For what value of unknown resistance $X$, the potential difference between $B$ and $D$ will be zero in the circuit shown in the figure …………… $\Omega$

During an experiment with a metre bridge, the galvanometer shows a null point when the joceky is pressed at $40.0 \ cm$ using a standard resistance of $90 \ \Omega$, as shown in the figure. The least count of the scale used in the meter bridge is $1 \ mm$. The unknown resistance is:

In a meter bridge experiment the balance point in obtained if the gaps are closed by $2\,\Omega$ and $3\,\Omega$. A shunt of $x\,\Omega$ is added to $3\,\Omega$ resistor to shift the balancing point by $22.5\,cm$. The value of $X$ is $…………….$

In a meter bridge, the wire of length $1\, m$ has a nonuniform cross-section such that, the variation $\frac{{dR}}{{d\ell }}$ of its resistance $R$ with length $\ell $ is $\frac{{dR}}{{d\ell }} \propto \frac{1}{{\sqrt \ell  }}$ Two equal resistances are connected as shown in the figure. The galvanometer has zero deflection when the jockey is at point $P$. What is the length $AP$ ? ……………. $m$

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