$6\mathrm{.}10.$ A converging$-$diverging nozzle has an area ratio of $7\mathrm{.}8$ but is not being operated at its design pressure ratio. Consequently, a normal shock is found in the diverging section at an area twice that of the throat. The fluid is oxygen.

$($a$)$ Find the Mach number at the exit and the operating pressure ratio.

$($b$)$ What is the entropy change through the nozzle if there is negligible friction?