Exercise 4 — Candidates who have NOT followed the specialisation course
For each of the following statements, indicate whether it is true or false, by justifying the answer.
One point is awarded for each correct answer that is properly justified. An unjustified answer is not taken into account. No answer is not penalised.
1. Statement 1: The equation $( 3 \ln x - 5 ) \left( e ^ { x } + 4 \right) = 0$ has exactly two real solutions.
2. Consider the sequence ( $u _ { n }$ ) defined by $$u _ { 0 } = 2 \text { and, for all natural number } n , u _ { n + 1 } = 2 u _ { n } - 5 n + 6 \text {. }$$ Statement 2: For all natural number $n , u _ { n } = 3 \times 2 ^ { n } + 5 n - 1$.
3. Consider the sequence ( $u _ { n }$ ) defined, for all natural number $n$, by $u _ { n } = n ^ { 2 } + \frac { 1 } { 2 }$.
Statement 3: The sequence $\left( u _ { n } \right)$ is geometric.
4. In a coordinate system of space, let $d$ be the line passing through point $\mathrm { A } ( - 3 ; 7 ; - 12 )$ and with direction vector $\vec { u } ( 1 ; - 2 ; 5 )$.
Let $d ^ { \prime }$ be the line with parametric representation $\left\{ \begin{array} { r l } x & = 2 t - 1 \\ y & = - 4 t + 3 \\ z & = 10 t - 2 . \end{array} , t \in \mathbf { R } \right.$
Statement 4: The lines $d$ and $d ^ { \prime }$ are coincident.
5. Consider a cube $A B C D E F G H$. The space is equipped with the orthonormal coordinate system ( $A$; $\overrightarrow { A B } , \overrightarrow { A D } , \overrightarrow { A E }$ ).
A parametric representation of the line (AG) is $\left\{ \begin{array} { l } x = t \\ y = t \\ z = t \end{array} \quad t \in \mathbf { R } \right.$.
Consider a point $M$ on the line (AG).
Statement 5: There are exactly two positions of point $M$ on the line (AG) such that the lines $( M \mathrm { ~B} )$ and $( M \mathrm { D } )$ are orthogonal.

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 or loses no points.
Space is referred to an orthonormal reference frame $(\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k})$.
We consider:

• The line $\mathscr{D}$ passing through the points $\mathrm{A}(1;1;-2)$ and $\mathrm{B}(-1;3;2)$.
• The line $\mathscr{D}'$ with parametric representation: $\left\{ \begin{aligned} x &= -4 + 3t \\ y &= 6 - 3t \\ z &= 8 - 6t \end{aligned} \right.$ with $t \in \mathbb{R}$.
• The plane $\mathscr{P}$ with Cartesian equation $x + my - 2z + 8 = 0$ where $m$ is a real number.

Question 1: Among the following points, which one belongs to the line $\mathscr{D}'$? a. $\mathrm{M}_1(-1;3;-2)$ b. $\mathrm{M}_2(11;-9;-22)$ c. $\mathrm{M}_3(-7;9;2)$ d. $\mathrm{M}_4(-2;3;4)$
Question 2: A direction vector of the line $\mathscr{D}'$ is: a. $\overrightarrow{u_1}\left(\begin{array}{c}-4\\6\\8\end{array}\right)$ b. $\overrightarrow{u_2}\left(\begin{array}{l}3\\3\\6\end{array}\right)$ c. $\overrightarrow{u_3}\left(\begin{array}{c}3\\-3\\-6\end{array}\right)$ d. $\overrightarrow{u_4}\left(\begin{array}{c}-1\\3\\2\end{array}\right)$
Question 3: The lines $\mathscr{D}$ and $\mathscr{D}'$ are: a. intersecting b. strictly parallel c. non-coplanar d. coincident
Question 4: The value of the real number $m$ for which the line $\mathscr{D}$ is parallel to the plane $\mathscr{P}$ is: a. $m = -1$ b. $m = 1$ c. $m = 5$ d. $m = -2$

Three lines $$\begin{array}{ll} L_1: & \vec{r} = \lambda\hat{i}, \lambda \in \mathbb{R}, \\ L_2: & \vec{r} = \hat{k} + \mu\hat{j}, \mu \in \mathbb{R} \text{ and} \\ L_3: & \vec{r} = \hat{i} + \hat{j} + v\hat{k}, v \in \mathbb{R} \end{array}$$ are given. For which point(s) $Q$ on $L_2$ can we find a point $P$ on $L_1$ and a point $R$ on $L_3$ so that $P$, $Q$ and $R$ are collinear?
(A) $\hat{k} - \frac{1}{2}\hat{j}$
(B) $\hat{k}$
(C) $\hat{k} + \frac{1}{2}\hat{j}$
(D) $\hat{k} + \hat{j}$

Let $L_1: \vec{r} = (\hat{i} - \hat{j} + 2\hat{k}) + \lambda(\hat{i} - \hat{j} + 2\hat{k})$, $\lambda \in R$, $L_2: \vec{r} = (\hat{j} - \hat{k}) + \mu(3\hat{i} + \hat{j} + p\hat{k})$, $\mu \in R$ and $L_3: \vec{r} = \delta(l\hat{i} + m\hat{j} + n\hat{k})$, $\delta \in R$ be three lines such that $L_1$ is perpendicular to $L_2$ and $L_3$ is perpendicular to both $L_1$ and $L_2$. Then the point which lies on $L_3$ is
(1) $(-1, 7, 4)$
(2) $(-1, -7, 4)$
(3) $(1, 7, -4)$
(4) $(1, -7, 4)$