In an AC circuit, supply voltage \( V_{\text{rms}} = 1000\,\text{V} \), \( R = 80\,\Omega \), \( X_L = 80\,\Omega \), and source frequency \( f = 50\,\text{Hz} \). Find the power factor.
Two resistors $2\,\Omega$ and $3\,\Omega$ are connected in the gaps of a bridge as shown in the figure. The null point is obtained with the contact of jockey at some point on wire $XY$. When an unknown resistor is connected in parallel with $3\,\Omega$ resistor, the null point is shifted by $22.5\,\text{cm}$ towards $Y$. The resistance of unknown resistor is ___ $\Omega$.