Correct option is (B)

The least count (LC) of a vernier caliper is calculated using the formula:

$\text{LC}=\frac{\text{One main scale division (MSD)}}{\text{Number of divisions on the vernier scale (VSD)}}$

Given that the least count of the instrument is $0.1\mathrm{mm}$ and the vernier scale has 20 divisions, we can plug these values into the formula to find the value of one main scale division (MSD).

$0.1=\frac{\text{MSD}}{20}$

To find the value of MSD, we solve for MSD:

$\text{MSD}=0.1\times 20=2\mathrm{mm}$

So, one main scale division is equal to $2\mathrm{mm}$. Therefore, the correct answer is Option B 2.

Correct option is (A)

The least count of a Vernier caliper is the smallest measurement that can be obtained with it and is determined by the difference in the measurement of one main scale division and one Vernier scale division.

In this case, 10 divisions on the main scale coincide with 11 divisions on the Vernier scale. This means that 11 divisions on the Vernier scale are equal in length to 10 divisions on the main scale. Since each division on the main scale is 5 units, this is equal to:

$10\text{ divisions on main scale}\times 5\text{ units per division}=50\text{ units}$

The length of 10 divisions on the main scale (or 50 units) is therefore equal to the length of 11 divisions on the Vernier scale. This means that:

$1\text{ division on the Vernier scale}=\frac{50\text{ units}}{11}$

Therefore, the least count of the Vernier caliper, which is the difference between one division on the main scale and one division on the Vernier scale, can be calculated as follows:

$\text{Least count}=\text{value of one main scale division}-\text{value of one Vernier scale division}$

$\text{Least count}=5\text{ units}-\frac{50\text{ units}}{11}$

$\text{Least count}=\frac{55\text{ units}}{11}-\frac{50\text{ units}}{11}$

$\text{Least count}=\frac{5\text{ units}}{11}$

Thus, the least count of the Vernier caliper is

$\frac{5\text{ units}}{11}$

Therefore, the correct answer is:

Option A: $\frac{5}{11}$

Correct option is (A)

$\begin{array}{rl}& 50\mathrm{V}+\mathrm{S}=49\mathrm{S}+\mathrm{S}\\ & \mathrm{S}=50(\mathrm{S}-\mathrm{V})\\ & 5=50(\mathrm{S}-\mathrm{V})\\ & \mathrm{S}-\mathrm{V}=\frac{0.5}{50}=\frac{1}{100}=0.01\mathrm{mm}\end{array}$

Correct option is (A)

$\begin{array}{rl}& 50\mathrm{V}+\mathrm{S}=49\mathrm{S}+\mathrm{S}\\ & \mathrm{S}=50(\mathrm{S}-\mathrm{V})\\ & 5=50(\mathrm{S}-\mathrm{V})\\ & \mathrm{S}-\mathrm{V}=\frac{0.5}{50}=\frac{1}{100}=0.01\mathrm{mm}\end{array}$

Correct option is (B)

The provided statements (A) Assertion and (R) Reason have to be analyzed to determine their correctness and also whether Reason (R) correctly explains Assertion (A).

Assertion (A) is addressing a scenario where a positive zero error occurs in a Vernier caliper. Positive zero error refers to the condition when the zero mark on the Vernier scale is to the right of the zero mark on the main scale when the jaws of the caliper are completely closed. In this case, even when measuring a zero length, the Vernier caliper will show a positive reading, indicating an error. This error is constant and will get added to actual measurements, causing the instrument to give readings that are more than the actual measurement. So, Assertion (A) is true.

Reason (R) explains the possible causes of zero error in Vernier calipers. Zero errors can indeed occur due to imperfect manufacturing processes, where the scales are not perfectly aligned. They can also happen due to rough handling, for instance, if the instrument is dropped, which might cause a permanent deformation leading to a continuous zero error. Hence, Reason (R) is true as well.

Finally, we need to analyze whether Reason (R) is the correct explanation of Assertion (A). Although both statements are correct, the Reason (R) does not explain why a positive zero error would lead to the reading being more than the actual reading, it merely states the possible causes of zero errors. Therefore, the correct relationship between the statements is that they are both true, but (R) does not provide the correct explanation for (A).

The correct option is :

Option B Both (A) and (R) are correct but (R) is not the correct explanation of (A)