c
(c)

Let tension be \(T\)

\(f_1=\sqrt{\frac{T}{\mu}} \times \frac{1}{2 l}=256\)

\(f_2=\sqrt{\frac{T+10}{\mu}} \times \frac{1}{2 l}=320\)

\(\sqrt{\frac{T}{T+10}}=\frac{256}{320}\)

\(\frac{T}{T+10}=\frac{(16)^2}{(16)^2 \times(20)^2}\)

or \(\frac{T}{T+10}=\frac{16^2}{(20)^2}\)

or \(400 T=256 T^2+2560\)

or \(144 T=2560\)

or \(T=\frac{2560}{144}\)

or \(T=\frac{2560}{16 \times 9}\)

or \(T=\frac{160}{9}\) Newton

\(=\frac{16}{9} \,kg -w t\)