\documentclass[a4paper,12pt]{article}

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% default_preamble.tex — packages + command definitions
% Everything here is placed BEFORE \begin{document}.
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\usepackage[left=1.5cm,right=1.5cm,top=2cm,bottom=1.5cm]{geometry}
\usepackage[utf8]{inputenc}
\usepackage{fontspec}
\usepackage{amsmath, amsfonts, amssymb, amsthm}
\usepackage{graphicx}
\usepackage{xcolor}
\usepackage{multicol}
\usepackage{enumitem}
\usepackage{mathrsfs}
\usepackage{colortbl}
\usepackage{varwidth}
\usepackage[most]{tcolorbox}
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\usepackage{tikz,tkz-euclide}
\usetikzlibrary{calc}
\usetikzlibrary{shadows}
\usepackage{tabularx, array, booktabs}
\usepackage{multirow}
\usepackage{array}
\usepackage{setspace}
\usepackage{hyperref}
\hypersetup{colorlinks=true, urlcolor=magenta}



\usepackage{fancyhdr}

% Language variables for French exams
\def\professor{R. MOSAID}  % Should be R. MOSAID not {{R. MOSAID}}
\def\classname{1BAC.SE}
\def\examtitle{Devoir 01 (1BAC.SE)}
\def\schoolname{Lycée Taghzirt}
\def\academicyear{2025/2026}
\def\subject{Mathématiques}
\def\duration{2h}
\def\secondtitle{\small{Notions de Logique et Généralités sur les fonctions }}
\def\province{}
\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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% Exercise Theme 6: Maroon title bar with triangles
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\newcommand{\printexo}[3]{%
  \if\relax\detokenize{#2}\relax
    \def\fulltitle{Exercice #1}%
  \else
    \def\fulltitle{Exercice #1~#2}%
  \fi
  \begin{exothemesix}[]{\fulltitle}#3\end{exothemesix}%
  \vspace{0.2cm}%
}



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\definecolor{bannerblue}{RGB}{210,240,240}

% Exam-specific settings and custom definitions
% This file is generated in the current working directory
% Edit this file to customize your exam settings



% Define column type for centered cells
\newcolumntype{Y}{>{\centering\arraybackslash}X}
\newcolumntype{M}[1]{@{}>{\centering\arraybackslash}m{#1}@{}}

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\newcommand{\circled}[1]{%
  \tikz[baseline=(char.base)]{
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}


% --- Margin offsets for note placement ---
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% Document spacing
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% Math display style
\everymath{\displaystyle}

\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
% \setlength{\parindent}{0pt}
% \setlength{\parskip}{1em}

% Custom colors for this exam
% \definecolor{myblue}{RGB}{0,100,200}
% \definecolor{mygreen}{RGB}{0,150,0}

\def\nex{01}

% ----------------------------------------------------------------------
% ADD YOUR CUSTOM DEFINITIONS BELOW THIS LINE
% ----------------------------------------------------------------------

% Example:
\newcommand{\dif}{\mathrm{d}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\C}{\mathbb{C}}


% ─────────────────────────────────────────────────────────────────
% default_wrapper.tex — the document body.
% Placed AFTER preamble / colors / definitions in _assemble().
% ─────────────────────────────────────────────────────────────────

\begin{document}



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% ── Header command (exam class only; article class uses fancyhdr)


% ── Two-column start


% ── Main exercises
\printexo{1}{Questions indépendantes (7,5 pts)
}{\noindent
\begin{tabular}{c|p{16cm}}
& \textbf{1.} Étudier la valeur de vérité des propositions suivantes puis donner leurs négations : \\
  & \hspace{1cm} \(P : "(\forall x \in \mathbb{R}) ; x > 0"\) \hfill \(Q : "\sqrt{2} \in \mathbb{Q} \quad \text{et} \quad \sqrt{9} = -3"\) \hspace{1cm} \\
  & \hspace{1cm} \(R : "(\exists n \in \mathbb{N}) : \frac{5}{n+3} \in \mathbb{N}"\) \newline \hspace*{1cm} \(S : "(\forall n \in \mathbb{N})(\exists k \in \mathbb{N}) ; (n = 2k \text{ ou } n = 2k + 1)"\) \hspace{1cm} \\
0,5 & \textbf{2.} On considère la fonction numérique \(f\) définie sur \(\mathbb{R}\) par : \(f(x) = 2x^2 - x + 1\) \\
0,5 & \hspace{1cm} \textbf{a)} Calculer \(f(1)\) et \(f(-1)\). \\
1   & \hspace{1cm} \textbf{b)} En déduire que la fonction \(f\) n'est ni paire ni impaire. \\
1 & \textbf{3.} Montrer que pour tout \(x\) et \(y\) de \(\mathbb{R}\) : \newline \hspace*{1cm}  \((x \neq y \text{ et } x + y \neq 3) \Rightarrow (x^2 - 3x + 5 \neq y^2 - 3y + 5)\) \\
0,5 & \textbf{4.} Soit \(x\) et \(y\) deux nombres réels. \\
0,5 & \hspace{1cm} \textbf{a)} Démontrer que : \((x + y)^2 \leq 2(x^2 + y^2)\) \\
1   & \hspace{1cm} \textbf{b)} En déduire que : \(x^2 + y^2 = 1 \Rightarrow |x + y| \leq \sqrt{2}\) \\
1 & \textbf{5.} Résoudre dans \(\mathbb{R}\) l'équation suivante : \quad \((E) : |-3x + 4| + |x - 5| = 10\) \\
1 & \textbf{6. a)} Montrer par récurrence que pour tout \(n \in \mathbb{N}^*\) : \newline \hspace*{1cm} \(1 + 2^2 + 3^2 + \dots + n^2 = \frac{n(n+1)(2n+1)}{6}\) \\
0,5 & \hspace{1cm} \textbf{b)} En déduire la valeur de la somme : \(1 + 2^2 + 3^2 + \dots + 10^2\)
\end{tabular}
}

\printexo{2}{(4,75 pts) \hrulefill
}{\noindent
\begin{tabular}{c|p{16cm}}
& \hspace{1cm} On considère la fonction numérique \(f\) définie par : \(f(x) = \frac{5x^2 + 1}{x^2 + 1}\) \\
0,75 & \textbf{1.} Déterminer \(D_f\) l'ensemble de définition de \(f\), puis montrer que \(f\) est paire. \\
1 & \textbf{2. a)} Montrer que \(f\) est majorée par le nombre 5 sur \(D_f\). \\
1 & \hspace{1cm} \textbf{b)} Est-ce que 5 est une valeur maximale de \(f\) sur \(D_f\) ? \\
1 & \textbf{3. a)} Montrer que \(f\) est minorée par le nombre 1 sur \(D_f\). \\
1 & \hspace{1cm} \textbf{b)} Est-ce que 1 est une valeur minimale de \(f\) sur \(D_f\) ?
\end{tabular}
}

\printexo{3}{(5,75 pts) \hrulefill
}{\noindent
\begin{tabular}{c|p{16cm}}
& \hspace{1cm} On considère la fonction numérique \(f\) définie sur \(\mathbb{R}^*\) par : \(f(x) = x + \frac{1}{x}\) \\
1 & \textbf{1.} Montrer que la fonction \(f\) est impaire. \\
1 & \textbf{2.} Soit \(a\) et \(b\) deux éléments distincts de \(\mathbb{R}^*\). Montrer que : \(T(a; b) = \frac{ab - 1}{ab}\) \\
1 & \textbf{3.} Montrer que \(f\) est décroissante sur \(]0; 1]\) et croissante sur \([1; +\infty[\). \\
0,5 & \textbf{4.} En déduire les variations de \(f\) dans chacun des intervalles suivants : \newline \hspace*{1cm}   \([-1; 0[\) et \(]-\infty; -1]\). \\
0,5 & \textbf{5.} Dresser le tableau des variations de \(f\) sur \(\mathbb{R}^*\). \\
  & \textbf{6.} Soit \(a\) un réel tel que \(a > 1\). \\
0,75 & \hspace{1cm} \textbf{a)} Montrer que pour tout \(a > 1\) : \(a > \sqrt{a}\) \\
1 & \hspace{1cm} \textbf{b)} En déduire la comparaison des deux nombres : \(\frac{a^2+1}{a}\) et \(\frac{a+1}{\sqrt{a}}\)
\end{tabular}
}

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\end{document}