JEE PYQ: Units & Measurements - Question ID cc0ef86a1935 (JEE Main 2026)

Core Formula & Concept:

In any vernier instrument (callipers, height gauge, etc.), the least count (LC) is the smallest length that can be measured with the instrument. It is defined as the difference between one main-scale division (MSD) and one vernier-scale division (VSD).

Mathematically, $\text{Least Count} = \text{Value of one MSD} - \text{Value of one VSD}$

If the vernier scale has $n$ divisions that coincide with $(n-1)$ main-scale divisions, then the value of one VSD is $\text{VSD} = \frac{(n-1)\times \text{MSD}}{n}$ However, in the given problem, 50 vernier divisions equal 48 main divisions. Hence, we can directly compute the value of one VSD and then find the least count.

Step-by-Step Derivation:

Step 1: Identify given data
- Number of vernier divisions, $N_v = 50$
- Number of main scale divisions they match, $N_m = 48$
- Value of one main scale division, $\text{MSD} = 0.05\ \text{mm}$

Step 2: Compute the value of one vernier scale division (VSD)
Since 50 VSD = 48 MSD, $\text{VSD} = \frac{48 \times \text{MSD}}{50} = \frac{48 \times 0.05\ \text{mm}}{50}$ $\text{VSD} = \frac{2.4\ \text{mm}}{50} = 0.048\ \text{mm}$

Step 3: Calculate the least count (LC)
The least count is the difference between one MSD and one VSD: $\text{LC} = \text{MSD} - \text{VSD} = 0.05\ \text{mm} - 0.048\ \text{mm} = 0.002\ \text{mm}$

Step 4: Match with the given options
The computed least count is $0.002\ \text{mm}$, which corresponds to option A.

Common Traps & Exam Tip:

1. Misidentifying the number of divisions: Students often confuse which scale has more divisions. Here, 50 vernier divisions match 48 main divisions, so the vernier scale is finer. 2. Incorrect formula application: Some mistakenly use $\text{LC} = \frac{\text{MSD}}{n}$ where $n$ is the number of vernier divisions. This shortcut only works when $n$ vernier divisions match $(n-1)$ main divisions. In this problem, 50 vernier divisions match 48 main divisions, so the shortcut does not apply directly. 3. Unit confusion: Ensure all values are in the same unit (mm here) before performing calculations.

Exam Tip: Always write down the formula for least count explicitly and substitute the values carefully. Double-check whether the vernier divisions are more or fewer than the main scale divisions to avoid sign errors.