JEE PYQ: Units & Measurements - Question ID 6f17cfceba1b (JEE Main 2024)

Core Formula & Concept:

In the chapter Units & Measurements, one of the key concepts is significant figures. Significant figures (or significant digits) indicate the precision of a measured or calculated quantity. When performing arithmetic operations (like calculating the mean), the result must be reported with the correct number of significant figures, determined by the least precise measurement in the data set.

The arithmetic mean (average) of a set of measurements is calculated as: $\text{Mean} = \frac{\sum_{i=1}^{n} x_i}{n}$ where $x_i$ are the individual measurements and $n$ is the number of measurements.

The rule for significant figures in addition/subtraction is:

• The result should be rounded to the least number of decimal places among the measurements.

This is because the precision of the result is limited by the least precise measurement.

In this problem, we are given four time period measurements of a simple pendulum: $4.62 \text{ s}, \quad 4.632 \text{ s}, \quad 4.6 \text{ s}, \quad 4.64 \text{ s}$ We are to compute their arithmetic mean and express it with the correct number of significant figures.

--- Step-by-Step Derivation:

Step 1: List the measurements and note their decimal places

• $4.62$ s → 2 decimal places
• $4.632$ s → 3 decimal places
• $4.6$ s → 1 decimal place
• $4.64$ s → 2 decimal places

The measurement with the least number of decimal places is $4.6$ s, which has only 1 decimal place. This determines the precision of the final result.

Step 2: Compute the sum of the measurements

$4.62 + 4.632 + 4.6 + 4.64$ Let's add them step by step:

• $4.62 + 4.632 = 9.252$
• $9.252 + 4.6 = 13.852$
• $13.852 + 4.64 = 18.492$

So, the total sum is $18.492$ s.

Step 3: Compute the arithmetic mean

There are 4 measurements, so: $\text{Mean} = \frac{18.492}{4} = 4.623 \text{ s}$

Step 4: Round the mean to the correct number of decimal places

Since the least precise measurement ($4.6$ s) has only 1 decimal place, the mean must be rounded to 1 decimal place. Now, $4.623$ rounded to 1 decimal place:

• The digit in the first decimal place is 6.
• The digit in the second decimal place is 2, which is less than 5, so we do not round up.

Thus, $4.623$ rounded to 1 decimal place is $4.6$ s.

Step 5: Verify significant figures

Although the question asks for the mean in "correct significant figure", in this context, it refers to the correct number of decimal places based on the precision of the measurements. The final answer should be $4.6$ s, which matches option B. --- Common Traps & Exam Tip:

Common Mistake 1: Students often confuse significant figures with decimal places. In addition/subtraction, it's the number of decimal places that matters, not the number of significant figures. For example, $4.6$ has 2 significant figures but only 1 decimal place.

Common Mistake 2: Some students compute the mean and report it as $4.623$ s (option D), ignoring the rounding rule. This is incorrect because the least precise measurement limits the precision of the result.

Common Mistake 3: Others might round each measurement to 1 decimal place before averaging, which is unnecessary and can introduce rounding errors. The correct approach is to compute the mean first and then round.

Exam Tip: Always identify the measurement with the least number of decimal places before performing addition or subtraction. This determines the precision of your final result.

Final Answer: The arithmetic mean in correct significant figure is 4.6 s, which corresponds to option B.