\textbf{Exercise 4 — Candidates who have NOT followed the specialization course}

For each of the following statements, say whether it is true or false by justifying the answer. One point is awarded for each correct justified answer. An unjustified answer will not be taken into account and the absence of an answer is not penalized.

\begin{itemize}
  \item In the diagram below, the density curve of a random variable $X$ following a normal distribution with mean $\mu = 20$ is represented. The probability that the random variable $X$ is between 20 and 21.6 is equal to 0.34.
\end{itemize}

Statement 1: The probability that the random variable $X$ belongs to the interval $[23.2; + \infty [$ is approximately 0.046.

\begin{itemize}
  \item Let $z$ be a complex number different from 2. We set:
\end{itemize}

$$Z = \frac { \mathrm { i } z } { z - 2 }$$

Statement 2: The set of points in the complex plane with affixe $z$ such that $| Z | = 1$ is a line passing through point $\mathrm { A } ( 1 ; 0 )$.\\
Statement 3: $Z$ is a pure imaginary number if and only if $z$ is real.

\begin{itemize}
  \item Let $f$ be the function defined on $\mathbb { R }$ by:
\end{itemize}

$$f ( x ) = \frac { 3 } { 4 + 6 \mathrm { e } ^ { - 2 x } }$$

Statement 4: The equation $f ( x ) = 0.5$ has a unique solution on $\mathbb { R }$.\\
Statement 5: The following algorithm displays as output the value 0.54.

\begin{center}
\begin{tabular}{|l|l|}
\hline
\begin{tabular}{l}
Variables: Initialization: \\
Processing: \\
Output: \\
\end{tabular} & \begin{tabular}{l}
$X$ and $Y$ are real numbers \\
$X$ takes the value 0 \\
$Y$ takes the value $\frac { 3 } { 10 }$ \\
While $Y < 0.5$ \\
$X$ takes the value $X + 0.01$ \\
$Y$ takes the value $\frac { 3 } { 4 + 6 \mathrm { e } ^ { - 2 X } }$ \\
End While \\
Display $X$ \\
\end{tabular} \\
\hline
\end{tabular}
\end{center}