In the circuit shown in figure, the reading of ideal ammeter connected as shown is same with both switches open as with both closed. Then, find the resistance $R$ (in $\mathrm{\Omega }$).

One of the circuits for the measurement of resistance by potentiometer is shown. The galvanometer is connected at point $A$ and zero deflection is observed at length $PJ=30 \mathrm{c}\mathrm{m}$. In second case, the secondary cell is changed. Take ${\epsilon }_{\mathrm{s}}=10 \mathrm{V}$ and $r=1 \mathrm{\Omega }$ in $1^{\mathrm{s}\mathrm{t}}$ reading and ${\epsilon }_{\text{s}}=5 \mathrm{V}$ and $r=2 \mathrm{\Omega }$ in $2^{\mathrm{n}\mathrm{d}}$ reading. In the second case, the zero deflection is observed at length $PJ=10 \mathrm{c}\mathrm{m}$. What is the resistance $R$ (in ohm)?

For the galvanometer of the potentiometer shown in the figure, the balance point is obtained at a distance $75 \mathrm{c}\mathrm{m}$ from $A$ when the key is open.

The second balance point is obtained at $60 \mathrm{c}\mathrm{m}$ from $A$ when the key is closed. Find the internal resistance (in $\mathrm{\Omega }$) of the battery ${\epsilon }_1$.

In the figure shown $PQ$ is a potentiometer wire. When the galvanometer is connected at $A$, it shows zero deflection when $PJ=x$. Now the galvanometer is connected to $B$ and it shows zero deflection when $PJ=3x$. Find the ratio of unknown resistance $R_x$ and $R$.