Calculating half power frequency (frequency response)

Question

I am having a doubt in calculating half power frequency for a given RLC AC circuit. I have attached images of two questions with their solutions. In the first question, the equation for $\cfrac{|V_2|}{|V_1|}$ came out to be:

$\cfrac{|V_2|}{|V_1|} = \cfrac{1}{\sqrt{4+(\omega RC)^2}}$

For calculating half power frequency, it was set equal $\cfrac{1}{\sqrt{2}}$ times the max. value which is $\cfrac{1}{2}$ at $\omega = 0$.

But, in the other problem, the equation came out to be:

$\cfrac{|V_2|}{|V_1|} = \cfrac{\sqrt{1+(\omega RC)^2}}{\sqrt{4 + (\omega RC)^2}}$

For calculating half power frequency, they set it equal to $\cfrac{1}{2}$ (which I think is the max. value at $\omega = 0$.

Can anyone please explain why this difference in solving the problems?

Thanks

Answer 1

The maximum of $$ \left|\cfrac{V_2}{V_1}\right| = \cfrac{\sqrt{1+(\omega RC)^2}}{\sqrt{4 + (\omega RC)^2}} $$ is $1$ at $\omega=\pm \infty$, and you find the half power frequency by solving: $$ \frac{1+(\omega RC)^2}{4+(\omega RC)^2}=\frac{1}{2} $$ which gives $\omega=\pm \sqrt{2}/RC$