\documentclass[a4paper,12pt, table]{/media/documents/Cours/Prof/Enseignements/2014-2015/tools/style/classDS}
\usepackage{/media/documents/Cours/Prof/Enseignements/2014-2015/2014_2015}
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% Title Page
\titre{4}
% \seconde \premiereS \PSTMG \TSTMG
\classe{\seconde}
\date{6 mai 2015}
%\duree{1 heure}
\sujet{\Var{infos.num}}
% DS DSCorr DM DMCorr Corr
\typedoc{DM}

%\printanswers

\begin{document}

\maketitle

Vous devez rendre le sujet avec la copie.

\begin{questions}

    \question
    \begin{parts}
        \part Développer et simplifier les expressions suivantes
        \Block{set A = Expression.random("({a}x + {b})({a} - {c}x)")}
        \Block{set B = Expression.random("({a}x + {b})^2 + {c}")}
        \Block{set C = Expression.random("{d}x + {c} + 4({a}x + {b})^2")}
        \begin{subparts}
            \begin{multicols}{2}
                \subpart $A = \Var{A}$
                \begin{solution}
                    \Block{set Ar = A.simplify()}
                    \begin{eqnarray*}
                        \Var{Ar.explain() | calculus(name = "A", sep = "=", end = "")}
                    \end{eqnarray*}
                \end{solution}
                \subpart $B = \Var{B}$
                \begin{solution}
                    \Block{set Br = B.simplify()}
                    \begin{eqnarray*}
                        \Var{Br.explain() | calculus(name = "A", sep = "=", end = "")}
                    \end{eqnarray*}
                \end{solution}
                \subpart $C = \Var{C}$
                \begin{solution}
                    \Block{set Ar = C.simplify()}
                    \begin{eqnarray*}
                        \Var{Ar.explain() | calculus(name = "A", sep = "=", end = "")}
                    \end{eqnarray*}
                \end{solution}
            \end{multicols}
        \end{subparts}

        \part Factoriser les expressions suivantes
        \Block{set A = Expression.random("{a}x^2 - x")}
        \Block{set B = Expression.random("{a*a}x^2 + {b*b} + {2*a*b}x ", ["{a}>0", "{b}>0"])}
        \Block{set C = Expression.random("{a*a}x^2 - {b*b}")}
        \Block{set D = Expression.random("{a*a}x^2 - {2*a*b}x + {b*b}", ["{a}>0", "{b}>0"])}
        \begin{subparts}
            \begin{multicols}{2}
                \subpart $A = \Var{A}$
                \subpart $B = \Var{B}$
                \subpart $C = \Var{C}$
                \subpart $D = \Var{D}$
                
            \end{multicols}
        \end{subparts}

        \part Résoudre les équations suivantes
        \Block{set A = Polynom.random(degree = 1)}
        \Block{set B1 = Polynom.random(degree = 1)}
        \Block{set B2 = Polynom.random(degree = 1)}
        \Block{set C1 = Polynom.random(degree = 2)}
        \Block{set D = Expression.random("({a}x + {b})({c}x - {d})", ["{b} > 0", "{d} > 0"])}
        \begin{subparts}
            \begin{multicols}{2}
                \subpart $\Var{A} = 0$
                \subpart $\Var{B1} = \Var{B2}$

                \columnbreak

                \subpart $\Var{C1} = \Var{C1.a}x^2$
                \subpart $\Var{D} = 0$
            \end{multicols}
        \end{subparts}
        
    \end{parts}
    
    \question
    \begin{parts}
        \Block{set Ax, Ay, Bx, By, Cx, Cy, Dx, Dy= random_str("{a},{b},{c},{d},{e},{f},{g},{h}", conditions = ["{g-e} != 0", "{h-f} != 0", "{c-a}/{g-e} == {d-b}/{h-f}"]).split(',')}
        \part Soit $A(\Var{Ax} ; \Var{Ay})$, $B(\Var{Bx} ; \Var{By})$, $C(\Var{Cx} ; \Var{Cy})$ et $D(\Var{Dx} ; \Var{Dy})$. Est-ce que les vecteurs $\vec{AB}$ et $\vec{CD}$ sont colinéaires?
        \Block{set Ax, Ay, Bx, By, Cx, Cy, Dx, Dy= random_str("{a},{b},{c},{d},{e},{f},{g},{h}", conditions = ["{g-e} != 0", "{h-f} != 0","{c-a}/{g-e} != {d-b}/{h-f}"]).split(',')}
        \part Soit $A(\Var{Ax} ; \Var{Ay})$, $B(\Var{Bx} ; \Var{By})$, $C(\Var{Cx} ; \Var{Cy})$ et $D(\Var{Dx} ; \Var{Dy})$. Est-ce que les vecteurs $\vec{AB}$ et $\vec{CD}$ sont colinéaires?
        \Block{set Ax, Ay, Bx, By, Cx, Cy, Dx, Dy= random_str("{a},{b},{c},{d},{e},{f},{g},{h}", conditions = ["{g-e} != 0", "{h-f} != 0","{c-a}/{g-e} == {d-b}/{h-f}"]).split(',')}
        \part Soit $A(\Var{Ax} ; \Var{Ay})$, $B(\Var{Bx} ; \Var{By})$, $C(\Var{Cx} ; \Var{Cy})$ et $D(\Var{Dx} ; \Var{Dy})$. Est-ce que les droites $(AC)$ et $(BD)$ sont colinéaires?
    \end{parts}

    \question
    \begin{parts}
        \part Faire les calculs suivants en \textbf{détaillant les étapes} et en simplifiant les fractions.
        \Block{set A = Expression.random("{a} / {b} * {c}", conditions = ["{b} not in [1, -1]"])}
        \Block{set B = Expression.random("{a} / {b} + {c} / {k*b}", conditions = ["{b} not in [1, -1]"])}
        \Block{set C = Expression.random("{a} / {b} + {c} / {d}", conditions = ["gcd({b},{d})==1"]) }
        \Block{set D = Expression.random("{a} / {b} + {c}", conditions = ["{b} not in [-1,1]"]) }
        \begin{multicols}{2}
            \begin{subparts}
                \subpart $\displaystyle A = \Var{A}$
                \begin{solution}
                    \Block{set Ar = A.simplify()}
                    \begin{eqnarray*}
                        \Var{Ar.explain() | calculus(name = "A", sep = "=", end = "")}
                    \end{eqnarray*}
                \end{solution}
                \subpart $\displaystyle B = \Var{B}$
                \begin{solution}
                    \Block{set Br = B.simplify()}
                    \begin{eqnarray*}
                        \Var{Br.explain() | calculus(name = "A", sep = "=", end = "")}
                    \end{eqnarray*}
                \end{solution}


                \subpart $\displaystyle C = \Var{C}$
                \begin{solution}
                    \Block{set Cr = C.simplify()}
                    \begin{eqnarray*}
                        \Var{Cr.explain() | calculus(name = "A", sep = "=", end = "")}
                    \end{eqnarray*}
                \end{solution}
                \subpart $\displaystyle D = \Var{D}$
                \begin{solution}
                    \Block{set Dr = D.simplify()}
                    \begin{eqnarray*}
                        \Var{Dr.explain() | calculus(name = "A", sep = "=", end = "")}
                    \end{eqnarray*}
                \end{solution}
            \end{subparts}
        \end{multicols}

        \part Mettre les expressions suivantes sur le même dénominateur
        \Block{set A = Expression.random("{a} / ({b}x) * {c}", conditions = ["{b} not in [1, -1]"])}
        \Block{set B = Expression.random("{a} / {b} + ({c}x) / {k*b}", conditions = ["{b} not in [1, -1]"])}
        \Block{set C = Expression.random("({a}x) / {b} + {c} / ({d}x)", conditions = ["gcd({b},{d})==1"]) }
        \Block{set D = Expression.random("{a} / ({b}x) + {c}", conditions = ["{b} not in [-1,1]"]) }
        \begin{multicols}{2}
            \begin{subparts}
                \subpart $\displaystyle A = \Var{A}$
                \subpart $\displaystyle B = \Var{B}$
                \subpart $\displaystyle C = \Var{C}$
                \subpart $\displaystyle D = \Var{D}$
            \end{subparts}
        \end{multicols}
    \end{parts}

\end{questions}

\end{document}

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