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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{Série: Droite dans le plan}
\def\schoolname{\textbf{Lycée :} 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 definitions ---

% Exam-specific settings and custom definitions
% This file is generated in the current working directory
% Edit this file to customize your exam settings

% Define column type for centered cells
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\newcolumntype{M}[1]{@{}>{\centering\arraybackslash}m{\#1}@{}}

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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:
\newcommand{\dif}{\mathrm{d}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\C}{\mathbb{C}}


% --- Section command ---
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\newcommand{\printexo}[3]{%
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% --- Header Style 9 (fr) ---
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      \textbf{\classname} & \centering \textbf{\examtitle}
      & \ddate \\
      \wsite & \centering \large\textbf{\secondtitle}
      &\hfill
        \begin{tabular}{|c}
          \hline
          \textbf{~~\professor}
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\begin{document}


\vspace*{-0.75cm}
% Exercise 1 (originally 1)
\printexo{1}{}{
On considère les points \(A(2 ; 4)\), \(B(-1 ; 3)\) et \(C(1 ; -2)\) et \(D(x + 1 ; 5)\)
\begin{enumerate}[tight]
    \item Calculer les coordonnées des vecteurs \(\overrightarrow{AB}\) et \(\overrightarrow{AC}\).
    \item Calculer les coordonnées de le milieu du segment \([AB]\).
    \item Déterminer le nombre réel \(x\) pour que :
          ~$
          \overrightarrow{CD} = \overrightarrow{AB}.
          $~
\end{enumerate}
}

% Exercise 2 (originally 2)
\printexo{2}{}{
\begin{enumerate}[tight]
    \item Soient \(\vec{u}\) et \(\vec{v}\) deux vecteurs tels que :
          ~$
          \vec{u}(2 ; -3) ; \vec{v}(-6 ; 9) \text{ et } \vec{w}(3x + 2 ; 12)
          $~
          \begin{enumerate}[tight,label=\alph*)]
              \item Montrer que les vecteurs \(\vec{u}\) et \(\vec{v}\) sont colinéaires.
              \item Déterminer la valeur de \(x\) tel que \(\vec{u}\) et \(\vec{w}\) sont colinéaires.
          \end{enumerate}
    \item On considère les points suivants :
          ~$
          A(1; -8), B(11; 7), C(5; -1) \text{ et } D(7; 2).
          $~\\
          Montrer que \(\overrightarrow{AB}\) et \(\overrightarrow{CD}\) sont colinéaires.
    \item Étudier l’alignement des points \(E, F\) et \(G\) dans les cas suivants :
          \begin{enumerate}[tight,label=\alph*)]
              \item \(E(-4; 2), F(5; 1) \text{ et } G(11; 3)\)
              \item \(E(-2; 3), F(0; -1) \text{ et } G(-1; 1)\)
          \end{enumerate}
\end{enumerate}
}

% Exercise 3 (originally 3)
\printexo{3}{}{
\begin{enumerate}[tight]
    \item Déterminer une équation cartésienne de la droite \(D(A, \vec{u})\) à chaque cas :
          \begin{enumerate}[tight,label=\alph*)]
              \item \(A(-2; 3) \text{ et } \vec{u}(-1; 3)\)
              \item \(A(3; 5) \text{ et } \vec{u}(2; -1)\)
              \item \(A(-1; 1) \text{ et } \vec{u}=2\vec{i} + \vec{j}\)
              \item \(A(2; 0) \text{ et } \vec{u}=3\vec{i} - \vec{j}\)
          \end{enumerate}
    \item Déterminer une équation cartésienne de la droite \((AB)\) à chaque cas :
          \begin{enumerate}[tight,label=\alph*)]
              \item \(A(-2; 1) \text{ et } B(-3; 2)\)
              \item \(A(4; -8) \text{ et } B(-2; 2)\)
          \end{enumerate}
    \item Déterminer l’équation cartésienne de la droite \((D)\) passant par \(A(1 ; -1)\) \\
      et parallèle à la droite \((A)\) d’équation cartésienne
          ~$
          2x - 3y + 5 = 0
          $~
\end{enumerate}
}

% Exercise 4 (originally 4)
\printexo{4}{}{
\begin{enumerate}[tight]
    \item Déterminer une représentation paramétrique de la droite \((D)\) passant par \(A\) \\
      et dirigé par \(\vec{u}\) dans les cas suivants :
          \begin{enumerate}[tight,label=\alph*)]
              \item \(A(2; -3) ; \vec{u} = \vec{i} - 2\vec{j}\)
              \item \(A(1; 0) ; \vec{u}(5; -7)\)
              \item \(A(3; -6) ; \vec{u}(-2; 3)\)
          \end{enumerate}
    \item Déterminer une représentation paramétrique de la droite \((D)\) passant par \(A\) et \(B\) à chaque cas :
          \begin{enumerate}[tight,label=\alph*)]
              \item \(A(2; -3) ; B(3; 2)\)
              \item \(A(1; -1) ; B(-2; 1)\)
              \item \(A(-4; 2) ; B(0; -1)\)
          \end{enumerate}
\end{enumerate}
}

% Exercise 5 (originally 5)
\printexo{5}{}{
\begin{enumerate}[tight]
    \item
    \begin{enumerate}[tight,label=\alph*)]
        \item Est-ce que le point \(E(-1; -11)\) appartient à \((D)\) à chaque cas :\\
              ~$
              (D) \begin{cases}
              x = 2 + 3t \\
              y = -4 + 7t
              \end{cases} (t \in \mathbb{R}) \quad
              (D) \begin{cases}
              x = -1 + t \\
              y = 2 - 2t
              \end{cases} (t \in \mathbb{R})
              $~
        \item Déterminer une équation cartésienne de la droite \((D)\) dans les cas précédents.
    \end{enumerate}
    \item
    \begin{enumerate}[tight,label=\alph*)]
        \item Est-ce que le point \(E(5; -3)\) appartient à \((D)\) à chaque cas :\\
              ~$
              (D) : x - 5 y + 2 = 0, \quad (D) : 2x + 3 y - 1 = 0.
              $~
        \item Déterminer une représentation paramétrique de la droite \((D)\) dans les cas précédents.
    \end{enumerate}
\end{enumerate}
}

% Exercise 6 (originally 6)
\printexo{6}{}{
Déterminer la position relative des droites \((D)\) et \((D')\)\\
(indiquer le point d’intersection s’il existe).
\begin{enumerate}[tight]
    \item \((D) : 2x + y - 3 = 0 ; (D') : x + y = 1\)
    \item \((D) : x + y = 2 ; (D') : 2x + y = 6\)
    \item \((D) : \frac{-1}{3}x + y - 1 = 0 ; (D') : x - 3y + 1 = 0\)
    \item \((D) : x + y - 1 = 0 ; (D') : \begin{cases} x = 2 - t \\ y = 3 + 2t \end{cases} (t \in \mathbb{R})\)
    \item
    ~$
    (D) : \begin{cases} x = -t \\ y = -5 + 2t \end{cases} (t \in \mathbb{R}), \quad
    (D') : \begin{cases} x = 2 - t \\ y = 3 + 2t \end{cases} (t \in \mathbb{R})
    $~
\end{enumerate}

%\textit{« Faites quelque chose et, si ça ne fonctionne pas, essayez autre chose. »}
}






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﴿بِسْمِ ٱللَّهِ ٱلرَّحْمَـٰنِ ٱلرَّحِيمِ ٱلْحَمْدُ لِلَّهِ ٱلَّذِىٓ أَنزَلَ عَلَىٰ عَبْدِهِ ٱلْكِتَـٰبَ وَلَمْ يَجْعَل لَّهُۥ عِوَجَا ۜ (1)•  قَيِّمًا لِّيُنذِرَ بَأْسًا شَدِيدًا مِّن لَّدُنْهُ وَيُبَشِّرَ ٱلْمُؤْمِنِينَ ٱلَّذِينَ يَعْمَلُونَ ٱلصَّـٰلِحَـٰتِ أَنَّ لَهُمْ أَجْرًا حَسَنًا (2)•  مَّـٰكِثِينَ فِيهِ أَبَدًا (3)•  وَيُنذِرَ ٱلَّذِينَ قَالُوا۟ ٱتَّخَذَ ٱللَّهُ وَلَدًا (4)•  مَّا لَهُم بِهِۦ مِنْ عِلْمٍ وَلَا لِـَٔابَآئِهِمْ ۚ كَبُرَتْ كَلِمَةً تَخْرُجُ مِنْ أَفْوَٰهِهِمْ ۚ إِن يَقُولُونَ إِلَّا كَذِبًا (5)•  فَلَعَلَّكَ بَـٰخِعٌ نَّفْسَكَ عَلَىٰٓ ءَاثَـٰرِهِمْ إِن لَّمْ يُؤْمِنُوا۟ بِهَـٰذَا ٱلْحَدِيثِ أَسَفًا (6)• ﴾ (الكهف الآيات 1-6)
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