An airplane flies at a speed of $993$ km$/$h at an altitude where the pressure conditions are $51$ kPa and the temperature is $-23\backslash$deg C$.$ The turbine compressor has pressures of $11$ and the temperature at the turbine inlet is $1019\backslash$deg C$.$ All processes are isentropic$,$ air can be considered an ideal gas. At the inlet before the compressor there is a diffuser and at the outlet of the engine a nozzle.

Calculate the following

$1)$ The pressure at the compressor inlet and at the turbine outlet for variable specific heats

$2)$ Exhaust velocity and propulsive efficiency

$3)$ The pressure at the inlet of the compressor and at the outlet of the turbine and constant specific heats

$4)$ The exhaust velocity at the exit nozzle for constant specific heats