% Dérivation 
%-----------------------

\question
    \begin{center}
        \textbf{Durée : 10min \hspace{3cm} Thème : Dérivation}
    \end{center}
    Calculer la dérivé des fonctions suivantes
    \begin{parts}
        \begin{multicols}{2}
            \Block{set A = Polynom.random(degree = 1, name = 'f')}
            \part $\Var{A.name}:x \mapsto \Var{A}$
            \begin{savedSolution}
                $ \Var{A.derivate().name}(x) = \Var{A.derivate()}$
            \end{savedSolution}

            \Block{set A = Polynom.random(degree = 1, name = 'g')}
            \part $\Var{A.name}:x \mapsto \Var{A}$
            \begin{savedSolution}
                $ \Var{A.derivate().name}(x) = \Var{A.derivate()}$
            \end{savedSolution}

            \Block{set A = Polynom.random(degree = 2, name = 'h')}
            \part $\Var{A.name}:x \mapsto \Var{A}$
            \begin{savedSolution}
                $ \Var{A.derivate().name}(x) = \Var{A.derivate()}$
            \end{savedSolution}

            \Block{set A = Polynom.random(degree = 2, name = 'i')}
            \part $\Var{A.name}:x \mapsto \Var{A}$
            \begin{savedSolution}
                $ \Var{A.derivate().name}(x) = \Var{A.derivate()}$
            \end{savedSolution}

            \Block{set A = Polynom.random(degree = 3, name = 'j')}
            \part $\Var{A.name}:x \mapsto \Var{A}$
            \begin{savedSolution}
                $ \Var{A.derivate().name}(x) = \Var{A.derivate()}$
            \end{savedSolution}

        \end{multicols}
    \end{parts}

\question
    \begin{center}
        \textbf{Durée : 10min \hspace{3cm} Thème : Dérivation}
    \end{center}
    Calculer la dérivé des fonctions suivantes
    \begin{parts}
        \begin{multicols}{2}
            \Block{set A = Polynom.random(degree = 2, name = 'f')}
            \part $\Var{A.name}:x \mapsto \Var{A}$
            \begin{savedSolution}
                $ \Var{A.derivate().name}(x) = \Var{A.derivate()}$
            \end{savedSolution}

            \Block{set A = Polynom.random(degree = 1, name = 'g')}
            \part $\Var{A.name}:x \mapsto \Var{A}$
            \begin{savedSolution}
                $ \Var{A.derivate().name}(x) = \Var{A.derivate()}$
            \end{savedSolution}

            \Block{set A = Polynom.random(degree = 3, name = 'h')}
            \part $\Var{A.name}:x \mapsto \Var{A}$
            \begin{savedSolution}
                $ \Var{A.derivate().name}(x) = \Var{A.derivate()}$
            \end{savedSolution}

            \Block{set A = Polynom.random(degree = 3, name = 'i')}
            \part $\Var{A.name}:x \mapsto \Var{A}$
            \begin{savedSolution}
                $ \Var{A.derivate().name}(x) = \Var{A.derivate()}$
            \end{savedSolution}

            \Block{set A = Polynom.random(degree = 2, name = 'j')}
            \part $\Var{A.name}:x \mapsto \Var{A}$
            \begin{savedSolution}
                $ \Var{A.derivate().name}(x) = \Var{A.derivate()}$
            \end{savedSolution}

        \end{multicols}
    \end{parts}

\question
    \begin{center}
        \textbf{Durée : 20min \hspace{3cm} Thème : Dérivation}
    \end{center}
    Calculer la dérivé des fonctions suivantes
    \begin{parts}
        \begin{multicols}{2}
            \Block{set A = Polynom.random(degree = 1, name = 'f')}
            \part $\Var{A.name}:x \mapsto \dfrac{1}{\Var{A}}$
            \begin{savedSolution}
                $ \Var{A.derivate().name}(x) = \dfrac{\Var{(-A.derivate())}}{\Var{Expression([A,2,"^"])}}$
            \end{savedSolution}

            \Block{set A = Polynom.random(degree = 1, name = 'g')}
            \part $\Var{A.name}:x \mapsto \dfrac{1}{\Var{A}}$
            \begin{savedSolution}
                $ \Var{A.derivate().name}(x) = \dfrac{\Var{(-A.derivate())}}{\Var{Expression([A,2,"^"])}}$
            \end{savedSolution}

            \Block{set A = Polynom.random(degree = 2, name = 'h')}
            \part $\Var{A.name}:x \mapsto \dfrac{1}{\Var{A}}$
            \begin{savedSolution}
                $ \Var{A.derivate().name}(x) = \dfrac{\Var{(-A.derivate())}}{\Var{Expression([A,2,"^"])}}$
            \end{savedSolution}

            \Block{set A = Polynom.random(degree = 2, name = 'i')}
            \part $\Var{A.name}:x \mapsto \dfrac{1}{\Var{A}}$
            \begin{savedSolution}
                $ \Var{A.derivate().name}(x) = \dfrac{\Var{(-A.derivate())}}{\Var{Expression([A,2,"^"])}}$
            \end{savedSolution}

            \Block{set A = Polynom.random(degree = 1, name = 'j')}
            \Block{set B = Polynom.random(degree = 1, name = 'f')}
            \part $\Var{A.name}:x \mapsto \dfrac{\Var{A}}{\Var{B}}$
            \begin{savedSolution}
                \Block{set num = Expression([A.derivate(), B, "*", A, B.derivate(), "*", "-"]).simplify()}%
                $ \Var{A.derivate().name}(x) = \dfrac{\Var{num}}{\Var{Expression([B,2,"^"])}}$
            \end{savedSolution}

            \Block{set A = Polynom.random(degree = 1, name = 'k')}
            \Block{set B = Polynom.random(degree = 1, name = 'f')}
            \part $\Var{A.name}:x \mapsto \dfrac{\Var{A}}{\Var{B}}$
            \begin{savedSolution}
                \Block{set num = Expression([A.derivate(), B, "*", A, B.derivate(), "*", "-"]).simplify()}%
                $ \Var{A.derivate().name}(x) = \dfrac{\Var{num}}{\Var{Expression([B,2,"^"])}}$
            \end{savedSolution}

        \end{multicols}
    \end{parts}

\question
    \begin{center}
        \textbf{Durée : 20min \hspace{3cm} Thème : Dérivation}
    \end{center}
    Calculer la dérivé des fonctions suivantes
    \begin{parts}
        \begin{multicols}{2}
            \Block{set A = Polynom.random(degree = 1, name = 'f')}
            \part $\Var{A.name}:x \mapsto \dfrac{1}{\Var{A}}$
            \begin{savedSolution}
                $ \Var{A.derivate().name}(x) = \dfrac{\Var{(-A.derivate())}}{\Var{Expression([A,2,"^"])}}$
            \end{savedSolution}

            \Block{set A = Polynom.random(degree = 1, name = 'g')}
            \part $\Var{A.name}:x \mapsto \dfrac{1}{\Var{A}}$
            \begin{savedSolution}
                $ \Var{A.derivate().name}(x) = \dfrac{\Var{(-A.derivate())}}{\Var{Expression([A,2,"^"])}}$
            \end{savedSolution}

            \Block{set A = Polynom.random(degree = 2, name = 'h')}
            \part $\Var{A.name}:x \mapsto \dfrac{1}{\Var{A}}$
            \begin{savedSolution}
                $ \Var{A.derivate().name}(x) = \dfrac{\Var{(-A.derivate())}}{\Var{Expression([A,2,"^"])}}$
            \end{savedSolution}

            \Block{set A = Polynom.random(degree = 2, name = 'i')}
            \part $\Var{A.name}:x \mapsto \dfrac{1}{\Var{A}}$
            \begin{savedSolution}
                $ \Var{A.derivate().name}(x) = \dfrac{\Var{(-A.derivate())}}{\Var{Expression([A,2,"^"])}}$
            \end{savedSolution}

            \Block{set A = Polynom.random(degree = 1, name = 'j')}
            \Block{set B = Polynom.random(degree = 1, name = 'f')}
            \part $\Var{A.name}:x \mapsto \dfrac{\Var{A}}{\Var{B}}$
            \begin{savedSolution}
                \Block{set num = Expression([A.derivate(), B, "*", A, B.derivate(), "*", "-"]).simplify()}%
                $ \Var{A.derivate().name}(x) = \dfrac{\Var{num}}{\Var{Expression([B,2,"^"])}}$
            \end{savedSolution}

            \Block{set A = Polynom.random(degree = 1, name = 'k')}
            \Block{set B = Polynom.random(degree = 1, name = 'f')}
            \part $\Var{A.name}:x \mapsto \dfrac{\Var{A}}{\Var{B}}$
            \begin{savedSolution}
                \Block{set num = Expression([A.derivate(), B, "*", A, B.derivate(), "*", "-"]).simplify()}%
                $ \Var{A.derivate().name}(x) = \dfrac{\Var{num}}{\Var{Expression([B,2,"^"])}}$
            \end{savedSolution}

        \end{multicols}
    \end{parts}

% Polynome 2nd degre
% ------------------

\question
    \begin{center}
        \textbf{Durée : 20min \hspace{3cm} Thème : 2nd degré}
    \end{center}
    Tracer le tableau de signe des fonctions suivantes
    \begin{parts}
        \begin{multicols}{2}

        \Block{set f = Polynom.random(degree = 2, name = 'f', conditions = ["{b**2-4*a*c} >0"] )}
        \part $\Var{f.name} : x \mapsto \Var{f}$
        \begin{savedSolution}
            \begin{tikzpicture}
                \tkzTabInit[espcl=2]{$x$/1,Signe de $\Var{f.name}(x)$/2}\Var{f.tbl_sgn_header()}
                \Var{f.tbl_sgn()}
            \end{tikzpicture}
        \end{savedSolution}

        \Block{set f = Polynom.random(degree = 2, name = 'g', conditions = ["{b**2-4*a*c} == 0"])}
        \part $\Var{f.name} : x \mapsto \Var{f}$
        \begin{savedSolution}
            \begin{tikzpicture}
                \tkzTabInit[espcl=2]{$x$/1,Signe de $\Var{f.name}(x)$/2}\Var{f.tbl_sgn_header()}
                \Var{f.tbl_sgn()}
            \end{tikzpicture}
        \end{savedSolution}

        \Block{set f = Polynom.random(degree = 2, name = 'h', conditions = ["{b**2-4*a*c} <0"])}
        \part $\Var{f.name} : x \mapsto \Var{f}$
        \begin{savedSolution}
            \begin{tikzpicture}
                \tkzTabInit[espcl=2]{$x$/1,Signe de $\Var{f.name}(x)$/2}\Var{f.tbl_sgn_header()}
                \Var{f.tbl_sgn()}
            \end{tikzpicture}
        \end{savedSolution}

        \Block{set f = Polynom.random(degree = 2, name = 'i')}
        \part $\Var{f.name} : x \mapsto \Var{f}$
        \begin{savedSolution}
            \begin{tikzpicture}
                \tkzTabInit[espcl=2]{$x$/1,Signe de $\Var{f.name}(x)$/2}\Var{f.tbl_sgn_header()}
                \Var{f.tbl_sgn()}
            \end{tikzpicture}
        \end{savedSolution}
            
        \end{multicols}
    \end{parts}

\question
    \begin{center}
        \textbf{Durée : 20min \hspace{3cm} Thème : 2nd degré}
    \end{center}
    Tracer le tableau de signe des fonctions suivantes
    \begin{parts}
        \begin{multicols}{2}

        \Block{set f = Polynom.random(degree = 2, name = 'f', conditions = ["{b**2-4*a*c} >0"] )}
        \part $\Var{f.name} : x \mapsto \Var{f}$
        \begin{savedSolution}
            \begin{tikzpicture}
                \tkzTabInit[espcl=2]{$x$/1,Signe de $\Var{f.name}(x)$/2}\Var{f.tbl_sgn_header()}
                \Var{f.tbl_sgn()}
            \end{tikzpicture}
        \end{savedSolution}

        \Block{set f = Polynom.random(degree = 2, name = 'g', conditions = ["{b**2-4*a*c} == 0"])}
        \part $\Var{f.name} : x \mapsto \Var{f}$
        \begin{savedSolution}
            \begin{tikzpicture}
                \tkzTabInit[espcl=2]{$x$/1,Signe de $\Var{f.name}(x)$/2}\Var{f.tbl_sgn_header()}
                \Var{f.tbl_sgn()}
            \end{tikzpicture}
        \end{savedSolution}

        \Block{set f = Polynom.random(degree = 2, name = 'h', conditions = ["{b**2-4*a*c} <0"])}
        \part $\Var{f.name} : x \mapsto \Var{f}$
        \begin{savedSolution}
            \begin{tikzpicture}
                \tkzTabInit[espcl=2]{$x$/1,Signe de $\Var{f.name}(x)$/2}\Var{f.tbl_sgn_header()}
                \Var{f.tbl_sgn()}
            \end{tikzpicture}
        \end{savedSolution}

        \Block{set f = Polynom.random(degree = 2, name = 'i')}
        \part $\Var{f.name} : x \mapsto \Var{f}$
        \begin{savedSolution}
            \begin{tikzpicture}
                \tkzTabInit[espcl=2]{$x$/1,Signe de $\Var{f.name}(x)$/2}\Var{f.tbl_sgn_header()}
                \Var{f.tbl_sgn()}
            \end{tikzpicture}
        \end{savedSolution}
            
        \end{multicols}
    \end{parts}

\question
    \begin{center}
        \textbf{Durée : 20min \hspace{3cm} Thème : 2nd degré}
    \end{center}
    Résoudre les équations suivantes
    \begin{parts}
        \begin{multicols}{2}

        \Block{set f = Polynom.random(degree = 2, name = 'f', conditions = ["{b**2-4*a*c} >0"] )}
        \part $\Var{f} = 0$
        \begin{savedSolution}
            $\mathcal{S} = \left\{ \Var{f.roots() | join(";")} \right\}$
        \end{savedSolution}

        \Block{set f = Polynom.random(degree = 2, name = 'g', conditions = ["{b**2-4*a*c} == 0"])}
        \part $\Var{f} = 0$
        \begin{savedSolution}
            $\mathcal{S} = \left\{ \Var{f.roots()[0]} \right\}$
        \end{savedSolution}

        \Block{set f = Polynom.random(degree = 2, name = 'h', conditions = ["{b**2-4*a*c} <0"])}
        \part $\Var{f} = 0$
        \begin{savedSolution}
            Il n'y a pas de solution.
        \end{savedSolution}

        \Block{set f = Polynom.random(degree = 2, name = 'f', conditions = ["{b**2-4*a*c} >0"] )}
        \part $\Var{f} = 0$
        \begin{savedSolution}
            $\mathcal{S} = \left\{ \Var{f.roots() | join(";")} \right\}$
        \end{savedSolution}
            
        \end{multicols}
    \end{parts}

\question
    \begin{center}
        \textbf{Durée : 20min \hspace{3cm} Thème : 2nd degré}
    \end{center}
    Résoudre les équations suivantes
    \begin{parts}
        \begin{multicols}{2}

        \Block{set f = Polynom.random(degree = 2, name = 'f', conditions = ["{b**2-4*a*c} >0"] )}
        \part $\Var{f} = 0$
        \begin{savedSolution}
            $\mathcal{S} = \left\{ \Var{f.roots() | join(";")} \right\}$
        \end{savedSolution}

        \Block{set f = Polynom.random(degree = 2, name = 'g', conditions = ["{b**2-4*a*c} == 0"])}
        \part $\Var{f} = 0$
        \begin{savedSolution}
            $\mathcal{S} = \left\{ \Var{f.roots()[0]} \right\}$
        \end{savedSolution}

        \Block{set f = Polynom.random(degree = 2, name = 'h', conditions = ["{b**2-4*a*c} <0"])}
        \part $\Var{f} = 0$
        \begin{savedSolution}
            Il n'y a pas de solution.
        \end{savedSolution}

        \Block{set f = Polynom.random(degree = 2, name = 'f', conditions = ["{b**2-4*a*c} >0"] )}
        \part $\Var{f} = 0$
        \begin{savedSolution}
            $\mathcal{S} = \left\{ \Var{f.roots() | join(";")} \right\}$
        \end{savedSolution}
            
        \end{multicols}
    \end{parts}

% Variations
% ----------

\question
    \begin{center}
        \textbf{Durée : 20min \hspace{3cm} Thème : Variations}
    \end{center}
    \Block{set f = Polynom.random(degree = 2, name = 'f')}
    Soit $\Var{f.name}$ la fonction définie par
    \begin{eqnarray*}
        \Var{f.name} : x \mapsto \Var{f}
    \end{eqnarray*}
    
    \begin{parts}
        \part Quel est le domaine de définition de $\Var{f.name}$? Quel est son domaine de dérivation?
        \begin{savedSolution}
            Domaine de définition: $\R$ \quad Domaine de dérivation: $\R$
        \end{savedSolution}
        \part Calculer la dérivée de $\Var{f.name}$.
        \begin{savedSolution}
            $\Var{f.derivate().name} (x) = \Var{f.derivate()}$
        \end{savedSolution}
        \part Déterminer le tableau de variations de $\Var{f.name}$.
        \begin{savedSolution}
            \begin{tikzpicture}
                \tkzTabInit[espcl=2]{$x$/1,Variations de $\Var{f.name}(x)$/2}{$-\infty$, $\Var{f.alpha}$, $+\infty$}
                \Var{f.tbl_variation()}
            \end{tikzpicture}
        \end{savedSolution}
    \end{parts}

\question
    \begin{center}
        \textbf{Durée : 20min \hspace{3cm} Thème : Variations}
    \end{center}
    \Block{set f = Polynom.random(degree = 2, name = 'f')}
    Soit $\Var{f.name}$ la fonction définie par
    \begin{eqnarray*}
        \Var{f.name} : x \mapsto \Var{f}
    \end{eqnarray*}
    
    \begin{parts}
        \part Quel est le domaine de définition de $\Var{f.name}$? Quel est son domaine de dérivation?
        \begin{savedSolution}
            Domaine de définition: $\R$ \quad Domaine de dérivation: $\R$
        \end{savedSolution}
        \part Calculer la dérivée de $\Var{f.name}$.
        \begin{savedSolution}
            $\Var{f.derivate().name} (x) = \Var{f.derivate()}$
        \end{savedSolution}
        \part Déterminer le tableau de variations de $\Var{f.name}$.
        \begin{savedSolution}
            \begin{tikzpicture}
                \tkzTabInit[espcl=2]{$x$/1,Variations de $\Var{f.name}(x)$/2}{$-\infty$, $\Var{f.alpha}$, $+\infty$}
                \Var{f.tbl_variation()}
            \end{tikzpicture}
        \end{savedSolution}
    \end{parts}

\question
    \begin{center}
        \textbf{Durée : 20min \hspace{3cm} Thème : Variations}
    \end{center}
    \Block{set f = Polynom.random(degree = 3, name = 'f', conditions = ["{a}>0", "{4*b**2-12*a*c}>0"])}
    Soit $\Var{f.name}$ la fonction définie par
    \begin{eqnarray*}
        \Var{f.name} : x \mapsto \Var{f}
    \end{eqnarray*}
    
    \begin{parts}
        \part Quel est le domaine de définition de $\Var{f.name}$? Quel est son domaine de dérivation?
        \begin{savedSolution}
            Domaine de définition: $\R$ \quad Domaine de dérivation: $\R$
        \end{savedSolution}
        \part Calculer la dérivée de $\Var{f.name}$.
        \begin{savedSolution}
            \Block{set fp = f.derivate()}%
            $\Var{fp.name} (x) = \Var{fp}$
        \end{savedSolution}
        \part Déterminer le tableau de variations de $\Var{f.name}$.
        \begin{savedSolution}
            \begin{tikzpicture}
                \tkzTabInit[espcl=2.5, lgt=3]{$x$/1,Variations de $\Var{f.name}(x)$/3}\Var{fp.tbl_sgn_header()}
                \tkzTabVar{-/{}, +/{$f(\Var{fp.roots()[0]})$}, -/{$f(\Var{fp.roots()[1]})$}, +/{} }
            \end{tikzpicture}
        \end{savedSolution}
    \end{parts}

\question
    \begin{center}
        \textbf{Durée : 20min \hspace{3cm} Thème : Variations}
    \end{center}
    \Block{set f = Polynom.random(degree = 3, name = 'f', conditions = ["{a}<0", "{4*b**2-12*a*c}>0"])}
    Soit $\Var{f.name}$ la fonction définie par
    \begin{eqnarray*}
        \Var{f.name} : x \mapsto \Var{f}
    \end{eqnarray*}
    
    Déterminer les variations de $\Var{f.name}$.
    \begin{savedSolution}
            \Block{set fp = f.derivate()}%
            \begin{tikzpicture}
                \tkzTabInit[espcl=2.5, lgt=3]{$x$/1,Variations de $\Var{f.name}(x)$/3}\Var{fp.tbl_sgn_header()}
                \tkzTabVar{+/{}, -/{$f(\Var{fp.roots()[0]})$}, +/{$f(\Var{fp.roots()[1]})$}, -/{} }
            \end{tikzpicture}
    \end{savedSolution}

\question
    \begin{center}
        \textbf{Durée : 20min \hspace{3cm} Thème : Variations}
    \end{center}
    \Block{set f = Polynom.random(degree = 3, name = 'f', conditions = ["{a}<0", "{4*b**2-12*a*c}<0"])}
    Soit $\Var{f.name}$ la fonction définie par
    \begin{eqnarray*}
        \Var{f.name} : x \mapsto \Var{f}
    \end{eqnarray*}
    
    Déterminer les variations de $\Var{f.name}$.
    \begin{savedSolution}
            \begin{tikzpicture}
                \tkzTabInit[espcl=2.5, lgt=3]{$x$/1,Variations de $\Var{f.name}(x)$/2}{$+\infty$, $-\infty$}
                \tkzTabVar{+/{}, -/{} }
            \end{tikzpicture}
    \end{savedSolution}

\question
    \begin{center}
        \textbf{Durée : 20min \hspace{3cm} Thème : Variations}
    \end{center}
    \Block{set f = Polynom.random(degree = 3, name = 'f', conditions = ["{a}<0", "{4*b**2-12*a*c}==0"])}
    Soit $\Var{f.name}$ la fonction définie par
    \begin{eqnarray*}
        \Var{f.name} : x \mapsto \Var{f}
    \end{eqnarray*}
    
    Déterminer les variations de $\Var{f.name}$.
    \begin{savedSolution}
            \Block{set fp = f.derivate()}%
            \begin{tikzpicture}
                \tkzTabInit[espcl=5, lgt=3]{$x$/1,Variations de $\Var{f.name}(x)$/2}{$+\infty$, $-\infty$}
                \tkzTabVar{+/{}, -/{} }
                \tkzTabVal{1}{2}{0.5}{$\Var{fp.roots()[0]}$}{$f(\Var{fp.roots()[0]})$}
            \end{tikzpicture}
    \end{savedSolution}

% Suites
% ------

\question
    \begin{center}
        \textbf{Durée : 20min \hspace{3cm} Thème : Suites}
    \end{center}
    Calculer les 3 premiers termes et le 10ième terme des 4 suites suivantes
    \begin{multicols}{2}
        \begin{parts}
            \Block{set u = Polynom.random(degree = 2, letter = "n", name = "u")}
            \part $\Var{u.name}_n = \Var{u}$
            \begin{savedSolution}
                $\Var{u.name}_0 = \Var{u(0)}$ \quad
                $\Var{u.name}_1 = \Var{u(1)}$ \quad
                $\Var{u.name}_2 = \Var{u(2)}$ \quad
                $\Var{u.name}_{10} = \Var{u(10)}$ \quad
            \end{savedSolution}
            \Block{set u = Polynom.random(degree = 2, letter = "n", name = "v")}
            \part $\Var{u.name}_n = \Var{u}$
            \begin{savedSolution}
                $\Var{u.name}_0 = \Var{u(0)}$ \quad
                $\Var{u.name}_1 = \Var{u(1)}$ \quad
                $\Var{u.name}_2 = \Var{u(2)}$ \quad
                $\Var{u.name}_{10} = \Var{u(10)}$ \quad
            \end{savedSolution}
            \Block{set u = Polynom.random(degree = 3, letter = "n", name = "w")}
            \part $\Var{u.name}_n = \Var{u}$
            \begin{savedSolution}
                $\Var{u.name}_0 = \Var{u(0)}$ \quad
                $\Var{u.name}_1 = \Var{u(1)}$ \quad
                $\Var{u.name}_2 = \Var{u(2)}$ \quad
                $\Var{u.name}_{10} = \Var{u(10)}$ \quad
            \end{savedSolution}
            \Block{set u = Polynom.random(degree = 2, letter = "n", name = "l")}
            \Block{set v = Polynom.random(degree = 2, letter = "n", name = "l")}
            \part $\Var{u.name}_n = \dfrac{\Var{u}}{\Var{v}}$
            \begin{savedSolution}
                $\Var{u.name}_0 = \Var{Expression([u(0),v(0),'/']).simplify()}$ \quad
                $\Var{u.name}_1 = \Var{Expression([u(1),v(1),'/']).simplify()}$ \quad
                $\Var{u.name}_2 = \Var{Expression([u(2),v(2),'/']).simplify()}$ \quad
                $\Var{u.name}_{10} = \Var{Expression([u(10),v(10),'/']).simplify()}$ \quad
            \end{savedSolution}
        \end{parts}
    \end{multicols}

\question
    \begin{center}
        \textbf{Durée : 20min \hspace{3cm} Thème : Suites}
    \end{center}
    Calculer les 3 premiers termes et le 10ième terme des 4 suites suivantes
    \begin{multicols}{2}
        \begin{parts}
            \Block{set u = Polynom.random(degree = 2, letter = "n", name = "u")}
            \part $\Var{u.name}_n = \Var{u}$
            \begin{savedSolution}
                $\Var{u.name}_0 = \Var{u(0)}$ \quad
                $\Var{u.name}_1 = \Var{u(1)}$ \quad
                $\Var{u.name}_2 = \Var{u(2)}$ \quad
                $\Var{u.name}_{10} = \Var{u(10)}$ \quad
            \end{savedSolution}
            \Block{set u = Polynom.random(degree = 2, letter = "n", name = "v")}
            \part $\Var{u.name}_n = \Var{u}$
            \begin{savedSolution}
                $\Var{u.name}_0 = \Var{u(0)}$ \quad
                $\Var{u.name}_1 = \Var{u(1)}$ \quad
                $\Var{u.name}_2 = \Var{u(2)}$ \quad
                $\Var{u.name}_{10} = \Var{u(10)}$ \quad
            \end{savedSolution}
            \Block{set u = Polynom.random(degree = 3, letter = "n", name = "w")}
            \part $\Var{u.name}_n = \Var{u}$
            \begin{savedSolution}
                $\Var{u.name}_0 = \Var{u(0)}$ \quad
                $\Var{u.name}_1 = \Var{u(1)}$ \quad
                $\Var{u.name}_2 = \Var{u(2)}$ \quad
                $\Var{u.name}_{10} = \Var{u(10)}$ \quad
            \end{savedSolution}
            \Block{set u = Polynom.random(degree = 2, letter = "n", name = "l")}
            \Block{set v = Polynom.random(degree = 2, letter = "n", name = "l")}
            \part $\Var{u.name}_n = \dfrac{\Var{u}}{\Var{v}}$
            \begin{savedSolution}
                $\Var{u.name}_0 = \Var{Expression([u(0),v(0),'/']).simplify()}$ \quad
                $\Var{u.name}_1 = \Var{Expression([u(1),v(1),'/']).simplify()}$ \quad
                $\Var{u.name}_2 = \Var{Expression([u(2),v(2),'/']).simplify()}$ \quad
                $\Var{u.name}_{10} = \Var{Expression([u(10),v(10),'/']).simplify()}$ \quad
            \end{savedSolution}
        \end{parts}
    \end{multicols}

\question
    \begin{center}
        \textbf{Durée : 20min \hspace{3cm} Thème : Suites}
    \end{center}
    Calculer les 3 premiers termes des 4 suites suivantes
    \begin{multicols}{2}
        \begin{parts}
            \Block{set u = Polynom.random(degree = 1, letter = "u_n", name = "u")}
            \Block{set a = Expression.random("{a}")}
            \part $\Var{u.name}_{n+1} = \Var{u}$ et $\Var{u.name}_0 = \Var{a}$.
            \begin{savedSolution}
                \Block{set a = u(a)}%
                $\Var{u.name}_1 = \Var{a}$ \quad
                \Block{set a = u(a)}%
                $\Var{u.name}_2 = \Var{a}$ \quad
                \Block{set a = u(a)}%
                $\Var{u.name}_3 =  \Var{a}$ \quad
            \end{savedSolution}

            \Block{set u = Polynom.random(degree = 1, letter = "u_n", name = "u")}
            \Block{set a = Expression.random("{a}")}
            \part $\Var{u.name}_{n+1} = \Var{u}$ et $\Var{u.name}_0 = \Var{a}$.
            \begin{savedSolution}
                \Block{set a = u(a)}%
                $\Var{u.name}_1 = \Var{a}$ \quad
                \Block{set a = u(a)}%
                $\Var{u.name}_2 = \Var{a}$ \quad
                \Block{set a = u(a)}%
                $\Var{u.name}_3 =  \Var{a}$ \quad
            \end{savedSolution}

            \Block{set u = Polynom.random(degree = 2, letter = "u_n", name = "u")}
            \Block{set a = Expression.random("{a}")}
            \part $\Var{u.name}_{n+1} = \Var{u}$ et $\Var{u.name}_0 = \Var{a}$.
            \begin{savedSolution}
                \Block{set a = u(a)}%
                $\Var{u.name}_1 = \Var{a}$ \quad
                \Block{set a = u(a)}%
                $\Var{u.name}_2 = \Var{a}$ \quad
                \Block{set a = u(a)}%
                $\Var{u.name}_3 =  \Var{a}$ \quad
            \end{savedSolution}

            \Block{set u = Polynom.random(degree = 1, letter = "u_n", name = "u")}
            \Block{set a = Expression.random("{a}")}
            \part $\Var{u.name}_{n+1} = \Var{u}$ et $\Var{u.name}_0 = \Var{a}$.
            \begin{savedSolution}
                \Block{set a = u(a)}%
                $\Var{u.name}_1 = \Var{a}$ \quad
                \Block{set a = u(a)}%
                $\Var{u.name}_2 = \Var{a}$ \quad
                \Block{set a = u(a)}%
                $\Var{u.name}_3 =  \Var{a}$ \quad
            \end{savedSolution}
        \end{parts}

    \end{multicols}

\question
    \begin{center}
        \textbf{Durée : 20min \hspace{3cm} Thème : Suites}
    \end{center}
    Calculer les 3 premiers termes des 4 suites suivantes
    \begin{multicols}{2}
        \begin{parts}
            \Block{set u = Polynom.random(degree = 1, letter = "u_n", name = "u")}
            \Block{set a = Expression.random("{a}")}
            \part $\Var{u.name}_{n+1} = \Var{u}$ et $\Var{u.name}_0 = \Var{a}$.
            \begin{savedSolution}
                \Block{set a = u(a)}%
                $\Var{u.name}_1 = \Var{a}$ \quad
                \Block{set a = u(a)}%
                $\Var{u.name}_2 = \Var{a}$ \quad
                \Block{set a = u(a)}%
                $\Var{u.name}_3 =  \Var{a}$ \quad
            \end{savedSolution}

            \Block{set u = Polynom.random(degree = 1, letter = "u_{n-1}", name = "u")}
            \Block{set a = Expression.random("{a}")}
            \part $\Var{u.name}_{n} = \Var{u}$ et $\Var{u.name}_0 = \Var{a}$.
            \begin{savedSolution}
                \Block{set a = u(a)}%
                $\Var{u.name}_1 = \Var{a}$ \quad
                \Block{set a = u(a)}%
                $\Var{u.name}_2 = \Var{a}$ \quad
                \Block{set a = u(a)}%
                $\Var{u.name}_3 =  \Var{a}$ \quad
            \end{savedSolution}

            \Block{set u = Polynom.random(degree = 1, letter = "u_{n-1}", name = "u")}
            \Block{set a = Expression.random("{a}")}
            \part $\Var{u.name}_n = \Var{u}$ et $\Var{u.name}_0 = \Var{a}$.
            \begin{savedSolution}
                \Block{set a = u(a)}%
                $\Var{u.name}_1 = \Var{a}$ \quad
                \Block{set a = u(a)}%
                $\Var{u.name}_2 = \Var{a}$ \quad
                \Block{set a = u(a)}%
                $\Var{u.name}_3 =  \Var{a}$ \quad
            \end{savedSolution}

            \Block{set u = Polynom.random(degree = 1, letter = "u_n", name = "u")}
            \Block{set P = Polynom.random([0,"{a}"], conditions=["{a}>0"], letter = 'n')}
            \Block{set a = Expression.random("{a}")}
            \part $\Var{u.name}_{n+1} = \Var{u} + \Var{P}$ et $\Var{u.name}_0 = \Var{a}$.
            \begin{savedSolution}
                \Block{set a = u(a) + P(1)}%
                $\Var{u.name}_1 = \Var{a}$ \quad
                \Block{set a = u(a) + P(2)}%
                $\Var{u.name}_2 = \Var{a}$ \quad
                \Block{set a = u(a) + P(3)}%
                $\Var{u.name}_3 =  \Var{a}$ \quad
            \end{savedSolution}
        \end{parts}

    \end{multicols}

\question
    \begin{center}
        \textbf{Durée : 20min \hspace{3cm} Thème : Suites}
    \end{center}
    Calculer les 3 premiers termes des 4 suites suivantes
    \begin{multicols}{2}
        \begin{parts}
            \Block{set u = Polynom.random(degree = 1, letter = "u_n", name = "u")}
            \Block{set a = Expression.random("{a}")}
            \part $\Var{u.name}_{n+1} = \Var{u}$ et $\Var{u.name}_0 = \Var{a}$.
            \begin{savedSolution}
                \Block{set a = u(a)}%
                $\Var{u.name}_1 = \Var{a}$ \quad
                \Block{set a = u(a)}%
                $\Var{u.name}_2 = \Var{a}$ \quad
                \Block{set a = u(a)}%
                $\Var{u.name}_3 =  \Var{a}$ \quad
            \end{savedSolution}

            \Block{set u = Polynom.random(degree = 1, letter = "u_{n-1}", name = "u")}
            \Block{set a = Expression.random("{a}")}
            \part $\Var{u.name}_{n} = \Var{u}$ et $\Var{u.name}_0 = \Var{a}$.
            \begin{savedSolution}
                \Block{set a = u(a)}%
                $\Var{u.name}_1 = \Var{a}$ \quad
                \Block{set a = u(a)}%
                $\Var{u.name}_2 = \Var{a}$ \quad
                \Block{set a = u(a)}%
                $\Var{u.name}_3 =  \Var{a}$ \quad
            \end{savedSolution}

            \Block{set u = Polynom.random(degree = 1, letter = "u_{n-1}", name = "u")}
            \Block{set a = Expression.random("{a}")}
            \part $\Var{u.name}_n = \Var{u}$ et $\Var{u.name}_0 = \Var{a}$.
            \begin{savedSolution}
                \Block{set a = u(a)}%
                $\Var{u.name}_1 = \Var{a}$ \quad
                \Block{set a = u(a)}%
                $\Var{u.name}_2 = \Var{a}$ \quad
                \Block{set a = u(a)}%
                $\Var{u.name}_3 =  \Var{a}$ \quad
            \end{savedSolution}

            \Block{set u = Polynom.random(degree = 1, letter = "u_n", name = "u")}
            \Block{set P = Polynom.random([0,"{a}"], conditions=["{a}>0"], letter = 'n')}
            \Block{set a = Expression.random("{a}")}
            \part $\Var{u.name}_{n+1} = \Var{u} + \Var{P}$ et $\Var{u.name}_0 = \Var{a}$.
            \begin{savedSolution}
                \Block{set a = u(a) + P(1)}%
                $\Var{u.name}_1 = \Var{a}$ \quad
                \Block{set a = u(a) + P(2)}%
                $\Var{u.name}_2 = \Var{a}$ \quad
                \Block{set a = u(a) + P(3)}%
                $\Var{u.name}_3 =  \Var{a}$ \quad
            \end{savedSolution}
        \end{parts}

    \end{multicols}