Let α, β, γ be the roots of the cubic x 3+ ax 2+ bx + c =0, which (taken in given order) are in G.P. If α and β are such that |2 1 2 1+α α β 4-β 3-β α+1|=0, then The value of a+b+c equals

Question

Let α, β, &gamma; be the roots of the cubic x 3+ ax 2+ bx + c =0, which (taken in given order) are in G.P. If α and β are such that |2 1 2 1+α α β 4-β 3-β α+1|=0, then The value of a+b+c equals (A) 0 (B) 1 (C) -1 (D) 2

Tardigrade Admin · Accepted Answer

c 1 arrow c 1- c 2, c 2 arrow c 2- c 1, c 3 arrow c 3-2 c 1 |2 1 2 1+α α β 4-β 3-β α+1|=|1 1 2 1 α β 1 3-β α+1|=|1 0 0 1 α-1 β-2 1 2-β α-1| =(α-1)2+(β-2)2=0 ⇒ α=1, β=2, &gamma;=4 ∴ the cubic equation is x 3-7 x 2+14 x -8=0

Q. Let α,β,γ be the roots of the cubic x3+ax2+bx+c=0, which (taken in given order) are in G.P. If α and β are such that ∣∣​21+α4−β​1α3−β​2βα+1​∣∣​=0, then The value of a+b+c equals

0

1

-1

2

Q. Let $α, β, γ$ be the roots of the cubic $x^{3} + a x^{2} + b x + c = 0$ , which (taken in given order) are in G.P. If $α$ and $β$ are such that $∣ ∣ 2 1 + α 4 - β 1 α 3 - β 2 β α + 1 ∣ ∣ = 0$ , then
The value of $a + b + c$ equals