Two cells of emf $2E$ and $E$ with internal resistance $r_1$and $r_2$ respectively are connected in series to an external resistor $R$ (see figure). The value of $R,$ at which the potential difference across the terminals of the first cell becomes zero is

$n$ identical cells, each of emf $E$ and internal resistance $r$, are joined in series to form a closed circuit. One of the cell $A$ is joined with reversed polarity. The potential difference across $A$ is :-

A group of N cells each of whose emf varies directly with the internal resistance as per the equation E_N = 1.5 r_N are connected as shown in the figure. The current in the circuit is :-

A meter bridge is set-up as shown, to determine an unknown resistance $\text{'}X\text{'}$ using a standard $10 \mathrm{ohm}$ resistor. The galvanometer shows null point when tapping-key is at $52 \mathrm{cm}$ mark. The end-corrections are $1 \mathrm{cm}$ and $2 \mathrm{cm}$ respectively for the ends $\mathrm{A}$ and $\mathrm{B}$. The determined value of ${\mathrm{X}}^{\text{'}}$ is: