omega is a complex cube root of unity. Match | vedclass

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(C) $(A) \ \omega^{1010} + \omega^{2000} = \omega^{3 	imes 336 + 2} + \omega^{3 	imes 666 + 2} = \omega^2 + \omega = -1$ (since $1 + \omega + \omega

Vedclass · Accepted Answer

$A-III, B-IV, C-II, D-V$

Vedclass · Answer

(C) $(A) \ \omega^{1010} + \omega^{2000} = \omega^{3 	imes 336 + 2} + \omega^{3 	imes 666 + 2} = \omega^2 + \omega = -1$ (since $1 + \omega + \omega^2 = 0$).$(B) \ (1 + \omega - \omega^2)(1 - \omega + \omega^2) = (-\omega^2 - \omega^2)(-\omega - \omega) = (-2\omega^2)(-2\omega) = 4\omega^3 = 4$.$(C) \ (2 + \omega^2 + \omega^4)^5 = (2 + \omega^2 + \omega)^5 = (1 + (1 + \omega + \omega^2))^5 = (1 + 0)^5 = 1$.$(D) \ (3 + 5\omega + 3\omega^2)^3 = (3(1 + \omega^2) + 5\omega)^3 = (3(-\omega) + 5\omega)^3 = (2\omega)^3 = 8\omega^3 = 8$.Thus,the correct match is $A-III, B-IV, C-II, D-V$.

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

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