When two resistance $$R_1$$ and $$R_2$$ 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 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 null point at $$40$$ cm from left end. The product of $$R_1 R_2$$ is ______ $$\Omega^2$$

Two resistances $$R_1$$ and $$R_2$$ are tested using a meter bridge in two configurations.

Case 1: Series configuration.

$$R_1 + R_2$$ in left gap, $$10 \Omega$$ in right gap. Null point at 60 cm.

$$\frac{R_1 + R_2}{10} = \frac{60}{40} = \frac{3}{2}$$

$$R_1 + R_2 = 15 \Omega$$ ... (i)

Case 2: Parallel configuration.

$$\frac{R_1 R_2}{R_1 + R_2}$$ in left gap, $$3 \Omega$$ in right gap. Null point at 40 cm.

$$\frac{R_1 R_2/(R_1 + R_2)}{3} = \frac{40}{60} = \frac{2}{3}$$

$$\frac{R_1 R_2}{R_1 + R_2} = 2 \Omega$$ ... (ii)

Finding $$R_1 R_2$$:

From (i) and (ii):

$$R_1 R_2 = 2 \times (R_1 + R_2) = 2 \times 15 = 30 \Omega^2$$

The product $$R_1 R_2 = \boxed{30} \Omega^2$$.