JEE PYQ: Units & Measurements - Question ID 8397f92f725a (JEE Main 2010)

In the chapter Units & Measurements, the concept of significant figures (also called significant digits) is used to express the precision of a measured or calculated quantity. Significant figures are the digits in a number that carry meaningful information about its precision, including all digits except:

• Leading zeros (zeros before the first non-zero digit), which are only placeholders.
• Trailing zeros in a number without a decimal point, unless they are explicitly measured (often indicated by a bar or scientific notation).

The key rules for determining significant figures are:

• All non-zero digits are significant.
• Zeros between non-zero digits are significant.
• Trailing zeros in a number containing a decimal point are significant.
• Leading zeros are not significant.
• In scientific notation, $a \times 10^n$, all digits in $a$ are significant; the exponent $n$ does not affect the count.

Number 1: 23.023

Let’s analyze the number $23.023$:

• The digits 2, 3, 0, 2, 3 are all present.
• All non-zero digits (2, 3, 2, 3) are significant.
• The zero is between two non-zero digits (3 and 2), so it is significant.
• There are no leading or trailing zeros that are non-significant.

Thus, all five digits are significant.
⇒ Significant figures = 5

Number 2: 0.0003

Analyze $0.0003$:

• The number has leading zeros: 0.000
• Leading zeros are not significant.
• The only non-zero digit is 3.

Therefore, only the digit 3 is significant.
⇒ Significant figures = 1

Number 3: $2.1 \times 10^{-3}$

This is in scientific notation. The rule states that all digits in the coefficient (the part before $\times 10^n$) are significant.

• The coefficient is $2.1$, which has two digits: 2 and 1.
• Both are non-zero and significant.
• The exponent $-3$ does not affect the count of significant figures.

Thus, the number of significant figures is 2.
⇒ Significant figures = 2

Summary of Results:

• $23.023$ → 5 significant figures
• $0.0003$ → 1 significant figure
• $2.1 \times 10^{-3}$ → 2 significant figures

Comparing with the options:
A: 5, 1, 2 → Matches our result
B: 5, 1, 5 → Incorrect (last number has 2, not 5)
C: 5, 5, 2 → Incorrect (second number has 1, not 5)
D: 4, 4, 2 → Incorrect (first and second numbers wrong)

Therefore, the correct answer is A.

Students often make the following mistakes:

• Counting leading zeros: Many students mistakenly count the zeros in $0.0003$ as significant. Remember: leading zeros are never significant.
• Misinterpreting scientific notation: Some confuse the exponent with significant digits. Only the digits in the coefficient matter. For example, $2.1 \times 10^{-3}$ has 2 significant figures, not 5 or 3.
• Overlooking embedded zeros: In $23.023$, the zero is between non-zero digits and is significant. Students sometimes skip it, leading to an incorrect count of 4 instead of 5.

Exam Tip: Always write the number in scientific notation if unsure. This helps isolate the significant digits clearly. For example:
$0.0003 = 3 \times 10^{-4}$ → only the 3 is significant.