$($a$)$ Write the characteristic equation for the control system shown in Figure below.

$($b$)$ Use the Routh test to determine if the system is stable for $Kc=10.$

$($c$)$ Determine the ultimate value of $Kc$ above which the system is unstable.

$($d$)$ Determine the value of $Kc$ for which two of the roots are on the imaginary axis, and determine the values of these imaginary roots and the remaining two roots.