1. Frequency of a sonometer wire

For a given wire vibrating in the fundamental mode,

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

Since the same wire and same length are used, we have

$f \propto \sqrt{T}$

2. At tension $T_1=6\,\text{N}$

The wire resonates with the tuning fork, so the wire frequency equals the fork frequency.

Let the tuning fork frequency be $f$.

Thus, at $6\,\text{N}$,

$f_1=f$

3. At tension $T_2=54\,\text{N}$

Using proportionality,

$\frac{f_2}{f_1}=\sqrt{\frac{T_2}{T_1}}=\sqrt{\frac{54}{6}}=\sqrt{9}=3$

So,

$f_2=3f$

4. Beat condition

When the tension is changed to $54\,\text{N}$, the beat frequency with the same tuning fork is $12\,\text{Hz}$.

Beat frequency is

$|f_2-f|=12$

Substitute $f_2=3f$:

$|3f-f|=12$ $2f=12$ $f=6\,\text{Hz}$

5. Final answer

The frequency of the tuning fork is

$\boxed{6\,\text{Hz}}$