JEE PYQ: Units & Measurements - Question ID 1b6c5fdf08ad (JEE Main 2025)

Core Formula & Concept:

In any vernier instrument (such as a travelling microscope), the least count (LC) is the smallest length that can be measured accurately. It is defined as the difference between one main-scale division and one vernier-scale division.

Key formulas:

• Value of one main-scale division (MSD): $\text{MSD} = \frac{\text{Total length of main scale}}{\text{Number of divisions on main scale}}$
• Value of one vernier-scale division (VSD): $\text{VSD} = \frac{\text{Number of main-scale divisions matched}}{\text{Number of vernier divisions}} \times \text{MSD}$
• Least count (LC): $\text{LC} = \text{MSD} - \text{VSD}$

Step-by-Step Derivation:

Step 1: Compute the value of one main-scale division (MSD)

Given: 300 divisions on the main scale correspond to 15 cm. Therefore, $\text{MSD} = \frac{15\ \text{cm}}{300} = 0.05\ \text{cm}$

Step 2: Compute the value of one vernier-scale division (VSD)

The vernier scale has 25 divisions that match 24 divisions of the main scale. Since each main-scale division is 0.05 cm, 24 main-scale divisions equal $24 \times 0.05\ \text{cm} = 1.2\ \text{cm}.$ This 1.2 cm is covered by 25 vernier divisions, so $\text{VSD} = \frac{1.2\ \text{cm}}{25} = 0.048\ \text{cm}.$

Step 3: Compute the least count (LC)

The least count is the difference between one main-scale division and one vernier-scale division: $\text{LC} = \text{MSD} - \text{VSD} = 0.05\ \text{cm} - 0.048\ \text{cm} = 0.002\ \text{cm}.$

Conclusion: The least count of the travelling microscope is 0.002 cm, which corresponds to option A.

Common Traps & Exam Tip:

1. Misidentifying matched divisions: Students often confuse which scale’s divisions match the other. Here, 25 vernier divisions match 24 main-scale divisions, not the other way around. 2. Incorrect unit conversion: Ensure all lengths are in the same unit (cm) before computing. 3. Arithmetic errors: Double-check the subtraction $0.05 - 0.048$ to avoid careless mistakes.

Exam Tip: Always write down the definitions of MSD, VSD, and LC explicitly before plugging in numbers. This keeps the logic clear and reduces errors.