The five questions of this exercise are independent. For each of the following statements, indicate whether it is true or false and justify the answer chosen. An unjustified answer is not taken into account. An absence of an answer is not penalized.
In the set $\mathbb{C}$ of complex numbers, we consider the equation $(E): z^2 - 2\sqrt{3}\,z + 4 = 0$. We denote $A$ and $B$ the points of the plane whose affixes are the solutions of $(E)$. We denote O the point with affix 0. Statement 1: The triangle $OAB$ is equilateral.
We denote $u$ the complex number: $u = \sqrt{3} + \mathrm{i}$ and we denote $\bar{u}$ its conjugate. Statement 2: $u^{2019} + \bar{u}^{2019} = 2^{2019}$
Let $n$ be a non-zero natural number. We consider the function $f_n$ defined on the interval $[0; +\infty[$ by: $$f_n(x) = x\,\mathrm{e}^{-nx+1}$$ Statement 3: For any natural number $n \geqslant 1$, the function $f_n$ admits a maximum.
We denote $\mathscr{C}$ the representative curve of the function $f$ defined on $\mathbb{R}$ by: $f(x) = \cos(x)\,\mathrm{e}^{-x}$. Statement 4: The curve $\mathscr{C}$ admits an asymptote at $+\infty$.
Let $A$ be a strictly positive real number. We consider the algorithm: $$\begin{array}{|l}
I \leftarrow 0 \\
\text{While } 2^I \leqslant A \\
\quad I \leftarrow I + 1 \\
\text{End While}
\end{array}$$ We assume that the variable $I$ contains the value 15 at the end of execution of this algorithm. Statement 5: $15\ln(2) \leqslant \ln(A) \leqslant 16\ln(2)$
\section*{Exercise 3}
The five questions of this exercise are independent.\\
For each of the following statements, indicate whether it is true or false and justify the answer chosen.\\
An unjustified answer is not taken into account. An absence of an answer is not penalized.
\begin{enumerate}
\item In the set $\mathbb{C}$ of complex numbers, we consider the equation $(E): z^2 - 2\sqrt{3}\,z + 4 = 0$.\\
We denote $A$ and $B$ the points of the plane whose affixes are the solutions of $(E)$.\\
We denote O the point with affix 0.\\
\textbf{Statement 1:} The triangle $OAB$ is equilateral.
\item We denote $u$ the complex number: $u = \sqrt{3} + \mathrm{i}$ and we denote $\bar{u}$ its conjugate.\\
\textbf{Statement 2:} $u^{2019} + \bar{u}^{2019} = 2^{2019}$
\item Let $n$ be a non-zero natural number. We consider the function $f_n$ defined on the interval $[0; +\infty[$ by:
$$f_n(x) = x\,\mathrm{e}^{-nx+1}$$
\textbf{Statement 3:} For any natural number $n \geqslant 1$, the function $f_n$ admits a maximum.
\item We denote $\mathscr{C}$ the representative curve of the function $f$ defined on $\mathbb{R}$ by: $f(x) = \cos(x)\,\mathrm{e}^{-x}$.\\
\textbf{Statement 4:} The curve $\mathscr{C}$ admits an asymptote at $+\infty$.
\item Let $A$ be a strictly positive real number.\\
We consider the algorithm:
$$\begin{array}{|l}
I \leftarrow 0 \\
\text{While } 2^I \leqslant A \\
\quad I \leftarrow I + 1 \\
\text{End While}
\end{array}$$
We assume that the variable $I$ contains the value 15 at the end of execution of this algorithm.\\
\textbf{Statement 5:} $15\ln(2) \leqslant \ln(A) \leqslant 16\ln(2)$
\end{enumerate}