To solve this problem, we need to find the resonant frequency $v_0$ of the LCR circuit and the frequency $v$ of the AC source.$\\$Given:$\\$• Inductance $L = 10 \,$H$\\$• Capacitance $C = 10 \, \mu$F$= 10 \times 10^{-6} \,$F$\\$• Resistance $R = 50 \, \Omega \\$• AC source voltage $V = 200 \sin(100t) \,$volt$\\$The resonant frequency $v_0$ of an LCR circuit is given by:$\\ v_0 = \frac{1}{2\pi \sqrt{LC}} \\$Substitute the given values:$\\ v_0 = \frac{1}{2\pi \sqrt{10 \times 10 \times 10^{-6}}} \\ v_0 = \frac{1}{2\pi \sqrt{10 \times 10^{-5}}} \\ v_0 = \frac{1}{2\pi \sqrt{10^{-4}}} \\ v_0 = \frac{1}{2\pi \times 10^{-2}} \\ v_0 = \frac{100}{2\pi} \\ v_0 = \frac{50}{\pi} \,$Hz$\\$The frequency of the AC source $v$ is given by the angular frequency $\omega = 100 \,$rad/s.$\\$Frequency $v$ is related to $\omega$ by:$\\ v = \frac{\omega}{2\pi} \\$Substitute the given value of $\omega: \\ v = \frac{100}{2\pi} \\ v = \frac{50}{\pi} \,$Hz$\\$Therefore, both the resonant frequency and the frequency of the AC source are $\frac{50}{\pi} \,$Hz.$\\$This corresponds to Option 2.