$(3\mathrm{.}5$ points$)$ Using the method of characteristics, compute flow for a two dimensional, minimum$-$length nozzle for the expansion of air to a design exit Mach number of $2.$

a$.$ Fill all the values of the table. An initial line of data is given. You MUST show all work.

b$.$ Compute the location of point $1($along symmetry plane$).$ Assume location of $b$ is $x=0,y=0$ and location of $a$ is $x=0,y=1.$

$\backslash$table$[[$Point$,K_{-}=+$

$u,K_{+}=-$

$u,=\frac{K_{-}+K_{+}}{2}[],$

$u=\frac{K_{-}-K_{+}}{2}[],M$