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(A) The resonant frequency of an $LC$ oscillator circuit is given by $\omega_0 = \frac{1}{\sqrt{LC}}$.Given that the new inductance $L' = 2L$ and the

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$\frac{1}{4}$

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(A) The resonant frequency of an $LC$ oscillator circuit is given by $\omega_0 = \frac{1}{\sqrt{LC}}$.Given that the new inductance $L' = 2L$ and the new capacitance $C' = 8C$.The new resonant frequency $\omega$ is given by $\omega = \frac{1}{\sqrt{L'C'}} = \frac{1}{\sqrt{(2L)(8C)}} = \frac{1}{\sqrt{16LC}}$.Simplifying this,we get $\omega = \frac{1}{4\sqrt{LC}}$.Since $\omega_0 = \frac{1}{\sqrt{LC}}$,we can substitute this into the expression for $\omega$ to get $\omega = \frac{\omega_0}{4}$.Comparing this with $\omega = x\omega_0$,we find that $x = \frac{1}{4}$.

In an $LC$ oscillator,if values of inductance and capacitance become twice and eight times,respectively,then the resonant frequency of the oscillator becomes $x$ times its initial resonant frequency $\omega_0$. The value of $x$ is:

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