TS EAMCET · Maths · Complex Number

$ω$ is a complex cube root of unity. Match the items of List-I to the items of List-II

List-I List-II

A) $ω^{1010} + ω^{2000}$ I) $0$

B) $(1 + ω - ω^{2}) (1 - ω + ω^{2})$ II) $1$

C) ${(2 + ω^{2} + ω^{4})}^{5}$ III) $- 1$

D) ${(3 + 5 ω + 3 ω^{2})}^{3}$ IV) $4$

V) $8$

The correct match is

A

A B C D

III IV I V
B

A B C D

I IV II V
C

A B C D

III IV II V
D

A B C D

III I II IV

Verified Solution

Answer & Solution

Correct Answer

(C)

A	B	C	D
III	IV	II	V

Step-by-step Solution

Detailed explanation

A ω1010+ω2000=ω3×336+2+ω3×666+2 =ω+ω2 ∵ω3n+2=ω2ω3n+1=ω =-1 B 1+ω-ω21-ω+ω2…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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

ω is a complex cube root of unity. Match the items of List-I to the items of List-II List-I List-II A) ω1010+ω2000 I) 0 B) 1+ω-ω21-ω+ω2 II) 1 C) 2+ω2+ω45 III) -1 D) 3+5ω+3ω23 IV) 4 V) 8 The correct match is

Step-by-step Solution