A variable capacitor with a range from $12$ to $402$ pF is used with a coil to form a variable$-$frequency LC circuit to tune the input to a radio. $($a$)$ What is the ratio of maximum frequency to minimum frequency that can be obtained with such a capacitor? If this circuit is to obtain frequencies from $0\mathrm{.}41$ MHz to $1\mathrm{.}03$ MHz $,$ the ratio computed in $($a$)$ is too large. By adding a capacitor in parallel to the variable capacitor, this range can be adjusted. To obtain the desired frequency range, $($b$)$ what capacitance in picofarads should be added and $($c$)$ what inductance should the coil have?

$($a$)$ Number $$ Units

$($b$)$ Number $$ Units $$

$($c$)$ Number $$ Units $$