Lifetime of an electric bulb is a random variable with density $$f\left( x \right) = k{x^2},$$ where $$x$$ is measured in years. If the minimum and maximum lifetimes of bulb are $$1$$ and $$2$$ years respectively, then the value of $$k$$ is ________.

A diode circuit feeds an ideal inductor as shown in the figure. Given $${V_s} = 100\,\,\sin \,\left( {\omega t} \right)V,$$ where $$\omega = 100\pi $$ rad/s, and $$L=31.83$$ $$mH$$. The initial value of inductor current is zero. Switch $$S$$ is closed at $$t=2.5$$ $$ms.$$ The peak value of inductor current $${i_L}\left( {in\,\,A} \right)$$ in the first cycle is __________.

A three-phase fully controlled bridge converter is fed through star-delta transformer as shown in the figure.

The converter is operated at a firing angle of $${30^0}.$$ Assuming the load current $$\left( {{{\rm I}_0}} \right)$$ to be virtually constant at $$1$$ $$p.u.$$ and transformer to be an ideal one, the input phase current waveform is

A single - phase voltage source inverter shown in figure is feeding power to load. The triggering pulses of the devices are also shown in the figure.

If the load current is sinusoidal and is zero at $$0,\,\,\pi ,2\pi ....,$$ the node voltage $${V_{AO}}$$ has the waveform