Two batteries,one with an $emf$ of $3\,V$ and internal resistance $1\,\Omega$,and the other with an $emf$ of $15\,V$ and internal resistance $2\,\Omega$,are connected in series with a resistance $R$ as shown in the figure. If the potential difference between points $a$ and $b$ is zero,find the resistance $R$ in $\Omega$.

Consider the following statements regarding the network shown in the figure.
$(1)$ The equivalent resistance of the network between points $A$ and $B$ is independent of the value of $G$.
$(2)$ The equivalent resistance of the network between points $A$ and $B$ is $\frac{4}{3} R$.
$(3)$ The current through $G$ is zero.
Which of the above statements is/are true?

When the switch $S$, in the circuit shown, is closed, then the value of current $i$ will be ................. $A$.

Determine the current in each branch of the network shown in Figure.

In the circuit shown in the figure,all the resistances are identical and each has the value $r \ \Omega$. The equivalent resistance of the combination between the points $A$ and $B$ will remain unchanged even when the following pairs of points marked in the figure are connected through a resistance $R$.

In the circuit shown below,the resistance of the galvanometer is $20 \, \Omega$. In which of the following alternatives are the currents arranged strictly in decreasing order?