In the given figure, find the values of ${\mathrm{I}}_1, {\mathrm{I}}_2 \mathrm{and} {\mathrm{I}}_3$ using Kirchhoff's law.

In the network shown, the equivalent resistance between $\mathrm{A}$ and $\mathrm{B}$ is $\frac{4}{3}\mathrm{\Omega }$, find the value of $\mathrm{r}$.

In the meter bridge shown in the figure, the balance point is found to be at $40\mathrm{cm}$ from the end $\mathrm{A}$, when the resistor $\mathrm{Y}$ is of $15 \mathrm{\Omega }$. Determine the 

$\left(\mathrm{i}\right)$ resistance of $\mathrm{X}$