Question

2. On any given arc of positive length on the unit circle
|z| = 1 in the complex plane:
A. there need not be any root of unity
B. there lies exactly one root of unity
C. there are more than one but finitely many roots of
unity
D. there are infinitely many roots of unity​

Answer 1

answer : option (D) there are infinitely many roots of unity.

explanation : as it is given that, |z| = 1

⇒ |z|ⁿ = 1ⁿ ⇔ zⁿ = 1

we can write , 1 = cos(2π) + isin(2π)

zⁿ = cos(2π) + isin(2π)

⇒z = [cos(2π) + isin(2π) ]^{1/n}

⇒z = [cos(2π/n) +isin(2π/n) ]

⇒z = $e^{i\frac{2k\pi}{n}}$ where k ∈ [0, (n - 1)]

hence, z = 1, $e^{i\frac{2\pi}{n}}$, $e^{i\frac{4\pi}{n}}$, $e^{i\frac{6\pi}{n}}$.......$e^{i\frac{2(n-1)\pi}{n}}$

all roots of unity lie on arc of circle .

therefore, there are infinitely many roots of unity.

hence, option (d) is correct choice.