$($b$)$ A $440$V$,50$Hz$,2-$ pole, Y$-$connected induction motor is rated at $100$hp$.$ Its equivalent circuit components have the following values:

R$1=0\mathrm{.}075$ ohm, X$1=0\mathrm{.}17$ohm. R$2=0\mathrm{.}065$ ohm, X$2=0\mathrm{.}17$ ohm. Xm$=7\mathrm{.}2$ ohm.

Rotor slip is $4\%$ at the rated voltage and frequency. Mechanical and core losses are $1\mathrm{.}0$ KW and $1\mathrm{.}10$ KW respectively. Find :

$($a$)$ rotor speed $($b$)$ line current and motor p$.$f $($c$)$ Air$-$gap and

converted power $($d$)$ Induced torque.