Consider the differential equation $$\,\,3y''\left( x \right) + 27y\left( x \right) = 0\,\,$$ with initial conditions $$y\left( 0 \right) = 0$$ and $$y'\left( 0 \right) = 2000.\,\,$$ The value of $$y$$ at $$x=1$$ is __________.

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 ____________.

A $$60$$ $$mm$$ $$-$$ diameter water jet strikes a plate containing a hole of $$40mm$$ diameter as shown in the figure. Part of the jet passes through the hole horizontally, and the remaining is deflected vertically. The density of water is $$1000$$ $$kg/{m^3}$$ . If velocities are as indicated in the figure, the magnitude of horizontal force (in $$N$$) required to hold the plate is _______________