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NVAEasyJEE 2025Significant Figures & Error Analysis

JEE Physics 2025 Question with Solution

A physical quantity C is related to four other quantities p, q, r and s as follows C=pq2r3sC = \frac{pq^2}{r^3 \sqrt{s}} The percentage errors in the measurement of p, q, r and s are 1%1\%, 2%2\%, 3%3\% and 2%2\% respectively. The percentage error in the measurement of C will be _____ %\%.

Answer

Correct answer:15

Step-by-step solution

Standard Method

Given: C=pq2r3sC = \frac{pq^2}{r^3\sqrt{s}}

The percentage errors are:

  • in pp = 1%1\%
  • in qq = 2%2\%
  • in rr = 3%3\%
  • in ss = 2%2\%

Find: The percentage error in CC.

For a quantity of the form C=paqbrcsdC = p^a q^b r^c s^d, the maximum percentage error is the sum of the absolute exponents multiplied by the corresponding percentage errors.

Rewrite the expression as

C=p1q2r−3s−1/2C = p^1 q^2 r^{-3} s^{-1/2}

Therefore,

% error in C=∣1∣(1%)+∣2∣(2%)+∣−3∣(3%)+∣−12∣(2%)\%\,\text{error in } C = |1|(1\%) + |2|(2\%) + |{-3}|(3\%) + \left|{-\frac{1}{2}}\right|(2\%)

Now calculate:

% error in C=1%+4%+9%+1%=15%\%\,\text{error in } C = 1\% + 4\% + 9\% + 1\% = 15\%

Therefore, the percentage error in the measurement of CC is 15%15\%. Hence the numerical answer is 15.

Using exponent form explicitly

Given: C=pq2r3sC = \frac{pq^2}{r^3\sqrt{s}}

Find: Maximum percentage error in CC.

Since s=s1/2\sqrt{s} = s^{1/2}, we write

C=pq2r3s1/2=p1q2r−3s−1/2C = \frac{pq^2}{r^3 s^{1/2}} = p^1 q^2 r^{-3} s^{-1/2}

Now use the rule:

ΔCC×100=∣1∣(Δpp×100)+∣2∣(Δqq×100)+∣−3∣(Δrr×100)+∣−12∣(Δss×100)\frac{\Delta C}{C} \times 100 = |1|\left(\frac{\Delta p}{p} \times 100\right) + |2|\left(\frac{\Delta q}{q} \times 100\right) + |{-3}|\left(\frac{\Delta r}{r} \times 100\right) + \left|{-\frac{1}{2}}\right|\left(\frac{\Delta s}{s} \times 100\right)

Substitute the given values:

ΔCC×100=1(1)+2(2)+3(3)+12(2)\frac{\Delta C}{C} \times 100 = 1(1) + 2(2) + 3(3) + \frac{1}{2}(2) =1+4+9+1=15= 1 + 4 + 9 + 1 = 15

Therefore, the maximum percentage error in CC is 15%15\%.

Common mistakes

  • Using the exponent of rr as negative and subtracting its contribution. In error analysis, percentage errors are added with the absolute values of exponents. So use ∣−3∣=3|{-3}| = 3, not −3-3.

  • Taking the error contribution of s\sqrt{s} as 2%2\% instead of half of it. Since s=s1/2\sqrt{s} = s^{1/2}, the contribution is 12×2%=1%\frac{1}{2} \times 2\% = 1\%.

  • Multiplying the percentage errors incorrectly for q2q^2. Because the power of qq is 22, its contribution is 2×2%=4%2 \times 2\% = 4\%, not 2%2\%.

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