Plot pressure $($P$)$ and Mach number $($M$)$ versus x along the duct for both cases $($a$)$ and $($b$).$ Here, you should consider different back pressure $($Pb$)$ at the exit of the nozzle. You should also include the $1$st$,2$nd$,$ and $3$rd critical points, as well as over$-$expansion and under$-$expansion conditions for both $($a$)$

$($b$)$pressure and Mach number plots.