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% --- Basic Settings ---
% Language variables for French exams
\def\professor{R. MOSAID}  % Should be R. MOSAID not {{R. MOSAID}}
\def\classname{1BAC.SF}
\def\examtitle{Devoir 02 - 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}
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\def\wsite{\href{https://www.mosaid.xyz}{\textcolor{magenta}{\texttt{www.mosaid.xyz}}}}
\def\ddate{\hfill \number\day/\number\month/\number\year~~}


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\begin{document}



\vspace*{-1cm}
% Exercise 1 (originally 1)
\printexo{1}{: ~ (12 points)}{
On considère les fonctions numériques \( f \) et \( g \) définies par :
~\(
f(x) = \sqrt{x-1} \quad \text{et} \quad g(x) = -x^2 + 4x
\)~\\
\( C_f \) et \( C_g \) les courbes représentatives de \( f \) et \( g \) dans un repère orthonormé \( (O ; \vec{i}, \vec{j}) \).

\begin{enumerate}[tight]
\item
\begin{enumerate}[label=\alph*)]
    \item Déterminer \( D_f \) et dresser le tableau de variations de \( f \). \dotfill (0,5 pt)
    \item Déterminer \( D_g \) et dresser le tableau de variations de \( g \). \dotfill (0,75 pt)
\end{enumerate}

\item
\begin{enumerate}[label=\alph*)]
    \item Déterminer les points d’intersection de la courbe \( C_g \) avec l’axe des abscisses. \dotfill (0,75 pt)
    \item Construire \( C_f \) et \( C_g \) dans le repère \( (O ; \vec{i}, \vec{j}) \). \dotfill (1,5 pt)
    \item Déterminer graphiquement : \( f([1; 5]) \) et \( f([5; +\infty[) \). \dotfill (1 pt)
\end{enumerate}

\item
\begin{enumerate}[label=\alph*)]
    \item Montrer que l’équation \( x^2 - 4x + \sqrt{x-1} = 0 \) admet une solution unique \( \alpha \) sur \( [1; +\infty[ \). \dotfill (1 pt)
    \item Résoudre graphiquement l’inéquation \( x^2 - 4x + \sqrt{x-1} \leq 0 \). \dotfill (1 pt)
    \item Montrer que : \( 3 < \alpha < 4 \). \dotfill (0,5 pt)
    \item Déduire une comparaison de \( -5 + 4\sqrt{5} \) et \( \sqrt{\sqrt5 - 1} \). \dotfill (0,5 pt)
\end{enumerate}

\item
\begin{enumerate}[label=\alph*)]
    \item Déterminer l’ensemble de définition de la fonction \( g \circ f \). \dotfill (0,5 pt)
    \item Calculer \( (g \circ f)(x) \) pour tout \( x \) de \( D_{g \circ f} \). \dotfill (0,5 pt)
    \item Étudier la monotonie de \( g \circ f \) sur \( [1; 5] \) et sur \( [5; +\infty[ \), et dresser le tableau de variations de \( g \circ f \).\\. \dotfill (1 pt)
    \item En déduire que : \( \forall x \in [1; +\infty[ : 4\sqrt{x-1} - x \leq 3 \). \dotfill (0,5 pt)
\end{enumerate}

\item
On considère la fonction numérique \( h \) définie par : \( h(x) = \sqrt{|x| - 1} \).
\begin{enumerate}[label=\alph*)]
    \item Montrer que : \( D_h = ]-\infty; -1] \cup [1; +\infty[ \). \dotfill (0,5 pt)
    \item Étudier la parité de \( h \). \dotfill (0,5 pt)
    \item Construire \( C_h \) la courbe représentative de \( h \) dans le même repère orthonormé \( (O ; \vec{i}, \vec{j}) \). \dotfill (1 pt)
\end{enumerate}
\end{enumerate}
}

% Exercise 2 (originally 2)
\printexo{2}{: ~ (8 points)}{
Soit ABC un triangle, et soient G et H deux points tels que :\\
\hspace*{2cm}
~\(
2\overrightarrow{AG} - 2\overrightarrow{AB} + \overrightarrow{CA} = \vec{0} \quad \text{et} \quad H = \text{bar} \{(A; -1) ; (B; 2)\}
\)~
\begin{enumerate}[tight]
    \item Montrer que : \( G = \text{bar} \{(A; -1) ; (B; 2) ; (C; 1)\} \). \dotfill (1 pt)
    \item Construire les points G et H. \dotfill (1 pt)
    \item Montrer que G est le milieu du segment \( [CH] \). \dotfill (1 pt)
    \item Soit K un point du plan tel que : \( \overrightarrow{BK} = \frac{1}{3} \overrightarrow{BC} \). Montrer que : \( K = \text{bar} \{(B; 2) ; (C; 1)\} \). \dotfill (1 pt)
    \item Montrer que les droites \( (CH) \) et \( (AK) \) se coupent au point G. \dotfill (1 pt)
    \item Sachant que \( A(-1; 2) , B(2; 1) \) et \( C(1; 3) \), déterminer les coordonnées de G. \dotfill (1 pt)
    \item Déterminer l’ensemble des points M du plan tels que :
    \begin{enumerate}[label=\alph*)]
        \item \( \|-\overrightarrow{MA} + 2\overrightarrow{MB} + \overrightarrow{MC}\| = \|2\overrightarrow{MA} - 4\overrightarrow{MB}\| \). \dotfill (1 pt)
        \item \(\| 2\overrightarrow{MB} + \overrightarrow{MC} \|= \|2\overrightarrow{MA} - \overrightarrow{MB} - \overrightarrow{MC} \|\). \dotfill (1 pt)
    \end{enumerate}
\end{enumerate}
}





% Bottom message
\begin{center}
  \normalsize{%
    \vskip 3pt \hrule height 3pt \vskip 5pt
    \RL{\arabicfont
    ﴿وَإِن تَعُدُّوا۟ نِعْمَةَ ٱللَّهِ لَا تُحْصُوهَآ ۗ إِنَّ ٱللَّهَ لَغَفُورٌ رَّحِيمٌ﴾ (النحل 18)
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\end{center}

\end{document}