1. Initial beat condition

For two tuning forks of frequencies $f_A$ and $340\,\text{Hz}$, the beat frequency is

$|f_A - 340| = 5$

So the possible frequencies of fork $A$ are:

$f_A = 345\,\text{Hz} \quad \text{or} \quad f_A = 335\,\text{Hz}$

2. Effect of filing a tuning fork

When a tuning fork is filed, its prongs become lighter, so its frequency increases.

Thus after filing, the frequency of fork $A$ increases.

3. Use the new beat frequency

After filing, the beat frequency decreases from $5$ to $2$.

That means the frequency of fork $A$ moved closer to $340\,\text{Hz}$ when it increased.

Let us test both possibilities:

• If initially $f_A = 335\,\text{Hz}$, then after filing it increases toward $340\,\text{Hz}$, so the beat frequency can decrease from $5$ to $2$. This is consistent.
• If initially $f_A = 345\,\text{Hz}$, then after filing it increases further away from $340\,\text{Hz}$, so the beat frequency would increase, not decrease. This is not consistent.

Hence,

$f_A = 335\,\text{Hz}$

4. Check options

• A: $335\,\text{Hz}$ ✅
• B: $345\,\text{Hz}$ ❌
• C: $338\,\text{Hz}$ ❌
• D: $342\,\text{Hz}$ ❌

Therefore, the correct option is A.