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).

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.

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.

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 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.

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

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

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).

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.

For each of the following statements, indicate whether it is true or false. Each answer must be justified. An unjustified answer earns no points. Space is referred to an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k})$. We consider the line $(d)$ whose parametric representation is: $$\left\{\begin{array}{rl} x & = 3 - 2t \\ y & = -1 \\ z & = 2 - 6t \end{array}, \text{ where } t \in \mathbb{R}\right.$$ We also consider the following points:

• $\mathrm{A}(3; -3; -2)$
• $\mathrm{B}(5; -4; -1)$
• C the point on line $(d)$ with x-coordinate 2
• H the orthogonal projection of point B onto the plane $\mathscr{P}$ with equation $x + 3z - 7 = 0$

Statement 1: The line $(d)$ and the y-axis are two non-coplanar lines.
Statement 2: The plane passing through $A$ and perpendicular to line $(d)$ has the Cartesian equation: $$x + 3z + 3 = 0$$
Statement 3: A measure, expressed in radians, of the geometric angle $\widehat{\mathrm{BAC}}$ is $\frac{\pi}{6}$.
Statement 4: The distance BH is equal to $\frac{\sqrt{10}}{2}$.

For each of the following statements, indicate whether it is true or false. Each answer must be justified. An unjustified answer earns no points.

PART A

ABCDEFGH is a cube with edge length 1. The points I, J, K, L and M are the midpoints respectively of the edges [AB], [BF], [AE], [CD] and [DH].
Statement 1: $\ll \overrightarrow { \mathrm { JH } } = 2 \overrightarrow { \mathrm { BI } } + \overrightarrow { \mathrm { DM } } - \overrightarrow { \mathrm { CB } } \gg$ Statement 2: ``The triplet of vectors ( $\overrightarrow { \mathrm { AB } } , \overrightarrow { \mathrm { AH } } , \overrightarrow { \mathrm { AG } }$ ) is a basis of space.'' Statement 3: ``$\overrightarrow { \mathrm { IB } } \cdot \overrightarrow { \mathrm { LM } } = - \frac { 1 } { 4 }$.''

PART B

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

• the plane $\mathscr { P }$ with Cartesian equation $2 x - y + 3 z + 6 = 0$
• the points $\mathrm { A } ( 2 ; 0 ; - 1 )$ and $\mathrm { B } ( 5 ; - 3 ; 7 )$

Statement 4: ``The plane $\mathscr { P }$ and the line ( AB ) are parallel.'' Statement 5: ``The plane $\mathscr { P } ^ { \prime }$ parallel to $\mathscr { P }$ passing through B has Cartesian equation $- 2 x + y - 3 z + 34 = 0$'' Statement 6: ``The distance from point A to plane $\mathscr { P }$ is equal to $\frac { \sqrt { 14 } } { 2 }$.'' We denote by (d) the line with parametric representation
$$\left\{ \begin{array} { r l } x & = - 12 + 2 k \\ y & = 6 \\ z & = 3 - 5 k \end{array} , \text { where } k \in \mathbb { R } \right.$$
Statement 7: ``The lines (AB) and (d) are not coplanar.''

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 $\Omega$.

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

• the points $\mathrm{C}(3; 0; 0)$, $\mathrm{D}(0; 2; 0)$, $\mathrm{H}(-6; 2; 2)$ and $\mathrm{J}\left(\frac{-54}{13}; \frac{62}{13}; 0\right)$;
• the plane $P$ with Cartesian equation $2x + 3y + 6z - 6 = 0$;
• the plane $P'$ with Cartesian equation $x - 2y + 3z - 3 = 0$;
• the line $(d)$ with a parametric representation: $\left\{\begin{array}{l} x = -8 + \frac{1}{3}t \\ y = -1 + \frac{1}{2}t \\ z = -4 + t \end{array}, t \in \mathbb{R}\right.$

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: The line $(d)$ is orthogonal to the plane $P$ and intersects this plane at $H$.
Statement 2: The measure in degrees of the angle $\widehat{\mathrm{DCH}}$, rounded to $10^{-1}$, is $17.3^{\circ}$.
Statement 3: The planes $P$ and $P'$ are secant and their intersection is the line $\Delta$ with a parametric representation: $\left\{\begin{array}{l} x = 3 - 3t \\ y = 0 \\ z = t \end{array}, t \in \mathbb{R}\right.$.
Statement 4: Point J is the orthogonal projection of point H onto the line (CD).

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: $N = \frac { 8 ^ { 2 } } { 2 }$.

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 $2 \sqrt { 3 }$.

For each of the following statements, indicate whether it is true or false. Each answer must be justified. An unjustified answer earns no points.

1. We consider the function $f$ defined on $]0; +\infty[$ by: $f(x) = x\ln(x)$.
   Statement 1: $$\int_1^{\mathrm{e}} f(x)\,\mathrm{d}x = \frac{\mathrm{e}^2 + 1}{4}$$
   
2. Let $n$ and $k$ be two non-zero natural integers such that $k \leqslant n$.
   Statement 2: $$n \times \binom{n-1}{k-1} = k \times \binom{n}{k}$$
   
3. For the three following statements, we consider that space is equipped with an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k})$.
   Let $d$ be the line with parametric representation: $\left\{\begin{array}{rl} x &= t + 1 \\ y &= 2t + 1 \\ z &= -t \end{array}\right., t \in \mathbb{R}$. Let $d'$ be the line with parametric representation: $\left\{\begin{array}{rl} x &= 2t' - 1 \\ y &= -t' + 2 \\ z &= t' + 1 \end{array}\right., t' \in \mathbb{R}$. Let $P$ be the plane with Cartesian equation: $2x + y - 2z + 18 = 0$. Let A be the point with coordinates $(-1; -3; 2)$ and B be the point with coordinates $(-5; -5; 6)$. We call the perpendicular bisector plane of segment $[\mathrm{AB}]$ the plane passing through the midpoint of segment $[\mathrm{AB}]$ and perpendicular to the line $(\mathrm{AB})$.
   Statement 3: Point A belongs to line $d$. Statement 4: Lines $d$ and $d'$ are secant. Statement 5: Plane $P$ is the perpendicular bisector plane of segment $[\mathrm{AB}]$.