a
(a) In figure \((b)\) current through \({R_2} = i - \frac{i}{{10}} = \frac{{9i}}{{10}}\)
Potential difference across \({R_2}\) = Potential difference across \(R\)

\(==>\) \({R_2} \times \frac{9}{{10}}i = R \times \frac{i}{{10}}\) i.e. \({R_2} = \frac{R}{9} = \frac{{11}}{9}\,\Omega \)

\({R_{eq}} = \frac{{{R_2} \times R}}{{({R_2} + R)}} = \frac{{\frac{{11}}{9} \times \frac{{11}}{1}}}{{\frac{{11}}{9} + \frac{{11}}{1}}} = \frac{{11}}{{10}}\,\Omega \)

Total circuit resistance \( = \frac{{11}}{{10}} + {R_1} = R = 11\) \(==>\) \({R_1} = 9.9\,\Omega \)