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% --- Basic Settings ---
\def\professor{R. MOSAID}
\def\classname{1BAC.SM}
\def\examtitle{Devoir 02 - S01, par Prof : FAHD HAOURECH}
\def\schoolname{\textbf{Lycée :} Taghzirt}
\def\academicyear{2025/2026}
\def\subject{Mathématiques}
\def\duration{2h}
\def\secondtitle{\small(Ensembles \& applications)}
\def\province{Direction provinciale de\\ Beni Mellal}
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\begin{document}

% Exercise 1
\printexo{1}{(2pts)}{
On considère les ensembles A et B suivants :\\
~$A = \{(x, y) \in \mathbb{Z}^2 / x^2 + y^2 = 1\} \quad \text{et} \quad B = \{n \in \mathbb{N} / n^2 - 7n + 6 < 0\}$~\\
Ecrire en extension les ensembles A et B.
}

% Exercise 2
\printexo{2}{(3pts)}{
On considère les ensembles A et B suivants :\\
~$A = \left\{ \frac{\pi}{3} + \frac{k\pi}{6} / k \in \mathbb{Z} \right\} \quad \text{et} \quad B = \left\{ \frac{\pi}{6} + \frac{k\pi}{3} / k \in \mathbb{Z} \right\}$~\\
Montrer que : \(B \subset A\) et \(A \not\subset B\)
}

% Exercise 3
\printexo{3}{(6pts)}{
On considère l'application \(f\) suivante :
\begin{tabular}{ll}
~$f : \mathbb{R} \backslash \{1\} \rightarrow \mathbb{R} $~\\
~$\quad\;\; x \mapsto \frac{x^2 - 2x + 3}{(x - 1)^2}$~
\end{tabular}


\begin{enumerate}
    \item
    \begin{enumerate}[label=(\alph*)]
        \item Montrer que : \((\forall x \in \mathbb{R} \backslash \{1\}) : f(x) > 1\)
        \item \(f\) est-elle surjective? Justifier.
    \end{enumerate}

    \item
    \begin{enumerate}[label=(\alph*)]
        \item Montrer que : \((\forall x \in IR \backslash \{1\}) : f(2 - x) = f(x)\)
        \item \(f\) est -elle injective? Justifier.
    \end{enumerate}

    \item On considère l'application \(g\) suivante :
\begin{tabular}{ll}
      ~$g : ]1, +\infty[\rightarrow ]1, +\infty[$ \\
      $\quad\;\; x \mapsto \frac{x^2 - 2x + 3}{(x - 1)^2}$~
\end{tabular}

    Montrer que \(g\) est bijective et déterminer sa bijection réciproque \(g^{-1}\).
\end{enumerate}
}

% Exercise 4
\printexo{4}{(5pts)}{
Soient A, B et C trois parties d'un ensemble E.
\begin{enumerate}
    \item Montrer que : \((A \cup B) \times C = (A \times C) \cup (B \times C)\)
    \item Montrer que : \(A \backslash (B \cap C) = (A \backslash B) \cup (A \backslash C)\)
    \item Montrer que : \(A \backslash (B \backslash C) = (A \backslash B) \cup (A \cap C)\)
    \item Montrer que : \(A \backslash B = B \backslash A \Leftrightarrow A = B\)
\end{enumerate}
}

% Exercise 5
\printexo{5}{(3pts)}{
  \vspace*{-0.75cm}
\begin{enumerate}
    \item Montrer que : \((\forall (a, b) \in IR^2) : a \times b \leq a^2 + b^2\)
    \item Soient les ensembles : \(A = \{x \in IR / x^2 - 2ax + b = 0\}\) Et \(B = \{x \in IR / x^2 - 2cx + d = 0\}\) où \(a, b, c\) et \(d\) sont des nombres réels tels que : \(b + d = ac\) Montrer que : \(A \neq \varnothing\) ou \(B \neq \varnothing\)
\end{enumerate}
}
\begin{center}
  \vskip 3pt \hrule height 3pt \vskip 5pt \RL{\arabicfont ﴿بِسْمِ ٱللَّهِ ٱلرَّحْمَـٰنِ ٱلرَّحِيمِ تَبَـٰرَكَ ٱلَّذِى بِيَدِهِ ٱلْمُلْكُ وَهُوَ عَلَىٰ كُلِّ شَىْءٍ قَدِيرٌ (1)•  ٱلَّذِى خَلَقَ ٱلْمَوْتَ وَٱلْحَيَوٰةَ لِيَبْلُوَكُمْ أَيُّكُمْ أَحْسَنُ عَمَلًا ۚ وَهُوَ ٱلْعَزِيزُ ٱلْغَفُورُ (2)• ﴾ (الملك الآيات 1-2) }
\end{center}

\end{document}