The power in an $AC$ circuit is given by $P = E_{rms}I_{rms} \cos \phi$. The value of $\cos \phi$ in a series $LCR$ circuit at resonance is:

An $ac$ source of variable frequency $f$ is connected to an $LCR$ series circuit. Which one of the graphs represents the variation of current $I$ in the circuit with frequency $f$?

An inductance $L$ having a resistance $R$ is connected to an alternating source of angular frequency $\omega$. The Quality factor $Q$ of the inductance is

$A$ series $LCR$ circuit is connected to a $45 \sin (\omega t) \text{ V}$ source. The resonant angular frequency of the circuit is $10^5 \text{ rad s}^{-1}$ and current amplitude at resonance is $I_0$. When the angular frequency of the source is $\omega = 8 \times 10^4 \text{ rad s}^{-1}$, the current amplitude in the circuit is $0.05 I_0$. If $L = 50 \text{ mH}$, match each entry in List-$I$ with an appropriate value from List-$II$ and choose the correct option.

List-$I$	List-$II$
$(P)$ $I_0$ in $\text{mA}$	$(1)$ $44.4$
$(Q)$ The quality factor of the circuit	$(2)$ $18$
$(R)$ The bandwidth of the circuit in $\text{rad s}^{-1}$	$(3)$ $400$
$(S)$ The peak power dissipated at resonance in $\text{Watt}$	$(4)$ $2250$
	$(5)$ $500$

When the frequency of the $AC$ voltage applied to a series $LCR$ circuit is gradually increased from a low value, the impedance of the circuit

An a.c. e.m.f. of peak value $230 \ V$ and frequency $50 \ Hz$ is connected to a circuit with $R=11.5 \ \Omega, L=2.5 \ H$ and a capacitor all in series. The value of capacitance is '$C$' for the current in the circuit to be maximum. The value of '$C$' and maximum current are respectively $(\pi^2=10)$