; 0; 3)$ and $B(4; 1; 0)$.
   A parametric representation of the line (AB) is: a. $\left\{ \begin{aligned} x & = 3 + t \\ y & = 1 \\ z & = -3 + 3t \end{aligned} \right.$ with $t \in \mathbb{R}$ b. $\left\{ \begin{array}{l} x = 1 + 4t \\ y = 3 \\ z = 3 \end{array} \right.$ with $t \in \mathbb{R}$ c. $\left\{ \begin{aligned} x & = 1 + 3t \\ y & = t \\ z & = 3 - 3t \end{aligned} \right.$ with $t \in \mathbb{R}$ d. $\left\{ \begin{aligned} x & = 4 + t \\ y & = 1 \\ z & = 3 - 3t \end{aligned} \right.$ with $t \in \mathbb{R}$
   
2. Consider the line (d) with parametric representation $\left\{ \begin{aligned} x & = 3 + 4t \\ y & = 6t \\ z & = 4 - 2t \end{aligned} \right.$ with $t \in \mathbb{R}$
   Among the following points, which one belongs to the line (d)? a. $M(7; 6; 6)$ b. $N(3; 6; 4)$ c. $P(4; 6; -2)$ d. $R(-3; -9; 7)$
   
3. Consider the line $(d')$ with parametric representation $\left\{ \begin{aligned} x & = -2 + 3k \\ y & = -1 - 2k \\ z & = 1 + k \end{aligned} \right.$ with $k \in \mathbb{R}$
   The lines $(d)$ and $(d')$ are: a. secant b. non-coplanar c. parallel d. coincident
   
4. Consider the plane $(P)$ passing through the point $I(2; 1; 0)$ and perpendicular to the line (d).
   An equation of the plane $(P)$ is: a. $2x + 3y - z - 7 = 0$ b. $-x + y - 4z + 1 = 0$ c. $4x + 6y - 2z + 9 = 0$ d. $2x + y + 1 = 0$

For each of the following statements, indicate whether it is true or false. Each answer must be justified. An unjustified answer earns no points.
In space with an orthonormal coordinate system, consider the following points: $$\mathrm{A}(2;0;0), \quad \mathrm{B}(0;4;3), \quad \mathrm{C}(4;4;1), \quad \mathrm{D}(0;0;4) \text{ and } \mathrm{H}(-1;1;2)$$
Statement 1: the points A, C and D define a plane $\mathscr{P}$ with equation $8x - 5y + 4z - 16 = 0$. Statement 2: the points A, B, C and D are coplanar. Statement 3: the lines $(\mathrm{AC})$ and $(\mathrm{BH})$ are secant. It is admitted that the plane (ABC) has the Cartesian equation $x - y + 2z - 2 = 0$. Statement 4: the point H is the orthogonal projection of point D onto the plane (ABC).