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JEE Main & Advanced Mathematics Complex Numbers and Quadratic Equations Question Bank De Moivre's theorem and Roots of unity

  • question_answer
    If \[n\] is a positive integer greater than unity and \[z\] is a complex number satisfying the equation \[{{z}^{n}}={{(z+1)}^{n}}\],  then

    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 \[\operatorname{Re}(z)<0\].


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