A feedback system is shown below.

$($a$)$ Sketch the root locus as $\backslash ($ a $\backslash )$ varies when $\backslash ($ a $\backslash$geq $0\backslash ).[10$pt$]$

$($b$)$ Determine if a non$-$negative value of $\backslash ($ a $\backslash )$ exists such that the dominant closed$-$loop poles have a damping ratio of $0\mathrm{.}5.$ If so$,$ calculate the value$($s$)$ of $\backslash ($ a $\backslash ).[5$pt$]$

$($c$)$ Find the maximum damping ratio of the dominant closed$-$loop poles for nonnegative $\backslash ($ a $\backslash ).[5$pt$]$