A balanced star-connected and purely resistive load is connected at the secondary of a star-delta transformer as shown in the figure. The line-to-line voltage rating of the transformer is 110 V/220 V. Neglecting the non-idealities of the transformer, the impedance 'Z' of the equivalent star-connected load, referred to the primary side of the transformer, is

A balanced three-phase voltage is applied to a star-connected induction motor, the phase to neutral voltage being V. The stator resistance, rotor resistance referred to the stator, stator leakage reactance, rotor leakage reactance referred to the stator, and the magnetizing reactance are denoted by $$r_s,\;r_r,\;r_s,\;r_r\;and\;X_m$$, respectively. The magnitude of the starting current of the motor is given by

Divergence of the three-dimentional radial vector field $$\overrightarrow F$$ is

Given $$f\left( t \right) = {L^{ - 1}}\left[ {{{3s + 1} \over {{s^3} + 4{s^2} + \left( {k - 3} \right)}}} \right].$$
$$\mathop {Lt}\limits_{t \to \propto } \,\,f\left( t \right) = 1$$ then value of $$k$$ is