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(C) The impedance $Z$ of a series $LCR$ circuit is given by the formula: $Z = \sqrt{R^2 + (X_L - X_C)^2} = \sqrt{R^2 + (\omega L - \frac{1}{\omega C})

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first decreases and then increases

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(C) The impedance $Z$ of a series $LCR$ circuit is given by the formula: $Z = \sqrt{R^2 + (X_L - X_C)^2} = \sqrt{R^2 + (\omega L - \frac{1}{\omega C})^2}$.At very low frequencies $(\omega 	o 0)$, the capacitive reactance $X_C = \frac{1}{\omega C}$ is very large, making the impedance $Z$ very high.As the frequency $\omega$ increases, the term $(\omega L - \frac{1}{\omega C})^2$ decreases until it reaches zero at the resonance frequency $\omega_0 = \frac{1}{\sqrt{LC}}$. At this point, $Z = R$, which is the minimum value.As the frequency increases further beyond $\omega_0$, the inductive reactance $X_L = \omega L$ becomes dominant, and the term $(\omega L - \frac{1}{\omega C})^2$ increases again, causing the impedance $Z$ to increase.Therefore, the impedance first decreases and then increases.

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

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