A) \[\operatorname{Re}(z)<0\]
B) \[\operatorname{Re}(z)>0\]
C) \[\operatorname{Re}(z)=0\]
D) None of these
Correct Answer: A
Solution :
We have \[{{z}^{n}}={{(1+z)}^{n}}\,\,\,\Rightarrow {{\left( \frac{z}{z+1} \right)}^{n}}=1\] Þ \[\frac{z}{z+1}={{1}^{1/n}}\]Þ \[\frac{z}{z+1}\]is a \[n\]th root of unity Þ \[\left| \frac{z}{z+1} \right|=1\]Þ\[\frac{|z|}{|z+1|}=1\]Þ \[|z|\,=\,|z+1|\] Þ \[x+\frac{1}{2}=0\]Þ \[x=\frac{-1}{2}\]Þ \[\operatorname{Re}(z)<0\].You need to login to perform this action.
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