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Question

Let p and q be real numbers such that p0, p32q and p3q. If α and β are non zero complex numbers satisfying α+β=p and α3+β3=q, then a quadratic equation having αβ and βα as its roots is

A
(p3+q)x2(p3+2q)x+(p3+q)=0
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B
(p3+q)x2(p32q)x+(p3+q)=0
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C
(p3q)x2(p32q)x+(p3+q)=0
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D
(p3q)x2(p3+2q)x+(p3q)=0
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Solution

The correct option is B (p3+q)x2(p32q)x+(p3+q)=0
Given that α+β=p and α3+β3=q(α+β)33αβ(α+β)=qp33αβ(p)=qαβ=p3+q3p
Now for required quadratic equation,
sum of roots =αβ+βα=α2+β2αβ=(α+β)22αβαβ=p22(p3+q3p)p3+q3p=3p32p32qp3+q=p32qp3+q
and product of roots =αβ.βα=1
Required equation is x2(p32qp3+q)x+1or (p3+q)x2(p32q)x+(p3+q)=0

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