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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 02 - S02 (2BAC.PC/SVT)}
\def\schoolname{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}
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\def\wsite{\href{https://www.mosaid.xyz}{\textcolor{magenta}{\texttt{www.mosaid.xyz}}}}
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% 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
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% ----------------------------------------------------------------------
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% ----------------------------------------------------------------------

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\begin{document}




\vspace*{-1cm}
% Exercise 1 (originally 1)
\printexo{1}{: (3 points)}{
\vspace*{-0.75cm}
\begin{enumerate}[label=\arabic*)]
    \item Résoudre l’équation différentielle $(E)$ telle que : $(E) : y'' - y' - 2y = 0$.
    \item Montrer $f$ la solution de l’équation $(E)$ vérifiant $f(0) = 0$ et $f'(0) = 3$ est : $f(x) = e^{2x} - e^{-x}$.
    \item Calculer $\displaystyle \int_{0}^{\ln(2)} f(x) \, dx$.
\end{enumerate}
}

% Exercise 2 (originally 2)
\printexo{2}{: (8 points)}{
\vspace*{-0.75cm}
\begin{enumerate}[label=\arabic*.]
    \item Calculer les intégrales suivantes :
    ~\(
    I = \int_{\frac{1}{e}}^{e} \frac{1}{x \ln x} \, dx \qquad J = \int_{1}^{2} \frac{\ln x}{x} \, dx
    \)~
    \item En utilisant une intégration par parties, montrer que :\\
      \hspace*{2cm}
    ~\(
    K = \int_{1}^{e} x^2 \ln x \, dx = \frac{2e^3 + 1}{9} \qquad;\qquad  M = \int_{0}^{1} (x - 1)e^x \, dx = -e
    \)~
    \item Soient la fonction $f$ définie par : $f(x) = x^2 e^{-x} - xe^{-x} + x$ et $(\mathcal{C}_f)$ sa courbe représentative dans un repère orthonormé $(O, \vec{i}, \vec{j})$.\quad
      ~$\lVert \vec{i} \rVert = 2\,cm$~
    \begin{enumerate}[label=\alph*), tight]
        \item Montrer que la fonction
        ~\(
        H : x \mapsto (x^2 + 2x + 2)e^{-x}
        \)~
        est une primitive de la fonction $h : x \mapsto -x^2 e^{-x}$.
    \item En déduire que :
    ~\(
    N = \int_{0}^{1} x^2 e^{-x} \, dx = \frac{2e - 5}{e}
    \)~
    \item En utilisant une intégration par parties montrer que :
    ~\(
    L = \int_{0}^{1} xe^{-x} \, dx = \frac{e - 2}{e}
    \)~
    \item Vérifier que $\displaystyle \int_{0}^{1} \left( f(x) - x \right) dx = \frac{e - 3}{e}$.
    \item Déduire en $\text{cm}^2$, l’aire du domaine du plan délimité par la courbe $(\mathcal{C}_f)$, la droite $(D)$ d’équation $y = x$ et les deux droites d’équations :
    ~\(
    x = 0 \text{ et } x = 1
    \)~
    \end{enumerate}
\end{enumerate}
}

% Exercise 3 (originally 3)
\printexo{3}{: (9 points)}{
Soient $(S)$ une sphère et $(P)$ un plan définient par les équations cartésiennes suivantes :\\
~\(
(S) : x^2 + y^2 + z^2 + 2x - 2y + 2z - 1 = 0, \quad (P) : 2x + y + 2z - 3 = 0.
\)~ respectivement
\begin{enumerate}[label=\arabic*)]
    \item
    \begin{enumerate}[label=\alph*)]
        \item Montrer que $(S)$ est une sphère du centre $\Omega(-1; 1; -1)$ et le rayon $R=2$.
        \item Calculer $d(\Omega, (P))$ la distance du point $\Omega$ au plan $(P)$.
        \item Déduire que le plan $(P)$ est tangent à la sphère $(S)$.
    \end{enumerate}
    \item
    \begin{enumerate}[label=\alph*)]
        \item Déterminer une représentation paramétrique de la droite $(\Delta)$ passant par le point $\Omega$ centre de la sphère $(S)$ et perpendiculaire au plan $(P)$.
        \item En déduire les coordonnées du point $H$ point d’intersection de la sphère $(S)$ et le plan $(P)$.
    \end{enumerate}
    \item
    \begin{enumerate}[label=\alph*)]
        \item Déterminer une équation cartésienne du plan $(Q)$ passant par le point $A(1; 2; -1)$, et dont $\vec{n}'(-1; 0; 1)$ est un vecteur normal à $(Q)$.
        \item Vérifier que les plans $(P)$ et $(Q)$ sont orthogonaux.
    \end{enumerate}
    \item Montrer que l’équation cartésienne de la sphère $(S')$ définie par son diamètre $[AB]$ où $A(0; 1; 5)$ et $B(2; 3; 1)$ est :
    ~\(
    (S') : x^2 + y^2 + z^2 - 2x - 4y - 6z + 8 = 0
    \)~
\end{enumerate}
}






% Bottom message
\begin{center}
  \normalsize{%
    \vskip 3pt \hrule height 3pt \vskip 5pt
    \RL{\arabicfont
    ﴿حُنَفَآءَ لِلَّهِ غَيْرَ مُشْرِكِينَ بِهِۦ ۚ وَمَن يُشْرِكْ بِٱللَّهِ فَكَأَنَّمَا خَرَّ مِنَ ٱلسَّمَآءِ فَتَخْطَفُهُ ٱلطَّيْرُ أَوْ تَهْوِى بِهِ ٱلرِّيحُ فِى مَكَانٍ سَحِيقٍ﴾ (الحج 31)
    }%
  }%
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\end{document}