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% Language variables for French exams
\def\professor{R. MOSAID}  % Should be R. MOSAID not {{R. MOSAID}}
\def\classname{2BAC.PC/SVT}
\def\examtitle{Devoir 02 - S01}
\def\schoolname{Lycée : Taghzirt}
\def\academicyear{2025/2026}
\def\subject{Mathématiques}
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\begin{document}



\vspace*{-0.75cm}
% Exercise 1 (originally 1)
\printexo{1}{: ~ (8 points)}{
On considère la suite numérique \((u_n)\) définie par :
~\(
u_0 = 2 \quad \text{et} \quad u_{n+1} = \frac{2u_n + 3}{u_n} \quad \text{pour tout } n \in \mathbb{N}
\)~
\begin{enumerate}[label=\arabic*.]
    \item Montrer que \(2 \leq u_n \leq \frac{7}{2}\) pour tout \(n \in \mathbb{N}\). \dotfill (1 pt)

    \item
    \begin{enumerate}[label=\alph*)]
        \item Montrer que \((\forall n \in \mathbb{N}) : |u_{n+1} - 3| \leq \frac{1}{2}|u_n - 3|\). \dotfill (1 pt)
        \item En déduire que \((\forall n \in \mathbb{N}) : |u_n - 3| \leq (\frac{1}{2})^n\). \dotfill (0,5 pt)
        \item Déterminer la limite de la suite \((u_n)\). \dotfill (0,25 pt)
    \end{enumerate}

    \item Soit \((v_n)\) la suite numérique telle que : \(v_n = \frac{u_n - 3}{u_n + 1}\) pour tout \(n \in \mathbb{N}\).
    \begin{enumerate}[label=\alph*)]
        \item Montrer que \((v_n)\) est une suite géométrique. \dotfill (1 pt)
        \item Exprimer \(v_n\) puis \(u_n\) en fonction de \(n\). \dotfill (1,5 pt)
    \end{enumerate}

    \item On pose : \(S_n = \frac{4}{u_0 + 1} + \frac{4}{u_1 + 1} + \cdots + \frac{4}{u_n + 1}\). \dotfill (0,75 pt)

    \item Montrer que \(S_n = n + \frac{5}{4} - \frac{1}{4} \Big(\frac{-1}{3}\Big)^{n+1}\) pour tout \(n \in \mathbb{N}\), puis calculer \(\lim_{n \to +\infty} S_n\). \dotfill (2 pts)
\end{enumerate}
}

% Exercise 2 (originally 2)
\printexo{2}{: ~ (12 points)}{
Soit \(f\) une fonction définie par : \(f(x) = \frac{x^2 + x + 2}{2(x-1)}\) et \((C_f)\) sa courbe représentative dans un repère orthonormé \((O; \vec{i}; \vec{j})\).

\begin{enumerate}[label=\arabic*.]
    \item
    \begin{enumerate}[label=\alph*)]
        \item Déterminer \(D_f\), le domaine de définition de \(f\). \dotfill (0,5 pt)
        \item Calculer \(\lim_{x \to +\infty} f(x)\) et \(\lim_{x \to -\infty} f(x)\). \dotfill (1 pt)
    \end{enumerate}

    \item Calculer \(\lim_{x \to 1^+} f(x)\) et \(\lim_{x \to 1^-} f(x)\), puis donner une interprétation géométrique des résultats.\\. \dotfill (1 pt)

    \item
    \begin{enumerate}[label=\alph*)]
        \item Montrer que pour tout \(x \in D_f\) :
        ~\(
        f(x) = \frac{1}{2}x + 1 + \frac{2}{x-1}
        \)~ \dotfill (1 pts)
        \item En déduire l’équation de l’asymptote oblique \((\Delta)\) de la courbe \((C_f)\). \dotfill (1 pt)
        \item Étudier la position relative de la courbe \((C_f)\) et l’asymptote \((\Delta)\). \dotfill (0,5 pt)
    \end{enumerate}

    \item
    \begin{enumerate}[label=\alph*)]
        \item Montrer que :
        ~\(
        (\forall x \in D_f); f'(x) = \frac{(x + 1)(x - 3)}{2(x - 1)^2}
        \)~ \dotfill (2 pts)
        \item Dresser le tableau de variations de \(f\). \dotfill (2 pts)
        \item Écrire l’équation de la tangente \((T)\) à la courbe \((C_f)\) au point d’abscisse 0. \dotfill (1 pt)
    \end{enumerate}

    \item Tracer la courbe \((C_f)\) dans le repère \((O; \vec{i}; \vec{j})\). \dotfill (2,5 pts)
\end{enumerate}
}





% Bottom message
\begin{center}
  \normalsize{%
    \RL{\arabicfont
    ﴿أَوَلَمْ يَتَفَكَّرُوا۟ فِىٓ أَنفُسِهِم ۗ مَّا خَلَقَ ٱللَّهُ ٱلسَّمَـٰوَٰتِ وَٱلْأَرْضَ وَمَا بَيْنَهُمَآ إِلَّا بِٱلْحَقِّ وَأَجَلٍ مُّسَمًّى ۗ وَإِنَّ كَثِيرًا مِّنَ ٱلنَّاسِ بِلِقَآئِ رَبِّهِمْ لَكَـٰفِرُونَ (8)•  أَوَلَمْ يَسِيرُوا۟ فِى ٱلْأَرْضِ فَيَنظُرُوا۟ كَيْفَ كَانَ عَـٰقِبَةُ ٱلَّذِينَ مِن قَبْلِهِمْ ۚ كَانُوٓا۟ أَشَدَّ مِنْهُمْ قُوَّةً وَأَثَارُوا۟ ٱلْأَرْضَ وَعَمَرُوهَآ أَكْثَرَ مِمَّا عَمَرُوهَا وَجَآءَتْهُمْ رُسُلُهُم بِٱلْبَيِّنَـٰتِ ۖ فَمَا كَانَ ٱللَّهُ لِيَظْلِمَهُمْ وَلَـٰكِن كَانُوٓا۟ أَنفُسَهُمْ يَظْلِمُونَ (9)• ﴾ (الروم الآيات 8-9)
    }%
  }%
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