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Question

# Let p and q be real numbers such that p≠0,p3≠q and p3≠–q. 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−(p3−2q)x+(p3+q)=0We have, α+β=−p α3+β3=q=(α+β)3−3αβ(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)2−2αβαβ)x+1=0 ⇒x2−p2−2p3+q3pp3+q3px+1=0 ⇒(p3+q)x2−(p3−2q)x+(p3+q)=0

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