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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{Devoir 01 - S01 (TCSF)}
\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}
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\def\ddate{\hfill \number\day/\number\month/\number\year~~}


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% Custom exercise environments (uncomment if needed)
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% Additional packages (uncomment if needed)
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\begin{document}




\vspace*{-1cm}
% Exercise 1 (originally 1)
\printexo{1}{(7 points)}{
Déterminer le domaine de définition de la fonction \( f \) dans les cas suivants :\\
~$f(x) = \frac{2x-1}{5x^2+8x-4}$~ \quad ; \quad
~$f(x) = \sqrt{5x^2+8x-4}$~ \quad ; \quad
~$f(x) = \frac{7x^2+2}{2\cos^2(x)-1}$~ \quad ; \quad
~$f(x) = \sqrt{\frac{2x-1}{5x^2+8x-4}}$~\\[0.5cm]
~$f(x) = \frac{x-2}{\sqrt{1-\cos^2(x)}}$~ \quad ; \quad
~$f(x) = \frac{2\sqrt{x-1}}{|3x-6|-4}$~\quad ; \quad
~\(
f(x) =
\begin{cases}
\dfrac{x-2}{x^2+x-12} & \text{si } x < 2 \\
\sqrt{x^2-4} & \text{si } x \ge 2
\end{cases}
\)~
}

\vspace*{-0.25cm}
% Exercise 2 (originally 2)
\printexo{2}{(4,5 points)}{
\vspace*{-0.75cm}

On considère la fonction \( f \) définie par \( f(x) = \dfrac{x^2}{|x|-1} \).
\begin{enumerate}[label=\arabic*), leftmargin=*]
    \item Déterminer \( D_f \) le domaine de définition de la fonction \( f \).
    \item Montrer que la fonction \( f \) est paire.
    \item
    \begin{enumerate}[label=\alph*., leftmargin=*]
        \item Soient \( a \) et \( b \) de \([0;1[ \cup ]1;+\infty[\) tels que : \( a \neq b \).
        Montrer que :
        \[
        \frac{f(a)-f(b)}{a-b} = \frac{(a-1)(b-1)-1}{(a-1)(b-1)} =1- \frac{1}{(a-1)(b-1)}
        \]
      \item Montrer que : \( f \) est décroissante sur \([0;1[\) et ~$]1;2]$~ et croissante sur \([2;+\infty[\).
        \item Dresser le tableau de variations de \( f \) sur \( D_f \).
    \end{enumerate}
    \item Déduire une comparaison de \( f\left(\sqrt{\frac{3}{2}}\right) \) et \( f\left(\sqrt{\frac{5}{3}}\right) \) (sans calcul).
\end{enumerate}
}

\vspace*{-0.5cm}
% Exercise 3 (originally 3)
\printexo{3}{(5,5 points)}{
\textbf{Les questions I et II sont indépendantes.}
\begin{enumerate}[label=\Roman*., leftmargin=*]
\item Simplifier les expressions suivantes :
\begin{enumerate}[label=\arabic*., leftmargin=*]
    \item \( E = \sin\left(\frac{6\pi}{16}\right) + \sin\left(\frac{4\pi}{16}\right) + \sin\left(\frac{2\pi}{16}\right) + \cos\left(\frac{3\pi}{8}\right) + \cos\left(\frac{2\pi}{8}\right) + \cos\left(\frac{\pi}{8}\right) \)
    \item \( F = \sin^2\left(\frac{\pi}{12}\right) + \sin^2\left(\frac{3\pi}{12}\right) + \sin^2\left(\frac{5\pi}{12}\right) + \sin^2\left(\frac{7\pi}{12}\right) + \sin^2\left(\frac{9\pi}{12}\right) + \sin^2\left(\frac{11\pi}{12}\right) \)
\end{enumerate}
\item Soit \( x \in \left[0;\frac{\pi}{2}\right[ \).
On pose :
~\(
A(x) = 3\sin\left(\frac{\pi}{2}-x\right) \cdot \cos(\pi-x) - 5\sin(\pi-x) \cdot \cos\left(\frac{\pi}{2}-x\right) + 2
\)~
\begin{enumerate}[label=\arabic*), leftmargin=*]
    \item Montrer que : \( A(x) = -3 + 2\cos^2(x) \).
    \item Montrer que : \( A(x) = \dfrac{-1-3\tan^2(x)}{1+\tan^2(x)} \).
    \item
    \begin{enumerate}[label=\alph*., leftmargin=*]
        \item Déterminer \( \cos(x) \) et \( \sin(x) \) sachant que : \( A(x) = -\dfrac{3}{2} \).
        \item Déterminer la valeur de \( x \).
    \end{enumerate}
\end{enumerate}
\end{enumerate}
}

% Exercise 4 (originally 4)
\printexo{4}{(3 points)}{
Soit \( x \in \mathbb{R} \).
\begin{enumerate}[label=\arabic*), leftmargin=*]
    \item Montrer que :
    ~\(
    2\sin^2\left(x + \frac{\pi}{2}\right) - \cos(5\pi + x) - 1 = (\cos(x) + 1)(2\cos(x) - 1)
    \)~
    \item Résoudre dans \( \mathbb{R} \) puis dans l’intervalle \( ]-\pi;2\pi[ \) l’équation \((E)\) :
    ~\(
    (E) : 2\sin^2\left(x + \frac{\pi}{2}\right) - \cos(5\pi + x) - 1 = 0
    \)~
    \item Résoudre dans l’intervalle \( ]-\pi;2\pi[ \) l’inéquation :
    ~\(
    2\sin^2\left(x + \frac{\pi}{2}\right) - \cos(5\pi + x) - 1 \ge 0
    \)~
\end{enumerate}
}







\end{document}