Two sitar strings, A and B playing the note 'Dha' are slightly out of tune and produce beats of frequency 5 Hz. The tension of the string B is slightly increased and the beat frequency is found to decrease by 3 Hz. If the frequency of A is 425 Hz. The original frequency of B is:

We are given that string A has frequency $$f_A = 425\ \text{Hz}$$ and, when it is sounded together with string B, a beat frequency of $$5\ \text{Hz}$$ is heard.

First, we recall the definition of beat frequency:

$$f_{\text{beat}} = \left|\,f_A - f_B\,\right|$$

Here $$f_{\text{beat}} = 5\ \text{Hz}$$, so

$$\left|\,425 - f_B\,\right| = 5$$

Solving this absolute-value equation gives two algebraic possibilities:

$$425 - f_B = 5 \quad \Longrightarrow \quad f_B = 420\ \text{Hz}$$ $$425 - f_B = -5 \quad \Longrightarrow \quad f_B = 430\ \text{Hz}$$

Thus, before we touch the tension, B could be either $$420\ \text{Hz}$$ or $$430\ \text{Hz}$$.

Next, the tension of B is slightly increased. The fundamental frequency of a stretched string is given by the formula

$$f = \frac{1}{2L}\sqrt{\frac{T}{\mu}}$$

where $$L$$ is the length, $$T$$ is the tension, and $$\mu$$ is the linear mass density. We see that $$f \propto \sqrt{T}$$, so increasing tension always increases the frequency of the string.

Experimentally, after increasing the tension, the beat frequency is observed to decrease by $$3\ \text{Hz}$$. The old beat frequency was $$5\ \text{Hz}$$, so the new beat frequency is

$$f'_{\text{beat}} = 5 - 3 = 2\ \text{Hz}$$

This means the new frequency difference satisfies

$$\left|\,f_A - f'_B\,\right| = 2\ \text{Hz}$$

Since increasing tension can only raise $$f_B$$, let us test each of the two earlier possibilities:

1. If the original $$f_B = 430\ \text{Hz}$$ (already higher than 425 Hz), raising the tension would further increase it, say to $$f'_B > 430\ \text{Hz}$$. The new difference $$|425 - f'_B|$$ would then be greater than $$|425 - 430| = 5\ \text{Hz}$$, so the beat frequency would rise above 5 Hz, not drop to 2 Hz. This contradicts the observation.

2. If the original $$f_B = 420\ \text{Hz}$$ (lower than 425 Hz), raising the tension lifts it closer to 425 Hz, perhaps to $$f'_B = 423\ \text{Hz}$$. The new difference $$|425 - 423| = 2\ \text{Hz}$$ is indeed smaller than 5 Hz, exactly matching the measured decrease to 2 Hz.

Therefore the only self-consistent choice is

$$f_B = 420\ \text{Hz}$$

Hence, the correct answer is Option D.