In the shown ‘AC’ circuit L, C and $v_0$ are fixed. What is the value of  $\omega$ such that reading of hot wire ammeter shown is independent of ‘R’?

‘Z’ of top branch $\sqrt{R^2+\frac{1}{{\omega }^2{C}^2}}$
Phase difference between currents of ‘RC’ branch and ‘L’ branch $={90}^{\circ }+\theta$ where $Tan\theta =\frac{1}{\omega CR}$

${(i in battery)}^2$

$={\left(\frac{V_0}{\omega L}\right)}^2+\frac{V_0^2}{Z^2}-\frac{2V_0^2\sin \theta }{\omega LZ}$

First term is independent of R.
So far i to be independent of R second and third terms are to be equal.
$$\begin{gathered} \Rightarrow 2Z\sin \theta =\omega L \\ \Rightarrow 2\frac{1}{\omega c}=\omega L\Rightarrow \omega =\sqrt{\frac{2}{LC}} \end{gathered}$$