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.
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 $\frac{2}{3}$.
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.
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).
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 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.
\begin{enumerate}
\item 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 $\frac{2}{3}$.
\item 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.
\item 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).
\item 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.
\end{enumerate}