Let $$\,\,\nabla .\left( {fV} \right) = {x^2}y + {y^2}z + {z^2}x,\,\,$$ where $$f$$ and $$V$$ are scalar and vector fields respectively. If $$V=yi+zj+xk,$$ then $$\,V.\left( {\nabla f} \right)$$ is

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 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 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 __________.