b
(b) First overtone of string \(A\)= Second overtone of string \(B.\)

\( \Rightarrow \) Second harmonic of \(A\) = Third harmonic of \(B\)

\( \Rightarrow \) \({n_2} = {n_3}\)\( \Rightarrow \) \({\left[ {2({n_1})} \right]_A} = {\left[ {3({n_1})} \right]_B}\) ( \(\because\) \({n_1} = \frac{1}{{2l}}\sqrt {\frac{T}{{\pi {r^2}\rho }}} \))

==>\(2\,\left[ {\frac{1}{{2{l_A}{r_A}}}\sqrt {\frac{T}{{\pi \rho }}} } \right] = 3\,\left[ {\frac{1}{{2{l_B}{r_B}}}\sqrt {\frac{T}{{\pi \rho }}} } \right]\)

\(\frac{{{l_A}}}{{{l_B}}} = \frac{2}{3}\frac{{{r_B}}}{{{r_A}}} \Rightarrow \frac{{{l_A}}}{{{l_B}}} = \frac{2}{3} \times \frac{{{r_B}}}{{(2{r_B})}} = \frac{1}{3}\)