Prove that\(\begin{vmatrix} a^2&bc &ac+c^2 \\ a^2+ab&b^2 &ac\\ ab&b^2+bc &c^2 \end{vmatrix}=4a^2b^2c^2\)
\(\Delta = \begin{vmatrix} a^2&bc &ac+c^2 \\ a^2+ab&b^2 &ac\\ ab&b^2+bc &c^2 \end{vmatrix}\)
Taking out common factors a,b and c from C1,C2 and C3 we have,
\(\Delta=abc\begin{vmatrix} a&c &a+c \\ a+b&b &a \\ b&b+c &c \end{vmatrix}\)
Applying R2\(\rightarrow\)R2-R1 and R3\(\rightarrow\)R3-R1,we have:
\(\Delta=abc\begin{vmatrix} a&c &a+c \\ b&b-c &-c \\ b-a&b &-a \end{vmatrix}\)
Applying R2\(\rightarrow\)R2+R1,we heve:
\(\Delta=abc\begin{vmatrix} a&c &a+c \\ a+b&b &a \\ 2b&2b &0 \end{vmatrix}\)
Applying R3\(\rightarrow\)R3+R2,we heve:
\(\Delta=abc\begin{vmatrix} a&c &a+c \\ a+b&b &a \\ 2b&2b &0 \end{vmatrix}\)
\(\Delta=2a^2bc\begin{vmatrix} a&c &a+c \\ a+b&b &a \\ 2b&2b &0 \end{vmatrix}\)
ApplyingC2\(\rightarrow\)C2-C1, we have:
\(\Delta=2a^2bc\begin{vmatrix} a&c-a &a+c \\ a-b&-a &a \\ 0&0 &0 \end{vmatrix}\)
Expanding along R3,we have:
Δ=2ab2c[a(c-a)+a(a+c)]
=2ab2c[ac-a2+a2+ac]
=2ab2c(2ac)
=4a2b2c2
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).