Answer
Please see the work below.
Work Step by Step
(a) We know that the resonating frequency of the LCR circuit is given as
$f_{\circ}=\frac{1}{2\pi \sqrt{L}{C}}$. When another capacitor is connected in series then the effective capacitance is $C_p=\frac{C}{2}$; that is, the capacitance of the new combination is decreased. Now the resonance frequency is given as $f_{\circ}=\frac{1}{2\pi \sqrt{LC/2}}=\sqrt{2}\times \frac{1}{2\pi\sqrt{LC}}=\sqrt{2}f$. Thus, the resonating frequency increases.
(b) We know that the best explanation is option (III) -- that is, adding a capacitor in series decreases the equivalent capacitance and this increases the resonance frequency.
