Determine the resonance frequency and the $$Q$$ $$-$$ factor of the circuit shown in Fig.

Data: $$\,\,\,\,\,R = 10\,\Omega ,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,C = 3\,\mu F,$$
$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{L_1} = 40\,mH,\,\,\,\,\,\,\,\,{L_2} = 10\,mH$$
and $$\,\,\,\,\,\,\,\,M = 10\,mH.$$

Determine the impedance seen by the source $${V_s} = 24\angle {0^0}$$ in the network shown in Fig.

A circuit consisting of a single resistor $$R$$ and an inductor $$L$$ in series is driven by a $$25$$ $$V$$ $$rms,$$ $$50$$ $$Hz$$ sinusoidal voltage source. A capacitor is to be placed in parallel with the source to improve the power factor. Given that the average power dissipated in the $$R$$ is $$100$$ $$W$$ and that the reactive power delivered to the $$L$$ is $$75$$ $$VAR,$$ what value of $$C$$ will yield a $$0.9$$ $$p.f.$$ lagging as seen by the source?