The capacitor of an oscillatory circuit is enclosed in a container. When the container is evacuated,the resonance frequency of the circuit is $10\, kHz$. When the container is filled with a gas,the resonance frequency changes by $50\, Hz$. The dielectric constant of the gas is

The resonant frequency of a series $LCR$ circuit is $f_R$. The circuit is connected to a sinusoidally alternating e.m.f. of frequency $2 f_R$. The inductive reactance becomes $X_{L_1}$ and capacitive reactance becomes $X_{C_1}$ after changing the frequency. $X_{C_1}$ is equal to:

$L=2 \text{ H}, C=5 \text{ mF}$ and $R=12 \text{ } \Omega$ are connected in series to an a.c. generator of frequency $50 \text{ Hz}$. Then:

Figure $(a)$ shows the plot of voltage across the capacitor as a function of the driving frequency for a sinusoidally driven electromagnetic $LCR$ oscillator circuit. Figure $(b)$ shows the phase angle $\phi$ (phase difference between voltage and current) vs $\omega/\omega_0$ graph for the same circuit,for three different quality factors corresponding to graphs $1, 2, 3$ of figure $(a)$. Each graph in figure $(a)$ can be matched by one of the graphs $a, b, c$ in figure $(b)$. Choose the correct statement:

Assertion: In a series $LCR$ circuit,resonance can take place.
Reason: Resonance takes place if inductive and capacitive reactances are equal and opposite.

In a series resonant $LCR$ circuit,for the power dissipated to become half of the maximum power dissipated,the current amplitude is