b
Figure of merit of a galvanometer is the current required to produce a deflection of one division in the galvancmeter i.e., figure of merit \(=\frac{\mathrm{I}}{\theta}\)

\(\mathrm{I}=\frac{\varepsilon}{\mathrm{R}+\mathrm{G}} \quad \mathrm{G}=\frac{1}{9} \,\mathrm{K} \,\Omega\)

\(\frac{1}{2}=\frac{\varepsilon}{R+\frac{G S}{G+\bar{S}}} \times \frac{S}{S+G} \Rightarrow \frac{1}{2}=\frac{\varepsilon S}{R(S+G)+G S}\)

\(S=\frac{R G \times \frac{1}{2}}{\varepsilon-\frac{(R+G) I}{2}}\)

\(S=\frac{11 \times 10^{3} \times \frac{1}{2} \times 10^{2} \times 270 \times 10^{-6}}{6-\left(\frac{6}{2}\right)}=110\, \Omega\)