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% --- Basic Settings ---
\def\professor{R. MOSAID}
\def\classname{TCSF}
\def\examtitle{Devoir 01 - S01}
\def\schoolname{\textbf{Lycée :} Taghzirt}
\def\academicyear{2025/2026}
\def\subject{Mathématiques}
\def\duration{2h}
\def\secondtitle{\small(Arithmétiques \& Calcul vectoriel)}
\def\province{Direction provinciale de\\ Beni Mellal}
\def\logo{\includegraphics[width=\linewidth]{images/logo-men.png}}
\def\bottommsg{Bonne chance!}
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\begin{document}

% Exercise 1
\printexo{1}{}{
Soient ~$ h $~ et ~$ k $~ les fonctions définies par:
~$ h : x \mapsto x^2 + 6x + 6 \quad \text{et} \quad k : x \mapsto \frac{2x+3}{x+2} $~

\begin{enumerate}
    \item Déterminer ~$ D_h $~ et ~$ D_k $~ les ensembles de définition de ~$ h $~ et de ~$ k $~.

    \item Déterminer ~$ T_h $~ le taux de variation de la fonction ~$ h $~.

    \item Déduire les variations de ~$ h $~ sur ~$] - \infty, -\frac{3}{2}]$~ et sur ~$[-\frac{3}{2}, +\infty[$~.

    \item Montrer que: ~$ (\forall x \in D_k),\quad k(x) = 2 - \frac{1}{x+2} $~.

    \item Déduire les variations de ~$ k $~ sur les intervalles de ~$ D_k $~.

    \item Tracer les courbes ~$ (C_h) $~ et ~$ (C_k) $~ de ~$ h $~ et de ~$ k $~.

    \item Déterminer géométriquement la solution de: ~$ h(x) \leq k(x) $~.

    \item Montrer que: ~$ (\forall x \in D_k) \quad h(x) = k(x) \iff (x + 2)(x^2 + 6x + 4) + 1 = 0 $~.

    \item Sur l’intervalle ~$[1, 2]$~:
    \begin{enumerate}
        \item déterminer les variations de ~$ h $~ sur ~$[1, 2]$~ et ~$ h([1, 2]) $~;
        \item déterminer les variations de ~$ k $~ sur ~$ h([1, 2]) $~;
        \item déduire les variations de ~$ k \circ h $~ sur ~$[1, 2]$~.
    \end{enumerate}
\end{enumerate}
}

% Exercise 2
\printexo{2}{}{
\begin{enumerate}
    \item Montrer que: ~$ |x| \leq 2 \Rightarrow \frac{3}{5} \leq \frac{3}{x+3} \leq 3 $~.

    \item Montrer que: ~$ (\forall n \in \mathbb{N}^* ) 4 $~ divise ~$ 5^n - 1 $~.

    \item Montrer que: ~$ (\forall n \in \mathbb{N}) $~ on a: ~$ 2 + 4 + \ldots + (2n) = n(n + 1) $~.

    \item Montrer que: ~$ x \neq 2 $~ et ~$ y \neq 1 \iff xy + 2 \neq x + 2y $~.

    \item Résoudre l’équation suivante: ~$ |3x + 2| = x $~.

    \item Montrer que l’équation: ~$ \frac{x^2 + 4x}{x^2 + 4x + 2} = 1 $~ n’admet pas de solution.

    \item Trouver ~$ a $~ et ~$ b \in \mathbb{N}^* $~ tel que: ~$ (a + 2)(b - 1) = 15 $~.

\end{enumerate}
}

\end{document}