To determine the value of one main scale division (MSD) for the given vernier calipers, we'll analyze the relationship between the main scale divisions, vernier scale divisions, and the least count of the instrument.

Given:

Number of vernier scale divisions (n): 20

Vernier scale coincides with the 19th division on the main scale.

Least count (LC): $ 0.1 \, \text{mm} $

Understanding the Vernier Calipers Configuration:

Relation between Vernier and Main Scale Divisions:

The total length of the vernier scale (comprising 20 divisions) coincides with the length of 19 main scale divisions.

Mathematically, this is expressed as:

$ n \times \text{VSD} = (n - 1) \times \text{MSD} $

where:

$ \text{VSD} $ is the value of one vernier scale division.

$ \text{MSD} $ is the value of one main scale division.

Calculating the Value of One Vernier Scale Division ($ \text{VSD} $):

$ \text{VSD} = \frac{(n - 1) \times \text{MSD}}{n} $

Least Count (LC):

The least count is the smallest measurement that can be accurately read using the instrument.

It is calculated as:

$ \text{LC} = \text{MSD} - \text{VSD} $

Substituting $ \text{VSD} $ from step 2:

$ \text{LC} = \text{MSD} - \left( \frac{(n - 1) \times \text{MSD}}{n} \right) $

$ \text{LC} = \text{MSD} \left( 1 - \frac{n - 1}{n} \right) $

$ \text{LC} = \text{MSD} \left( \frac{n - (n - 1)}{n} \right) $

$ \text{LC} = \text{MSD} \left( \frac{1}{n} \right) $

Solving for One Main Scale Division ($ \text{MSD} $):

Rearranging the equation:

$ \text{LC} = \frac{\text{MSD}}{n} $

$ \text{MSD} = \text{LC} \times n $

Substituting the given values ($ \text{LC} = 0.1 \, \text{mm}, n = 20 $):

$ \text{MSD} = 0.1 \, \text{mm} \times 20 $

$ \text{MSD} = 2 \, \text{mm} $

Conclusion:

The value of one main scale division is 2 mm, which corresponds to Option B.