A separately excited $$dc$$ machine is coupled to a $$50Hz,$$ three-phase, 4-pole induction machine as shown in the figure. The dc machine is energized first and the machines rotate at $$1600$$ $$rpm.$$ Subsequently the induction machine is also connected to a $$50Hz,$$ three-phase source, the phase sequence being consistent with the direction of rotation. In steady state

Divergence of the three-dimentional radial vector field $$\overrightarrow F$$ is

For the set of equations $$${x_1} + 2{x_2} + {x_3} + 4{x_4} = 2,$$$ $$$3{x_1} + 6{x_2} + 3{x_3} + 12{x_4} = 6.$$$
The following statement is true

An eigen vector of $$p = \left[ {\matrix{ 1 & 1 & 0 \cr 0 & 2 & 2 \cr 0 & 0 & 3 \cr } } \right]$$ is