JEE PYQ: Units & Measurements - Question ID 24243f0d904d (JEE Main 2018)

Density ($\rho$) of a material is defined as the mass ($m$) per unit volume ($V$). For a cube, the volume is calculated as the cube of its side length ($L$). Thus, the formula for density is:

When dealing with errors in measurements, the relative error (or percentage error) in a derived quantity like density depends on the relative errors in the measured quantities (mass and length). The key concept here is the propagation of errors for multiplication/division and exponentiation:

• For a product or quotient of quantities, the relative errors add up.
• If a quantity is raised to a power ($n$), its relative error is multiplied by $|n|$.

Mathematically, if $Z = \frac{X}{Y^n}$, then the relative error in $Z$ is:

Given:

• Relative error in mass ($\frac{\Delta m}{m}$) = 1.5% = 0.015
• Relative error in length ($\frac{\Delta L}{L}$) = 1% = 0.01

The density is given by:

To find the relative error in density ($\frac{\Delta \rho}{\rho}$), we apply the error propagation rules:

1. The relative error in $m$ is $\frac{\Delta m}{m}$. Since mass is in the numerator, its relative error contributes directly.
2. The relative error in $L^3$ is $3 \cdot \frac{\Delta L}{L}$ because $L$ is raised to the power of 3. Since $L^3$ is in the denominator, its relative error also contributes directly (but with a positive sign due to division).

Thus, the total relative error in density is:

Substitute the given values:

Convert the relative error to a percentage:

Therefore, the maximum error in determining the density is 4.5%.

Students often make the following mistakes in this question:

1. Ignoring the exponent in volume: Some students forget that the side length is cubed in the volume formula and only multiply the length error by 1 instead of 3. This leads to an incorrect answer of 2.5% (Option B).
2. Sign errors in error propagation: While the relative errors add up for division, some students mistakenly subtract the errors, leading to incorrect results.
3. Confusing absolute and relative errors: The question provides relative errors, but some students treat them as absolute errors, leading to confusion in calculations.
4. Rounding off errors prematurely: Students might round off intermediate steps (e.g., 0.015 + 0.03 = 0.045 as 0.05), which can lead to selecting the wrong option (e.g., 5%, which is not listed).

Exam Tip: Always write down the formula for the derived quantity first, then apply the error propagation rules systematically. For quantities raised to a power, remember to multiply the relative error by the absolute value of the exponent. Double-check your calculations to avoid arithmetic mistakes.