Let $${\rm I} = c\int {\int {_Rx{y^2}dxdy,\,\,} } $$ where $$R$$ is the region shown in the figure and $$c = 6 \times {10^{ - 4}}.\,\,$$ The value of $${\rm I}$$ equals ___________.

A function $$f(x)$$ is defined as
$$f\left( x \right) = \left\{ {\matrix{ {{e^x},x < 1} \cr {\ln x + a{x^2} + bx,x \ge 1} \cr } \,\,,\,\,} \right.$$ where $$x \in R.$$

Which one of the following statements is TRUE?

Consider the differential equation $$\left( {{t^2} - 81} \right){{dy} \over {dt}} + 5ty = \sin \left( t \right)\,\,$$ with $$y\left( 1 \right) = 2\pi .$$ There exists a unique solution for this differential equation when $$t$$ belongs to the interval

The figure below shows an uncontrolled diode bridge rectifier supplied form a $$220$$ Z$$V,$$ $$50$$ $$Hz,$$ $$1$$-phase $$ac$$ source. The load draws a constant $${{\rm I}_0}$$ $$=14A.$$ The conduction angle of the diode $${D_1}$$ in degrees (rounded off to two decimal places) is ______________.