$A$ $10 \, \Omega$ resistor, a $5 \, \text{mH}$ inductor, and a $10 \, \mu\text{F}$ capacitor are connected in series. When a suitable frequency alternating current source is connected to this combination, the circuit resonates. If the resistance is halved, the resonance frequency:

The plot given below shows the average power delivered to an $LRC$ circuit versus frequency. The quality factor of the circuit is

The power factor in an $L-C-R$ circuit at resonance is

$A$ parallel combination of a pure inductor and a capacitor is connected across a source of alternating e.m.f. '$e$'. The currents flowing through the inductor and the capacitor are $i_{L}$ and $i_{C}$ respectively. In this parallel resonant circuit,the condition for the currents $i$,$i_{L}$,and $i_{C}$ is ($i =$ net r.m.s. current in the circuit).

To increase the resonant frequency in a series $LCR$ circuit,

$A$ series $LCR$ circuit driven by $300 \, V$ at a frequency of $50 \, Hz$ contains a resistance $R = 3 \, k\Omega$, an inductor of inductive reactance $X_{L} = 250 \pi \, \Omega$ and an unknown capacitor. The value of capacitance to maximize the average power should be: (Take $\pi^{2} = 10$) (in $\mu F$)