$8\mathrm{.}1$: The resistance, inductance, and capacitance in a parallel RLC circuit are $5000$ ohm, $1\mathrm{.}25$ H$,$ and $8$ nF$,$ respectively.

a$)$ Calculate the roots of the characteristic equation that describe the voltage response of the circuit.

b$)$ Will the response be over$-,$ under$-,$ or critically damped?

c$)$ What value of R will yield a damped frequency of $6$ krad$/$s$?$

d$)$ What are the roots of the characteristic equation for the value of R found in $($c$)?$

e$)$ What value of R will result in a critically damped response?