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% French language support
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% Bottom message in Arabic requires bidi package
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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}
\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~~}


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% Document spacing
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% Math display style
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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]{%
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% --- Header Style 9 (fr) ---
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      & \ddate \\
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\begin{document}



\vspace*{-0.5cm}
% Exercise 1 (originally 1)
\printexo{1}{: ~ (3 points)}{
\begin{enumerate}[tight]
  \item Déterminer une équation cartésienne de la droite \( (D) \) passant par \( A(2; -2) \) \\et de vecteur normal \( \vec{n}(1; 2) \). \dotfill (1 pt)
  \item Montrer que : \( d(B; (D)) = \sqrt{5} \) tels que : \( B(-1; 2) \) et \( D : x + 2y + 2 = 0 \). \dotfill (0.5 pt)
    \item Déterminer la valeur de \( m \) pour que les deux droites \( (\Delta_1) \) et \( (\Delta_2) \) soient perpendiculaires :\\
      \hspace*{1cm}
    ~\(
    (\Delta_1) : 3x + (2m - 1)y + 1 = 0 \quad \text{et} \quad (\Delta_2) : mx + 3y + 1 = 0.
    \)~ \dotfill (1 pt)
    \item Donner une équation cartésienne du cercle \( \mathcal{C} \) de centre \( \Omega(-2; 1) \) et de rayon \( R = \sqrt{3} \). \dotfill (0.5 pt)
\end{enumerate}
}

% Exercise 2 (originally 2)
\printexo{2}{: ~ (9 points)}{
Dans le plan rapporté à un repère orthonormé \( (O, \vec{i}, \vec{j}) \), on considère les points \( A(-1; 1) \), \( B(2; 2) \) et \( C(0; 3) \).
\vspace*{-0.5cm}
\begin{enumerate}[tight]
    \item Déterminer les coordonnées de : \( \overrightarrow{AB} \), \( \overrightarrow{AC} \), \( \overrightarrow{BC} \). \dotfill (0.75 pt)
    \item Calculer les distances : \( AB \), \( AC \), \( BC \). \dotfill (0.75 pt)
    \item Calculer \( \overrightarrow{CA} \cdot \overrightarrow{CB} \) puis déterminer la nature du triangle \( ABC \). \dotfill (1 pt)
    \item
    \begin{enumerate}[tight]
        \item Calculer : \( \overrightarrow{AB} \cdot \overrightarrow{AC} \) et \( \det(\overrightarrow{AB}, \overrightarrow{AC}) \). \dotfill (1 pt)
        \item Calculer : \( \sin(\overrightarrow{AB}, \overrightarrow{AC}) \) et \( \cos(\overrightarrow{AB}, \overrightarrow{AC}) \). \dotfill (1 pt)
        \item Déduire les mesures de l’angle orienté : \( (\overrightarrow{AB}, \overrightarrow{AC}) \). \dotfill (1 pt)
        \item Calculer la surface du triangle \( ABC \). \dotfill (0.5 pt)
    \end{enumerate}
    \item On considère le cercle \( \mathcal{C} \) d’équation cartésienne : \( x^2 + y^2 + 2x - 2y - 8 = 0 \).
    \begin{enumerate}[tight]
        \item Montrer que le cercle \( \mathcal{C} \) est de centre \( A \) et de rayon \( R = \sqrt{10} \). \dotfill (0.5 pt)
        \item Déduire que : \( B \in \mathcal{C} \) et que le point \( C \) est à l’intérieur du cercle \( \mathcal{C} \). \dotfill (0.5 pt)
    \end{enumerate}
    \item Résoudre géométriquement le système :
    ~\(
    (S):
    \begin{cases}
    x^2 + y^2 + 2x - 2y - 2 \geq 0 \\
    x - 2y + 1 > 0
    \end{cases}.
    \)~ \dotfill (1 pt)
    \item Déterminer \( E \) l’ensemble des points \( M(x; y) \) du plan tel que :\\
      \hspace*{2cm}
    ~\(
    (E): x^2 + y^2 - 4x - 4y + 8 = 0.
    \)~ \dotfill (0.5 pt)
\end{enumerate}
}

% Exercise 3 (originally 3)
\printexo{3}{: ~ (8 points)}{
Dans le plan rapporté à un repère orthonormé \( (O, \vec{i}, \vec{j}) \), soit \( ABC \) un triangle et \( I \) le milieu du segment \( [AB] \), on considère \( G \) et \( G' \) deux points tels que :\\\hspace*{1cm}
~\(
G = \text{bar}\{(A; -1), (B; 5)\} \quad \text{et} \quad G' = \text{bar}\{(A; -1), (B; 5), (C; -1)\}.
\)~
\begin{enumerate}[tight]
    \item Montrer que : \( \overrightarrow{AG} = \frac{5}{4} \overrightarrow{AB} \) et que : \( \overrightarrow{AG'} = \frac{5}{3} \overrightarrow{AB} - \frac{1}{3} \overrightarrow{AC} \). \dotfill (2 pt)
    \item Construire le triangle \( ABC \) et les points : \( G \) et \( G' \). \dotfill (2 pt)
    \item En utilisant la propriété d’associativité du barycentre, montrer que les points \( G ,\;G'\) et \( C \) sont alignés.\\. \dotfill (1 pt)
    \item Déterminer l’ensemble des points \( M(x; y) \) du plan tels que :\\
      \hspace*{2cm}
    ~\(
    \| -\overrightarrow{MA} + 5\overrightarrow{MB} \| = \| 2\overrightarrow{MA} + 2\overrightarrow{MB} \|.
    \)~ \dotfill (1.5 pt)
    \item On donne les coordonnées des points \( A(1; 1) \) et \( B(1; 3) \). Déterminer les coordonnées du point \( G(x_G; y_G) \). \dotfill (1.5 pt)
\end{enumerate}
}





% Bottom message
\vspace*{-0.5cm}
\begin{center}
  \normalsize{%
    \vskip 3pt \hrule height 3pt \vskip 5pt
    \RL{\arabicfont
    ﴿بِسْمِ ٱللَّهِ ٱلرَّحْمَـٰنِ ٱلرَّحِيمِ الٓر ۚ تِلْكَ ءَايَـٰتُ ٱلْكِتَـٰبِ وَقُرْءَانٍ مُّبِينٍ﴾ (الحجر 1)
    }%
  }%
\end{center}

\end{document}