$3)$ For the circuit shown below, the initial value of capacitor is $-100$ V at $\backslash ($ t$=0\backslash )-.$

a$.$ Find the $\backslash (\backslash$mathrm$\{$V$\}\_\{0\}(\backslash$mathrm$\{$~s$\})\backslash )$ for the circuit. $($Points $13)$

b$.$ Show that initial value and final value theorem provides the expected outcome. $($Points $10)$

c$.$ Use the partial fraction expansion to find the coefficients corresponding to its roots. $($Points $5)$

d$.$ Use the inverse Laplace table to find $\backslash (\backslash$mathrm$\{$v$\}\_\{\backslash$mathrm$\{$o$\}\}(\backslash$mathrm$\{$t$\})\backslash ).($Points $7)$