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% French language support
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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 Libre 01 - S02 (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}
\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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% 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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% Custom colors for this exam
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% ----------------------------------------------------------------------
% 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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% --- Style d'en-tête 9 (fr)  ---
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\begin{document}




% Exercise 1 (originally 1)
\printexo{1}{(5pts)}{
\vspace*{-0.5cm}
\begin{enumerate}
\item Résoudre dans \(\mathbb{R}\) les équations :
~\(
2x^2 + x - 1 = 0 \quad ; \quad 2x^2 - 2\sqrt{2} + 1 = 0
\)~
\item Résoudre dans \(\mathbb{R}\) l’inéquation :
~\(
2x^2 + x - 1 > 0
\)~
\item Résoudre dans \(\mathbb{R}\) l’équation :
~\(
2\sin^2(x) - 2\sqrt{2}\sin(x) + 1 = 0
\)~
\item Résoudre dans \(\mathbb{R}^2\) le système suivant en utilisant la méthode du déterminant :
~\(
\begin{cases}
-x + 2y = 3 \\
3x - 4y = -5
\end{cases}
\)~
\end{enumerate}
\vspace*{-0.5cm}
}

% Exercise 2 (originally 2)
\printexo{2}{(6pts)}{
Soit \(x\) un nombre réel, on pose :
~\(
A(x) = 4\cos^2(x) + \sin^4(x)
\)~
\begin{enumerate}
\item Montrer que pour tout \(x\) de \(\mathbb{R}\) :
~\(
A(x) = \bigl(2 - \sin^2(x)\bigr)^2
\)~

\item Calculer :
~\(
A(0) \quad ; \quad A\left(\frac{\pi}{2}\right)
\)~

\item
\begin{enumerate}
\item Déterminer l’abscisse curviligne principale du point \(M\) d’abscisse curviligne :
~\(
-\frac{2005\pi}{3}
\)~
\item Déduire la valeur de :
~\(
A\left(-\frac{2005\pi}{2}\right)
\)~
\end{enumerate}

\item
\begin{enumerate}
\item Montrer que :
~\(
A(x) = \left(1 + \frac{1}{1+\tan^2(x)}\right)^2
\)~
\item Soit \(x \in \left[0; \frac{\pi}{2}\right[\), Calculer \(A(x)\) sachant que :
~\(
\tan(x) = 1
\)~
\end{enumerate}
\end{enumerate}
}

% Exercise 3 (originally 3)
\printexo{3}{(5pts)}{
Soit \(x\) un nombre réel.
\begin{enumerate}
\item Calculer :
~\(
\sin\left(\frac{52\pi}{3}\right), \quad
\cos\left(\frac{-39\pi}{2}\right), \quad
\tan\left(\frac{-413\pi}{4}\right)
\)~

\item Simplifier le nombre :
~\(
B = \cos\left(3x + \frac{41\pi}{2}\right) + \sin\left(3x + 305\pi\right) + \cos\left(3x + \frac{19\pi}{2}\right)
\)~

\item Déduire la valeur de :
~\(
B\left(\frac{\pi}{9}\right)
\)~
\end{enumerate}
}

% Exercise 4 (originally 4)
\printexo{4}{(4pts)}{
Soit \(x \in \mathbb{R}\) ; on pose :
~\(
F(x) = \sqrt{2}\cos^2(x) - (\sqrt{2} + 1)\cos(x) + 1
\)~
\begin{enumerate}
\item Montrer que :
~\(
F(x) = (\cos(x) - 1)\bigl(\sqrt{2}\cos(x) - 1\bigr)
\)~

\item Résoudre dans \([-\pi; \pi]\) l’équation :
~\(
F(x) = 0
\)~

\item Étudier le signe de \(F(x)\) sur \([-\pi; \pi]\) (dresser le tableau de signe de \(F(x)\)).

\item Déduire l’ensemble des solutions de l’inéquation :
~\(
F(x) \leq 0
\)~
\end{enumerate}
}

% Exercise 5 (originally 5)
\printexo{5}{(2pts)}{
Résoudre dans \(]-\pi; \pi[\) l’inéquation :
~\(
\frac{2\sin^2(x) + \sin(x) - 1}{4\cos^2(x) - 1} \geq 0
\)~
}







\end{document}