In the density measurement of a cube, the mass and edge length are measured as (10.00 ± 0.10) kg and (0.10 ± 0.01) m, respectively. The error in the measurement of density is :

Units and Measurements PYQ — JEE Main 2019 · 9 Apr · Shift 1 · Q72 with Solution

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Correct answer: C

For a cube of edge length $a$ and mass $m$ ,

\rho = \frac{m}{a^3}

m = (10.00 \pm 0.10)\ \text{kg}, \qquad a = (0.10 \pm 0.01)\ \text{m}

\rho = \frac{10.00}{(0.10)^3} = \frac{10.00}{0.001} = 10000\ \text{kg/m}^3

Since

\rho = m a^{-3},

the fractional error is

\frac{\Delta \rho}{\rho} = \frac{\Delta m}{m} + 3\frac{\Delta a}{a}

Substitute the values:

\frac{\Delta m}{m} = \frac{0.10}{10.00} = 0.01

\frac{\Delta a}{a} = \frac{0.01}{0.10} = 0.10

So,

\frac{\Delta \rho}{\rho} = 0.01 + 3(0.10) = 0.31

Thus, the fractional error is $0.31$ , i.e. 31%.

If needed,

\Delta \rho = 0.31 \times 10000 = 3100\ \text{kg/m}^3

The options are numerical values without percent sign, and the intended answer is clearly the fractional error:

0.31

So the correct option is C.