KVPY2018MathematicsComplex Number
On any given arc of positive length on the unit circle | z | = 1 in the complex plane.
Options
- Athere need not be any root of unity
- Bthere lies exactly one root of unity
- Cthere are more than one but finitely many roots of unity
- Dthere are infinitely many roots of unity
Correct answer
D. there are infinitely many roots of unity
Step-by-step solution
We have, | z | = 1 ⇒ z n = 1 ⇒ z n = e i 2 k π ; k ∈ I ⇒ k ∈ 0 , n - 1 ⇒ z = e i 2 k π 2 n z = 1 , e i 2 π n , e i 4 π n , … , e i 2 n - 1 π n All roots of unity lie on arc of circle. ∴ There are infinitely many roots of unity.