All the roots of the equation `1 lz^(10) + 10iz^(9) + 10iz -11=0` lie

Options

A. inside `|z|=1`
B. one `|z|=1`
C. outside `|z|=1`
D. cannot say

Answer

B

Explanation

`11z^(10)+10iz ^(9)+10iz-11=0`
or `z^(9)(11z+10i)=11-10iz`
`or z^(9)=(11-10iz)/(11z+10i)`
`or |z^(9)|=(|11i-10z|)/(|11z+10i|)`
Now `|11i-10z|^(2)-|11z+10i|^(2)=21(1-|z|^(2))`
For `|z|lt1`
`|11i-10z|^(2)-|11z+10i|^(2)gt0`
`rArr |z^(9)|=(|11i-10z|)/(|11z+10i|)gt1`
i.e. `|z^(9)|gt1` which contradicts with `|z| lt 1`
For `|z|gt 1` we get `|z^(9)|lt1`
`rArr |z| = 1`