If omega is the complex cube root of unity,th | vedclass

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(D) We know that $1+\omega+\omega^2 = 0$,which implies $1+\omega^2 = -\omega$ and $\omega^3 = 1$.Given expression: $E = (3+5\omega+3\omega^2)^2 + (3+3

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-$4$

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(D) We know that $1+\omega+\omega^2 = 0$,which implies $1+\omega^2 = -\omega$ and $\omega^3 = 1$.Given expression: $E = (3+5\omega+3\omega^2)^2 + (3+3\omega+5\omega^2)^2$Rearranging terms using $3(1+\omega+\omega^2) = 0$:$E = (3(1+\omega+\omega^2) + 2\omega)^2 + (3(1+\omega+\omega^2) + 2\omega^2)^2$$E = (0 + 2\omega)^2 + (0 + 2\omega^2)^2$$E = 4\omega^2 + 4\omega^4$Since $\omega^3 = 1$,we have $\omega^4 = \omega$.$E = 4\omega^2 + 4\omega = 4(\omega^2 + \omega)$Since $1+\omega+\omega^2 = 0$,we have $\omega^2 + \omega = -1$.$E = 4(-1) = -4$.

List-$I$ (Expression)	List-$II$ (Value)
$A$. $\omega^{1010} + \omega^{2000}$	$I$. $0$
$B$. $(1 + \omega - \omega^2)(1 - \omega + \omega^2)$	$II$. $1$
$C$. $(2 + \omega^2 + \omega^4)^5$	$III$. $-1$
$D$. $(3 + 5\omega + 3\omega^2)^3$	$IV$. $4$
	$V$. $8$

If $\omega$ is the complex cube root of unity,then the value of $\left(3+5 \omega+3 \omega^2\right)^2+\left(3+3 \omega+5 \omega^2\right)^2$ is:

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