$\mathrm{\omega }$ is a complex cube root of unity. Match the items of List-I to the items of List-II

	List I		List II
A	${\mathrm{\omega }}^{1010} + {\mathrm{\omega }}^{2020}$	I)	0
B	$(1 + \mathrm{\omega } - {\mathrm{\omega }}^2)(1 - \mathrm{\omega } + {\mathrm{\omega }}^2)$	II)	1
C	${(2 + {\mathrm{\omega }}^2 + {\mathrm{w}}^4)}^5$	III)	-1
D	$\hspace{0.17em}{(3 + 5\mathrm{\omega } + 3{\mathrm{\omega }}^2)}^3$	IV)	4
		V)	8

 

The correct match is

A)   $\begin{array}{rcl} & & {\mathrm{w}}^{1010}+{\mathrm{w}}^{2020}\\ & & {\left({\mathrm{w}}^3\right)}^{336}·{\mathrm{w}}^2+{\left({\mathrm{w}}^3\right)}^{673}\mathrm{w}\\ & =& {\mathrm{w}}^2+\mathrm{w}\\ & =& -1\end{array}$

B)   $\begin{array}{rcl} & & \left(1+\mathrm{w}-{\mathrm{w}}^2\right) \left(1-\mathrm{w}+{\mathrm{w}}^2\right)\\ & & \left(-{\mathrm{w}}^2-{\mathrm{w}}^2\right) \left(-\mathrm{w}-\mathrm{w}\right)\\ & =& \left(-2{\mathrm{w}}^2\right) \left(-2\mathrm{w}\right)\\ & =& 4{\mathrm{w}}^3\\ & =& 4\end{array}$

C)   $\begin{array}{rcl} & & {\left(2+{\mathrm{w}}^2+{\mathrm{w}}^4\right)}^5\\ & =& {\left(2+\mathrm{w}+{\mathrm{w}}^2\right)}^5\\ & =& {\left(1+\left(1+\mathrm{w}+{\mathrm{w}}^2\right)\right)}^5\\ & =& {\left(1+0\right)}^5\\ & =& 1\end{array}$

D)   $\begin{array}{rcl} & & {\left(3+5\mathrm{w}+3{\mathrm{w}}^2\right)}^3\\ & & {\left(3\left(1+\mathrm{w}+{\mathrm{w}}^2\right)+2\mathrm{w}\right)}^3\\ & & {\left(3\left(0\right)+2\mathrm{w}\right)}^3\\ & =& 2^3 {\mathrm{w}}^3\\ & =& 8\left(1\right)=8\\ \mathrm{A} & \to & \mathrm{III} \mathrm{B}\to \mathrm{IV} \mathrm{C}\to \mathrm{II} \mathrm{D}\to \mathrm{V}\end{array}$

Option (3) correct.