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Question

# If α,β,γ are the roots of ax3+bx2+cx+d=0 and ∣∣ ∣ ∣∣αβγβγαγαβ∣∣ ∣ ∣∣=0,α≠β≠γ, then find the equation whose roots are α+β−γ,γ+α−β,β+γ−α

A
ay32by24cy8d=0
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B
ax32bx2+4cx+8d=0
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C
ax32bx2+4cx8d=0
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D
ay32by2+4cy8d=0
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Solution

## The correct option is D ay3−2by2+4cy−8d=0Given α,β,γ are the roots of ax3+bx2+cx+d=0 and ∣∣ ∣ ∣∣αβγβγαγαβ∣∣ ∣ ∣∣=0,α≠β≠γ⇒(α+β+γ)∣∣ ∣∣111βγαγαβ∣∣ ∣∣=0⇒(α+β+γ)(αβ+βγ+γα−α2−β2−γ2)=0⇒−12(α+β+γ)((α−β)2+(β−γ)2+(γ−α)2)=0∴α+β+γ=0 as α≠β≠γy=α+β−γ=−2γ⇒γ=−y2∴ Equation whose roots are α+β−γ,γ+α−β,β+γ−α isa(−y2)3+b(−y2)2+c(−y2)+d=0ay3−2by2+4cy−8d=0Hence, options D.

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