A cell $E_1$ of emf $6 \mathrm{V}$ and internal resistance $2\mathrm{\Omega }$ is connected with another cell $E_2$ of emf $4 \mathrm{V}$ and internal resistance $8 \mathrm{\Omega }$ (as shown in the figure). The potential difference across points $X$ and $Y$ is:

A current of $6 \mathrm{A}$ enters one corner $P$ of an equilateral triangle $PQR$ having $3$ wires of resistance $2 \mathrm{\Omega }$ each and leaves by the corner $R.$ The currents $i_1$ in ampere is

Five equal resistances are connected in a network as shown in figure. The net resistance between the points $A$ and $B$ is :

A potentiometer wire $PQ$ of $1 \mathrm{m}$ length is connected to a standard cell $E_1$. Another cell $E_2$ of emf $1.02 \mathrm{V}$ is connected with a resistance $r$ and switch $S$ (as shown in figure). With switch $S$ open, the null position is obtained at a distance of $49 \mathrm{cm}$ from $\mathrm{Q}$. The potential gradient in the potentiometer wire is :