$\mathrm{n}=\frac{1}{2 \mathrm{L}} \sqrt{\frac{\mathrm{T}}{\mathrm{m}}}$

The tuning fork produces $5$ beats per second with lengths $20 \mathrm{\,cm}$ and $21 \mathrm{\,cm}$. If $\mathrm{n}$ be frequency of fork, then

$\mathrm{n}+5=\frac{1}{2 \times 0.20} \sqrt{\frac{\mathrm{T}}{\mathrm{m}}}$          .........$(i)$

$\mathrm{n}-5=\frac{1}{2 \times 0.21} \sqrt{\frac{\mathrm{T}}{\mathrm{m}}}$           .........$(ii)$

Dividing eqn. $(i)$ by $(ii)$ we get:

hence, $\quad \frac{n+5}{n-5}=\frac{0.21}{0.20}$

or  $0.2 n+1=0.21 n-1.05$

or  $1+1.05=0.21 n-0.2 n^{\prime}$

or  $0.05=0.01\, n$

or  $\mathrm{n}=205 \mathrm{\,Hz}$