If $1, W, W^{2}$ are the cubs roots of unity then $w^{2} (1 + W)^{3} - (1 + W^{2}) W =$

1

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i

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0

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Solution

The correct option is D $0$
Given, $1, ω, ω^{2}$ are the cube roots of unity.
$∴ 1 + ω + ω^{2} = 0 - - - - - (1)$
Now, $ω^{2} {(1 + ω)}^{3} - (1 + ω^{2}) ω$
$= ω^{2} {(- ω^{2})}^{3} - (- ω) ω [f r o m (1)]$
$- ω^{2} . ω^{6} + ω^{2}$
$= - ω^{2} . {(ω^{3})}^{2} + ω^{2}$
$= - ω^{2} .1 + ω^{2} [∵ ω^{3} = 1] = - ω^{2} + ω^{2} = 0$ Ans.

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