A $$2m \times 2m$$ square operating in a vertical wall is covered with a metalic plate of the same dimensions as shown in the figure below. Consider the acceleration due to gravity to be $$10.0\,\,m/{s^2}.$$ The force (in kN) exerted by water on the plate is _____________.

A long slender metallic rod of length $$L$$ is used as a fin. As shown in the figure below, the left end of the fin is kept at a constant temperature $${t_b}.$$ The fin loses heat by convection to the atmosphere which is at a temperature ta $${t_a}\,\,\left( {{t_a} < {t_b}} \right).$$ Four options of temperature profiles are shown. Identify the correct option.

The throughput rate of a production system is $$20$$ units per hour. The average flow time is $$30$$ minutes and the cycle time is $$3$$ minutes. The average inventory (in units) in the system is

Consider a network with nodes $$1, 2, 3, 4, 5$$ and $$6.$$ The nodes are connected with directed arcs as shown in the table below. The respective costs (in $$INR$$) incurred while traversing the directed arcs are also mentioned.

The second shortest path from node $$1$$ to node $$6$$ (i.e. the path that has the second least total cost and does not use any part of the shortest path) has a total cost (in $$INR$$) of