The cube ABCDEFGH has edge length 1 cm. The point I is the midpoint of segment [AB] and the point J is the midpoint of segment [CG].
We place ourselves in 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. Show that the vector $\overrightarrow{\mathrm{EJ}}$ is normal to the plane (FHI).
3. Show that a Cartesian equation of the plane (FHI) is $-2x - 2y + z + 1 = 0$.
4. Determine a parametric representation of the line (EJ).
5. 1. [a.] We denote K the orthogonal projection of point E onto the plane $(\mathrm{FHI})$. Calculate its coordinates.
   2. [b.] Show that the volume of the pyramid EFHI is $\frac{1}{6}\mathrm{~cm}^3$.
   We may use the point L, midpoint of segment $[\mathrm{EF}]$. We admit that this point is the orthogonal projection of point I onto the plane (EFH).
   1. [c.] Deduce from the two previous questions the area of triangle FHI.

Space is equipped with an orthonormal coordinate system ( $\mathrm { O } ; \vec { \imath } , \vec { \jmath } , \vec { k }$ ). Consider the points $\mathrm { A } ( 5 ; 5 ; 0 ) , \mathrm { B } ( 0 ; 5 ; 0 ) , \mathrm { C } ( 0 ; 0 ; 10 )$ and $\mathrm { D } \left( 0 ; 0 ; - \frac { 5 } { 2 } \right)$.

1. a. Show that $\overrightarrow { n _ { 1 } } \left( \begin{array} { c } 1 \\ - 1 \\ 0 \end{array} \right)$ is a normal vector to the plane (CAD). b. Deduce that the plane (CAD) has the Cartesian equation: $x - y = 0$.
2. Consider the line $\mathscr { D }$ with parametric representation $\left\{ \begin{aligned} x & = \frac { 5 } { 2 } t \\ y & = 5 - \frac { 5 } { 2 } t \text { where } t \in \mathbb { R } \text { . } \\ z & = 0 \end{aligned} \right.$ a. We admit that the line $\mathscr { D }$ and the plane (CAD) intersect at a point H. Justify that the coordinates of H are $\left( \frac { 5 } { 2 } ; \frac { 5 } { 2 } ; 0 \right)$. b. Prove that the point H is the orthogonal projection of B onto the plane (CAD).
3. a. Prove that the triangle ABH is right-angled at H. b. Deduce that the area of triangle ABH is equal to $\frac { 25 } { 4 }$.
4. a. Prove that ( CO ) is the height of the tetrahedron ABCH from C. b. Deduce the volume of the tetrahedron ABCH.

We recall that the volume of a tetrahedron is given by: $V = \frac { 1 } { 3 } \mathscr { B } h$, where $\mathscr { B }$ is the area of a base and h the height relative to this base.
5. We admit that the triangle ABC is right-angled at B. Deduce from the previous questions the distance from point H to the plane (ABC).

Exercise 4

The objective of this exercise is to determine the distance between two non-coplanar lines. By definition, the distance between two non-coplanar lines in space, $( d _ { 1 } )$ and $( d _ { 2 } )$ is the length of the segment $[\mathrm { EF }]$, where E and F are points belonging respectively to $\left( d _ { 1 } \right)$ and to $( d _ { 2 } )$ such that the line (EF) is orthogonal to $( d _ { 1 } )$ and $( d _ { 2 } )$. The space is equipped with an orthonormal coordinate system $( \mathrm { O } ; \vec { \imath } , \vec { \jmath } , \vec { k } )$. Let $\left( d _ { 1 } \right)$ be the line passing through $\mathrm { A } ( 1 ; 2 ; - 1 )$ with direction vector $\overrightarrow { u _ { 1 } } \left( \begin{array} { l } 1 \\ 2 \\ 0 \end{array} \right)$ and $\left( d _ { 2 } \right)$ the line with parametric representation: $\left\{ \begin{array} { l } x = 0 \\ y = 1 + t \\ z = 2 + t \end{array} , t \in \mathbb { R } \right.$.

1. Give a parametric representation of the line $\left( d _ { 1 } \right)$.
2. Prove that the lines $\left( d _ { 1 } \right)$ and $\left( d _ { 2 } \right)$ are non-coplanar.
3. Let $\mathscr { P }$ be the plane passing through A and directed by the non-collinear vectors $\overrightarrow { u _ { 1 } }$ and $\vec { w } \left( \begin{array} { c } 2 \\ - 1 \\ 1 \end{array} \right)$. Justify that a Cartesian equation of the plane $\mathscr { P }$ is: $- 2 x + y + 5 z + 5 = 0$.
4. a. Without seeking to calculate the coordinates of the intersection point, justify that the line $( d _ { 2 } )$ and the plane $\mathscr { P }$ are secant. b. We denote F the intersection point of the line $( d _ { 2 } )$ and the plane $\mathscr { P }$. Verify that the point F has coordinates $\left( 0 ; - \frac { 5 } { 3 } ; - \frac { 2 } { 3 } \right)$. Let $( \delta )$ be the line passing through F with direction vector $\vec { w }$. It is admitted that the lines $( \delta )$ and $( d _ { 1 } )$ are secant at a point E with coordinates $\left( - \frac { 2 } { 3 } ; - \frac { 4 } { 3 } ; - 1 \right)$.
5. a. Justify that the distance EF is the distance between the lines $\left( d _ { 1 } \right)$ and $\left( d _ { 2 } \right)$. b. Calculate the distance between the lines $\left( d _ { 1 } \right)$ and $\left( d _ { 2 } \right)$.

Space is equipped with an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k})$. We consider the three points $\mathrm{A}(3;0;0)$, $\mathrm{B}(0;2;0)$ and $\mathrm{C}(0;0;2)$.
The objective of this exercise is to demonstrate the following property: ``The square of the area of triangle ABC is equal to the sum of the squares of the areas of the three other faces of the tetrahedron OABC''.
Part 1: Distance from point O to the plane (ABC)

1. Demonstrate that the vector $\vec{n}(2;3;3)$ is normal to the plane (ABC).
2. Demonstrate that a Cartesian equation of the plane (ABC) is: $2x + 3y + 3z - 6 = 0$.
3. Give a parametric representation of the line $d$ passing through O and with direction vector $\vec{n}$.
4. We denote H the point of intersection of the line $d$ and the plane (ABC). Determine the coordinates of point H.
5. Deduce that the distance from point O to the plane (ABC) is equal to $\dfrac{3\sqrt{22}}{11}$.

Part 2: Demonstration of the property

1. Demonstrate that the volume of the tetrahedron OABC is equal to 2.
2. Deduce that the area of triangle ABC is equal to $\sqrt{22}$.
3. Demonstrate that for the tetrahedron OABC, ``the square of the area of triangle ABC is equal to the sum of the squares of the areas of the three other faces of the tetrahedron''. Recall that the volume of a tetrahedron is given by $V = \dfrac{1}{3}B \times h$ where $B$ is the area of a base of the tetrahedron and $h$ is the height relative to this base.

In an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k})$ of space, we consider the plane $(P)$ with equation:
$$(P) : \quad 2x + 2y - 3z + 1 = 0 .$$
We consider the three points $\mathrm{A}$, $\mathrm{B}$ and $\mathrm{C}$ with coordinates:
$$\mathrm{A}(1;0;1), \quad \mathrm{B}(2;-1;1) \quad \text{and} \quad \mathrm{C}(-4;-6;5).$$
The purpose of this exercise is to study the ratio of areas between a triangle and its orthogonal projection onto a plane.
Part A

1. For each of the points $\mathrm{A}$, $\mathrm{B}$ and $\mathrm{C}$, verify whether it belongs to the plane $(P)$.
2. Show that the point $\mathrm{C}^{\prime}(0;-2;-1)$ is the orthogonal projection of point $\mathrm{C}$ onto the plane $(P)$.
3. Determine a parametric representation of the line (AB).
4. We admit the existence of a unique point H satisfying the two conditions $$\left\{ \begin{array}{l} \mathrm{H} \in (\mathrm{AB}) \\ (\mathrm{AB}) \text{ and } (\mathrm{HC}) \text{ are orthogonal.} \end{array} \right.$$ Determine the coordinates of point H.

Part B
We admit that the coordinates of the vector $\overrightarrow{\mathrm{HC}}$ are: $\overrightarrow{\mathrm{HC}} \left( \begin{array}{c} -\frac{11}{2} \\ -\frac{11}{2} \\ 4 \end{array} \right)$.

1. Calculate the exact value of $\| \overrightarrow{\mathrm{HC}} \|$.
2. Let $S$ be the area of triangle ABC. Determine the exact value of $S$.

Part C
We admit that $\mathrm{HC}^{\prime} = \sqrt{\frac{17}{2}}$.

1. Let $\alpha = \widehat{\mathrm{CHC}^{\prime}}$. Determine the value of $\cos(\alpha)$.
2. a. Show that the lines $(\mathrm{C}^{\prime}\mathrm{H})$ and (AB) are perpendicular. b. Calculate $S^{\prime}$ the area of triangle $\mathrm{ABC}^{\prime}$, give the exact value. c. Give a relationship between $S$, $S^{\prime}$ and $\cos(\alpha)$.

Space is equipped with an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k})$. Consider:

• the points $\mathrm{A}(-2; 0; 2)$, $\mathrm{B}(-1; 3; 0)$, $\mathrm{C}(1; -1; 2)$ and $\mathrm{D}(0; 0; 3)$.
• the line $\mathscr{D}_1$ whose parametric representation is $\left\{ \begin{aligned} x &= t \\ y &= 3t \\ z &= 3 + 5t \end{aligned} \right.$ with $t \in \mathbb{R}$.
• the line $\mathscr{D}_2$ whose parametric representation is $\left\{ \begin{aligned} x &= 1 + 3s \\ y &= -1 - 5s \\ z &= 2 - 6s \end{aligned} \right.$ with $s \in \mathbb{R}$.

1. Prove that the points $\mathrm{A}$, $\mathrm{B}$ and $\mathrm{C}$ are not collinear.
2. 1. [a.] Prove that the vector $\vec{n}\begin{pmatrix} 1 \\ 3 \\ 5 \end{pmatrix}$ is orthogonal to the plane (ABC).
   2. [b.] Justify that a Cartesian equation of the plane (ABC) is: $$x + 3y + 5z - 8 = 0$$
   3. [c.] Deduce that the points $\mathrm{A}$, $\mathrm{B}$, $\mathrm{C}$ and $\mathrm{D}$ are not coplanar.
3. 1. [a.] Justify that the line $\mathscr{D}_1$ is the altitude of the tetrahedron ABCD from D. It is admitted that the line $\mathscr{D}_2$ is the altitude of the tetrahedron ABCD from C.
   2. [b.] Prove that the lines $\mathscr{D}_1$ and $\mathscr{D}_2$ are secant and determine the coordinates of their point of intersection.
4. 1. [a.] Determine the coordinates of the orthogonal projection H of point D onto the plane (ABC).
   2. [b.] Calculate the distance from point D to the plane (ABC). Round the result to the nearest hundredth.

Consider a cube ABCDEFGH and the space is referred to the orthonormal coordinate system $(\mathrm{A}; \overrightarrow{\mathrm{AB}}, \overrightarrow{\mathrm{AD}}, \overrightarrow{\mathrm{AE}})$. For any real $m$ belonging to the interval $[0; 1]$, we consider the points $K$ and $L$ with coordinates: $$K(m; 0; 0) \text{ and } L(1-m; 1; 1).$$

1. Give the coordinates of points E and C in this coordinate system.
2. In this question, $m = 0$. Thus, the point $\mathrm{L}(1; 1; 1)$ coincides with point G, the point $\mathrm{K}(0; 0; 0)$ coincides with point A and the plane (LEK) is therefore the plane (GEA). a. Justify that the vector $\overrightarrow{\mathrm{DB}} \left(\begin{array}{c} 1 \\ -1 \\ 0 \end{array}\right)$ is normal to the plane (GEA). b. Determine a Cartesian equation of the plane (GEA).
3. In this question, $m$ is any real number in the interval $[0; 1]$. a. Prove that $\mathrm{CKEL}$ is a parallelogram. b. Justify that $\overrightarrow{KC} \cdot \overrightarrow{KE} = m(m-1)$. c. Prove that $\mathrm{CKEL}$ is a rectangle if, and only if, $m = 0$ or $m = 1$.
4. In this question, $m = \frac{1}{2}$. Thus, L has coordinates $\left(\frac{1}{2}; 1; 1\right)$ and K has coordinates $\left(\frac{1}{2}; 0; 0\right)$. a. Prove that the parallelogram CKEL is then a rhombus. b. Using question 3.b., determine an approximate value to the nearest degree of the measure of the angle $\widehat{\mathrm{CKE}}$.

Space is referred to an orthonormal coordinate system ( $\mathrm { O } ; \vec { \imath } , \vec { \jmath } , \vec { k }$ ). We consider:

• $\alpha$ any real number;
• the points $\mathrm { A } ( 1 ; 1 ; 0 ) , \mathrm { B } ( 2 ; 1 ; 0 )$ and $\mathrm { C } ( \alpha ; 3 ; \alpha )$;
• (d) the line with parametric representation:

$$\left\{ \begin{array} { l } x = 1 + t \\ y = 2 t , \quad t \in \mathbb { R } \\ z = - t \end{array} \right.$$ For each of the following statements, specify whether it is true or false, then justify the answer given. An answer without justification will not be taken into account. Statement 1: For all values of $\alpha$, the points $A , B$ and $C$ define a plane and a normal vector to this plane is $\vec { J } \left( \begin{array} { l } 0 \\ 1 \\ 0 \end{array} \right)$. Statement 2: There exists exactly one value of $\alpha$ such that the lines ( $A C$ ) and (d) are parallel. Statement 3: A measure of the angle $\widehat { \mathrm { OAB } }$ is $135 ^ { \circ }$. Statement 4: The orthogonal projection of point $A$ onto the line (d) is the point $\mathrm { H } ( 1 ; 2 ; 2 )$. Statement 5: The sphere with center $O$ and radius 1 intersects the line $( d )$ at two distinct points. Recall that the sphere with center $\Omega$ and radius $r$ is the set of points in space at distance $r$ from $\Omega$.

For each of the four following statements, indicate whether it is true or false, by justifying the answer. An unjustified answer is not taken into account. An absence of answer is not penalised.
Consider a cube ABCDEFGH with edge length 1 and the point I defined by $\overrightarrow { \mathrm { FI } } = \frac { 1 } { 3 } \overrightarrow { \mathrm { FB } }$. One may place oneself in the orthonormal coordinate system of space $( \mathrm { A } ; \overrightarrow { \mathrm { AB } } , \overrightarrow { \mathrm { AD } } , \overrightarrow { \mathrm { AE } } )$.

1. Consider the triangle HAC.

Statement 1: The triangle HAC is a right-angled triangle.
2. Consider the lines (HF) and (DI).
Statement 2: The lines (HF) and (DI) are secant.
3. Consider a real number $\alpha$ belonging to the interval $] 0 ; \pi [$.
Consider the vector $\vec { u }$ with coordinates $\left( \begin{array} { c } \sin ( \alpha ) \\ \sin ( \pi - \alpha ) \\ \sin ( - \alpha ) \end{array} \right)$. Statement 3: The vector $\vec { u }$ is a normal vector to the plane (FAC).
4. The cube ABCDEFGH has 8 vertices. We are interested in the number $N$ of segments that can be constructed by connecting 2 distinct vertices of the cube. Statement 4: $N = \frac { 8 ^ { 2 } } { 2 }$.

This exercise is a multiple choice questionnaire. For each question, only one of the three propositions is correct.
In all the following questions, space is referred to an orthonormal coordinate system.

1. Consider the line $\Delta_1$ with parametric representation $\left\{ \begin{aligned} x &= 1 - 3t \\ y &= 4 + 2t \\ z &= t \end{aligned} \right.$, where $t \in \mathbb{R}$ as well as the line $\Delta_2$ with parametric representation $\left\{ \begin{aligned} x &= -4 + s \\ y &= 2 + 2s \\ z &= -1 + s \end{aligned} \right.$, where $s \in \mathbb{R}$. a. The lines $\Delta_1$ and $\Delta_2$ are parallel. b. The lines $\Delta_1$ and $\Delta_2$ are orthogonal. c. The lines $\Delta_1$ and $\Delta_2$ are secant.
2. Consider the line $d$ with parametric representation $\left\{ \begin{aligned} x &= 1 + t \\ y &= 3 - t \\ z &= 1 + 2t \end{aligned} \right.$, where $t \in \mathbb{R}$, and the plane $P$ with Cartesian equation: $4x + 2y - z + 3 = 0$. a. The line $d$ is contained in the plane $P$. b. The line $d$ is strictly parallel to the plane $P$. c. The line $d$ is secant to the plane $P$.
3. Consider the points $\mathrm{A}(3;2;1)$, $\mathrm{B}(7;3;1)$, $\mathrm{C}(-1;4;5)$ and $\mathrm{D}(-3;3;5)$. a. The points $\mathrm{A}$, $\mathrm{B}$, $\mathrm{C}$ and D are not coplanar. b. The points $\mathrm{A}$, $\mathrm{B}$ and C are collinear. c. $\overrightarrow{\mathrm{AB}}$ and $\overrightarrow{\mathrm{CD}}$ are collinear.
4. Consider the planes $Q$ and $Q'$ with respective Cartesian equations $3x - 2y + z + 1 = 0$ and $4x + y - z + 3 = 0$. a. The point $\mathrm{R}(1;1;-2)$ belongs to both planes. b. The two planes are orthogonal. c. The two planes are secant with intersection the line with parametric representation $$\left\{ \begin{aligned} x &= t \\ y &= 7t + 4, \text{ where } t \in \mathbb{R}. \\ z &= 11t + 7 \end{aligned} \right.$$

In space with respect to an orthonormal coordinate system $(O; \vec{\imath}, \vec{\jmath}, \vec{k})$, we consider the points:
$$A(4; -4; 4), \quad B(5; -3; 2), \quad C(6; -2; 3), \quad D(5; 1; 1)$$

1. Prove that triangle ABC is right-angled at $B$.
2. Justify that a Cartesian equation of the plane (ABC) is: $$x - y - 8 = 0.$$
3. We denote $d$ the line passing through point $D$ and perpendicular to the plane (ABC). a. Determine a parametric representation of the line $d$. b. We denote H the orthogonal projection of point $D$ onto the plane $(ABC)$. Determine the coordinates of point H. c. Show that $DH = 2\sqrt{2}$.
4. a. Show that the volume of the pyramid ABCD is equal to 2. We recall that the volume V of a pyramid is calculated using the formula: $$V = \frac{1}{3} \times \mathscr{B} \times h$$ where $\mathscr{B}$ is the area of a base of the pyramid and $h$ is the corresponding height. b. We admit that the area of triangle BCD is equal to $\frac{\sqrt{42}}{2}$. Deduce the exact value of the distance from point A to the plane (BCD).

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 earns neither points nor deducts points.
Throughout the exercise, we consider that space is equipped with an orthonormal reference frame $(\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k})$. We consider:

• the points $\mathrm{A}(-3; 1; 4)$ and $\mathrm{B}(1; 5; 2)$
• the plane $\mathscr{P}$ with Cartesian equation $4x + 4y - 2z + 3 = 0$
• the line $(d)$ with parametric representation $\left\{\begin{aligned} x &= -6 + 3t \\ y &= 1 \\ z &= 9 - 5t \end{aligned}\right.$, where $t \in \mathbb{R}$.

1. The lines $(\mathrm{AB})$ and $(d)$ are: a. secant and non-perpendicular. b. perpendicular. c. non-coplanar. d. parallel.
2. The line $(\mathrm{AB})$ is: a. included in the plane $\mathscr{P}$. b. strictly parallel to the plane $\mathscr{P}$. c. secant and non-orthogonal to the plane $\mathscr{P}$. d. orthogonal to the plane $\mathscr{P}$.
3. We consider the plane $\mathscr{P}'$ with Cartesian equation $2x + y + 6z + 5 = 0$. The planes $\mathscr{P}$ and $\mathscr{P}'$ are: a. secant and non-perpendicular. b. perpendicular. c. identical. d. strictly parallel.
4. We consider the point $\mathrm{C}(0; 1; -1)$. The value of the angle $\widehat{\mathrm{BAC}}$ rounded to the nearest degree is: a. $90^\circ$ b. $51^\circ$ c. $39^\circ$ d. $0^\circ$

Exercise 2
Consider the cube ABCDEFGH. We place the point M such that $\overrightarrow{\mathrm{BM}} = \overrightarrow{\mathrm{AB}}$.
Part A

1. Show that the lines (FG) and (FM) are perpendicular.
2. Show that the points A, M, G and H are coplanar.

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

1. Determine the coordinates of the vectors $\overrightarrow{\mathrm{GM}}$ and $\overrightarrow{\mathrm{AH}}$ and show that they are not collinear.
2. 1. [a.] Justify that a parametric representation of the line (GM) is: $$\left\{\begin{aligned} x &= 1+t \\ y &= 1-t \quad \text{with } t \in \mathbb{R}. \\ z &= 1-t \end{aligned}\right.$$
   2. [b.] We admit that a parametric representation of the line (AH) is: $$\left\{\begin{aligned} x &= 0 \\ y &= k \\ z &= k \end{aligned}\right. \text{ with } k \in \mathbb{R}.$$ Show that the intersection point of (GM) and (AH), which we will call N, has coordinates $(0;2;2)$.
3. 1. [a.] Show that the triangle AMN is a right-angled triangle at A.
   2. [b.] Calculate the area of this triangle.
4. Let J be the centre of the face BCGF.
   1. [a.] Determine the coordinates of point J.
   2. [b.] Show that the vector $\overrightarrow{\mathrm{FJ}}$ is a normal vector to the plane (AMN).
   3. [c.] Show that J belongs to the plane (AMN). Deduce that it is the orthogonal projection of point F onto the plane (AMN).
5. We recall that the volume $V$ of a tetrahedron or a pyramid is given by the formula: $$V = \frac{1}{3} \times \mathscr{B} \times h$$ $\mathscr{B}$ being the area of a base and $h$ the height relative to this base. Show that the volume of the tetrahedron AMNF is twice the volume of the pyramid BCGFM.

Two aircraft are approaching an airport. We equip space with an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k})$ whose origin O is the base of the control tower, and the ground is the plane $P_0$ with equation $z = 0$. The unit of the axes corresponds to 1 km. We model the aircraft as points.
Aircraft Alpha transmits to the tower its position at $\mathrm{A}(-7; 1; 7)$ and its trajectory is directed by the vector $\vec{u}\begin{pmatrix} 2 \\ -1 \\ -3 \end{pmatrix}$.
Aircraft Beta transmits a trajectory defined by the line $d_{\mathrm{B}}$ passing through point B with a parametric representation: $$\left\{\begin{aligned} x &= -11 + 5t \\ y &= -5 + t \\ z &= 11 - 4t \end{aligned}\right. \text{ where } t \text{ describes } \mathbb{R}.$$

1. If it does not deviate from its trajectory, determine the coordinates of point S where aircraft Beta will touch the ground.
2. a. Determine a parametric representation of the line $d_{\mathrm{A}}$ characterizing the trajectory of aircraft Alpha. b. Can the two aircraft collide?
3. a. Prove that aircraft Alpha passes through position $\mathrm{E}(-3; -1; 1)$. b. Justify that a Cartesian equation of the plane $P_{\mathrm{E}}$ passing through E and perpendicular to the line $d_{\mathrm{A}}$ is: $$2x - y - 3z + 8 = 0.$$ c. Verify that the point $\mathrm{F}(-1; -3; 3)$ is the intersection point of the plane $P_{\mathrm{E}}$ and the line $d_{\mathrm{B}}$. d. Calculate the exact value of the distance EF, then verify that this corresponds to a distance of 3464 m, to the nearest 1 m.
4. Air traffic regulations stipulate that two aircraft on approach must be at least 3 nautical miles apart at all times (1 nautical mile equals 1852 m). If aircraft Alpha and Beta are respectively at E and F at the same instant, is their safety distance respected?

For each of the following statements, indicate whether it is true or false. Justify each answer. An unjustified answer earns no points.
We equip space with an orthonormal coordinate system ( $\mathrm { O } ; \vec { \imath } , \vec { \jmath } , \vec { k }$ ).

1. We consider the points $\mathrm { A } ( - 1 ; 0 ; 5 )$ and $\mathrm { B } ( 3 ; 2 ; - 1 )$.
   Statement 1: A parametric representation of the line (AB) is $$\left\{ \begin{aligned} x & = 3 - 2 t \\ y & = 2 - t \text { with } t \in \mathbb { R } \\ z & = - 1 + 3 t \end{aligned} \right.$$
   Statement 2: The vector $\vec { n } \left( \begin{array} { c } 5 \\ - 2 \\ 1 \end{array} \right)$ is normal to the plane (OAB).
   
2. We consider:
   • the line $d$ with parametric representation $\left\{ \begin{aligned} x & = 15 + k \\ y & = 8 - k \\ z & = - 6 + 2 k \end{aligned} \right.$ with $k \in \mathbb { R }$;
   • the line $d ^ { \prime }$ with parametric representation $\left\{ \begin{array} { l } x = 1 + 4 s \\ y = 2 + 4 s \\ z = 1 - 6 s \end{array} \right.$ with $s \in \mathbb { R }$.
   
   Statement 3: The lines $d$ and $d ^ { \prime }$ are not coplanar.
   
3. We consider the plane $\mathscr { P }$ with equation $x - y + z + 1 = 0$.
   Statement 4: The distance from point $\mathrm { C } ( 2 ; - 1 ; 2 )$ to the plane $\mathscr { P }$ is equal to $2 \sqrt { 3 }$.

Exercise 2 (5 points)
The space is equipped with an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k})$. We consider:

• the points $\mathrm{A}(-1; 2; 1)$, $\mathrm{B}(1; -1; 2)$ and $\mathrm{C}(1; 1; 1)$;
• the line $d$ whose parametric representation is given by: $$d : \left\{ \begin{aligned} x &= \frac{3}{2} + 2t \\ y &= 2 + t \\ z &= 3 - t \end{aligned} \quad \text{with } t \in \mathbb{R}; \right.$$
• the line $d'$ whose parametric representation is given by: $$d' : \left\{ \begin{aligned} x &= s \\ y &= \frac{3}{2} + s \\ z &= 3 - 2s \end{aligned} \quad \text{with } s \in \mathbb{R}. \right.$$

Part A

1. Show that the lines $d$ and $d'$ intersect at the point $\mathrm{S}\left(-\frac{1}{2}; 1; 4\right)$.
2. 1. [a.] Show that the vector $\vec{n}\begin{pmatrix}1\\2\\4\end{pmatrix}$ is a normal vector to the plane (ABC).
   2. [b.] Deduce that a Cartesian equation of the plane (ABC) is: $$x + 2y + 4z - 7 = 0$$
3. Prove that the points $\mathrm{A}$, $\mathrm{B}$, $\mathrm{C}$ and S are not coplanar.
4. 1. [a.] Prove that the point $\mathrm{H}(-1; 0; 2)$ is the orthogonal projection of S onto the plane (ABC).
   2. [b.] Deduce that there is no point $M$ in the plane (ABC) such that $\mathrm{S}M < \frac{\sqrt{21}}{2}$.

Part B
We consider a point $M$ belonging to the segment [CS]. We thus have $\overrightarrow{\mathrm{CM}} = k\overrightarrow{\mathrm{CS}}$ with $k$ a real number in the interval $[0; 1]$.

1. Determine the coordinates of point $M$ as a function of $k$.
2. Does there exist a point $M$ on the segment [CS] such that the triangle $(MAB)$ is right-angled at $M$?

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

1. Show that the points $\mathrm{A}$, $\mathrm{B}$ and $\mathrm{C}$ determine a plane.
2. Show that the triangle ABC is right-angled at A.
3. Let $\Delta$ be the line passing through point D and with direction vector $\vec{u}\begin{pmatrix} 2 \\ -1 \\ 1 \end{pmatrix}$. a. Prove that the line $\Delta$ is orthogonal to the plane (ABC). b. Justify that the plane (ABC) admits the Cartesian equation: $$2x - y + z + 1 = 0$$ c. Determine a parametric representation of the line $\Delta$.
4. We call H the point with coordinates $\left(-\dfrac{2}{3}; \dfrac{4}{3}; \dfrac{5}{3}\right)$. Verify that $H$ is the orthogonal projection of point $D$ onto the plane (ABC).
5. We recall that the volume of a tetrahedron is given by $V = \dfrac{1}{3} B \times h$, where $B$ is the area of a base of the tetrahedron and $h$ is its height relative to this base. a. Show that $\mathrm{DH} = \dfrac{2\sqrt{6}}{3}$. b. Deduce the volume of the tetrahedron ABCD.
6. We consider the line $d$ with parametric representation: $$\left\{\begin{aligned} x &= 1 - 2k \\ y &= -3k \\ z &= 1 + k \end{aligned}\right. \text{ where } k \text{ describes } \mathbb{R}.$$ Are the line $d$ and the plane (ABC) secant or parallel?

Space is equipped with an orthonormal coordinate system ( $\mathrm { O } ; \vec { \imath } , \vec { \jmath } , \vec { k }$ ). We consider the points
$$\mathrm { A } ( 2 \sqrt { 3 } ; 0 ; 0 ) , \quad \mathrm { B } ( 0 ; 2 ; 0 ) , \quad \mathrm { C } ( 0 ; 0 ; 1 ) \quad \text { and } \quad \mathrm { K } \left( \frac { \sqrt { 3 } } { 2 } ; \frac { 3 } { 2 } ; 0 \right) .$$

1. Justify that a parametric representation of the line (CK) is:

$$\left\{ \begin{aligned} x & = \frac { \sqrt { 3 } } { 2 } t \\ y & = \frac { 3 } { 2 } t \quad ( t \in \mathbb { R } ) \\ z & = 1 - t \end{aligned} \right.$$

1. Let $\mathrm { M } ( t )$ be a point on the line (CK) parametrized by a real number $t$. Establish that $\mathrm { OM} ( t ) = \sqrt { 4 t ^ { 2 } - 2 t + 1 }$.
2. Let $f$ be the function defined and differentiable on $\mathbb { R }$ by $f ( t ) = \mathrm { OM } ( t )$. a. Study the variations of the function $f$ on $\mathbb { R }$. b. Deduce the value of $t$ for which $f$ reaches its minimum.
3. Deduce that the point $\mathrm { H } \left( \frac { \sqrt { 3 } } { 8 } ; \frac { 3 } { 8 } ; \frac { 3 } { 4 } \right)$ is the orthogonal projection of point O onto the line (CK).
4. Prove, using the dot product tool, that point H is the orthocenter (intersection of the altitudes of a triangle) of triangle ABC.
5. a. Prove that the line $( \mathrm { OH } )$ is orthogonal to the plane $( \mathrm { ABC } )$. b. Deduce an equation of the plane (ABC).
6. Calculate, in square units, the area of triangle ABC.

Exercise 4

We place ourselves in an orthonormal frame $(\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k})$ of space. We consider the points $\mathrm{A}(1; 0; 3)$, $\mathrm{B}(-2; 1; 2)$ and $\mathrm{C}(0; 3; 2)$.

1. a. Show that the points $\mathrm{A}$, $\mathrm{B}$ and $\mathrm{C}$ are not collinear. b. Let $\vec{n}$ be the vector with coordinates $\left(\begin{array}{c} -1 \\ 1 \\ 4 \end{array}\right)$. Verify that the vector $\vec{n}$ is orthogonal to the plane (ABC). c. Deduce that the plane $(\mathrm{ABC})$ has for Cartesian equation $-x + y + 4z - 11 = 0$.

We consider the plane $\mathscr{P}$ with Cartesian equation $3x - 3y + 2z - 9 = 0$ and the plane $\mathscr{P}'$ with Cartesian equation $x - y - z + 2 = 0$.

1. a. Prove that the planes $\mathscr{P}$ and $\mathscr{P}'$ are secant. We denote by (d) their line of intersection. b. Determine whether the planes $\mathscr{P}$ and $\mathscr{P}'$ are perpendicular.
2. Show that the line (d) is directed by the vector $\vec{u}\left(\begin{array}{l} 1 \\ 1 \\ 0 \end{array}\right)$.
3. Show that the point $\mathrm{M}(2; 1; 3)$ belongs to the planes $\mathscr{P}$ and $\mathscr{P}'$. Deduce a parametric representation of the line (d).
4. Show that the line (d) is also included in the plane (ABC). What can we say about the three planes (ABC), $\mathscr{P}$ and $\mathscr{P}'$?

Space is referred to an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k})$. We consider the points $$\mathrm{A}(4; -1; 3), \quad \mathrm{B}(-1; 1; -2), \quad \mathrm{C}(0; 4; 5) \text{ and } \mathrm{D}(-3; -4; 6).$$

1. a. Verify that the points $\mathrm{A}, \mathrm{B}, \mathrm{C}$ are not collinear.
   We admit that a Cartesian equation of the plane (ABC) is: $29x + 30y - 17z = 35$. b. Are the points A, B, C, D coplanar? Justify.
   
2. Let $P _ { 1 }$ be the perpendicular bisector plane of the segment $[\mathrm{AB}]$. a. Determine the coordinates of the midpoint of the segment $[\mathrm{AB}]$. b. Deduce that a Cartesian equation of $P _ { 1 }$ is: $5x - 2y + 5z = 10$.
   
3. We denote by $P _ { 2 }$ the perpendicular bisector plane of the segment $[\mathrm{CD}]$. a. Let M be a point of the plane $P _ { 2 }$ with coordinates $(x; y; z)$.
   Express $\mathrm{MC}^{2}$ and $\mathrm{MD}^{2}$ as functions of the coordinates of M. Deduce that a Cartesian equation of the plane $P _ { 2 }$ is: $-3x - 8y + z = 10$. b. Justify that the planes $P _ { 1 }$ and $P _ { 2 }$ are secant.
   
4. Let $\Delta$ be the line with a parametric representation: $$\left\{ \begin{array} { r l r l } x & = & -2 - 1.9t \\ y & = & t & \text{ where } t \in \mathbb{R} \\ z & = & 4 + 2.3t \end{array} \right.$$ Demonstrate that $\Delta$ is the line of intersection of $P _ { 1 }$ and $P _ { 2 }$.
   We denote by $P _ { 3 }$ the perpendicular bisector plane of the segment $[\mathrm{AC}]$. We admit that a Cartesian equation of the plane $P _ { 3 }$ is: $8x - 10y - 4z = -15$.
   
5. Demonstrate that the line $\Delta$ and the plane $P _ { 3 }$ are secant.
   
6. Justify that the point of intersection between $\Delta$ and $P _ { 3 }$ is the point H.

4. Aviation regulations stipulate that two aircraft on approach must be at least 3 nautical miles apart at all times (1 nautical mile equals 1852 m). If aircraft Alpha and Beta are respectively at $E$ and $F$ at the same instant, is their safety distance respected?

Exercise 3. (5 points)

The plane is equipped with an orthonormal coordinate system. For every natural number $n$, we consider the function $f_n$ defined on $[0; +\infty[$ by:
$$f_0(x) = \mathrm{e}^{-x} \text{ and, for } n \geq 1, f_n(x) = x^n \mathrm{e}^{-x}.$$
For every natural number $n$, we denote $C_n$ the representative curve of function $f_n$. Parts A and B are independent.

Part A: Study of functions $\boldsymbol{f}_{\boldsymbol{n}}$ for $\boldsymbol{n} \geq \mathbf{1}$

We consider a natural number $n \geq 1$.

1. a. We admit that function $f_n$ is differentiable on $[0; +\infty[$.

Show that for all $x \geq 0$,
$$f_n'(x) = (n - x)x^{n-1}\mathrm{e}^{-x}$$
b. Justify all elements of the table below:

$x$	0		$n$		$+\infty$
$f_n'(x)$		+	0	-
			$\left(\frac{n}{\mathrm{e}}\right)^n$
$f(x)$
	0			0

1. Justify by calculation that point $A\left(1; \mathrm{e}^{-1}\right)$ belongs to curve $C_n$.

Part B: Study of integrals $\int_0^1 \boldsymbol{f}_{\boldsymbol{n}}(\boldsymbol{x})\mathrm{d}\boldsymbol{x}$ for $\boldsymbol{n} \geq \mathbf{0}$

In this part, we study functions $f_n$ on $[0; 1]$ and we consider the sequence $(I_n)$ defined for every natural number $n$ by:
$$I_n = \int_0^1 f_n(x)\mathrm{d}x = \int_0^1 x^n \mathrm{e}^{-x}\mathrm{d}x$$

1. On the graph in APPENDIX (page 9/9), curves $C_0, C_1, C_2, C_{10}$ and $C_{100}$ are represented. a. Give a graphical interpretation of $I_n$. b. By reading this graph, what conjecture can be made about the limit of sequence $(I_n)$?
2. Calculate $I_0$.
3. a. Let $n$ be a natural number.

Prove that for all $x \in [0; 1]$,
$$0 \leq x^{n+1} \leq x^n.$$
b. Deduce that for every natural number $n$, we have:
$$0 \leq I_{n+1} \leq I_n$$

1. Prove that sequence $(I_n)$ is convergent, towards a limit greater than or equal to zero that we denote $\ell$.
2. Using integration by parts, prove that for every natural number $n$ we have:

$$I_{n+1} = (n+1)I_n - \frac{1}{\mathrm{e}}$$

1. a. Prove that if $\ell > 0$, the equality from question 5 leads to a contradiction. b. Prove that $\ell = 0$. You may use question 6.a.

Below is the script of the mystere function, written in Python. The constant e has been imported.
\begin{verbatim} def mystere(n): I = 1 - 1/e L = [I] for i in range(n): I = (i + 1)*I - 1/e L.append(I) return L \end{verbatim}

1. What does mystere(100) return in the context of the exercise?

Exercise 4. (5 points)

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.

1. We consider the differential equation (E):

$$y' = \frac{1}{2}y + 4.$$
Statement 1: The solutions of (E) are the functions $f$ defined on $\mathbb{R}$ by:
$$f(x) = k\mathrm{e}^{\frac{1}{2}x} - 8, \text{ with } k \in \mathbb{R}$$

1. In a final year class, there are 18 girls and 14 boys.

A volleyball team is formed by randomly choosing 3 girls and 3 boys. Statement 2: There are 297024 possibilities for forming such a team.

Let P be the plane containing the vectors $(6,6,9)$ and $(7,8,10)$. Find a unit vector that is perpendicular to $(2,-3,4)$ and that lies in the plane P. (Note: all vectors are considered as line segments starting at the origin $(0,0,0)$. In particular the origin lies in the plane P.)

As shown in the figure on the right, in a cube ABCD-EFGH with edge length 3, there are three points $\mathrm { P } , \mathrm { Q } , \mathrm { R }$ on the three edges AD, BC, FG such that $\overline { \mathrm { DP } } = \overline { \mathrm { BQ } } = \overline { \mathrm { GR } } = 1$. The angle between plane PQR and plane CGHD is $\theta$. What is the value of $\cos \theta$? (where $0 < \theta < \frac { \pi } { 2 }$) [3 points]
(1) $\frac { \sqrt { 10 } } { 5 }$
(2) $\frac { \sqrt { 10 } } { 10 }$
(3) $\frac { \sqrt { 11 } } { 11 }$
(4) $\frac { 2 \sqrt { 11 } } { 11 }$
(5) $\frac { 3 \sqrt { 11 } } { 11 }$

In coordinate space, there are two points $\mathrm { A } ( 3,1,1 ) , \mathrm { B } ( 1 , - 3 , - 1 )$. For a point P on the plane $x - y + z = 0$, what is the minimum value of $| \overrightarrow { \mathrm { PA } } + \overrightarrow { \mathrm { PB } } |$? [4 points]
(1) $\frac { 4 \sqrt { 3 } } { 3 }$
(2) $\frac { 5 \sqrt { 3 } } { 3 }$
(3) $2 \sqrt { 3 }$
(4) $\frac { 7 \sqrt { 3 } } { 3 }$
(5) $\frac { 8 \sqrt { 3 } } { 3 }$

A sphere with center $\mathrm { C } ( 0,1,1 )$ and radius $2 \sqrt { 2 }$ intersects the line $\frac { x } { 2 } = y = - z$ at two points A and B. Let $S$ be the area of triangle CAB. Find the value of $S ^ { 2 }$.