a
(a) The frequency of vibration of a string \(n = \frac{p}{{2l}}\sqrt {\frac{T}{m}} \)

Also number of loops = Number of antinodes.

Hence, with \(5\) antinodes and hanging mass of \(9 kg\).

We have \(p = 5\) and \(T = 9g\) ==> \({n_1} = \frac{5}{{2l}}\sqrt {\frac{{9g}}{m}} \)

With \(3\) antinodes and hanging mass \(M\)

We have \(p = 3\) and \(T = Mg\)

==> \({n_2} = \frac{3}{{2l}}\sqrt {\frac{{Mg}}{m}} \)

\( \because n_1 = n_2\) 

==> \(\frac{5}{{2l}}\sqrt {\frac{{9g}}{m}} = \frac{3}{{2l}}\sqrt {\frac{{Mg}}{m}} \)==> \(M = 25\, kg.\)