The line integral $$\int\limits_{{P_1}}^{{P_2}} {\left( {ydx + xdy} \right)} $$ from $${P_1}\left( {{x_1},{y_1}} \right)$$ to $${P_2}\left( {{x_2},{y_2}} \right)$$ along the semi-circle $${P_1}$$ $${P_2}$$ shown in the figure is

If $$A$$ $$(0,4,3),$$ $$B(0,0,0)$$ and $$C(3,0,4)$$ are there points defined in $$x, y, z$$ coordinate system, then which one of the following vectors is perpendicular to both the vectors $$\overrightarrow {AB} $$ and $$\overrightarrow {BC} $$.

The eigen values of the following matrix $$\left[ {\matrix{ {10} & { - 4} \cr {18} & { - 12} \cr } } \right]$$ are

Water is flowing through a horizontal pipe of constant diameter and the flow is laminar. If the diameter of the pipe is increased by $$50\% $$ keeping the volume flow rate constant, then the pressure drop in the pipe due to friction will decrease by