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 matrix $$A = \left[ {\matrix{ 2 & 4 \cr 1 & 3 \cr } } \right]$$ and matrix $$B = \left[ {\matrix{ 4 & 6 \cr 5 & 9 \cr } } \right]$$ the transpose of product of these two matrices i.e., $${\left( {AB} \right)^T}$$ is equal to

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

Cold water flowing at $$0.1$$ $$kg/s$$ is heated from $${20^ \circ }C$$ to $${70^ \circ }C$$ in a counter-flow type heat exchanger by a hot water stream flowing at $$0.1$$ $$kg/s$$ and entering at $${90^ \circ }C$$. The specific heat of water is $$4200$$ $$J$$($$kg$$ $$K$$) and density is $$1000$$ $$kg/{m^3}.$$ If the overall heat transfer coefficient $$U$$ for the heat exchanger is $$2000$$ $$W/\,\,\left( {{m^2}\,\,K} \right),$$ the required heat exchange area (in $${{m^2}}$$) is