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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.SM}
\def\examtitle{Devoir 02 - S02 (1BAC.SM)}
\def\schoolname{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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% 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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% ----------------------------------------------------------------------
% ADD YOUR CUSTOM DEFINITIONS BELOW THIS LINE
% ----------------------------------------------------------------------

% Example:
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\newcommand{\Z}{\mathbb{Z}}
\newcommand{\Q}{\mathbb{Q}}
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\begin{document}




% Exercise 1 (originally 1)
\printexo{1}{(4 Points)}{
\begin{enumerate}[label=\arabic*.]
\item Calculer la limite suivante :
~\(
\lim_{x\to -1} \frac{(2x+3)^{2026}-1}{x + 1} \dotfill (2\text{ Pts})
\)~

\item
\begin{enumerate}[label=\alph*)]
\item Résoudre l'équation différentielle : \((E) : y'' + 16y = 0\) \dotfill (1 Pts)
\item Déterminer la solution particulière de l'équation différentielle \((E)\) qui vérifie:\\ \(y(0)=1\) et \(y\!\left(\frac{\pi}{8}\right)=1\). \dotfill (1 Pts)
\end{enumerate}
\end{enumerate}
}

% Exercise 2 (originally 2)
\printexo{2}{(12 Points)}{
Soit \(f\) la fonction définie sur \(\mathbb{R}\) par :
~\(
f(x)=x^3-3x^2+4
\)~
et soit \((\mathcal{C}_f)\) sa courbe représentative dans un repère orthonormé \((O,\vec{i},\vec{j})\).

\begin{enumerate}[label=\arabic*.]
\item
\begin{enumerate}[label=\alph*)]
\item Calculer \(\displaystyle\lim_{x\to -\infty}f(x)\), \(\displaystyle\lim_{x\to +\infty}f(x)\), \(\displaystyle\lim_{x\to -\infty}\frac{f(x)}{x}\) et \(\displaystyle\lim_{x\to +\infty}\frac{f(x)}{x}\). \dotfill (2 Pts)
\item En déduire les branches infinies de \((\mathcal{C}_f)\). \dotfill (1 Pts)
\end{enumerate}

\item Calculer \(f'(x)\) pour tout \(x\in\mathbb{R}\) et dresser le tableau de variation de \(f\). \dotfill (1,5 Pts)

\item
\begin{enumerate}[label=\alph*)]
\item Étudier la concavité de \((\mathcal{C}_f)\) et déterminer ses points d'inflexion. \dotfill (1 Pts)
\item Montrer que le point \(I(1,2)\) est un centre de symétrie de \((\mathcal{C}_f)\). \dotfill (1 Pts)
\item Déterminer l'équation de la tangente \((T)\) à la courbe \((\mathcal{C}_f)\) au point \(I\). \dotfill (1 Pts)
\end{enumerate}

\item
\begin{enumerate}[label=\alph*)]
\item Vérifier que : \((\forall x\in\mathbb{R})\; f(x)=(x-2)^2(x+1)\). \dotfill (0,5 Pts)
\item En déduire les points d'intersection de \((\mathcal{C}_f)\) avec l'axe des abscisses. \dotfill (1 Pts)
\end{enumerate}

\item Construire la courbe \((\mathcal{C}_f)\). \dotfill (1,5 Pts)

\item Discuter suivant les valeurs du paramètre \(m\) le nombre de solutions de l'équation :\\
  \hspace*{2cm}
~\(
(E_m) : x^3-3x^2-m+4=0 \dotfill (1,5\text{ Pts})
\)~
\end{enumerate}
}

% Exercise 3 (originally 3)
\printexo{3}{(4 Points)}{
On considère la fonction \(g\) définie sur l'intervalle \(I=\left[0,\frac{\pi}{2}\right[\) par :
~\(
g(x)=2\sin x+\tan x-3x
\)~\\
Pour tout \(t\in\mathbb{R}\), on pose \(P(t)=2t^3-3t^2+1\).

\begin{enumerate}[label=\arabic*.]
\item Vérifier que : \((\forall t\in\mathbb{R})\;\quad  P(t)=(t-1)(2t^2-t-1)\). \dotfill (0,5 Pts)

\item
\begin{enumerate}[label=\alph*)]
\item Montrer que : \((\forall x\in I)\;\quad  g'(x)=\dfrac{P(\cos x)}{\cos^2 x}\). \dotfill (1 Pts)
\item En déduire les variations de la fonction \(g\). \dotfill (1 Pts)
\end{enumerate}

\item Montrer que : \((\forall x\in I)\;\quad  3x \le 2\sin x + \tan x\). \dotfill (1,5 Pts)
\end{enumerate}
}






% Bottom message
\begin{center}
  \normalsize{%
    \vskip 3pt \hrule height 3pt \vskip 5pt
    \RL{\arabicfont
    ﴿خُلِقَ ٱلْإِنسَـٰنُ مِنْ عَجَلٍ ۚ سَأُو۟رِيكُمْ ءَايَـٰتِى فَلَا تَسْتَعْجِلُونِ (37)•  وَيَقُولُونَ مَتَىٰ هَـٰذَا ٱلْوَعْدُ إِن كُنتُمْ صَـٰدِقِينَ (38)•  لَوْ يَعْلَمُ ٱلَّذِينَ كَفَرُوا۟ حِينَ لَا يَكُفُّونَ عَن وُجُوهِهِمُ ٱلنَّارَ وَلَا عَن ظُهُورِهِمْ وَلَا هُمْ يُنصَرُونَ (39)•  بَلْ تَأْتِيهِم بَغْتَةً فَتَبْهَتُهُمْ فَلَا يَسْتَطِيعُونَ رَدَّهَا وَلَا هُمْ يُنظَرُونَ (40)• ﴾ (الأنبياء الآيات 37-40)
    }%
  }%
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\end{document}