KVPY SX SB Exam 2018 Question Paper

Question : 2 of 120

Marks: +1, -0

On any given arc of positive length on the unit circle |z| = 1 in the complex plane.

there need not be any root of unity
there lies exactly one root of unity
there are more than one but finitely many roots of unity
there are infinitely many roots of unity

Solution:

\n^\th

root of unity

=e^{i{2π}/n},e^{i{4π}/n}

All these roots lie on the unit circle |z| = 1 and however small the arc of this circle, we may find infinite n for which the roots of unity lie on circle

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