Let $$y(x)$$ be the solution of the differential equation $$\,\,{{{d^2}y} \over {d{x^2}}} - 4{{dy} \over {dx}} + 4y = 0\,\,$$ with initial conditions $$y(0)=0$$ and $$\,\,{\left. {{{dy} \over {dx}}} \right|_{x = 0}} = 1.\,\,$$ Then the value of $$y(1)$$ is __________.

Consider the function $$f\left( z \right) = z + {z^ * }$$ where $$z$$ is a complex variable and $${z^ * }$$ denotes its complex conjugate. Which one of the following is TRUE?

The solution of the differential equation, for
$$t > 0,\,\,y''\left( t \right) + 2y'\left( t \right) + y\left( t \right) = 0$$ with initial conditions $$y\left( 0 \right) = 0$$ and $$y'\left( 0 \right) = 1,$$ is $$\left[ {u\left( t \right)} \right.$$ denotes the unit step function$$\left. \, \right]$$,

A full-bridge converter supplying an RLE load is shown in figure. The firing angle of the bridge converter is 120º. The supply voltage $$${v_m}\left( t \right) = 200\pi \sin \left( {100\pi t} \right)\,V,$$$ $$$R = 20 Ω, E = 800 V$$$The inductor L is large enough to make the output current I_L a smooth dc current. Switches are lossless. The real power fed back to the source, in kW, is __________.