Source Latex
de la correction du devoir
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pdfauthor={Yoann Morel},
pdfsubject={Correction du devoir de math�matiques: nombres complexes},
pdftitle={Corrig� du devoir de math�matiques: nombres complexes},
pdfkeywords={d�riv�e, nombre d�riv�, tangente, sens de variation,
�tude de fonction, STI2D,
STI, premi�re, Math�matiques}
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\lfoot{Y. Morel - \url{xymaths.free.fr/Lycee/1STI/}}
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\rfoot{Corrig� du devoir de math�matiques - 1STI2D - \thepage/\pageref{LastPage}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}
\ct{\bf\LARGE{Corrig� du devoir de math\'ematiques}}
\vspq
\bgex
Soit $z=2-3i$, alors:
$\Re e(z)=2\quad;\
\Im m(z)=-3\quad;\
|z|=\sqrt{2^2+3^2}=\sqrt{13}
$.
\enex
\bgex
\[
z_1
=\dfrac{2+i}{3-2i}
=\dfrac{(2+i)(3+2i)}{(3-2i)(3+2i)}
=\dfrac{4+7i}{13}
=\dfrac{4}{13}+i\dfrac{7}{13}
\]
\[
z_2
=\dfrac{-2+3i}{-1+i}
=\dfrac{(-2+3i)(-1-i)}{(-1+i)(-1-i)}
=\dfrac{5-i}{2}
=\dfrac52-i\dfrac12
\]
\enex
\bgex
Calculer le module des nombres complexes suivants:
\[
|z_1|=\left|\dfrac12+2i\right|
=\sqrt{\lp\dfrac12\rp^2+2^2}
=\sqrt{\dfrac14+4}
=\sqrt{\dfrac{17}{4}}
=\dfrac{\sqrt{17}}{2}
\]
\[
|z_2|=|i(1-i)|=|i|\tm|1-i|=1\tm\sqrt{2}=\sqrt{2}
\quad;\quad
|z_3|=\left|\dfrac{1+i}{-3-4i}\right|
=\dfrac{|1+i|}{|-3-4i|}
=\dfrac{\sqrt{2}}{\sqrt{25}}
=\dfrac{\sqrt{2}}{5}
\]
\enex
\bgex
\bgen
\item $A$, $B$ et $C$ sont les points d'affixes respectives
$z_A=1-i$, $z_B=-2+i$ et $z_C=3+2i$.
\[
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\multido{\i=-5+1}{11}{
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}
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%
\rput[c](1,-1){\bf\large$\tm$}\rput(1.25,-1.25){$A$}
\rput[c](-2,1){\bf\large$\tm$}\rput(-2.25,1.25){$B$}
\rput[c](3,2){\bf\large$\tm$}\rput(3.25,2.25){$C$}
\rput[c](-0.5,0){\bf\large$\tm$}\rput(-0.75,0.25){$I$}
% et la m�diatrice
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\]
\item
$AB=|z_B-z_A|=|-2+i-(1-i)|=|-3+2i|=\sqrt{13}$
$BC=|z_C-z_B|=|3+2i-(-2+i)|=|5+i|=\sqrt{26}$
\item Soit $D(z_D)$, alors $ABCD$ est un parall�logramme si et
seulement si
\[\bgar{ll}
\V{AB}=\V{DC}
&\iff
z_{\V{AB}}=z_{\V{DC}}
\iff
z_B-z_A=z_C-z_D \\[0.3cm]
&\iff
z_D
=z_C-z_B+z_A=3+2i-(-2+i)+1-i
=6-2i
\enar\]
\item L'affixe $z_I$ du milieu $I$ de $[AB]$ est
$z_I=\dfrac{z_A+z_B}{2}=-\dfrac12$.
\item On a: $|z-1+i|=|z+2-i|\iff |z-z_A|=|z-z_B| \iff AM=BM$.
L'ensemble $\mathcal{E}$ des points $M$ recherch�s est donc la m�diatrice de
$[AB]$.
\enen
\enex
%\bgex
%Ecrire sous forme trigonom�trique le nombre complexe
%$z=1-i\sqrt{3}$.
%\enex
\label{LastPage}
\end{document}
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