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% French language support
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% Bottom message in Arabic requires bidi package
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% --- Basic Settings ---
% 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 - S01 (2BAC.PC/SVT) Par Pr. MOHAMED JOUAHRI}
\def\schoolname{Lycée Qualifiant 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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% Edit this file to customize your exam settings

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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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% \definecolor{mygreen}{RGB}{0,150,0}

% ----------------------------------------------------------------------
% 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 settings
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\begin{document}


\noindent\shadowhead

% Exercise 1 (originally 1)
\printexo{1}{: ~ (4 points)}{
{
\renewcommand{\thequestion}{\arabic{question}}
\vspace*{-0.5cm}
\begin{questions}
  \question[2] Résoudre dans $]0, +\infty[$ :
             $(E_1): \ln(x) - 3 = 0$ \quad ;\quad
             $(E_2): \ln(x^2 + 8) = \ln(x + 4) + \ln(x)$
    \question[2] Déterminer une primitive de chacune des fonctions suivantes :\\
        ~\(
        f_1(x) = \frac{1}{x \ln(x)} \quad ; \quad f_2(x) = \frac{3x^2}{x^3 + 5} + \frac{1}{x}
        \)~
\end{questions}
}
}

% Exercise 2 (originally 2)
\printexo{2}{: ~ (18 points)}{
\vspace*{-0.75cm}
\begin{questions}

% ===================== QUESTION 1 =====================
\question
Soit $g$ la fonction numérique définie sur $]0,+\infty[$ par
~\(
g(x)=x-1+\ln(x).
\)~

\begin{parts}
\part[1]
Montrer que
~\(
\lim_{x\to0^+} g(x)=-\infty
\)~
et Calculer
~\(
\lim_{x\to+\infty} g(x).
\)~

\part[1]
Montrer que, pour tout $x\in]0,+\infty[$,
~\(
g'(x)=\frac{x+1}{x}.
\)~

\part[1]
Montrer que $g$ est strictement croissante sur $]0,+\infty[$.

\part[1]
Dresser le tableau de variations de la fonction $g$ sur $]0,+\infty[$.

\part[1]
Calculer $g(1)$ puis montrer que
~\(
g(x)\le 0 \quad \forall x\in]0,1]
\)~
et que
~\(
g(x)\ge 0 \quad \forall x\in[1,+\infty[.
\)~
\end{parts}

% ===================== QUESTION 2 =====================
\question
Considérons la fonction $f$ définie sur $]0,+\infty[$ par
~\(
f(x)=x+\frac12-\ln(x)+\frac12(\ln(x))^2,
\)~\\
et $(C_f)$ sa courbe représentative dans un repère orthonormé $(O,\vec{i},\vec{j})$.

\begin{parts}
\part[1]
Montrer que
~\(
\lim_{x\to0^+} f(x)=+\infty
\)~
puis interpréter graphiquement ce résultat.

\part[1]
Vérifier que, pour tout $x\in\mathbb{R}_+^*$,
~\(
f(x)=x+\frac12+\left(\frac12\ln(x)-1\right)\ln(x),
\)~\\
et montrer que
~\(
\lim_{x\to+\infty} f(x)=+\infty.
\)~

\part[1]
Montrer que
~\(
\lim_{x\to+\infty}\frac{\ln^2(x)}{x}=0
\)~
(on pourra poser $\sqrt{x} = t$). Et que
~\(
\lim_{x\to+\infty}\frac{f(x)}{x}=1.
\)~

\part[1]
Montrer que $(C_f)$ admet, au voisinage de $+\infty$, une branche parabolique\\
de direction asymptotique la droite $(\Delta)$ d’équation $y=x$.

\part[1]
Montrer que, pour tout $x\in]0,+\infty[$,
~\(
f'(x)=\frac{g(x)}{x}.
\)~

\part[1]
Dresser le tableau de variations de la fonction $f$ sur $]0,+\infty[$.

\part[1]
Montrer que, pour tout $x\in]0,+\infty[$,
~\(
f''(x)=\frac{2-\ln(x)}{x}.
\)~

\part[1]
En déduire que $(C_f)$ admet un point d’inflexion et déterminer ses coordonnées.

\part[1]
Vérifier que
\(
\forall x\in\R^*_+\;
f(x)-x=\frac12(\ln(x)-1)^2
\)
et déduire la position relative de $(C_f)$ et $(\Delta)$.

\part[1]
Tracer $(C_f)$ et $(\Delta)$ dans le même repère orthonormé.
\end{parts}

% ===================== QUESTION 3 =====================
\question
On considère la suite $(U_n)$ définie par
~\(
U_0=1, \qquad~\text{et}~\quad  U_{n+1}=f(U_n).
\)~

\begin{parts}
\part[1]
Montrer que, pour tout $n\in\mathbb{N}$,
~\(
1\le U_n\le e.
\)~

\part[1]
Montrer que la suite $(U_n)$ est croissante.

\part[1]
En déduire que la suite $(U_n)$ est convergente puis déterminer
~\(
\lim U_n.
\)~
\end{parts}

\end{questions}

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}






% Bottom message
\vspace*{-1cm}
\begin{center}
  \normalsize{%
    \vskip 2pt \hrule height 3pt \vskip 2pt
    \RL{\arabicfont
    ﴿ٱللَّهُ يَعْلَمُ مَا تَحْمِلُ كُلُّ أُنثَىٰ وَمَا تَغِيضُ ٱلْأَرْحَامُ وَمَا تَزْدَادُ ۖ وَكُلُّ شَىْءٍ عِندَهُۥ بِمِقْدَارٍ﴾ (الرعد 8)
    }%
  }%
\end{center}

\end{document}