Prove that the determinant \(\begin{vmatrix} x &sin\theta &cos\theta \\ -sin\theta&-x &1 \\ cos\theta&1 &x \end{vmatrix}\) is independent of θ.
\(\Delta = \begin{vmatrix} x &sin\theta &cos\theta \\ -sin\theta&-x &1 \\ cos\theta&1 &x \end{vmatrix}\)
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).