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% French language support
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% --- Basic Settings ---
% Language variables for French exams
\def\professor{R. MOSAID}  % Should be R. MOSAID not {{R. MOSAID}}
\def\classname{2BAC.PC/SVT}
\def\examtitle{Devoir 03 - S01}
\def\schoolname{Lycée : Taghzirt}
\def\academicyear{2025/2026}
\def\subject{Mathématiques}
\def\duration{2h}
\def\secondtitle{\small{visit ~~\wsite~~ for more!}}
\def\province{Direction provinciale\newline de Beni Mellal}
\def\logo{\includegraphics[width=\linewidth]{images/logo-men.png}}
\def\wsite{\href{https://www.mosaid.xyz}{\textcolor{magenta}{\texttt{www.mosaid.xyz}}}}
\def\ddate{\hfill \number\day/\number\month/\number\year~~}


% Exam-specific settings and custom definitions
% This file is generated in the current working directory
% Edit this file to customize your exam settings



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% Custom theorem environments (uncomment if needed)
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% \newtheorem{lemma}{Lemma}
% \newtheorem{corollary}{Corollary}

% Custom exercise environments (uncomment if needed)
% \newenvironment{solution}{\begin{proof}[Solution]}{\end{proof}}
% \newenvironment{remark}{\begin{proof}[Remark]}{\end{proof}}

% Custom commands for this exam (add your own below)
% \newcommand{\answerline}[1]{\underline{\hspace{\#1}}}
% \newcommand{\points}[1]{\hfill(\mbox{\#1 points})}

% Additional packages (uncomment if needed)
% \usepackage{siunitx} % For SI units
% \usepackage{chemformula} % For chemical formulas
% \usepackage{circuitikz} % For circuit diagrams

% Page layout adjustments
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% Custom colors for this exam
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% ----------------------------------------------------------------------
% ADD YOUR CUSTOM DEFINITIONS BELOW THIS LINE
% ----------------------------------------------------------------------

% Example:
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\newcommand{\R}{\mathbb{R}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\C}{\mathbb{C}}

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  \fi
  \exothemeone{\fulltitle}%
  \noindent #3%
  \vspace{0.2cm}%
}

% --- Exam Header Style 9 (fr) ---
\newcommand{\printheadnine}{%
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  \begin{tabular}{m{0.22\textwidth} m{0.52\textwidth} m{0.22\textwidth}}
      \textbf{\classname} & \centering \textbf{\examtitle}
      & \ddate \\
      \wsite & \centering \large\textbf{\secondtitle}
      &\hfill
        \begin{tabular}{|c}
          \hline
          \textbf{~~\professor}
        \end{tabular}\\
      \hline
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}

\begin{document}

\noindent\printheadnine

\VMargin{1.7cm}
\vspace*{-0.8cm}
% Exercise 1 (originally 1)
\printexo{1}{}{
\vspace*{-0.75cm}
\begin{questions}
    \question[1 ] Montrer que :
    ~\(
    \ln(4) + \ln\left(\frac{2}{7}\right) + 2\ln\left(\sqrt{7}\right) + \ln\left(\frac{e}{8}\right) = 1
    \)~
    \question[1 ] Résoudre dans \(\mathbb{R}\) l’équation suivante :
    ~\(
    (\mathrm{E}): \ln(x - 1) + \ln(x + 2) = \ln(x + 7)
    \)~
    \question Résoudre dans \(\mathbb{R}\) l’inéquation suivante :
    ~\(
    (\mathrm{I}): \log_{\frac{1}{2}}(3x - 2) > 2
    \)~
    \question On considère la fonction \(g\) définie sur l’intervalle \(D = ]2; +\infty[\) par :
    ~\(
    g(x) = \frac{2x^2 - 4x + 7}{x - 1}
    \)~
    \begin{parts}
        \part[0.5 ] Vérifier que \(\forall x \in D, \; g(x) = 2x - 2 + \frac{5}{x - 1}\).
        \part[1.5 ] Déterminer \(G\) la fonction primitive de \(g\) sur l’intervalle \(I\) telle que \(G(3) = 5\ln(3)\).
    \end{parts}
\end{questions}
}

\vspace*{-0.25cm}
% Exercise 2 (originally 2)
\printexo{2}{}{
\noindent\textbf{Partie I}
On considère la fonction \(g\) définie sur l’intervalle \(]0; +\infty[\) par :
~\(
g(x) = x^2 + 2 - 2\ln(x)
\)~
\begin{questions}
    \question[1 ] Calculer \(g'(x)\) puis dresser le tableau de variations de \(g\).
    \question[1 ] Calculer \(g(1)\) puis déduire que \(\forall x \in ]0; +\infty[, \; g(x) > 0\).
\end{questions}
\noindent\textbf{Partie II}
Soit \(f\) la fonction définie sur l’intervalle \(]0; +\infty[\) par :
~\(
f(x) = x - 1 + 2\frac{\ln x}{x}
\)~
et \((C_f)\) sa courbe représentative dans un repère orthonormé \((O; \vec{i}, \vec{j})\).
\begin{questions}
    \question[1 ] Calculer \(\lim_{x \to 0^+} f(x)\) puis interpréter le résultat graphiquement.
    \question
    \begin{parts}
        \part Calculer \(\lim_{x \to +\infty} f(x)\).
        \part[0.5 ] Montrer que la droite \((\Delta)\) d'équation \(y = x - 1\) est une asymptote oblique à \((C_f)\) au voisinage de \(+\infty\).
        \part[1 ] Déterminer la position relative de \((C_f)\) et de la droite \((\Delta)\).
    \end{parts}
    \item
    \begin{parts}
        \part[1 ] Vérifier que \(\forall x \in ]0; +\infty[, \; f'(x) = \frac{g(x)}{x^2}\).
        \part[0.5 ] Dresser le tableau de variation de la fonction \(f\) sur l’intervalle \(]0; +\infty[\).
    \end{parts}
    \question[1 ] Donner l’équation de la tangente \((T)\) à \((C_f)\) au point d’abscisse \(1\).
    \question[1 ] Tracer la droite \((\Delta)\) et la courbe de \((C_f)\) dans le repère \((O; \vec{i}, \vec{j})\).
\end{questions}
}

\vspace*{-0.25cm}
% Exercise 3 (originally 3)
\printexo{3}{}{
\vspace*{-1cm}
\begin{questions}
  \question[1.5]
  \noindent\textbf{Partie A:~}
  Déterminer la forme algébrique des nombres complexes suivants :\\
~\(
z_1 = (3 - 6i) - 3 + 2i, \quad z_2 = (1 + i)(-5 + 3i), \quad z_3 = \frac{(1+i)(4 - 5i)}{3 + 4i}
\)~
\question
\noindent\textbf{Partie B:~}
Le plan \(\mathcal{P}\) est rapporté à un repère orthonormé direct \((O, \vec{u}, \vec{v})\). On considère les points \(A(1 + 2i)\), \(B(3 + 4i)\) et \(C(3)\).
\begin{parts}
    \part[0.75 ] Placer les points \(A\), \(B\) et \(C\).
    \part[1 ] Déterminer les affixes des vecteurs \(\overrightarrow{AB}\) et \(\overrightarrow{BC}\).
    \part[1 ] Déterminer l’affixe du point \(D\) tel que le quadrilatère \(ABCD\) soit un parallélogramme.
    \part[0.75 ] Déterminer l’affixe du point \(I\) centre du parallélogramme \(ABCD\).
    \part[1 ] Soit \(E\) un point du plan complexe d’affixe \(z_E = \frac{5}{2}(1 + i)\). Les points \(D\), \(B\) et \(E\) sont-ils alignés ?
\end{parts}
\end{questions}
}






\end{document}