TS EAMCET · Maths · Complex Number

If $α, β$ are the roots of the equation $1 + x + x^{2} = 0,$ then $(2 - α) (2 - β) (2 - α^{10}) (2 - α^{20}) =$

A $36$
B $64$
C $49$
D $81$

Verified Solution

Answer & Solution

Correct Answer

(C) $49$

Step-by-step Solution

Detailed explanation

Given α, β are the roots of the equation x2+x+1=0. The given equation is of the form ω2+ω+1=0, which is the condition for ω & ω2 to be the imaginary cube roots of unity. We know that, if ω is the cube root if unity, then…

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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If α, β are the roots of the equation 1+x+x2=0, then (2-α)(2-β)2-α102-α20=

Step-by-step Solution

If $α, β$ are the roots of the equation $1 + x + x^{2} = 0,$ then $(2 - α) (2 - β) (2 - α^{10}) (2 - α^{20}) =$