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Question

If α,β,γ are non zero roots of x3+px2+qx+r=0, then the equation whose roots are α(β+γ),β(γ+α),γ(α+β)

A
x32qx2+(pr+q2)x+(r2pqr)=0
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B
x32qx2+(pr+q2)x+(r2+pqr)=0
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C
x3+2qx2+(pr+q2)x+(r2pqr)=0
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D
x3+2qx2+(pr+q2)x+(r2+pqr)=0
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Solution

The correct option is A x32qx2+(pr+q2)x+(r2pqr)=0
Given equation:
x3+px2+qx+r=0
We know that,
α+β+γ=pαβ+βγ+αγ=qαβγ=rr0 (α,β,γ0)
Let
y=α(β+γ) =qαβγα =q(r)α
yq=rαα=ryq
which is a root of given equation.

Therefore, the equation whose roots are α(β+γ),β(γ+α),γ(α+β) is,

r3(yq)3+p(ryq)2+q(ryq)+r=0
r3+pr2(yq)+qr(yq)2+r(yq)3=0
y32qy2+(pr+q2)y+(r2pqr)=0(r0)

Hence, the transformed equation is
x32qx2+(pr+q2)x+(r2pqr)=0

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