In the figure shown,for given values of $R_1$ and $R_2$,the balance point for the jockey is at $40\,cm$ from $A$. When $R_2$ is shunted by a resistance of $10\, \Omega$,the balance point shifts to $50\,cm$. Find $R_1$ and $R_2$. $(AB = 1\,m)$

In a meter bridge experiment to determine the value of an unknown resistance,first the resistances $2 \Omega$ and $3 \Omega$ are connected in the left and right gaps of the bridge and the null point is obtained at a distance $l \ cm$ from the left. Now when an unknown resistance $x \ \Omega$ is connected in parallel to $3 \ \Omega$ resistance,the null point is shifted by $10 \ cm$ to the right of the wire. The value of unknown resistance $x$ is . . . . . . $\Omega$.

The resistance of the meter bridge wire $AB$ in the given figure is $4\,\Omega$. With a cell of emf $\varepsilon_1 = 0.5\,\text{V}$ and rheostat resistance $R_{h1} = 2\,\Omega$,the null point is obtained at point $J$. When the cell is replaced by another one of emf $\varepsilon_2$,the same null point $J$ is found for $R_{h2} = 6\,\Omega$. The emf $\varepsilon_2$ is ................. $\text{V}$.

In a metre bridge experiment,the ratio of the left gap resistance to the right gap resistance is $2:3$. The balance point from the left is: (in $cm$)

$A$ meter bridge is set up as shown to determine the end corrections at $A$ and $B$. When a resistance of $15\,\Omega$ is used in the left gap and $20\,\Omega$ in the right gap,the null point is obtained at a distance of $42\,cm$ from $A$. When these resistances are interchanged,the null point is obtained at a distance of $57\,cm$ from $A$. The values of the end corrections are:

$A$ resistance of $2\,\Omega$ is connected across one gap of a meter-bridge and an unknown resistance,greater than $2\,\Omega$,is connected across the other gap. When these resistances are interchanged,the balance point shifts by $20\,cm$. Neglecting any end correction,the unknown resistance is ................ $\Omega$.