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% --- Basic Settings ---
% Language variables for French exams
\def\professor{R. MOSAID}  % Should be R. MOSAID not {{R. MOSAID}}
\def\classname{TCSF}
\def\examtitle{Devoir 02 - S02 (TCSF)}
\def\schoolname{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}
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% Additional packages (uncomment if needed)
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\begin{document}




\vspace*{-0.75cm}
% Exercise 1 (originally 1)
\printexo{1}{: ~ (6 points)}{
Determiner les domaines de définition des fonctions suivantes:\\

~$f(x) = \frac{3x^2}{x^2 - 9}$~\quad ;\quad
~$g(x) = \frac{\sqrt{3}x}{x^4 + 5}$~\quad ;\quad
~$h(x) = \sqrt{3x^2 - 5x + 2}$~\quad ;\quad
~$k(x) = \sqrt{\frac{x^3 - 1}{-x^2 - x + 6}}$~
}

% Exercise 2 (originally 2)
\printexo{2}{: ~ (5.5 points)}{
\hspace*{0.5cm}Soit ~$f$~ une fonction numérique définie sur ~$[-2,4]$~ et sa représentation graphique est la suivante
\begin{tikzpicture}[overlay, remember picture, xshift=13.5cm, yshift=-4cm]
%\begin{tikzpicture}[overlay, remember picture, xshift=-5cm, yshift=-4.5cm]
    \coordinate (A) at (-2,0);
    \coordinate (B) at (-1,3);
    \coordinate (C) at (1,-1);
    \coordinate (D) at (2,3);
    \coordinate (E) at (4,1);
    \draw[red!75!black, looseness=0.3, thick] (A) to[out=0,in=180] (B)
                                  to[out=0,in=180] (C)
                                  to[out=0,in=-110] (D)
                                  -- (E)
                                  node[above right] {$C_f$};
    \draw[thin, gray, dashed] (-2.5,-1.5) grid (4,4);
    \draw[thick] (0,-1.5) -- (0,3.5);
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    \draw (\x,-0.1) -- (\x,0.1) node [below left, yshift=-0.1cm,font=\fontsize{6}{8}\selectfont] {$\x$} ;
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    \draw (-0.1,\x) -- (0.1,\x) node [left, yshift=0.1cm,font=\fontsize{6}{8}\selectfont] {$\x$} ;
    }
    \fill (0.38,0) circle (1.5pt) node[above, xshift=0.1cm,font=\fontsize{6}{8}\selectfont]{$ \frac{4}{10}$};
    \fill (1.45,0) circle (1.5pt) node[above, xshift=-0.1cm,font=\fontsize{6}{8}\selectfont]{$ \frac{14}{10}$};
\end{tikzpicture}
\begin{enumerate}
  \item
  Compléter le tableau suivant\\
  \begin{tabular}{| *{7}{ >{~~\(} c < {\)~~}| }   }
    \hline
    x & -1 & 0 & 1 & 2 & 3 &4 \\
    \hline
    f(x) &&&&&&\\
    \hline
  \end{tabular}
  \item
  Déterminer les extremums de la fonctionc ~$f$~
  \item
  Dresser le tableau des variations de ~$f$~
  \item
  Résoudre graphiquement sur $[-2,4]$\\
l'équation ~$f(x)=3$~ et l'inéquation ~$f(x)<0$
\end{enumerate}
}

% Exercise 3 (originally 3)
\printexo{3}{: ~ (8.5 points)}{
\begin{enumerate}
    \item On considère la fonction $g(x) = -\dfrac{2}{x}$.

    \begin{enumerate}
        \item Déterminer l'ensemble de définition $D_g$.

        \item Déterminer la nature de la courbe représentative $(\zeta_g)$.
    \end{enumerate}

    \item Construire $(\zeta_g)$ dans un repère orthonormé $(O, \vec{i}, \vec{j})$.

    \item On considère maintenant la fonction $f(x) = \dfrac{x-4}{x-2}$.

    \begin{enumerate}
        \item Déterminer l'ensemble de définition $D_f$.

        \item Vérifier que :
        ~\(
        f(x)  = 1 -\frac{2}{x-2}
        \)~

        \item Déterminer une transformation permettant de construire $(\zeta_f)$ à partir de $(\zeta_g)$.

        \item Construire $(\zeta_f)$ dans un repère orthonormé $(O, \vec{i}, \vec{j})$.
    \end{enumerate}

    \item Déterminer graphiquement le tableau de variations de $f$.

    \item Déterminer graphiquement le signe de $f(x)$.
\end{enumerate}
}






% Bottom message
\vspace*{-0.5cm}
\begin{center}
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    ﴿ٱللَّهُ نُورُ ٱلسَّمَـٰوَٰتِ وَٱلْأَرْضِ ۚ مَثَلُ نُورِهِۦ كَمِشْكَوٰةٍ فِيهَا مِصْبَاحٌ ۖ ٱلْمِصْبَاحُ فِى زُجَاجَةٍ ۖ ٱلزُّجَاجَةُ كَأَنَّهَا كَوْكَبٌ دُرِّىٌّ يُوقَدُ مِن شَجَرَةٍ مُّبَـٰرَكَةٍ زَيْتُونَةٍ لَّا شَرْقِيَّةٍ وَلَا غَرْبِيَّةٍ يَكَادُ زَيْتُهَا يُضِىٓءُ وَلَوْ لَمْ تَمْسَسْهُ نَارٌ ۚ نُّورٌ عَلَىٰ نُورٍ ۗ يَهْدِى ٱللَّهُ لِنُورِهِۦ مَن يَشَآءُ ۚ وَيَضْرِبُ ٱللَّهُ ٱلْأَمْثَـٰلَ لِلنَّاسِ ۗ وَٱللَّهُ بِكُلِّ شَىْءٍ عَلِيمٌ﴾ (النور 35)
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\end{document}