gate 2017 EE(Set-B) q38-solution

gate 2017 EE(Set-B) q38-solution

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To get the maximum power the load Ckt must be at resonance i.e. imaginary part of load
impedance is zero.
$$Z_L=j\omega L+\frac{R\times \frac{1}{j\omega C}}{R+\frac{1}{j\omega C}}=j\omega L+\frac{R}{1+j\omega RC}=j\omega L+\frac{R(1-j\omega RC)}{1+\omega ^2+R^2+C^2}$$
$$j term=0$$
$$\Rightarrow \omega L=\frac{\omega R^2C}{1+\omega ^2+R^2C^2}\Rightarrow L=\frac{R^2C}{1+\omega ^2R^2C^2}\Rightarrow 5\times 10^{-3}=\frac{C}{1+10^4C^2}$$
From options C =10mF will satisfy the about equation