\section*{Exercise 3 -- Common to all candidates}
In a vast plain, a network of sensors makes it possible to detect lightning and to produce an image of storm phenomena. The radar screen is divided into forty sectors denoted by a letter and a number between 1 and 5. The lightning sensor is represented by the center of the screen; five concentric circles corresponding to the respective radii $20, 40, 60, 80$ and 100 kilometres delimit five zones numbered from 1 to 5, and eight segments starting from the sensor delimit eight portions of the same angular opening, named in the trigonometric direction from A to H.

We assimilate the radar screen to a part of the complex plane by defining an orthonormal coordinate system $(\mathrm{O}, \vec{u}, \vec{v})$:
\begin{itemize}
  \item the origin O marks the position of the sensor;
  \item the abscissa axis is oriented from West to East;
  \item the ordinate axis is oriented from South to North;
  \item the chosen unit is the kilometre.
\end{itemize}
In the following, a point on the radar screen is associated with a point with affix $z$.

\section*{Part A}
\begin{enumerate}
  \item We denote $z_{P}$ the affix of the point P located in sector B3 on the graph. We call $r$ the modulus of $z_{P}$ and $\theta$ its argument in the interval $]-\pi ; \pi]$. Among the four following propositions, determine the only one that proposes a correct bound for $r$ and for $\theta$ (no justification is required):

\begin{center}
\begin{tabular}{ | c | c | c | c | }
\hline
Proposition A & Proposition B & Proposition C & Proposition D \\
\hline
\begin{tabular}{ c }
$40 < r < 60$ \\
and \\
$0 < \theta < \frac{\pi}{4}$
\end{tabular} & \begin{tabular}{ c }
$20 < r < 40$ \\
and \\
$\frac{\pi}{2} < \theta < \frac{3\pi}{4}$
\end{tabular} & \begin{tabular}{ c }
$40 < r < 60$ \\
and \\
$\frac{\pi}{4} < \theta < \frac{\pi}{2}$
\end{tabular} & \begin{tabular}{ c }
$0 < r < 60$ \\
and \\
$-\frac{\pi}{2} < \theta < -\frac{\pi}{4}$
\end{tabular} \\
\hline
\end{tabular}
\end{center}

  \item A lightning impact is materialized on the screen at a point with affix $z$. In each of the two following cases, determine the sector to which this point belongs:\\
a. $z = 70 \mathrm{e}^{-\mathrm{i}\frac{\pi}{3}}$;\\
b. $z = -45\sqrt{3} + 45\mathrm{i}$.
\end{enumerate}

\section*{Part B}
We assume in this part that the sensor displays an impact at point P with affix $50\mathrm{e}^{\mathrm{i}\frac{\pi}{3}}$. When the sensor displays the impact point P with affix $50\mathrm{e}^{\mathrm{i}\frac{\pi}{3}}$, the affix $z$ of the actual lightning impact point admits:
\begin{itemize}
  \item a modulus that can be modeled by a random variable $M$ following a normal distribution with mean $\mu = 50$ and standard deviation $\sigma = 5$;
  \item an argument that can be modeled by a random variable $T$ following a normal distribution with mean $\frac{\pi}{3}$ and standard deviation $\frac{\pi}{12}$.
\end{itemize}
We assume that the random variables $M$ and $T$ are independent.\\
In the following, probabilities will be rounded to $10^{-3}$ near.

\begin{enumerate}
  \item Calculate the probability $P(M < 0)$ and interpret the result obtained.
  \item Calculate the probability $P(M \in ]40 ; 60[)$.
  \item We admit that $P\left(T \in \left]\frac{\pi}{4} ; \frac{\pi}{2}\right[\right) = 0.819$. Deduce the probability that the lightning actually struck sector B3 according to this modeling.
\end{enumerate}