Q.The power factor of L.C.R. circuit at resonance is

Three equivalent definitions. The power factor of an AC circuit can be defined, equivalently, as (i) $\cos\phi$, the cosine of the phase angle between voltage and current; (ii) R/Z, the ratio of resistance to impedance; or (iii) $P_{av}/(V_{RMS}I_{RMS})$, the ratio of true (average) power to apparent power -- all three give identical numerical values for any given circuit.

Special cases. Purely resistive circuit: $\phi=0$, power factor $=\cos0^{\circ}=1$ -- the maximum possible, meaning true power equals apparent power exactly. Purely inductive or capacitive circuit: $\phi=\pm90^{\circ}$, power factor $=\cos(\pm90^{\circ})=0$ -- the minimum possible, meaning zero true power despite a genuine current flowing (all of it wattless). Series RLC circuit generally: power factor lies strictly BETWEEN 0 and 1, given by $\cos\phi=R/Z$. Series RLC circuit AT RESONANCE: $Z=R$ (since $X_L=X_C$), so power factor $=R/R=1$ -- the maximum value, achieved specifically at the resonant frequency and nowhere else for a fixed RLC combination. …