If $$f\left( z \right) = \left( {{x^2} + a{y^2}} \right) + ibxy$$ is a complex analytic function of $$z=x+iy,$$
where $${\rm I} = \sqrt { - 1} ,$$ then

The rod PQ of length L = 2 m, and uniformly distributed mass of M = 10 kg, is released from rest at the position shown in the figure. The ends slide along the frictionless faces OP and OQ. Assume acceleration due to gravity, g = 10 m/s² . The mass moment of inertia of the rod about its centre of mass and an axis perpendicular to the plane of the figure is (ML² /12). At this instant, the magnitude of angular acceleration (in radian/s² ) of the rod is ____________.

The arrangement shown in the figure measures the velocity $$V$$ of a gas of density $$1kg/{m^3}$$ flowing through a pipe. The acceleration due to gravity is $$9.81m/{s^2}.$$ If the manometric fluid is water (density $${1000kg/{m^3}}$$ ) and the velocity $$V$$ is $$20 m/s,$$ the differential head $$h$$ (in $$mm$$) between the two arms of the manometer is ______.

For the stability of a floating body the