The total derivative of the function $$'xy'$$ is

The value of $${x_3}$$ obtained by solving the following system of linear equations is $$${x_1} + 2{x_2} - 2{x_3} = 4$$$ $$$2{x_1} + {x_2} + {x_3} = - 2$$$ $$$ - {x_1} + {x_2} - {x_3} = 2$$$

Consider steady flow of water in a situation where two pipe lines (pipe $$1$$ and pipe $$2$$) combine into a single pipe line $$(pipe-3)$$ as shown in the figure. The cross-sectional areas of all three pipelines are constant. The following data is given :

Assuming the water properties and the velocities to be uniform across the cross section of the inlets and the outlet, the exit velocity (in $$m/s$$) in pipe $$3$$ is

The pressure drop for laminar flow of a liquid in a smooth pipe at normal temperature and pressure is