A $$3$$ phase, $$4$$ pole, $$400$$ $$V,$$ $$50$$ $$Hz,$$ star connected induction motor has following circuit parameters $${r_1} = 1.0\Omega ,\,{r_2} = 0.5\Omega ,$$ $${\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {x_1} = x{'_2} = 1.2\Omega ,{\mkern 1mu} {x_m} = 35\Omega $$
The starting torque when the motor is started direct-on-line is (use approximate equivalent circuit model)

The speed of a $$4$$-pole induction motor is controlled by varying the supply frequency while maintaining the ratio of supply voltage to supply frequency $$(V/f)$$ constant. At rated frequency of $$50$$ $$Hz$$ and rated voltage of $$400$$ $$V$$ its speed is $$1440$$ $$rpm$$. Find the speed at $$30$$ $$Hz,$$ if the load torque is constant

A $$300$$ $$kVA$$ transformer has $$95\% $$ efficiency at full load $$0.8$$ $$pf$$ lagging and $$96\% $$ efficiency at half load, unity $$pf.$$

What is the maximum efficiency (in %) at unity $$pf$$ load?

Two transformers are to be operated in parallel such that they share load in proportion to their $$kVA$$ ratings. The rating of the first transformer is $$500$$ $$kVA$$ and its $$pu$$ leakage impedance is $$0.05$$ $$pu.$$ If the rating of second transformer is $$250$$ $$kVA,$$ its $$pu$$ leakage impedance is