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\usepackage{amsmath, amsfonts, amssymb, amsthm}
\usepackage{graphicx}
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\usepackage{multicol}
\usepackage{enumitem}
\usepackage{mathrsfs}
\usepackage{colortbl}
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% French language support
\usepackage[french]{babel}



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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 01 - S02 (1BAC.SF)}
\def\schoolname{Taghzirt}
\def\academicyear{2025/2026}
\def\subject{Mathématiques}
\def\duration{2h}
\def\secondtitle{\small{Par Prof TABRART Abdelaziz}}
\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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% Document spacing
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% Math display style
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%\mathversion{bold}

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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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% \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}}
\newcommand{\C}{\mathbb{C}}


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% --- Header Style 9 (fr)  ---
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\begin{document}




% Exercise 1 (originally 1)
\printexo{1}{}{
Calculer les limites suivantes:\\
\begin{minipage}[t]{0.3\textwidth}
\begin{enumerate}[label=\arabic*.]
    \item \(\displaystyle \lim_{x \to -\infty} \bigl(3x^4 + 2x^3 - 4\bigr).\)
    \item \(\displaystyle \lim_{x \to +\infty} \frac{x^2 + 4x - 3}{3x^4 - 5x^2 + x}.\)
\end{enumerate}
\end{minipage}
\begin{minipage}[t]{0.4\textwidth}
\begin{enumerate}[label=\arabic*.,start=3]
    \item \(\displaystyle \lim_{x \to -2} \frac{x^3 + 8}{x^2 + 3x + 2}.\)
    \item \(\displaystyle \lim_{x \to 1} \frac{\sqrt{3x+1} - 2}{x-1}.\)
    \item \(\displaystyle \lim_{x \to +\infty} \bigl(\sqrt{x^2 + 4x - 1} - 3x\bigr).\)
\end{enumerate}
\end{minipage}
\begin{minipage}[t]{0.3\textwidth}
\begin{enumerate}[label=\arabic*.,start=6]
\item \(\displaystyle \lim_{\substack{x \to \sqrt{3} \\ x > \sqrt{3}}} \frac{2x^2 - 7}{3 - x^2}.\)
\item \(\displaystyle \lim_{x \to 0} \frac{\cos(x) - 1}{x^3} \tan(2x).\)
\end{enumerate}
\end{minipage}
}

% Exercise 2 (originally 2)
\printexo{2}{}{
\begin{enumerate}
    \item Soit \(f\) la fonction définie sur \(]0;+\infty[\) par
    ~\( f(x) = 2 + \frac{\sqrt{x} \cos(x)}{x^2}. \)~
    \begin{enumerate}[label=\alph*.]
        \item Montrer que \(\forall x \in ]0;+\infty[ :\; |f(x) - 2| \leq \dfrac{1}{x\sqrt{x}}.\)
        \item En déduire \(\displaystyle \lim_{x \to +\infty} f(x).\)
    \end{enumerate}
    \item
    \begin{enumerate}[label=\alph*.]
        \item Vérifier que \(\forall x \in \mathbb{R} :\; \sin(x) - \sqrt{3} \cos(x) = 2 \sin\left(x - \frac{\pi}{3}\right).\)
        \item En déduire la limite \(\displaystyle \lim_{x \to \frac{\pi}{3}} \frac{\sin(x) - \sqrt{3} \cos(x)}{3x - \pi}.\)
    \end{enumerate}
\end{enumerate}
}

% Exercise 3 (originally 3)
\printexo{3}{(5pts)}{
On considère les deux triangles \(OAB\) et \(OCD\) isocèles et rectangles en \(O\) (voir la figure ci-contre). Soit \(R\) la rotation de centre \(O\) et d'angle \(\frac{\pi}{2}\).
\begin{tikzpicture}[scale=1.2]

% Center
\coordinate (O) at (0,0);

% Points using polar coordinates
\coordinate (A) at (30:2);
\coordinate (B) at ({30+90}:2);

\coordinate (D) at ({30+90+25}:3);
\coordinate (C) at ({30+90+25+90}:3);

% Fill triangles
\fill[blue!12] (O)--(A)--(B)--cycle;
\fill[green!12] (O)--(C)--(D)--cycle;

% Draw triangles
\draw[thick] (O)--(A)--(B)--cycle;
\draw[thick] (O)--(C)--(D)--cycle;

% Right angle marks at O
\draw ($(O)+(30:0.4)$) --
      ($(O)+(30:0.4)+(120:0.4)$) --
      ($(O)+(120:0.4)$);

\draw ($(O)+(145:0.4)$) --
      ($(O)+(145:0.4)+(235:0.4)$) --
      ($(O)+(235:0.4)$);

% Single ticks for OA = OB
\draw ($(O)!0.5!(A)+(120:0.1)$) -- ($(O)!0.5!(A)+(300:0.1)$);
\draw ($(O)!0.5!(B)+(30:0.1)$)  -- ($(O)!0.5!(B)+(210:0.1)$);

% Double ticks for OC = OD
% First tick
\draw ($(O)!0.5!(C)+(0,-0.2)$) -- ($(O)!0.5!(C)+(0,0.2)$);
\draw ($(O)!0.53!(C)+(0,-0.2)$) -- ($(O)!0.53!(C)+(0,0.2)$);

\draw ($(O)!0.5!(D)+(0,-0.2)$) -- ($(O)!0.5!(D)+(0,0.2)$);
\draw ($(O)!0.53!(D)+(0,-0.2)$) -- ($(O)!0.53!(D)+(0,0.2)$);

% Labels
\node[right] at (A) {$A$};
\node[above left] at (B) {$B$};
\node[left] at (C) {$C$};
\node[above] at (D) {$D$};
\node[below right] at (O) {$O$};

\end{tikzpicture}
\begin{enumerate}
    \item Montrer que \(R(A) = B\) et \(R(D) = C.\)
    \item Montrer que \(BC = AD\) et que \(\bigl(\overline{\overrightarrow{AD}; \overrightarrow{BC}}\bigr) \equiv \frac{\pi}{2}\;[2\pi].\)
    \item Soit \(M\) un point tel que \(R(C) = M.\)
    \begin{enumerate}[label=\alph*.]
        \item Recopie la figure puis construire le point \(M.\)
        \item Montrer que les deux droites \((AC)\) et \((BM)\) sont perpendiculaires.
        \item Montrer que \(O\) est le milieu de \([DM].\)
    \end{enumerate}
    \item Soit \((\zeta)\) un cercle de centre \(D\) tel que \(C \in (\zeta)\). Déterminer l'image de \((\zeta)\) par la rotation \(R\).
\end{enumerate}
}







\end{document}