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A root of unity is a complex number that...

A root of unity is a complex number that is a solution to the equation, `z^n=1` for some positive integer nNumber of roots of unity that are also the roots of the equation `z^2+az+b=0`, for some integer a and b is

A

`6`

B

`8`

C

`9`

D

`10`

Text Solution

Verified by Experts

The correct Answer is:
B
`(b)` Let `alpha(|alpha|=1)` is a root of unity that is also a root of the equation `z^(2)+az+b=0`
`:. Alpha+baralpha=-a`
`:. |a|=|alpha+baralpha| le |alpha|+|baralpha|=2`
and `b=alphabaralpha=1`
Hence, we must check those equations for which `-2 le a le 2` and `b=1`.
i.e., `z^(2)+2z+1=0` , `z^(2)+z+1=0` `z^(2)+1=0`
`z^(2)-2z+1=0` , `z^(2)-z+1=0`
Hence roots are `+-1`, `+-i` , `(-1+-sqrt(-3))/(2)`, `(1+-sqrt(-3))/(2)`
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