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

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

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.

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 $\mathrm { F } \left( \frac { 4 } { 3 } ; - \frac { 1 } { 3 } ; \frac { 22 } { 3 } \right)$.

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.

This exercise is a multiple choice questionnaire. For each of the following questions, only one of the four proposed answers is correct. A correct answer earns one point. An incorrect answer, multiple answers, or no answer to a question earns or loses no points.
In space with respect to an orthonormal coordinate system $(O; \vec{\imath}, \vec{\jmath}, \vec{k})$, we consider the points $A(1; 0; 2)$, $B(2; 1; 0)$, $C(0; 1; 2)$ and the line $\Delta$ whose parametric representation is: $$\left\{\begin{array}{rl}x & = 1 + 2t \\ y & = -2 + t \\ z & = 4 - t\end{array}, t \in \mathbb{R}\right.$$

1. Which of the following points belongs to the line $\Delta$?
   Answer A: $M(2; 1; -1)$; Answer B: $N(-3; -4; 6)$; Answer C: $P(-3; -4; 2)$; Answer D: $Q(-5; -5; 1)$.
   
2. The vector $\overrightarrow{AB}$ has coordinates:
   $$\begin{array}{ll} \text{Answer A}: \left(\begin{array}{c} 1.5 \\ 0.5 \\ 1 \end{array}\right); & \text{Answer B}: \left(\begin{array}{c} -1 \\ -1 \\ 2 \end{array}\right); \\ \text{Answer C}: \left(\begin{array}{c} 1 \\ 1 \\ -2 \end{array}\right) & \text{Answer D}: \left(\begin{array}{l} 3 \\ 1 \\ 2 \end{array}\right). \end{array}$$
   
3. A parametric representation of the line (AB) is:
   $$\begin{array}{ll} \text{Answer A}: \left\{\begin{array}{l} x = 1 + 2t \\ y = t \\ z = 2 \end{array}, t \in \mathbb{R}\right. & \text{Answer B}: \left\{\begin{array}{l} x = 2 - t \\ y = 1 - t \\ z = 2t \end{array}, t \in \mathbb{R}\right. \\ \text{Answer C}: \left\{\begin{array}{l} x = 2 + t \\ y = 1 + t \\ z = 2t \end{array}, t \in \mathbb{R}\right. & \text{Answer D}: \left\{\begin{array}{l} x = 1 + t \\ y = 1 + t \\ z = 2 - 2t \end{array}, t \in \mathbb{R}\right. \end{array}$$
   
4. A Cartesian equation of the plane passing through point C and orthogonal to the line $\Delta$ is: Answer A: $x - 2y + 4z - 6 = 0$; Answer B: $2x + y - z + 1 = 0$; Answer C: $2x + y - z - 1 = 0$; Answer D: $y + 2z - 5 = 0$.
   
5. We consider the point D defined by the vector relation $\overrightarrow{OD} = 3\overrightarrow{OA} - \overrightarrow{OB} - \overrightarrow{OC}$.
   Answer A: $\overrightarrow{AD}$, $\overrightarrow{AB}$, $\overrightarrow{AC}$ are coplanar; Answer B: $\overrightarrow{AD} = \overrightarrow{BC}$; Answer C: D has coordinates $(3; -1; -1)$; Answer D: the points A, B, C and D are collinear.

This exercise is a multiple choice questionnaire. For each of the following questions, only one of the four proposed answers is correct.
A correct answer earns one point. An incorrect answer, a multiple answer, or the absence of an answer to a question earns neither points nor deducts points. To answer, indicate on your paper the question number and the letter of the chosen answer. No justification is required.
SABCD is a regular pyramid with square base ABCD in which all edges have the same length. The point I is the center of the square ABCD. We assume that: $\mathrm{IC} = \mathrm{IB} = \mathrm{IS} = 1$. The points K, L and M are the midpoints of edges [SD], [SC] and [SB] respectively.

1. The following lines are not coplanar: a. (DK) and (SD) b. (AS) and (IC) c. (AC) and (SB) d. (LM) and (AD)

For the following questions, we place ourselves in the orthonormal coordinate system of space $(\mathrm{I}; \overrightarrow{\mathrm{IC}}, \overrightarrow{\mathrm{IB}}, \overrightarrow{\mathrm{IS}})$. In this coordinate system, we are given the coordinates of the following points: $$\mathrm{I}(0;0;0) \quad ; \quad \mathrm{A}(-1;0;0) \quad ; \quad \mathrm{B}(0;1;0) \quad ; \quad \mathrm{C}(1;0;0) \quad ; \quad \mathrm{D}(0;-1;0) \quad ; \quad \mathrm{S}(0;0;1).$$

1. The coordinates of the midpoint N of [KL] are: a. $\left(\frac{1}{4};\frac{1}{4};\frac{1}{4}\right)$ b. $\left(\frac{1}{4};-\frac{1}{4};\frac{1}{2}\right)$ c. $\left(-\frac{1}{4};\frac{1}{4};\frac{1}{2}\right)$ d. $\left(-\frac{1}{2};\frac{1}{2};1\right)$
2. The coordinates of the vector $\overrightarrow{\mathrm{AS}}$ are: a. $\left(\begin{array}{l}1\\1\\0\end{array}\right)$ b. $\left(\begin{array}{l}1\\0\\1\end{array}\right)$ c. $\left(\begin{array}{c}2\\1\\-1\end{array}\right)$ d. $\left(\begin{array}{l}1\\1\\1\end{array}\right)$
3. A parametric representation of the line (AS) is: a. $\left\{\begin{array}{rl}x &= -1-t\\y &= t\\z &= -t\end{array}\right.$ b. $\left\{\begin{aligned}x =& -1+2t\\y =& 0\\z =& 1+2t\end{aligned}\right.$ c. $\left\{\begin{aligned}x &= t\\y &= 0\\z &= 1+t\end{aligned}\right.$ d. $\left\{\begin{aligned}x &= -1-t\\y &= 1+t\\z &= 1-t\end{aligned}\right. (t \in \mathbb{R})$
4. A Cartesian equation of the plane (SCB) is: a. $y+z-1=0$ b. $x+y+z-1=0$ c. $x-y+z=0$ d. $x+z-1=0$

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 neither awards nor deducts points. The five questions are independent. Space is equipped with an orthonormal reference frame $(\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k})$. We consider the points $\mathrm{A}(-1; 2; 5)$, $\mathrm{B}(3; 6; 3)$, $\mathrm{C}(3; 0; 9)$ and $\mathrm{D}(8; -3; -8)$. We admit that points $\mathrm{A}$, $\mathrm{B}$ and $\mathrm{C}$ are not collinear.

1. Triangle ABC is: a. isosceles right-angled at A b. isosceles right-angled at B c. isosceles right-angled at C d. equilateral
   
2. A Cartesian equation of plane (BCD) is: a. $2x + y + z - 15 = 0$ b. $9x - 5y + 3 = 0$ c. $4x + y + z - 21 = 0$ d. $11x + 5z - 73 = 0$
   
3. We admit that plane $(\mathrm{ABC})$ has Cartesian equation $x - 2y - 2z + 15 = 0$. We call H the orthogonal projection of point D onto plane (ABC). We can affirm that: a. $\mathrm{H}(-2; 17; 12)$ b. $\mathrm{H}(3; 7; 2)$ c. $\mathrm{H}(3; 2; 7)$ d. $\mathrm{H}(-15; 1; -1)$
   
4. Let the line $\Delta$ with parametric representation $\left\{\begin{array}{l} x = 5 + t \\ y = 3 - t \\ z = -1 + 3t \end{array}\right.$, with $t$ real. Lines (BC) and $\Delta$ are: a. coincident b. strictly parallel c. intersecting d. non-coplanar
   
5. We consider the plane $\mathscr{P}$ with Cartesian equation $2x - y + 2z - 6 = 0$. We admit that plane (ABC) has Cartesian equation $x - 2y - 2z + 15 = 0$. We can affirm that: a. planes $\mathscr{P}$ and (ABC) are strictly parallel b. planes $\mathscr{P}$ and (ABC) are intersecting and their intersection is line (AB) c. planes $\mathscr{P}$ and (ABC) are intersecting and their intersection is line (AC) d. planes $\mathscr{P}$ and (ABC) are intersecting and their intersection is line (BC)

This exercise is a multiple choice questionnaire. For each of the following questions, only one of the four proposed answers is correct. To answer, indicate on your paper the question number and the letter of the chosen answer. No justification is required.
A wrong answer, no answer, or multiple answers, neither earn nor lose points.

1. We consider the function $f$ defined on $\mathbb{R}$ by: $f(x) = (x + 1)e^x$.
   An antiderivative $F$ of $f$ on $\mathbb{R}$ is defined by: a. $F(x) = 1 + xe^x$ b. $F(x) = (1 + x)e^x$ c. $F(x) = (2 + x)e^x$ d. $F(x) = \left(\frac{x^2}{2} + x\right)e^x$.
   
2. We consider the lines $(d_1)$ and $(d_2)$ whose parametric representations are respectively: $$\left(d_1\right) \left\{\begin{array}{l} x = 2 + r \\ y = 1 + r \\ z = -r \end{array} \quad (r \in \mathbb{R}) \quad \text{and} \quad (d_2) \left\{\begin{array}{rl} x & = 1 - s \\ y & = -1 + s \\ z & = 2 - s \end{array} \quad (s \in \mathbb{R})\right.\right.$$ The lines $(d_1)$ and $(d_2)$ are: a. secant. b. strictly parallel. c. coincident. d. non-coplanar.
   
3. We consider the plane $(P)$ whose Cartesian equation is: $$2x - y + z - 1 = 0$$ We consider the line $(\Delta)$ whose parametric representation is: $$\left\{\begin{array}{l} x = 2 + u \\ y = 4 + u \quad (u \in \mathbb{R}) \\ z = 1 - u \end{array}\right.$$ The line $(\Delta)$ is: a. secant and non-orthogonal to the plane $(P)$. b. included in the plane $(P)$. c. strictly parallel to the plane $(P)$. d. orthogonal to the plane $(P)$.
   
4. We consider the plane $(P_1)$ whose Cartesian equation is $x - 2y + z + 1 = 0$, as well as the plane $(P_2)$ whose Cartesian equation is $2x + y + z - 6 = 0$. The planes $(P_1)$ and $(P_2)$ are: a. secant and perpendicular. b. coincident. c. secant and non-perpendicular. d. strictly parallel.
   
5. We consider the points $E(1; 2; 1)$, $F(2; 4; 3)$ and $G(-2; 2; 5)$.
   We can affirm that the measure $\alpha$ of the angle $\widehat{FEG}$ satisfies: a. $\alpha = 90°$ b. $\alpha > 90°$ c. $\alpha = 0°$ d. $\alpha \approx 71°$

This exercise is a multiple choice questionnaire. For each of the following questions, only one of the four proposed answers is correct. A correct answer earns one point. An incorrect answer, a multiple answer, or the absence of an answer to a question earns neither points nor deducts points.
Space is referred to an orthonormal coordinate system $(O; \vec{\imath}, \vec{\jmath}, \vec{k})$.
We consider:

• the points $A(-1; -2; 3)$, $B(1; -2; 7)$ and $C(1; 0; 2)$;
• the line $\Delta$ with parametric representation: $\left\{\begin{array}{l} x = 1 - t \\ y = 2 \\ z = -4 + 3t \end{array}\right.$, where $t \in \mathbb{R}$;
• the plane $\mathscr{P}$ with Cartesian equation: $3x + 2y + z - 4 = 0$;
• the plane $\mathscr{Q}$ with Cartesian equation: $-6x - 4y - 2z + 7 = 0$.

1. Which of the following points belongs to the plane $\mathscr{P}$? a. $R(1; -3; 1)$; b. $S(1; 2; -1)$; c. $T(1; 0; 1)$; d. $U(2; -1; 1)$.
2. Triangle ABC is: a. equilateral; b. right isosceles; c. isosceles non-right; d. right non-isosceles.
3. The line $\Delta$ is: a. orthogonal to the plane $\mathscr{P}$; b. secant to the plane $\mathscr{P}$; c. included in the plane $\mathscr{P}$; d. strictly parallel to the plane $\mathscr{P}$.
4. We are given the dot product $\overrightarrow{BA} \cdot \overrightarrow{BC} = 20$.
   A measure to the nearest degree of the angle $\widehat{ABC}$ is: a. $34°$; b. $120°$; c. $90°$; d. $0°$.
5. The intersection of planes $\mathscr{P}$ and $\mathscr{Q}$ is: a. a plane; b. the empty set; c. a line; d. reduced to a point.

Space is equipped with an orthonormal coordinate system $(O; \vec{\imath}, \vec{\jmath}, \vec{k})$ in which we consider:

• the points $A(6; -6; 6)$, $B(-6; 0; 6)$ and $C(-2; -2; 11)$.
• the line $(d)$ orthogonal to the two secant lines $(AB)$ and $(BC)$ and passing through point A;
• the line $(d')$ with parametric representation:

$$\left\{\begin{aligned} x &= -6 - 8t \\ y &= 4t, \text{ with } t \in \mathbb{R}. \\ z &= 6 + 5t \end{aligned}\right.$$
This exercise is a multiple choice questionnaire. For each of the following questions, only one of the four proposed answers is correct. A wrong answer, multiple answers or absence of answer to a question neither awards nor deducts points. No justification is required.
Question 1 Among the following vectors, which is a direction vector of the line $(d)$? a. $\overrightarrow{u_1}\left(\begin{array}{c}-6 \\ 3 \\ 0\end{array}\right)$ b. $\overrightarrow{u_2}\left(\begin{array}{l}1 \\ 2 \\ 6\end{array}\right)$ c. $\overrightarrow{u_3}\left(\begin{array}{c}1 \\ 2 \\ 0.2\end{array}\right)$ d. $\overrightarrow{u_4}\left(\begin{array}{l}1 \\ 2 \\ 0\end{array}\right)$
Question 2 Among the following equations, which is a parametric representation of the line (AB)? a. $\left\{\begin{aligned}x &= 2t + 6 \\ y &= -6 \text{ with } t \in \mathbb{R} \\ z &= t + 6\end{aligned}\right.$ b. $\left\{\begin{aligned}x &= 2t - 6 \\ y &= -6 \text{ with } t \in \mathbb{R} \\ z &= -t - 6\end{aligned}\right.$ c. $\left\{\begin{aligned}x &= 2t + 6 \\ y &= -t - 6 \text{ with } t \in \mathbb{R} \\ z &= 6\end{aligned}\right.$ d. $\left\{\begin{aligned}x &= 2t + 6 \\ y &= t - 6 \text{ with } t \in \mathbb{R} \\ z &= 6\end{aligned}\right.$
Question 3
A direction vector of the line $(d')$ is: a. $\overrightarrow{v_1}\left(\begin{array}{c}-6 \\ 0 \\ 6\end{array}\right)$ b. $\overrightarrow{v_2}\left(\begin{array}{c}-14 \\ 4 \\ 11\end{array}\right)$ c. $\overrightarrow{v_3}\left(\begin{array}{c}8 \\ -4 \\ -5\end{array}\right)$ d. $\overrightarrow{v_4}\left(\begin{array}{l}8 \\ 4 \\ 5\end{array}\right)$
Question 4 Which of the following four points belongs to the line $(d')$? a. $M_1(50; -28; -29)$ b. $M_2(-14; -4; 1)$ c. $M_3(2; -4; -1)$ d. $M_4(-3; 0; 3)$
Question 5 The plane with equation $x = 1$ has as normal vector: a. $\overrightarrow{n_1}\left(\begin{array}{l}1 \\ 0 \\ 0\end{array}\right)$ b. $\overrightarrow{n_2}\left(\begin{array}{l}0 \\ 1 \\ 1\end{array}\right)$ c. $\overrightarrow{n_3}\left(\begin{array}{l}0 \\ 1 \\ 0\end{array}\right)$ d. $\overrightarrow{n_4}\left(\begin{array}{l}1 \\ 0 \\ 1\end{array}\right)$

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 $$\left\{ \begin{aligned} x &= t \\ y &= 7t + 4, \text{ where } t \in \mathbb{R}. \\ z &= 11t + 7 \end{aligned} \right.$$

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. $0^\circ$

In the right triangular prism $ABC - A_1B_1C_1$, let $D$ be the midpoint of $BC$. Then
A. $AD \perp A_1C$
B. $BC \perp$ plane $AA_1D$
C. $AD \parallel A_1B_1$
D. $CC_1 \parallel$ plane $AA_1D$

Let $\gamma \in \mathbb { R }$ be such that the lines $L _ { 1 } : \frac { x + 11 } { 1 } = \frac { y + 21 } { 2 } = \frac { z + 29 } { 3 }$ and $L _ { 2 } : \frac { x + 16 } { 3 } = \frac { y + 11 } { 2 } = \frac { z + 4 } { \gamma }$ intersect. Let $R _ { 1 }$ be the point of intersection of $L _ { 1 }$ and $L _ { 2 }$. Let $O = ( 0,0,0 )$, and $\hat { n }$ denote a unit normal vector to the plane containing both the lines $L _ { 1 }$ and $L _ { 2 }$.
Match each entry in List-I to the correct entry in List-II.
List-I
(P) $\gamma$ equals
(Q) A possible choice for $\hat { n }$ is
(R) $\overrightarrow { O R _ { 1 } }$ equals
(S) A possible value of $\overrightarrow { O R _ { 1 } } \cdot \hat { n }$ is
List-II
(1) $- \hat { i } - \hat { j } + \hat { k }$
(2) $\sqrt { \frac { 3 } { 2 } }$
(3) 1
(4) $\frac { 1 } { \sqrt { 6 } } \hat { i } - \frac { 2 } { \sqrt { 6 } } \hat { j } + \frac { 1 } { \sqrt { 6 } } \hat { k }$
(5) $\sqrt { \frac { 2 } { 3 } }$
The correct option is
(A) $(\mathrm{P}) \rightarrow (3)$, $(\mathrm{Q}) \rightarrow (4)$, $(\mathrm{R}) \rightarrow (1)$, $(\mathrm{S}) \rightarrow (2)$
(B) $(\mathrm{P}) \rightarrow (5)$, $(\mathrm{Q}) \rightarrow (4)$, $(\mathrm{R}) \rightarrow (1)$, $(\mathrm{S}) \rightarrow (2)$
(C) $(\mathrm{P}) \rightarrow (3)$, $(\mathrm{Q}) \rightarrow (4)$, $(\mathrm{R}) \rightarrow (1)$, $(\mathrm{S}) \rightarrow (5)$
(D) $(\mathrm{P}) \rightarrow (3)$, $(\mathrm{Q}) \rightarrow (1)$, $(\mathrm{R}) \rightarrow (4)$, $(\mathrm{S}) \rightarrow (5)$

The image of the line $\frac { x - 1 } { 3 } = \frac { y - 3 } { 1 } = \frac { z - 4 } { - 5 }$ in the plane $2 x - y + z + 3 = 0$ is the line
(1) $\frac { x - 3 } { 3 } = \frac { y + 5 } { 1 } = \frac { z - 2 } { - 5 }$
(2) $\frac { x - 3 } { - 3 } = \frac { y + 5 } { - 1 } = \frac { z - 2 } { 5 }$
(3) $\frac { x + 3 } { 3 } = \frac { y - 5 } { 1 } = \frac { z - 2 } { - 5 }$
(4) $\frac { x + 3 } { - 3 } = \frac { y - 5 } { - 1 } = \frac { z + 2 } { 5 }$