An alternating current is flowing through a series $L-C-R$ circuit. It is found that the current reaches a value of $1 \ mA$ at both $200 \ Hz$ and $800 \ Hz$ frequency. What is the resonance frequency of the circuit (in $Hz$)?

The resonant frequency of an $L-C$ circuit is

With a gradual increase in the frequency of an $A.C.$ supply,the impedance of an $LCR$ series circuit

The quality factor of an $LCR$ circuit having resistance $(R)$ and inductance $(L)$ at resonance frequency $(\omega)$ is given by:

$A$ telegraph line of length $100 \, km$ has a capacity of $0.01 \, \mu F/km$ and it carries an alternating current at $0.5 \, kHz$. If minimum impedance is required, then the value of the inductance that needs to be introduced in series is . . . . . . $mH$. (Take $\pi = \sqrt{10}$)

$A$ $10 \Omega$ resistance,$5 \ mH$ coil,and $10 \ \mu F$ capacitor are joined in series. When an alternating current source of suitable frequency is joined to this combination,the circuit resonates. If the resistance is halved,the resonant frequency is