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Question

Let p and q be real numbers such that p0,p3q 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 βαas its roots is

A
(p3+q)x2(p3+2q)x+(p3+q)=0
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B
(p3q)x2(5p32q)x+(p3q)=0
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C
(p3q)x2(5p3+2q)x+(p3q)=0
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D
(p3+q)x2(p32q)x+(p3+q)=0
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Solution

The correct option is D (p3+q)x2(p32q)x+(p3+q)=0
We have,
α+β=p
α3+β3=q=(α+β)33αβ(α+β)=p3+3p(αβ)
αβ=p3+q3p
The quadratic equation with αβ and βα as roots is
x2(αβ+βα)x+αβ.βα=0
x2((α+β)22αβαβ)x+1=0
x2p22p3+q3pp3+q3px+1=0
(p3+q)x2(p32q)x+(p3+q)=0

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