If α, β, γ are the roots of x3 + px2 + q = 0, where q≠0, then Δ = \(\begin{vmatrix}\frac{1}{α} & \frac{1}{β} & \frac{1}{γ} \\\frac{1}{β} & \frac{1}{γ} & \frac{1}{α} \\ \frac{1}{γ} & \frac{1}{α} & \frac{1}{β} \end{vmatrix}\) equals to which of the following?
Answer & explanation
Correct answer: option 4
We have βγ + γα + αβ = 0
Δ = \(\frac{1}{α^2β^2γ^2}\)\(\begin{vmatrix}βγ & γα & αβ \\γα & αβ & βγ\\αβ & βγ & γα \end{vmatrix}\)
Δ = \(\frac{1}{α^2β^2γ^2}\)\(\begin{vmatrix}βγ+γα+αβ & γα & αβ \\γα+αβ+βγ & αβ & βγ\\αβ+βγ+γα & βγ & γα \end{vmatrix}\)
(using C1 → C1 + C2 + C3)
Δ = \(\frac{1}{α^2β^2γ^2}\)\(\begin{vmatrix}0 & γα & αβ \\0 & αβ & βγ\\0 & βγ & γα \end{vmatrix}\) = 0 [all zero property]