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If z is a complex number satisfying the equation `z^6 +z^3 + 1 = 0`. If this equation has a root `re^(itheta)` with `90^@<0<180^@` then the value of `theta` is

A

`100^(@)`

B

`110^(@)`

C

`160^(@)`

D

`170^(@)`

Text Solution

Verified by Experts

The correct Answer is:
C
`(c )` Given equation is `Z^(6)+Z^(3)+1=0`
Let `Z^(3)=t`
Hence, equation becomes
`t^(2)+t+1=0`
`implies t=omega` or `omega^(2)`
`:. Z^(3)=cos"(2pi)/(3)+isin"(2pi)/(3)=e^((2npi+(2pi)/(3))i)`
`:. Z=e^(((2npi+(2pi)/(3)))/(3)i)`
Putting `n=1`, we get `theta=(8pi)/(9)(=160^(@)) in (90^(@),180^(@))`
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