$5.$ A $3-$phase, $4-$pole, Y$-$connected induction motor is supplied with $380$ V and $50$ Hz frequency. When the motor is fully loaded its mechanical power is $12$ kW $,$ its line current is $25$ A $,$ power factor $0\mathrm{.}8$ and it rotates at $1400$ rpm $.$

a$)$ Find the rotor slip s$,$ rotor frequency and load torque.

b$)$ Determine the power crossing the air$-$gap, power losses in the rotor winding and the motor efficiency if the mechanical power losses are negligible.

c$)$ Find the starting torque if the ratio $\backslash (\backslash$mathrm$\{$T$\}\_\{(\backslash$mathrm$\{$a$\})\}/\backslash$mathrm$\{$T$\}\_\{\backslash$max $\}=0\mathrm{.}4\backslash ),$ where $\backslash (\backslash$mathrm$\{$T$\}\_\{(\backslash$mathrm$\{$a$\})\}\backslash )$ is the torque found in question $($a$).$