\documentclass[a4paper,12pt,landscape]{exam}

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% default_preamble.tex — packages + command definitions
% Everything here is placed BEFORE \begin{document}.
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\usepackage[left=1.3cm,right=1.3cm,top=1cm,bottom=1cm]{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{pgfplots}
\pgfplotsset{compat=1.18}
\usepackage{tabularx, array, booktabs}
\usepackage{multirow}
\usepackage{array}
\usepackage{setspace}
\usepackage{hyperref}
\hypersetup{colorlinks=true, urlcolor=magenta}





% Language variables for French exams
\def\professor{R. MOSAID}  % Should be R. MOSAID not {{R. MOSAID}}
\def\classname{2BAC.SM}
\def\examtitle{Série Exercices}
\def\schoolname{Lycée Taghzirt}
\def\academicyear{2026/2027}
\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~~}



\newcommand{\headerstyleone}{\academicyear\hfill\textbf{\examtitle}\hfill \professor\par}


% Exercise Theme 7: Simple underline style
\newcommand{\exothemeseven}[1]{%
\par\vspace{0pt}\noindent\leavevmode
\underline{\textbf{\textit{#1}}}\\[0.3cm]
\noindent
}

\newcommand{\printexo}[3]{%
  \if\relax\detokenize{#2}\relax
    \def\fulltitle{Exercice #1}%
  \else
    \def\fulltitle{Exercice #1~#2}%
  \fi
  \exothemeseven{\fulltitle}%
  \noindent #3%
  \vspace{0.2cm}%
}



% --- Colors ---
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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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% --- Note number placed 1cm from page edge (left margin) ---
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% Document spacing
\setstretch{1.1}

% Math display style
\everymath{\displaystyle}

\mathversion{bold}

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

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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}}

% ── Exam class settings ──────────────────────────────────────
\pointsinmargin
\pointname{}
\pointformat{\textcolor{red}{\circled{\thepoints}}}
\colorfillwithdottedlines
\definecolor{FillWithDottedLinesColor}{gray}{0.7}
\unframedsolutions
\renewcommand{\solutiontitle}{%
  \noindent\textbf{\textcolor{green!70!black}{Solution}}\enspace}
\SolutionEmphasis{\itshape\small\color{red}}
% ─────────────────────────────────────────────────────────────


% ─────────────────────────────────────────────────────────────────
% 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
\setlength{\columnsep}{0.5cm}
\setlength{\columnseprule}{2pt}
\renewcommand{\columnseprulecolor}{\color{black}}
\begin{multicols}{2}[\raggedcolumns]
\vspace*{-\topskip}
% Header for first column
\noindent\headerstyleone

% ── Main exercises
\printexo{1}{}{Soit \(f\) la fonction définie par :
\[
f(x) =
\begin{cases}
\frac{(x+2)^3 - 1}{x^2 + x} & \text{si } x < -1 \\[1.5ex]
\frac{x - 3}{x - 1 - \sqrt{x+1}} & \text{si } x \ge -1 \text{ et } x \ne 3.
\end{cases}
\]
avec \(f(3) = a\), (\(a \in \mathbb{R}\)).
\begin{enumerate}[tight]
    \item Déterminer l'ensemble de définition de \(f\).
    \item Déterminer \(a\) pour que \(f\) soit continue au point 3.
    \item Étudier la continuité de \(f\) en \(-1\).
\end{enumerate}

}

\printexo{2}{}{Soit la fonction \(f\) définie sur \(\mathbb{R} \setminus \{-1\}\) par :
\[
f(x) =
\begin{cases}
\frac{(3 - x)|x|}{x^2 + x} & \text{si } x \le 3 \text{ et } x \ne 0 \text{ et } x \ne -1 \\[1.5ex]
\frac{\sqrt{x^2 + 2x - 15} - 3}{2x^2 - 7x - 4} & \text{si } x > 3 \text{ et } x \ne 4.
\end{cases}
\]
avec \(f(0) = 3\) et \(f(4) = a\).

\begin{enumerate}[tight]
    \item Étudier la continuité de \(f\) en 0 et en 3.
    \item
    \begin{enumerate}[tight]
        \item[a)] Montrer que pour \(x > 3\) et \(x \ne 4\) :
        \[
        f(x) = \frac{x + 6}{(2x + 1)(3 + \sqrt{x^2 + 2x - 15})}
        \]
        \item[b)] Déterminer le réel \(a\) pour que \(f\) soit continue en 4.
        \item[c)] Pour la valeur de \(a\) trouvée, montrer que \(f\) est continue sur \(]3, +\infty[\).
    \end{enumerate}
    \item Montrer que l'équation \(f(x) = \frac{1}{2}\) admet au moins une solution dans \([1, 2]\).
\end{enumerate}

}

\printexo{3}{}{\noindent On considère la fonction \(g\) définie par :
\[
f(x) =
\begin{cases}
\frac{x^3 + x^2 + x + 1}{x^2 + 3x - 2} & \text{si } x < -1 \\[1.5ex]
(x + 1)E(x) + 2 & \text{si } -1 \le x < 1 \\[1.5ex]
\frac{x - 1}{\sqrt{x^2 + 3} - 2} & \text{si } x > 1.
\end{cases}
\]

\begin{enumerate}[tight]
    \item
    \begin{enumerate}[tight]
        \item[a)] Calculer \(\lim\limits_{x \to -1^-} f(x)\) puis déduire que \(f\) est continue en \(-1\).
        \item[b)] \(f\) est-elle continue en 0.
    \end{enumerate}
    \item
    \begin{enumerate}[tight]
        \item[a)] Montrer que \(\forall x \in ]1, +\infty[ : f(x) = \frac{\sqrt{x^2 + 3} + 2}{x + 1}\).
        \item[b)] \(f\) est-elle prolongeable par continuité en 1 ?
    \end{enumerate}
    \item
    \begin{enumerate}[tight]
        \item[a)] Montrer que \(f\) est continue sur \(]1, +\infty[\).
        \item[b)] En déduire que l'équation \(f(x) = x - 1\) admet dans l'intervalle \([2, 3]\) une solution.
    \end{enumerate}
    \item Calculer \(\lim\limits_{x \to -3} \frac{f(x) - f(-3)}{x + 3}\).
\end{enumerate}

}

\printexo{4}{}{\vspace*{-0.5cm}
\begin{enumerate}[tight]
    \item Soit \(f\) la fonction définie sur \(]-\infty, 2] \setminus \{1\}\) par :
    \[
    f(x) = \frac{\sqrt{2x^2 - 3x + 2} - 1}{x - 1}.
    \]
    \vspace*{-0.5cm}
    \begin{enumerate}[tight]
        \item[a)] Montrer que \(f\) est prolongeable par continuité en 1.
        \item[b)] Soit \(F\) le prolongement par continuité de \(f\) en 1. Montrer que \(F\) est continue sur \(]-\infty, 2]\).
    \end{enumerate}
    \item Soit \(g\) la fonction définie sur \(\mathbb{R}\) par :
    \[
    g(x) =
    \begin{cases}
    \frac{1 - 2x}{\sqrt{2x^2 - 3x + 2} + 1} & \text{si } x \in ]-\infty, 2] \\[1.5ex]
    \frac{x^3 + x^2 - 12x + 12}{x^2 - 4} & \text{si } x \in ]2, +\infty[.
    \end{cases}
    \]
    \begin{enumerate}[tight]
        \item[a)] Calculer \(\lim\limits_{x \to 2^+} g(x)\).
        \item[b)] \(g\) est-elle continue en 2 ?
    \end{enumerate}
\end{enumerate}
}

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% ── Repeat section



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\end{multicols}

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\end{document}