Consider an ideal jet propulsion cycle with a diffuser. Air enters the diffuser at $80$ kPa and $-40C$ with a velocity of $200\frac{m}{s}.$ The air velocity leaving the diffuser $($point a$)$ is negligible. The pressure leaving the compressor is $800$ kPa $.$ The air expands in the turbine to a pressure at which the turbine work is just equal to the compressor work. On leaving the turbine, the air expands in a nozzle to $80$ kPa $.$ The nozzle exit velocity is $900\frac{m}{s}.$

a$)$ Show the process on a T$-$s diagram.

b$)$ Find the pressure at the compressor inlet $($point a$).$

c$)$ Find the maximum temperature in the cycle.