\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{Par: Prof M. JOUAHRI }}
\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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      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
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  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{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}@{}}

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}


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


% --- Margin offsets for note placement ---
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% --- Note number placed 1cm from page edge (right margin) ---
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% Document spacing
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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
% \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
\vspace*{-1cm}

\printexo{1}{}{\begin{enumerate}[label=\arabic*.]
    \item Mettre en ordre croissant les nombres \(\sqrt[6]{7}\) ; \(\sqrt[5]{5}\)
    \item Résoudre dans \(\mathbb{R}\) : \(\sqrt[3]{x^2 + 9} = \sqrt[3]{6x}\)
      \quad ;\quad \(\sqrt[3]{5x + 7} \le 3\)
    \item Calculer les limites suivantes : \(\lim_{x \to 2} \sqrt[3]{x^2 + x + 2}\)
      ; \(\lim_{x \to +\infty} \sqrt[3]{x^3 + 2x} - x\)
    \item Calculer \(f'(x)\) dans les cas suivants : \(f(x) = \sin(x^2 - 5x)\) ; \(f(x) = \sqrt[3]{x^6 + 3x^2 + 1}\)
\end{enumerate}
}

\printexo{2}{}{Soit \(f\) la fonction numérique définie sur \(\mathbb{R}\) et (C) sa courbe représentative dans un\\ repère orthonormé (voir la figure).\\
\begin{minipage}[b]{0.6\textwidth}
\begin{enumerate}[label=\arabic*.]
    \item Déterminer \(f(0)\) et \(\lim_{x \to -\infty} f(x)\)
    \item Dresser le tableau de variations de \(f\) sur \(\mathbb{R}\)
    \item Montrer que l'équation \(f(x) = 0\) admet une solution unique \(\alpha\) tel que : \(0 < \alpha < 1\)
    \item Montrer que \(h\) la restriction de \(f\) sur \(I = [0, +\infty[\) admet une fonction réciproque définie sur un intervalle \(J\) à Déterminer.
\end{enumerate}
\end{minipage}
\begin{minipage}[t]{0.28\textwidth}
\begin{tikzpicture}[>=stealth, scale=1.3]

%================================================
% 1. Coordinate grid
%================================================
\draw[step=0.1, gray!12, very thin] (-3,-1.5) grid (2.2,3);
\draw[step=0.5, gray!25, thin] (-3,-1.5) grid (2.2,3);
\draw[step=1, gray!40, thin] (-3,-1.5) grid (2.2,3);

%================================================
% 2. Coordinate axes
%================================================
\draw[->, thick] (-3,0) -- (2.15,0);
\draw[->, thick] (0,-1.5) -- (0,3.15);

% Origin and graduations
\node[below left] at (0,0) {$O$};

\foreach \x in {-2,-1,1,2}
    \node[below] at (\x,0) {$\x$};

\node[below] at (0.5,0) {$\frac12$};
\node[left] at (0,2) {$2$};

% Unit vectors
\draw[->, thick] (0,0.15) -- (0,1)
    node[above left] {$\vec{j}$};

\draw[->, thick] (0.15,0) -- (1,0)
    node[above right] {$\vec{i}$};

%================================================
% 3. Asymptotes
%================================================

% Horizontal asymptote y=0 (coincides with x-axis)

% Vertical asymptote x=1
\draw[dashed, gray!70, thick] (1,-1.5) -- (1,3);


%================================================
% 4. Curve (C)
%================================================

% Model function:
% f(x) = ((x-0.7)/0.7)*exp(13x/14)/sqrt(1-x)
%
% f(0) = -1
% f'(0) = 0
% lim f(x) = 0 as x -> -infinity
% lim f(x) = +infinity as x -> 1-

\begin{scope}
    \clip (-3,-1.5) rectangle (3,3);

    \draw[
        red!75!magenta,
        thick,
        domain=-3:0.999,
        samples=400,
        variable=\x
    ]
    plot (
        {\x},
        {((\x-0.7)/0.7)*exp(13*\x/14)/sqrt(1-\x)}
    );
\end{scope}

% Curve label
\node at (1.25,2.8) {$(C)$};

%================================================
% 5. Tangent at the minimum
%================================================
\draw[<->, thick] (-0.35,-1) -- (0.35,-1);

%================================================
% 6. Intersection with x-axis
%================================================
\node[above left] at (0.72,0) {$\alpha$};

\end{tikzpicture}
\end{minipage}
\begin{enumerate}[label=\arabic*.,start=5, before={\vspace*{-0.3cm}}]
    \item Construire dans le même repère la courbe de la fonction \((C_{h^{-1}})\) et déterminer \(\lim_{x \to -1^+} \frac{h^{-1}(x)}{x+1}\)
\end{enumerate}



}

\printexo{3}{}{\begin{enumerate}[label=\arabic*., before={\vspace*{-0.2cm}}]
    \item Soit \(f\) la fonction numérique définie sur \(\mathbb{R}\) par :
    ~\(
    \begin{cases}
    f(x) = \sqrt[3]{x-1} + 2 & ; x > 1 \\
    f(x) = x^3 + 6x - 5 & ; x \leq 1
    \end{cases}
    \)~
    \begin{enumerate}[label=\arabic*., before={\vspace*{-0.2cm}}]
        \item Montrer que la fonction \(f\) est continue en 1.
        \item Etudier la continuité de la fonction \(f\) sur \(\mathbb{R}\).
        \item Etudier la dérivabilité de \(f\) en 1 puis interpréter géométriquement les résultats obtenus.
        \item Montrer que pour tout \(x \in ]-\infty, 1]\) : \(f'(x) = 3(x^2 + 2)\).
        \item Dresser le tableau de variations de la fonction \(f\) sur \(]-\infty, 1]\).
        \item Montrer que l'équation \(f(x) = 0\) admet une solution unique \(\alpha\) dans \(]-\infty, 1]\).
        \item Vérifier que : \(0 < \alpha < 1\).
        \item Résoudre dans \(]-\infty, 1]\) l'inéquation : \(xf(x) \leq 0\).
    \end{enumerate}
    \item Soit \(g\) la fonction définie sur l'intervalle \(I = ]-\infty, 1]\) par : \(g(x) = f(x)\).
    \begin{enumerate}[label=\arabic*.]
        \item Montrer que \(g\) admet une fonction réciproque \(g^{-1}\) définie sur un intervalle \(J\) à Déterminer.
        \item Dresser le tableau de variations de la fonction \(g^{-1}\).
        \item Montrer que \(g^{-1}\) est dérivable en 0 et donner \((g^{-1})'(0)\). (remarquer que \(g(\alpha) = 0\))
    \end{enumerate}
\end{enumerate}

}

% ── Between columns (if two-column)


% ── Repeat section



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


% ── Two-column end


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




\end{document}