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

Options

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

D

Explanation

|Z|=1
`Z^(n)=1`
`Z^(n)=e^(i2mpi),m in I, m in[0,n-1)`
`Z=e^(i(2mpi)/(n)) " " |Z^(n)|=|1|`
`Z=e^(i(0))" "|Z|^(n)=1`
`Z=e^(i(2pi)/(n))` So, |Z|=1
`Z=e^(i(4 pi)/(n))`
All roots of unity will always lies on arc of circle.
So,we can say, there are infinitely many roots of unity on any given cases are of positive lenth on the unit circle |Z|=1