The Wheatstone bridge shown in the figure below, gets balanced when the carbon resistor used as $R_1$ has the colour code (orange, red, brown). The resistors $R_2$ and $R_4$ are $80 \mathrm{\Omega }$ and $40 \mathrm{\Omega }$, respectively. Assuming that the colour code for the carbon resistors gives their accurate values, the colour code for the carbon resistor, used as $R_3$, would be

The galvanometer deflection, when key $K_1$ is closed but $K_2$ is open, equals ${\theta }_0$ (see figure). On closing $K_2$ also and adjusting $R_2$ to $5\Omega ,$ the deflection in galvanometer becomes $\frac{{\theta }_0}{5}.$ The resistance of the galvanometer is, then, given by [Neglect the internal resistance of battery]:

In a meter bridge, the wire of length $1 m$ has a non-uniform cross-section such that, the variation $\frac{dR}{dl}$ of its resistance $R$ with length $l$ is $\frac{dR}{dl}\propto \frac{1}{\sqrt{l}}.$ Two equal resistances are connected as shown in the figure. The galvanometer has zero deflection when the jockey is at point $P.$ What is the length $AP?$