$A$ uniform metallic wire has a resistance of $18\,\Omega$ and is bent into an equilateral triangle. Then,the resistance between any two vertices of the triangle is .................. $\Omega$.

$A$ ring is made of a wire having a total resistance $R_0 = 12\,\Omega$. Find the ratio of the lengths $\frac{\ell_1}{\ell_2}$ of the two arcs between points $A$ and $B$ as shown in the figure,such that the equivalent resistance $R$ of the circuit between these points is $\frac{8}{3}\,\Omega$.

For the mixed connection shown in the figure,derive the equation for the equivalent resistance.

In which type of connection of resistors will the equivalent resistance increase?

Consider the following two circuits:
$A$: $20$ bulbs are connected in series to a power supply line.
$B$: $20$ bulbs identical to $A$ are connected in a parallel circuit to an identical power supply line.
Identify which of the following is not true.

$A$ copper wire of resistance $R$ is cut into ten parts of equal length. Two pieces each are joined in series and then five such combinations are joined in parallel. The new combination will have a resistance