JEE PYQ: Units & Measurements - Question ID 7042af7c351d (JEE Main 2024)

Core Formula & Concept:

In the chapter Units & Measurements, the Vernier calliper is a precision instrument used to measure lengths with an accuracy finer than the main scale. Its key components are:

• Main scale (MS) – fixed, graduated in millimetres.
• Vernier scale (VS) – sliding, with divisions slightly smaller than the main scale.

The least count (LC) of the Vernier calliper is defined as: $LC = \text{Value of 1 MS division} - \text{Value of 1 VS division}$ Typically, 1 MS division = 1 mm, and 10 VS divisions = 9 mm, so $LC = 0.1$ mm.

Zero error occurs when the zero of the Vernier scale does not coincide with the zero of the main scale when the jaws are fully closed. It can be:

• Positive zero error: The zero of the Vernier scale lies to the right of the main scale zero. This means the instrument reads a positive value even when no object is being measured.
• Negative zero error: The zero of the Vernier scale lies to the left of the main scale zero, resulting in a negative apparent reading when no object is present.

The actual measurement is obtained by correcting the observed reading: $\text{Actual reading} = \text{Observed reading} - \text{Zero error}$

--- Step-by-Step Derivation:

Step 1: Analyze Assertion (A)

Assertion (A) states: “In Vernier calliper if positive zero error exists, then while taking measurements, the reading taken will be more than the actual reading.”

Let’s denote:

• $R_{\text{obs}}$ = Observed reading (what the instrument shows)
• $R_{\text{actual}}$ = True length of the object
• $Z$ = Zero error (positive in this case)

When the jaws are closed (no object), the instrument shows a non-zero reading $Z$, where $Z > 0$. This $Z$ is the positive zero error.

When measuring an object, the observed reading $R_{\text{obs}}$ includes this zero error. The actual length of the object is: $R_{\text{actual}} = R_{\text{obs}} - Z$ Rearranging: $R_{\text{obs}} = R_{\text{actual}} + Z$ Since $Z > 0$, it follows that: $R_{\text{obs}} > R_{\text{actual}}$ Thus, the observed reading is more than the actual reading. So, Assertion (A) is true.

Step 2: Analyze Reason (R)

Reason (R) states: “The zero error in Vernier Calliper might have happened due to manufacturing defect or due to rough handling.”

This is factually correct. Zero error can arise from:

• Manufacturing defect: Imperfect alignment of the Vernier scale during production.
• Rough handling: Misalignment caused by dropping, excessive force, or wear and tear.

So, Reason (R) is true.

Step 3: Check if (R) explains (A)

While Reason (R) correctly identifies causes of zero error, it does not explain why a positive zero error leads to an observed reading greater than the actual reading.

The explanation for Assertion (A) lies in the correction formula and the definition of positive zero error, not in the cause of the error. Therefore, (R) is not the correct explanation of (A).

Step 4: Match with Options

• A: Both (A) and (R) correct, and (R) explains (A) → Incorrect
• B: Both (A) and (R) correct, but (R) does not explain (A) → Correct
• C: (A) true, (R) false → Incorrect
• D: (A) false, (R) true → Incorrect

Hence, the correct answer is B.

--- Common Traps & Exam Tip:

Students often confuse:

• The sign of zero error: Positive vs. negative. Remember: Positive zero error means the instrument reads more than actual.
• The role of Reason (R): Just because a reason is true doesn’t mean it explains the assertion. Always check if the reason logically supports the assertion’s mechanism.
• Misapplying the correction: Some students subtract the observed reading from zero error, leading to incorrect conclusions.

Exam Tip: Always write down the correction formula: $\text{Actual} = \text{Observed} - \text{Zero Error}$ and substitute values to verify the relationship. This avoids sign errors and conceptual confusion.