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MCQEasyJEE 2024Significant Figures & Error Analysis

JEE Physics 2024 Question with Solution

A physical quantity QQ is found to depend on quantities aa, bb, cc by the relation Q=a4b3c2Q = \frac{a^4b^3}{c^2}. The percentage error in aa, bb, cc are 3%3\%, 4%4\%, and 5%5\% respectively. Then, the percentage error in QQ is:

  • A

    66%66\%

  • B

    43%43\%

  • C

    34%34\%

  • D

    14%14\%

Answer

Correct answer:C

Step-by-step solution

Standard Method

Given: Q=a4b3c2Q = \frac{a^4 b^3}{c^2} and the percentage errors in a,b,ca, b, c are 3%3\%, 4%4\%, and 5%5\% respectively.

Find: The percentage error in QQ.

For a derived quantity, the percentage error is obtained by adding the percentage errors multiplied by the absolute values of their powers.

So,

ΔQQ=4Δaa+3Δbb+2Δcc\frac{\Delta Q}{Q} = 4\frac{\Delta a}{a} + 3\frac{\Delta b}{b} + 2\frac{\Delta c}{c}

Substituting the given percentage errors,

ΔQQ×100=4×3%+3×4%+2×5%\frac{\Delta Q}{Q} \times 100 = 4 \times 3\% + 3 \times 4\% + 2 \times 5\%

Therefore,

12%+12%+10%=34%12\% + 12\% + 10\% = 34\%

Therefore, the percentage error in QQ is 34%34\%. The correct option is C.

Detailed Breakdown

Given: Q=a4b3c2Q = \frac{a^4 b^3}{c^2}.

Find: Percentage error in QQ.

Contribution due to aa:

4×3%=12%4 \times 3\% = 12\%

Contribution due to bb:

3×4%=12%3 \times 4\% = 12\%

Contribution due to cc:

2×5%=10%2 \times 5\% = 10\%

Adding all contributions,

12%+12%+10%=34%12\% + 12\% + 10\% = 34\%

Hence, the percentage error in QQ is 34%34\%.

Common mistakes

  • Taking the denominator term with a negative sign in error calculation is incorrect. In percentage error propagation, powers are added by magnitude, so for c−2c^{-2} the contribution is 2×5%2 \times 5\%, not negative.

  • Using the original percentage errors directly without multiplying by the powers is wrong. Since aa and bb have powers 44 and 33, their contributions become 12%12\% and 12%12\% respectively.

  • Confusing fractional error with percentage error can lead to mistakes. First apply the exponent rule to relative errors, then convert consistently into percentage form before adding.

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