b
Least count of vernier calipers \(=\frac{1}{20 \mathrm{~N}} \mathrm{~cm}\)

\(\because\) Least count\(=1 \mathrm{MSD}-1 \mathrm{VSD}\)

let \(\mathrm{x}\) no. of divisions of main scale coincides with \(\mathrm{N}\) division of vernier scale, then

\(1 \mathrm{VSD}=\frac{\mathrm{x} \times 1 \mathrm{~mm}}{\mathrm{~N}}\)

\(\therefore \frac{1}{20 \mathrm{~N}} \mathrm{~cm}=1 \mathrm{~mm}-\frac{\mathrm{x} \times 1 \mathrm{~mm}}{\mathrm{~N}}\)

\(\frac{1}{2 \mathrm{~N}} \mathrm{~mm}=1 \mathrm{~mm}-\frac{\mathrm{x}}{\mathrm{N}} \mathrm{mm}\)

\(\mathrm{x}=\left(1-\frac{1}{2 \mathrm{~N}}\right) \mathrm{N}\)

\(\mathrm{x}=\frac{2 \mathrm{~N}-1}{2}\)