In an amplitude modulation, a modulating signal having amplitude of $$X$$ V is superimposed with a carrier signal of amplitude $$Y$$ V in first case. Then, in second case, the same modulating signal is superimposed with different carrier signal of amplitude $$2Y$$ V. The ratio of modulation index in the two case respectively will be :

In amplitude modulation, the first case involves a modulating signal amplitude of $$X$$ V and a carrier amplitude of $$Y$$ V, while in the second case the same modulating signal amplitude $$X$$ V is used with a carrier amplitude of $$2Y$$ V.

Since the modulation index ($$\mu$$) is defined as $$\mu = \frac{\text{Amplitude of modulating signal}}{\text{Amplitude of carrier signal}} = \frac{A_m}{A_c}$$, substituting the given amplitudes yields $$\mu_1 = \frac{X}{Y}$$ for the first case and $$\mu_2 = \frac{X}{2Y}$$ for the second case.

Next, finding the ratio of these indices gives $$\frac{\mu_1}{\mu_2} = \frac{X/Y}{X/2Y} = \frac{X}{Y} \times \frac{2Y}{X} = 2$$, which means $$\mu_1 : \mu_2 = 2 : 1$$.

The correct answer is Option 3: 2 : 1.