AP EAMCET · Maths · Complex Number

If $α$ and $β$ are the roots of the quadratic equation $x^{2} + x + 1 = 0$ , then the equation whose roots are $α^{2021}, β^{2021}$ is given by ______

A $x^{2} - x + 1 = 0$
B $x^{2} + x - 1 = 0$
C $x^{2} - x - 1 = 0$
D $x^{2} + x + 1 = 0$

Verified Solution

Answer & Solution

Correct Answer

(D) $x^{2} + x + 1 = 0$

Step-by-step Solution

Detailed explanation

For quadratic equation, x2+x+1=0 ⇒x=-1±1-42=-1±3i2 ⇒α=-1+3i2, β=-1-3i2 By using cube roots of unity, we can say α=ω & β=ω2 So, α2021=ω2021=ω3673.ω2=ω2, as ω3=1…

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 and β are the roots of the quadratic equation x2+x+1=0, then the equation whose roots are α2021,β2021 is given by ______

Step-by-step Solution

If $α$ and $β$ are the roots of the quadratic equation $x^{2} + x + 1 = 0$ , then the equation whose roots are $α^{2021}, β^{2021}$ is given by ______