If `z=(1)/(2)(sqrt(3)-i)`, then the least possible integral value of `m` such that `(z^(101)+i^(109))^(106)=z^(m+1)` is

Options

A. `11`
B. `7`
C. `8`
D. `9`

Answer

D

Explanation

`(d)` `z=(1)/(2)(sqrt(3)-i)=iw^(2)`, where `w` is imaginary cube root of unity.
`implies(z^(101)+i^(109))^(106)`
`=((iomega^(2))^(101)+i^(109))^(106)`
`=(iomega^(2)+i)^(106)`
`=(-iomega)^(106)`
`=-omega^(2)`
Also `z^(m+1)=(omega^(2)i)^(m+1)`
`implies m=9` satisfies.