$(25$ points$)$ I connect a $100$ resistor, a $470F$ capacitor, and a $2\mathrm{.}50$ H ideal inductor in series to an AC power supply with a voltage output of $*$ t $).$

a$.$ Determine the voltage across the resistor as a function of time.

b$.$ Determine the voltage across the capacitor as a function of time.

c$.$ Determine the voltage across the inductor as a function of time.

d$.$ Determine the voltage across the power supply as a function of time.

e$.$ If I were to double the frequency, would the power supply current increase or decrease, assuming the peak voltage remains $12\mathrm{.}0$ V $?$ Explain.