Q33 / 91 · KVPY 2018 (SB/SX) · Mathematics

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

Solution

We have,

$| z | = 1$

$\Rightarrow z^{n} = 1$

$\Rightarrow z^{n} = e^{i 2 k π}; k \in I$

$\Rightarrow k \in (0, n - 1)$

$\Rightarrow z = e^{\frac{i 2 k π^{2}}{n}}$

$z = 1, e^{\frac{i 2 π}{n}}, e^{\frac{i 4 π}{n}}, \dots, e^{\frac{i 2 (n - 1) π}{n}}$

All roots of unity lie on arc of circle.

$∴$ There are infinitely many roots of unity.

Hence, the correct answer is option: 'D'.