A series LR circuit connected with an ac source E = ( 25 1000t ) V...

For $LR$ circuit, $cosϕ = \dfrac{R}{\sqrt{{\left({X}_{L}\right)}^{2} + {R}^{2}}} = \dfrac{1}{\sqrt{2}}$

$⇒ R = {X}_{L}$

Now, ${\omega}^{'} = 2\omega$

Therefore, ${\left({X}_{L}\right)}^{'} = {\omega}^{'}L = 2\omegaL = 2{X}_{L}$

Hence, $cos{ϕ}^{'} = \dfrac{R}{\sqrt{{\left(2{X}_{L}\right)}^{2} + {R}^{2}}} = \dfrac{1}{\sqrt{5}}$