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% French language support
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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~~}


% Exam-specific settings and custom definitions
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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
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% ----------------------------------------------------------------------
% ADD YOUR CUSTOM DEFINITIONS BELOW THIS LINE
% ----------------------------------------------------------------------

% Example:
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\newcommand{\Z}{\mathbb{Z}}
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      & \ddate \\
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\begin{document}



% Section for Exercise 1
\vspace*{-1cm}
\noindent
\hbox to \textwidth{
  \hfil
  \fcolorbox{red}{yellow!25}{%
    \begin{minipage}{0.6\textwidth}
      Le plan est rapporté à un repère orthonormé direct $(O; \vec i, \vec j)$
    \end{minipage}%
  }%
  \hfil
}%

% Exercise 1 (originally 1)
\printexo{1}{: ~ (8 points)}{
On considère dans le plan, les points \( A(3;4) \), \( B(5;3) \) et \( C(6;6) \).\\
Et soit \( G \) le barycentre du système des trois points pondérés \(\{(A;-2); (B;6); (C;-1)\}\).

\begin{enumerate}
\item
\textbf{a)} Montrer que :
~\(
\overrightarrow{AG} = 2\overrightarrow{AB} - \frac{1}{3}\overrightarrow{AC}
\)~
\dotfill (1\ \text{pt})

\textbf{b)} Placer dans le repère \((O;\vec{i},\vec{j})\) les points \(A, B\) et \(C\) puis construire le point \(G\).
\dotfill (1\ \text{pt})

\textbf{c)} Déterminer les coordonnées du barycentre \(G\).
\dotfill (1\ \text{pt})

\item
Soit \(E\) un point du plan défini par :
~\(
4\overrightarrow{AE} = 6\overrightarrow{AB}
\)~
\dotfill (1\ \text{pt})

\textbf{a)} Montrer que :
~\(
E = \text{bar}\{(A;-2); (B;6)\}
\)~
\dotfill (1\ \text{pt})

\textbf{b)} Déduire que :
~\(
G = \text{bar}\{(E;4); (C;-1)\}
\)~
\dotfill (1\ \text{pt})

\textbf{c)} Déduire que les points \(G, E\) et \(C\) sont alignés.
\dotfill (0,5\ \text{pt})

\item
Soit \(F\) le barycentre du système des deux points pondérés \(\{(B;6); (C;-1)\}\).\\
Montrer que les deux droites \((AF)\) et \((EC)\) se coupent en \(G\).
\dotfill (1\ \text{pt})

\item
Déterminer et construire l’ensemble des points \(M\) du plan tels que :\\\hspace*{2cm}
~\(
\|-2\overrightarrow{MA} + 6\overrightarrow{MB} - \overrightarrow{MC}\| = 3\|\overrightarrow{AB}\|
\dotfill (1,5\ \text{pt})
\)~
\end{enumerate}
}

% Exercise 2 (originally 2)
\printexo{2}{: ~ (4 points)}{
On considère dans le plan, les points \(A(2;4)\), \(B(4;-2)\) et \(C(3+3\sqrt{3}; 1+\sqrt{3})\).
\begin{enumerate}
\item
Calculer
~\(
    \cos(\overline{\overrightarrow{AB}, \overrightarrow{AC}}) \quad \text{et} \quad \sin(\overline{\overrightarrow{AB}, \overrightarrow{AC}})
\)~
\dotfill (3\ \text{pts})

\item
Déduire une mesure de l’angle orienté \((\overrightarrow{AB}, \overrightarrow{AC})\) et déduire la nature du triangle \(ABC\).
\dotfill (1\ \text{pt})
\end{enumerate}
}

% Exercise 3 (originally 3)
\printexo{3}{: ~ (8 points)}{
\vspace*{-1cm}
\begin{enumerate}
\item
Soit \((\varphi)\) le cercle défini par la représentation paramétrique :
~\(
\begin{cases}
x = 3 + \sqrt{2} \cos\theta \\
y = -2 + \sqrt{2} \sin\theta
\end{cases}, \quad \theta \in \mathbb{R}
\)~

\textbf{a)} Montrer que
~\(
x^2 + y^2 - 6x + 4y + 11 = 0
\)~
est une équation cartésienne du cercle \((\varphi)\).
\dotfill (1\ \text{pt})

\textbf{b)} Vérifier que \(K(2;-1) \in (\varphi)\) et montrer que
~\(
x - y - 3 = 0
\)~
est une équation cartésienne de la droite \((T)\) qui est tangente à \((\varphi)\) au point \(K\).
\dotfill (0,25 pt + 1 pt)

\item
Soit \((C)\) l’ensemble des points \(M(x;y)\) du plan tel que
~\(
\overrightarrow{AM} \cdot \overrightarrow{BM} = 0
\)~
avec \(A(2;-1)\) et \(B(2;-5)\).
%\dotfill (1\ \text{pt})

\textbf{a)} Montrer que
~\(
\overrightarrow{AM} \cdot \overrightarrow{BM} = 0 \iff x^2 + y^2 - 4x + 6y + 9 = 0
\)~
\dotfill (1\ \text{pt})

\textbf{b)} Déduire que \((C)\) est le cercle de centre \(\Omega(2;-3)\) et de rayon \(R = 2\).
\dotfill (1\ \text{pt})

\textbf{c)} Montrer que la droite \((T)\) coupe le cercle \((\varphi)\) en deux points distincts.
\dotfill (1\ \text{pt})

\item
\textbf{a)} Construire dans le repère \((O;\vec{i},\vec{j})\) les deux cercles \((\varphi)\) et \((C)\) et la droite \((T)\)
%~\(
%\begin{cases}
%x^2 + y^2 - 6x + 4y + 11 \geq 0 \\
%x^2 + y^2 - 4x + 6y + 9 \leq 0 \\
%x - y - 3 > 0
%\end{cases}
%\)~
\dotfill (0,75\ \text{pt})

\textbf{b)} Résoudre graphiquement le système suivant :
~\(
\begin{cases}
x^2 + y^2 - 6x + 4y + 11 \geq 0 \\
x^2 + y^2 - 4x + 6y + 9 \leq 0 \\
x - y - 3 > 0
\end{cases}
\)~
\dotfill (1\ \text{pt})

\item
Soit \((D_m): 2y - 3mx + \sqrt{5} = 0\) une droite dans le plan tels que \(m\) est un paramètre réel.\\
Déterminer la valeur du paramètre \(m\) sachant que \((D_m) \perp (T)\).
\dotfill (1\ \text{pt})
\end{enumerate}
}






\end{document}