\documentclass[a4paper,12pt]{article}

% ─────────────────────────────────────────────────────────────────
% default_preamble.tex — packages + command definitions
% Everything here is placed BEFORE \begin{document}.
% ─────────────────────────────────────────────────────────────────

\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}
\tcbuselibrary{skins,breakable}
\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{2BAC.PC/SVT}
\def\examtitle{Devoir 01 (2BAC.PC/SVT)}
\def\schoolname{Lycée Taghzirt}
\def\academicyear{2025/2026}
\def\subject{Mathématiques}
\def\duration{2h}
\def\secondtitle{\small{}}
\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~~}

\fancyhf{}
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\renewcommand{\footrulewidth}{0pt}
\setlength{\headheight}{47pt}
\setlength{\headsep}{0pt}
\fancyhead[C]{\printheadnine}
\pagestyle{fancy}

\newcommand{\printheadnine}{%
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      M{0.48\textwidth}
      M{0.25\textwidth}
      }
      \textbf{\classname} & \textbf{\examtitle} & \ddate \\
      \wsite & \secondtitle & \hfill \begin{tabular}{|c|}\hline\professor\end{tabular}  \\
      \hline
  \end{tabularx}
  \arrayrulecolor{black}
}


% Exercise Theme 6: Maroon title bar with triangles
\newtcolorbox{exothemesix}[2][]{%
  before=\centering,
  after={},
  before skip=0pt,
  after skip=0pt,
  boxsep=0pt,
  left=4pt,
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  enhanced,
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    -- +(0,-0.3) -- +(0.4,0) -- cycle;
  },
  title=#2,
  #1
}

\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{JungleGreen}{HTML}{29AB87}

\definecolor{myblue3}{RGB}{36, 57, 126}
\definecolor{mygray}{RGB}{112,121,139}
\definecolor{LemonGlacier}{RGB}{253,255,0}
\colorlet{myblue3}{red!75!black}

\definecolor{lightgray}{gray}{0.6}
\definecolor{arrowblue}{RGB}{0,140,180}
\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}@{}}

\newcommand{\tb}{\tikz[baseline=-0.6ex]{\fill (0,0) circle (2pt);}~}

\newcommand{\ccc}[1]{
    \begin{tikzpicture}[overlay, remember picture]
        \node[circle, inner sep=3pt, draw=black, outer sep=0pt] at (0.5,0.2) {#1};
    \end{tikzpicture}
}


\newcommand{\circled}[1]{%
  \tikz[baseline=(char.base)]{
    \node[shape=circle,draw,inner sep=1pt, font=\bf] (char) {#1};}%
}


% --- Margin offsets for note placement ---
\def\rnotemargin{-1.5}   % distance from right edge of text block
\def\lnotemargin{0.5}   % distance from left edge of text block


% --- Note number placed 1cm from page edge (right margin) ---
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    % a tiny anchor at the current line baseline
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    % coordinate 1cm left of the physical page right edge
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% --- Note number placed 1cm from page edge (left margin) ---
\newcommand{\note}[1]{%
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      inner sep=0pt      % prevent expansion
    ] at (marg |- text.base) {$#1$};%
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% Document spacing
\setstretch{1.25}

% Math display style
\everymath{\displaystyle}

\mathversion{bold}

%\AtBeginDocument{\fontsize{15}{17}\selectfont}

%\newcommand{\fff}[1]{\makebox[#1cm]{\dotfill}}

\newcommand{\dfp}[1]{\dotfill (\bf{#1 pt})}

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\newcommand{\VMargin}[1]{%
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      ([xshift=#1]current page.north west) --
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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}



% ── Top of document (user block) ────────────────────────────────


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


% ── Two-column start


% ── Main exercises
\printexo{1}{(4.5 points)
}{\noindent
\begin{tabularx}{\textwidth}{>{\centering\arraybackslash}p{.5cm}|X}
0,5 & Soit \(f\) une fonction définit sur \(\mathbb{R}\) par : \(\begin{cases} f(x) = \frac{2x^2-5x-3}{x-3} & ; x \neq 3 \\ f(3) = 7 \end{cases}\) \\
1 & \textbf{1)} a) Vérifier que \((\forall x \in \mathbb{R}): 2x^2 - 5x - 3 = (x - 3)(2x + 1)\) \\
1 & \quad b) Montrer que \(f\) est continue en 3 \\
1 & \textbf{2)} a) Montrer que \(f\) est dérivable en 3 \\
1 & \quad b) Déterminer l'équation de la tangente de la courbe de f en 3
\end{tabularx}
}

\printexo{2}{(4 points)
}{\noindent
\begin{tabularx}{\textwidth}{>{\centering\arraybackslash}p{.5cm}|X}
  2& \textbf{1)} Calculer les limites suivantes : \(\lim\limits_{x \to 5} \frac{x-5}{\sqrt[3]{3x-7}-2}\) \quad ; \quad \(\lim\limits_{x \to -\infty} \sqrt[4]{16x^4+2}-x\) \\[0.5cm]
1 & \textbf{2)} Résoudre dans \(\mathbb{R}\) l'équation ; \((E) : 2\sqrt[3]{x} - 5\sqrt[6]{x} - 3 = 0\) \\
1 & \textbf{3)} Soit ~$a$~ un réel, déterminer la valeur de ~$a$~ sachant que \(\lim\limits_{x \to 2} (x-a)^3 = -27\)
\end{tabularx}
}

\printexo{3}{(7.5 points)
}{\noindent
\begin{tabularx}{\textwidth}{>{\centering\arraybackslash}p{0.5cm}|X}
0,5 & Soit \(g\) une fonction définit sur \(\mathbb{R}\) par : \(g(x) = x^3 + x^2 - 3\) \\
0,5 & \textbf{1)} Calculer \(\lim\limits_{x \to -\infty} g(x)\) et \(\lim\limits_{x \to +\infty} g(x)\) \\
1 & \textbf{2)} a) Vérifier que \(g'(x) = x(3x + 2)\) pour tout \(x \in \mathbb{R}\) \\
1 & \quad b) Dresser le tableau de variations de g sur \(\mathbb{R}\) en justifiant votre réponse \\
1 & \textbf{3)} a) Montrer que l'équation \(g(x) = 0\) admet une unique solution \(\alpha\) dans l'intervalle \(]1 ; 2[\) \\
1 & \quad b) Vérifier que \(\alpha = \frac{3}{\alpha^2+\alpha}\) puis en déduire que \(\alpha < \frac{3}{2}\) \\
1 & \quad c) Résoudre dans \(\mathbb{R}\) l'équation \((E): x^{12} + x^8 - 3 = 0\) \\
0,5 & \textbf{4)} Soit \(f\) une fonction définit sur \(\mathbb{R}\) par : \(f(x) = \frac{1}{4}x^4 + \frac{1}{3}x^3 - 3x + 1\) \\
1 & \quad a) Vérifier que \(f'(x) = g(x)\) pour tout \(x \in \mathbb{R}\) \\
1 & \quad b) Etudier le sens variations de f sur \(\mathbb{R}\) en justifiant votre réponse \\
1 & \quad c) Montrer que \(f(\alpha) < 0\) puis déterminer le nombre de solution de l'équation \(f(x) = 0\)
\end{tabularx}
}

\printexo{4}{(5 points)
}{\noindent
\begin{tabularx}{\textwidth}{>{\centering\arraybackslash}p{.5cm}|X}
1 & Soit \(f\) la fonction définie par : \(f(x) = 3 + 2\sqrt{2 - x}\) \\
1 & \textbf{1)} Vérifier que \(D_f = ]-\infty ; 2]\) et calculer \(\lim\limits_{x \to -\infty} f(x)\) \\
0,5 & \textbf{2)} a) Etudier la dérivabilité de f à gauche de 2 \\
0,5 & \quad b) Donner une interprétation géométrique du résultat \\
0,5 & \textbf{3)} a) Calculer \(f'(x)\) pour tout ~$x$~ dans \(]-\infty ; 2[\) \\
0,5 & \quad b) Dresser le tableau de variations de f en justifiant votre réponse \\
1 & \textbf{4)} Montrer que f admet une fonction réciproque \(f^{-1}\) définie sur un intervalle J que l'on déterminera \\
1 & \quad c) Déterminer \(f^{-1}(x)\) pour tout \(x \in J\)
\end{tabularx}
}

% ── Between columns (if two-column)


% ── Repeat section



% ── Bottom message — inside column 2 (last column)


% ── Two-column end


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\end{document}