In order to double the frequency of the fundamental note emitted by a stretched string, the length is reduced to $\frac{3}{4}$$^{th}$ of the original length and the tension is changed. The factor by which the tension is to be changed, is
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(d) $n = \frac{1}{{2l}}\sqrt {\frac{T}{m}} \Rightarrow n \propto \frac{{\sqrt T }}{l}$
==> $\frac{{{T_2}}}{{{T_1}}} = {\left( {\frac{{{n_2}}}{{{n_1}}}} \right)^2}{\left( {\frac{{{l_2}}}{{{l_1}}}} \right)^2} = {(2)^2}{\left( {\frac{3}{4}} \right)^2} = \frac{9}{4}$
==> $\frac{{{T_2}}}{{{T_1}}} = {\left( {\frac{{{n_2}}}{{{n_1}}}} \right)^2}{\left( {\frac{{{l_2}}}{{{l_1}}}} \right)^2} = {(2)^2}{\left( {\frac{3}{4}} \right)^2} = \frac{9}{4}$
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