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% This file now contains only packages and settings
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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 surveillé - 2-S1, par Pr : A. SOUFIANE}
\def\schoolname{\textbf{Lycée :} 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 definitions ---

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

% Define column type for centered cells
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% Custom theorem environments (uncomment if needed)
% \newtheorem{theorem}{Theorem}
% \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
% \setlength{\parindent}{0pt}
% \setlength{\parskip}{1em}

% Custom colors for this exam
% \definecolor{myblue}{RGB}{0,100,200}
% \definecolor{mygreen}{RGB}{0,150,0}

% ----------------------------------------------------------------------
% 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}}
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\newcommand{\printexo}[3]{%
  \if\relax\detokenize{\#2}\relax
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    \def\fulltitle{Exercice #1~#2}%
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% --- Header Style 9 (fr) ---
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      \textbf{\classname} & \centering \textbf{\examtitle}
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\begin{document}



% Exercise 1 (originally 1)
\printexo{1}{}{
On considère la fonction numérique \( f \) définie par : \( f(x) = \dfrac{x+1}{2\sqrt{x}} \) et \( (C_f) \) sa courbe représentative dans un repère orthonormé \( (O; \vec{i}, \vec{j}) \) telles que : \( \|\vec{i}\| = \|\vec{j}\| = 1\text{cm} \)
\begin{enumerate}[label=\arabic*)]
    \item Vérifier que \( D_f = ]0; +\infty[ \). \dotfill (0,5 pt)
    \item Montrer que : \( \displaystyle\lim_{x \to 0^+} f(x) = +\infty \), puis interpréter le résultat géométriquement. \dotfill (1 pt)
    \item Calculer : \( \displaystyle\lim_{x \to +\infty} f(x) \) et \( \displaystyle\lim_{x \to +\infty} \dfrac{f(x)}{x} \), puis en déduire la nature de la branche infinie de \( (C_f) \) au voisinage de \( +\infty \). \dotfill (1 pt)
    \item
      \begin{enumerate}
          \item
      Montrer que : \( (\forall x \in D_f) : f(x) - x = \dfrac{(1-\sqrt{x})(2x+\sqrt{x}+1)}{2\sqrt{x}} \). \dotfill (1 pt)
    \item En déduire la position relative de \( (C_f) \) et la droite \( (\Delta) : y = x \). \dotfill (1 pt)
      \end{enumerate}
    \item
      \begin{enumerate}
    \item Montrer que : \( (\forall x \in ]0; +\infty[) : f'(x) = \dfrac{x-1}{4\sqrt{x^3}} \). \dotfill (1 pt)
    \item En déduire que \( f \) est strictement croissante sur \( [1; +\infty[ \) et décroissante sur \( ]0; 1] \). \dotfill (1 pt)
      \end{enumerate}
    \item
      \begin{enumerate}
    \item Montrer que : \( (\forall x \in ]0; +\infty[) : f''(x) = \dfrac{3-x}{8\sqrt{x^5}} \). \dotfill (1 pt)
    \item Déduire la concavité de \( (C_f) \) et déterminer son point d'inflexion s'il existe. \dotfill (1 pt)
      \end{enumerate}
    \item
      \begin{enumerate}
        \item Déterminer les primitives de \( f \) sur \( ]0; +\infty[ \) (\textbf{remarquez que} : \( f(x) = \dfrac{\sqrt{x}}{2} + \dfrac{1}{2\sqrt{x}} \)). \dotfill (1 pt)
    \item En déduire la fonction primitive \( F \) de \( f \) vérifiant : \( F(1) = 0 \). \dotfill (1 pt)
      \end{enumerate}
    \item Construire dans le même repère la droite \( (\Delta) \) et la courbe \( (C_f) \). \dotfill (1 pt)
    \item On considère la suite \( (u_n) \) définie par : \( u_0 = 4 \), \( u_{n+1} = f(u_n) \) avec \( n \in \mathbb{N} \).
    \begin{enumerate}[label=\alph*)]
        \item Montrer que : \( (\forall n \in \mathbb{N}) : u_n \geq 1 \). \dotfill (1 pt)
        \item Montrer que \( (u_n) \) est décroissante. \dotfill (0,5 pt)
        \item En déduire que la suite \( (u_n) \) est convergente, puis calculer \( \displaystyle\lim_{n \to +\infty} u_n \). \dotfill (1 pt)
    \end{enumerate}
\end{enumerate}
}

% Exercise 2 (originally 2)
\printexo{2}{}{
On considère la suite numérique \( (u_n) \) définie par : \( u_0 = 4 \), \( u_{n+1} = \dfrac{4u_n-2}{1+u_n}, \forall n \in \mathbb{N} \). \\
Et soit \( (v_n) \) la suite numérique définie par : \( v_n = \dfrac{2-u_n}{1-u_n} \) pour tout \( n \in \mathbb{N} \).
\begin{enumerate}[label=\arabic*)]
    \item Montrer que \( (v_n) \) est une suite géométrique de raison \( \dfrac{2}{3} \), puis calculer \( v_0 \). \dotfill (1,5 + 0,5 pt)
    \item Écrire \( v_n \) en fonction de \( n \), puis déduire que : \( u_n = 1 + \dfrac{1}{1-\left(\dfrac{3}{2}\right)^{n+1}}, \forall n \in \mathbb{N} \). \dotfill (1 $\times$ 2 pt)
    \item Calculer \( \displaystyle\lim_{n \to +\infty} u_n \), puis en déduire la convergence de la suite \( (u_n) \). \dotfill (1 $\times$ 2 pt)
\end{enumerate}
}






% Bottom message
\begin{center}
  \normalsize{ \vskip 3pt \hrule height 3pt \vskip 5pt \RL{\arabicfont
  ﴿قُل لَّوْ كَانَ ٱلْبَحْرُ مِدَادًا لِّكَلِمَـٰتِ رَبِّى لَنَفِدَ ٱلْبَحْرُ قَبْلَ أَن تَنفَدَ كَلِمَـٰتُ رَبِّى وَلَوْ جِئْنَا بِمِثْلِهِۦ مَدَدًا﴾ (الكهف 109)
  }  }
\end{center}


\end{document}