$A$ 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:

In the meter bridge experiment,for given values of $R_1$ and $R_2$,the balance point is $20 \ cm$ from point $A$. When $R_2$ is shunted by a $20 \ \Omega$ resistance,the balancing point shifts to $50 \ cm$ from point $A$. Find the value of $R_2$ in $\Omega$.

In the meter bridge experiment, the length $AB$ of the wire is $1 \, m$. The resistors $X$ and $Y$ have values $5 \, \Omega$ and $2 \, \Omega$ respectively. When a shunt resistance $S$ is connected in parallel to $X$, the balancing point is found to be $0.625 \, m$ from $A$. Then, the resistance of the shunt $S$ is

In the experimental setup of a meter bridge shown in the figure,the null point is obtained at a distance of $40\,cm$ from $A$. If a $10\,\Omega$ resistor is connected in series with $R_1$,the null point shifts by $10\,cm$. The resistance that should be connected in parallel with $(R_1 + 10)\,\Omega$ such that the null point shifts back to its initial position is .............. $\Omega$.

When two resistances $R_1$ and $R_2$ are connected in series and introduced into the left gap of a meter bridge and a resistance of $10 \ \Omega$ is introduced into the right gap,a null point is found at $60 \ cm$ from the left side. When $R_1$ and $R_2$ are connected in parallel and introduced into the left gap,a resistance of $3 \ \Omega$ is introduced into the right gap to get a null point at $40 \ cm$ from the left end. The product $R_1 R_2$ is $............. \ \Omega$.

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