$4.(30$ points$)$ Consider the information signal $\backslash (\backslash$mathrm$\{$m$\}(\backslash$mathrm$\{$t$\})\backslash )$ shown below. The AM$-$DSB$-$C is employed with a carrier frequency of $100$ kHz $,$ a scaling factor of $2,$ and a dc offset of $3$ volts.

$($a$)$ Give the math expression of the transmitted signal.

$($b$)$ Calculate the modulation index a

$($c$)$ Draw the transmitted signal.