$5.$ A $440-$V$,3-$phase, $4-$pole, $50$ Hz slip$-$ring star$-$ connected induction motor has a voltage of $80$ V between slip$-$rings when full$-$voltage is applied to the stator and the slip$-$rings are open$-$ circuited with the rotor stationary. The stator current at no$-$load is $2$ A at a p$.$f of $0\mathrm{.}21$ lagging. The rotor is star$-$connected with a per phase standstill leakage impedance of $0\mathrm{.}05+$ j$0\mathrm{.}250$mega referred to rotor. For the motor running with the slip$-$rings short$-$circuited and at a slip of $5\backslash \%,$ calculate

$($a$)$ The torque developed in Nm $,$

$($b$)$ The mechanical power developed,

$($c$)$ The rotor ohmic loss and

$($d$)$ The stator current and power factor.

Neglect stator leakage impedance and rotational losses.