Using properties of determinants, prove that:
\(\begin{vmatrix} \alpha &\alpha^2 &\beta+\gamma \\ \beta&\beta^2 &\gamma+\alpha \\ \gamma&\gamma^2 &\alpha+\beta \end{vmatrix}\)=(\(\beta-\gamma\))( \(\gamma-\alpha\))(\(\alpha-\beta\))(\(\alpha+\beta+\gamma\))
Δ=\(\begin{vmatrix} \alpha &\alpha^2 &\beta+\gamma \\ \beta&\beta^2 &\gamma+\alpha \\ \gamma&\gamma^2 &\alpha+\beta \end{vmatrix}\)
Applying R2\(\rightarrow\)R2-R1 and R3\(\rightarrow\)R3-R1,we have
=\(\begin{vmatrix} \alpha &\alpha^2 &\beta+\gamma \\ \beta+\alpha&\beta^2-\alpha^2 &\alpha+\beta \\ \gamma-\alpha&\gamma^2-\alpha^2 &\alpha-\gamma \end{vmatrix}\)
Applying R3\(\rightarrow\)R3-R2, we have:
Δ=(β-α)(γ-α)\(\begin{vmatrix} \alpha &\alpha^2 &\beta+\gamma \\ 1&\beta+\alpha &-1 \\ 0&\gamma-\beta &0 \end{vmatrix}\)
Expanding along R3,we have:
Δ=(β-α)(γ-α)[-(γ-β)(-α-β-γ)]
=(β-α)(γ-α)(γ-β)(α+β+γ)
=(β-γ)( γ-α)(α-β)(α+β+γ)
Hence,the given result is proved.
A settling chamber is used for the removal of discrete particulate matter from air with the following conditions. Horizontal velocity of air = 0.2 m/s; Temperature of air stream = 77°C; Specific gravity of particle to be removed = 2.65; Chamber length = 12 m; Chamber height = 2 m; Viscosity of air at 77°C = 2.1 × 10\(^{-5}\) kg/m·s; Acceleration due to gravity (g) = 9.81 m/s²; Density of air at 77°C = 1.0 kg/m³; Assume the density of water as 1000 kg/m³ and Laminar condition exists in the chamber.
The minimum size of particle that will be removed with 100% efficiency in the settling chamber (in $\mu$m is .......... (round off to one decimal place).