$(30$ Points$)$ A converging$-$diverging nozzle is designed to operate with an exit Mach number of $3\mathrm{.}0.$ The nozzle is supplied from an air reservoir at $1000$ psia $.$ Assuming one dimensional flow, calculate:

a$)$ Maximum back pressure to choke the nozzle.

b$)$ The back pressure for which a normal shock will appear at the nozzle exit plane.

c$)$ The back pressure required to produce a normal shock within the nozzle at $\backslash ($ A$=1\mathrm{.}5$ A$\_\{$t$\}\backslash ).$

d$)$ The back pressure for the nozzle to be perfectly expanded to the design Mach number.

e$)$ The flow conditions $($i$.$e$.,$ Mach number, pressure and the flow direction$)$ immediately outside the nozzle for a back pressure of $100$ psia $.$