A) \[({{2}^{99}},\,{{2}^{99}}\sqrt{3})\]
B) \[({{2}^{99}},\,-{{2}^{99}}\sqrt{3})\]
C) \[(-{{2}^{99}},\,{{2}^{99}}\sqrt{3})\]
D) None of these
Correct Answer: C
Solution :
We have, \[{{(1-i\sqrt{3})}^{100}}={{2}^{100}}{{\left( -\frac{1}{2}+\frac{i\sqrt{3}}{2} \right)}^{100}}\] \[={{2}^{100}}{{\omega }^{100}}\] \[={{2}^{100}}\cdot \omega \] \[={{2}^{100}}\left( -\frac{1}{2}+\frac{\sqrt{3}\,i}{2} \right)\] \[=-{{2}^{99}}+{{2}^{99}}\sqrt{3}\,i\] \[\therefore \] \[x+iy={{(1-i\sqrt{3})}^{100}}\] \[=-{{2}^{99}}+{{2}^{99}}\sqrt{3}\,i\] \[\therefore \] \[x=-\,{{2}^{99}},\,y={{2}^{99}}\sqrt{3}\,\] \[\therefore \] \[(x,\,y)=(-{{2}^{99}},\,{{2}^{99}}\sqrt{3})\]
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