An AC circuit has an inductor and a resistor of resistance R in series, such that X L = 3 R . Now, a…

First Recall the formula of power factor, $\cos \varphi =\frac{R}{Z}$$\Rightarrow \cos {\varphi }_{\mathrm{initial}} =\frac{R}{Z}=\frac{R}{\sqrt{{\left(X_{\mathrm{L}}\right)}^2+R^2}}=\frac{R}{\sqrt{9R^2+R^2}}=\frac{1}{\sqrt{10}}$. 

$\Rightarrow \cos {\varphi }_{final} =\frac{R}{Z}=\frac{R}{\sqrt{{\left(X_{\mathrm{L}}-X_C\right)}^2+R^2}}=\frac{R}{\sqrt{R^2+R^2}}=\frac{1}{\sqrt{2}}$

$\Rightarrow \frac{\cos {\varphi }_{final} }{\cos {\varphi }_{\mathrm{initial}} }=\frac{\sqrt{5}}{x}\Rightarrow x=1$