206 lines
6.8 KiB
TeX
206 lines
6.8 KiB
TeX
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\documentclass[a4paper,12pt, table]{/media/documents/Cours/Prof/Enseignements/2014-2015/tools/style/classDS}
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\usepackage{/media/documents/Cours/Prof/Enseignements/2014-2015/2014_2015}
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%\geometry{left=10mm,right=10mm, top=10mm, bottom=10mm}
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% Title Page
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\titre{4}
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% \seconde \premiereS \PSTMG \TSTMG
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\classe{\seconde}
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\date{6 mai 2015}
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%\duree{1 heure}
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\sujet{15}
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% DS DSCorr DM DMCorr Corr
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\typedoc{DM}
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%\printanswers
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\begin{document}
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\maketitle
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Vous devez rendre le sujet avec la copie.
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\begin{questions}
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\question
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\begin{parts}
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\part Développer et simplifier les expressions suivantes
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\begin{subparts}
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\begin{multicols}{2}
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\subpart $A = ( 1 x + 7 ) ( 1 - 6 x )$
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\begin{solution}
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\begin{eqnarray*}
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A & = & ( 1 x + 7 ) ( 1 - 6 x ) \\
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A & = & ( x + 7 ) ( 1 - 6 x ) \\
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A & = & ( x + 7 ) ( 1 - 6 x ) \\
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A & = & - 6 x^{ 2 } - 41 x + 7
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\end{eqnarray*}
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\end{solution}
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\subpart $B = ( -3 x + 10 )^{ 2 } - 3$
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\begin{solution}
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\begin{eqnarray*}
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A & = & ( -3 x + 10 )^{ 2 } - 3 \\
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A & = & ( - 3 x + 10 )^{ 2 } - 3 \\
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A & = & ( - 3 x + 10 ) ( - 3 x + 10 ) - 3 \\
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A & = & -3 \times ( -3 ) x^{ 2 } + ( 10 \times ( -3 ) - 3 \times 10 ) x + 10 \times 10 - 3 \\
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A & = & 9 x^{ 2 } + ( -30 - 30 ) x + 100 - 3 \\
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A & = & 9 x^{ 2 } - 60 x + 97
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\end{eqnarray*}
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\end{solution}
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\subpart $C = -4 x + 7 + 4 ( 5 x + 3 )^{ 2 }$
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\begin{solution}
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\begin{eqnarray*}
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A & = & -4 x + 7 + 4 ( 5 x + 3 )^{ 2 } \\
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A & = & - 4 x + 7 + 4 ( 5 x + 3 )^{ 2 } \\
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A & = & - 4 x + 7 + 4 ( 5 x + 3 ) ( 5 x + 3 ) \\
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A & = & - 4 x + 7 + 4 ( 5 \times 5 x^{ 2 } + ( 3 \times 5 + 5 \times 3 ) x + 3 \times 3 ) \\
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A & = & - 4 x + 7 + 4 ( 25 x^{ 2 } + ( 15 + 15 ) x + 9 ) \\
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A & = & - 4 x + 7 + 4 ( 25 x^{ 2 } + 30 x + 9 ) \\
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A & = & - 4 x + 7 + 4 \times 25 x^{ 2 } + 4 \times 30 x + 4 \times 9 \\
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A & = & 100 x^{ 2 } + 116 x + 43
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\end{eqnarray*}
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\end{solution}
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\end{multicols}
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\end{subparts}
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\part Factoriser les expressions suivantes
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\begin{subparts}
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\begin{multicols}{2}
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\subpart $A = 1 x^{ 2 } - x$
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\subpart $B = 100 x^{ 2 } + 9 + 60 x$
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\subpart $C = 9 x^{ 2 } - 16$
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\subpart $D = 81 x^{ 2 } - 144 x + 64$
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\end{multicols}
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\end{subparts}
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\part Résoudre les équations suivantes
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\begin{subparts}
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\begin{multicols}{2}
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\subpart $- 2 x + 2 = 0$
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\subpart $- x + 1 = 6 x + 1$
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\columnbreak
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\subpart $2 x^{ 2 } + 4 x + 1 = 2x^2$
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\subpart $( -2 x + 9 ) ( -8 x - 9 ) = 0$
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\end{multicols}
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\end{subparts}
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\end{parts}
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\question
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\begin{parts}
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\part Soit $A(-9 ; 6)$, $B(-8 ; 7)$, $C(-6 ; -8)$ et $D(-4 ; -6)$. Est-ce que les vecteurs $\vec{AB}$ et $\vec{CD}$ sont colinéaires?
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\part Soit $A(9 ; 10)$, $B(-10 ; -9)$, $C(-4 ; -6)$ et $D(-3 ; 9)$. Est-ce que les vecteurs $\vec{AB}$ et $\vec{CD}$ sont colinéaires?
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\part Soit $A(-7 ; 8)$, $B(-6 ; 7)$, $C(2 ; -7)$ et $D(-7 ; 2)$. Est-ce que les droites $(AC)$ et $(BD)$ sont colinéaires?
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\end{parts}
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\question
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\begin{parts}
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\part Faire les calculs suivants en \textbf{détaillant les étapes} et en simplifiant les fractions.
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\begin{multicols}{2}
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\begin{subparts}
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\subpart $\displaystyle A = \frac{ -5 }{ -6 } \times ( -8 )$
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\begin{solution}
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\begin{eqnarray*}
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A & = & \frac{ -5 }{ -6 } \times ( -8 ) \\
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A & = & \frac{ -5 \times 4 \times ( -2 ) }{ 3 \times ( -2 ) } \\
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A & = & \frac{ -20 \times ( -2 ) }{ -6 } \\
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A & = & \frac{ 40 }{ -6 } \\
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A & = & \frac{ -40 }{ 6 } \\
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A & = & \frac{ -20 \times 2 }{ 3 \times 2 } \\
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A & = & \frac{ -20 }{ 3 }
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\end{eqnarray*}
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\end{solution}
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\subpart $\displaystyle B = \frac{ -3 }{ 6 } + \frac{ 9 }{ -12 }$
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\begin{solution}
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\begin{eqnarray*}
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A & = & \frac{ -3 }{ 6 } + \frac{ 9 }{ -12 } \\
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A & = & \frac{ -3 \times ( -2 ) }{ 6 \times ( -2 ) } + \frac{ 9 \times 1 }{ -12 \times 1 } \\
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A & = & \frac{ 6 }{ -12 } + \frac{ 9 }{ -12 } \\
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A & = & \frac{ 6 + 9 }{ -12 } \\
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A & = & \frac{ 15 }{ -12 } \\
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A & = & \frac{ -15 }{ 12 } \\
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A & = & \frac{ -5 \times 3 }{ 4 \times 3 } \\
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A & = & \frac{ -5 }{ 4 }
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\end{eqnarray*}
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\end{solution}
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\subpart $\displaystyle C = \frac{ -8 }{ 9 } + \frac{ 1 }{ -4 }$
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\begin{solution}
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\begin{eqnarray*}
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A & = & \frac{ -8 }{ 9 } + \frac{ 1 }{ -4 } \\
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A & = & \frac{ -8 \times ( -4 ) }{ 9 \times ( -4 ) } + \frac{ 1 \times 9 }{ -4 \times 9 } \\
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A & = & \frac{ 32 }{ -36 } + \frac{ 9 }{ -36 } \\
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A & = & \frac{ 32 + 9 }{ -36 } \\
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A & = & \frac{ 41 }{ -36 } \\
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A & = & \frac{ -41 }{ 36 }
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\end{eqnarray*}
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\end{solution}
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\subpart $\displaystyle D = \frac{ 2 }{ 6 } - 2$
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\begin{solution}
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\begin{eqnarray*}
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A & = & \frac{ 2 }{ 6 } - 2 \\
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A & = & \frac{ 2 \times 1 }{ 6 \times 1 } + \frac{ -2 \times 6 }{ 1 \times 6 } \\
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A & = & \frac{ 2 }{ 6 } + \frac{ -12 }{ 6 } \\
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A & = & \frac{ 2 - 12 }{ 6 } \\
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A & = & \frac{ -10 }{ 6 } \\
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A & = & \frac{ -5 \times 2 }{ 3 \times 2 } \\
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A & = & \frac{ -5 }{ 3 }
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\end{eqnarray*}
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\end{solution}
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\end{subparts}
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\end{multicols}
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\part Mettre les expressions suivantes sur le même dénominateur
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\begin{multicols}{2}
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\begin{subparts}
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\subpart $\displaystyle A = \frac{ -5 }{ -6 x } \times 1$
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\subpart $\displaystyle B = \frac{ -2 }{ -6 } + \frac{ 5 x }{ 54 }$
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\subpart $\displaystyle C = \frac{ 3 x }{ 3 } + \frac{ -10 }{ 2 x }$
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\subpart $\displaystyle D = \frac{ 9 }{ 10 x } + 8$
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\end{subparts}
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\end{multicols}
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\end{parts}
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\end{questions}
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\end{document}
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%%% Local Variables:
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%%% mode: latex
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%%% TeX-master: "master"
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%%% End:
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