The energy of a system is given as E(t)=α 3 e -β t , where t is the time and β=0.3 s -1 . The errors in the measurement of α and t are 1.2 \% and 1.6 \% , respectively. At t =5 s , maximum percentage error in the energy is :

Physics · Units, Dimensions & Error Analysis

JEE Main 2025 — 23 January, Evening Shift — Question 52

The energy of a system is given as $E(t)=\alpha^{3} e^{-\beta t}$ , where $t$ is the time and $\beta=0.3 \mathrm{~s}^{-1}$ . The errors in the measurement of $\alpha$ and t are $1.2 \%$ and $1.6 \%$ , respectively. At $\mathrm{t}=5 \mathrm{~s}$ , maximum percentage error in the energy is :

Option A:
0.04
Option B:
0.116
Option C:
0.06
Correct
Option D:
0.084

Answer: C

Step-by-step solution

$\alpha^{3} e^{-\beta t}$

$\ln \mathrm{E}=3 \ln \alpha-\beta \mathrm{t}$

$\left(\frac{\mathrm{dE}}{\mathrm{E}}\right)_{\text {max }}=\frac{3 \mathrm{~d} \alpha}{\alpha}+\beta \frac{\mathrm{dt}}{\mathrm{t}} \times \mathrm{t}$

$=3 \times 1.2 \%+(0.3 \times 1.6 \times 5) \%$

$=6 \%$

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Exam: JEE Main 2025
Paper: 23 January, Evening Shift
Subject: Physics
Chapter: Units, Dimensions & Error Analysis
Topic: Significant Figures and Error Analysis

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Answer: C

Step-by-step solution

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