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?

None of these

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]