205 lines
6.7 KiB
TeX
205 lines
6.7 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{9}
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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 $A = ( 1 x - 3 ) ( 1 - 6 x )$
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\begin{solution}
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\begin{eqnarray*}
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A & = & ( 1 x - 3 ) ( 1 - 6 x ) \\
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A & = & ( x - 3 ) ( 1 - 6 x ) \\
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A & = & ( x - 3 ) ( 1 - 6 x ) \\
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A & = & x - 3 ( 1 - 6 x ) \\
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A & = & - 6 x^{ 2 } + 19 x - 3
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\end{eqnarray*}
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\end{solution}
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\subpart $B = ( -3 x + 4 )^{ 2 } + 5$
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\begin{solution}
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\begin{eqnarray*}
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A & = & ( -3 x + 4 )^{ 2 } + 5 \\
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A & = & ( - 3 x + 4 )^{ 2 } + 5 \\
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A & = & ( - 3 x + 4 ) ( - 3 x + 4 ) + 5 \\
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A & = & -3 \times ( -3 ) x^{ 2 } + ( 4 \times ( -3 ) - 3 \times 4 ) x + 4 \times 4 + 5 \\
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A & = & 9 x^{ 2 } + ( -12 - 12 ) x + 16 + 5 \\
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A & = & 9 x^{ 2 } - 24 x + 21
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\end{eqnarray*}
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\end{solution}
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\subpart $C = -3 x + 10 + 4 ( 8 x + 5 )^{ 2 }$
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\begin{solution}
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\begin{eqnarray*}
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A & = & -3 x + 10 + 4 ( 8 x + 5 )^{ 2 } \\
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A & = & - 3 x + 10 + 4 ( 8 x + 5 )^{ 2 } \\
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A & = & - 3 x + 10 + 4 ( 8 x + 5 ) ( 8 x + 5 ) \\
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A & = & - 3 x + 10 + 4 ( 8 \times 8 x^{ 2 } + ( 5 \times 8 + 8 \times 5 ) x + 5 \times 5 ) \\
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A & = & - 3 x + 10 + 4 ( 64 x^{ 2 } + ( 40 + 40 ) x + 25 ) \\
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A & = & - 3 x + 10 + 4 ( 64 x^{ 2 } + 80 x + 25 ) \\
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A & = & - 3 x + 10 + 4 \times 64 x^{ 2 } + 4 \times 80 x + 4 \times 25 \\
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A & = & 256 x^{ 2 } + 317 x + 110
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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 $A = -3 x^{ 2 } - x$
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\subpart $B = 81 x^{ 2 } + 9 + 54 x$
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\subpart $C = 64 x^{ 2 } - 1$
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\subpart $D = 36 x^{ 2 } - 120 x + 100$
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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 $- 4 x + 10 = 0$
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\subpart $6 x + 2 = - 2 x - 7$
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\columnbreak
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\subpart $3 x^{ 2 } + 8 x + 5 = 3x^2$
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\subpart $( 4 x + 1 ) ( 8 x - 9 ) = 0$
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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 $A(2 ; -3)$, $B(9 ; -10)$, $C(-1 ; -8)$ et $D(-3 ; -6)$. Est-ce que les vecteurs $\vec{AB}$ et $\vec{CD}$ sont colinéaires?
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\part Soit $A(3 ; -8)$, $B(-10 ; 8)$, $C(-9 ; -7)$ et $D(-1 ; 10)$. Est-ce que les vecteurs $\vec{AB}$ et $\vec{CD}$ sont colinéaires?
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\part Soit $A(1 ; -1)$, $B(-5 ; -4)$, $C(3 ; 3)$ et $D(9 ; 6)$. 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 $\displaystyle A = \frac{ 5 }{ -9 } \times ( -7 )$
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\begin{solution}
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\begin{eqnarray*}
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A & = & \frac{ 5 }{ -9 } \times ( -7 ) \\
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A & = & \frac{ 5 \times 7 \times ( -1 ) }{ 9 \times ( -1 ) } \\
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A & = & \frac{ 35 \times ( -1 ) }{ -9 } \\
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A & = & \frac{ -35 }{ -9 } \\
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A & = & \frac{ 35 }{ 9 }
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\end{eqnarray*}
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\end{solution}
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\subpart $\displaystyle B = \frac{ -3 }{ -9 } + \frac{ 6 }{ -45 }$
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\begin{solution}
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\begin{eqnarray*}
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A & = & \frac{ -3 }{ -9 } + \frac{ 6 }{ -45 } \\
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A & = & \frac{ -3 \times 5 }{ -9 \times 5 } + \frac{ 6 \times 1 }{ -45 \times 1 } \\
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A & = & \frac{ -15 }{ -45 } + \frac{ 6 }{ -45 } \\
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A & = & \frac{ -15 + 6 }{ -45 } \\
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A & = & \frac{ -9 }{ -45 } \\
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A & = & \frac{ 9 }{ 45 } \\
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A & = & \frac{ 1 \times 9 }{ 5 \times 9 } \\
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A & = & \frac{ 1 }{ 5 }
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\end{eqnarray*}
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\end{solution}
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\subpart $\displaystyle C = \frac{ 7 }{ -2 } + \frac{ 5 }{ 5 }$
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\begin{solution}
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\begin{eqnarray*}
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A & = & \frac{ 7 }{ -2 } + \frac{ 5 }{ 5 } \\
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A & = & \frac{ 7 \times 5 }{ -2 \times 5 } + \frac{ 5 \times ( -2 ) }{ 5 \times ( -2 ) } \\
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A & = & \frac{ 35 }{ -10 } + \frac{ -10 }{ -10 } \\
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A & = & \frac{ 35 - 10 }{ -10 } \\
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A & = & \frac{ 25 }{ -10 } \\
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A & = & \frac{ -25 }{ 10 } \\
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A & = & \frac{ -5 \times 5 }{ 2 \times 5 } \\
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A & = & \frac{ -5 }{ 2 }
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\end{eqnarray*}
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\end{solution}
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\subpart $\displaystyle D = \frac{ -4 }{ 9 } - 7$
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\begin{solution}
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\begin{eqnarray*}
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A & = & \frac{ -4 }{ 9 } - 7 \\
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A & = & \frac{ -4 \times 1 }{ 9 \times 1 } + \frac{ -7 \times 9 }{ 1 \times 9 } \\
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A & = & \frac{ -4 }{ 9 } + \frac{ -63 }{ 9 } \\
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A & = & \frac{ -4 - 63 }{ 9 } \\
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A & = & \frac{ -67 }{ 9 }
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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 $\displaystyle A = \frac{ -4 }{ -10 x } \times 2$
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\subpart $\displaystyle B = \frac{ 3 }{ 6 } + \frac{ -1 x }{ 42 }$
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\subpart $\displaystyle C = \frac{ 4 x }{ 5 } + \frac{ 9 }{ -3 x }$
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\subpart $\displaystyle D = \frac{ 4 }{ 10 x } - 10$
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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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