Q$3($a$)$ Consider the first$-$order RC circuit as shown in Figure Q$3\mathrm{.}1.$ Determine the output voltage, $v_o(t)$ by using Laplace Transform. Given that $v(t)=t{e}^{-3t}u(t),R=2$ and $C=0\mathrm{.}5F.$

$($b$)$ Given a signal $z(t)$ as shown in Figure Q$3\mathrm{.}2.$

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$($i$)$ Write the mathematical expression for $z(t)$ in term of unit step and unit ramp.

$($ii$)$ Find the Laplace transform for the $z(t)$ using Laplace definition. $($Hint : refer Laplace Transform Properties Table to solve ${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}t{e}^{-st}dt)$

$($iii$)$ Proof your answer in Q$3($b$)($ii$)$ using the mathematical expression in Q$3($b$)($i$).$

$($c$)$ Determine an appropriate impulse response, $h(t)$ for a stable system, if the system function, $H(s)$ is given as:

$H(s)=\frac{s+3}{(s-4)(s-3)}=\frac{7}{s-4}-\frac{6}{s-3}$