JEE PYQ: Units & Measurements - Question ID 314851dc22a7 (JEE Main 2022)

Core Formula & Concept:

In experiments and measurements, every physical quantity is associated with some uncertainty or error. When quantities are combined through mathematical operations (like multiplication, division, or exponentiation), the errors propagate in a specific manner. The key concept here is error propagation in functions of multiple variables.

For a general function of the form: $Z = f(A, B, C, \dots)$ the relative error in $Z$, denoted as $\frac{\Delta Z}{Z}$, is determined using the logarithmic differentiation method. This method is particularly useful when $Z$ is expressed as a product or ratio of powers of measured quantities.

The fundamental rule is: If $Z = A^m B^n C^p \cdots$ then the relative error in $Z$ is: $\frac{\Delta Z}{Z} = |m| \frac{\Delta A}{A} + |n| \frac{\Delta B}{B} + |p| \frac{\Delta C}{C} + \cdots$ where $\Delta A, \Delta B, \Delta C$ are the absolute errors in $A, B, C$, respectively.

Note: The exponents become multiplicative factors, and the signs of the exponents do not affect the error (since errors are always added in magnitude).

--- Step-by-Step Derivation:

We are given: $Z = \frac{A^2 B^3}{C^4}$

Step 1: Take the natural logarithm of both sides

$\ln Z = \ln\left(\frac{A^2 B^3}{C^4}\right) = \ln(A^2) + \ln(B^3) - \ln(C^4)$ $\Rightarrow \ln Z = 2\ln A + 3\ln B - 4\ln C$

Step 2: Differentiate both sides with respect to the variables

Differentiating implicitly (treating $\Delta A, \Delta B, \Delta C$ as small changes): $\frac{dZ}{Z} = 2 \cdot \frac{dA}{A} + 3 \cdot \frac{dB}{B} - 4 \cdot \frac{dC}{C}$ In the context of errors, we interpret $dA \rightarrow \Delta A$, $dB \rightarrow \Delta B$, $dC \rightarrow \Delta C$, and take absolute values to ensure errors add up (since errors are always positive in magnitude): $\frac{\Delta Z}{Z} = 2 \frac{\Delta A}{A} + 3 \frac{\Delta B}{B} + 4 \frac{\Delta C}{C}$ Note: The negative sign from $-4\ln C$ becomes positive in the error expression because we are concerned with the magnitude of the error, not its direction.

Step 3: Compare with the given options

The derived expression is: $\frac{\Delta Z}{Z} = \frac{2\Delta A}{A} + \frac{3\Delta B}{B} + \frac{4\Delta C}{C}$ This matches Option C.

--- Common Traps & Exam Tip:

Trap 1: Sign Confusion
Many students mistakenly keep the negative sign from the denominator term $C^4$. They write: $\frac{\Delta Z}{Z} = \frac{2\Delta A}{A} + \frac{3\Delta B}{B} - \frac{4\Delta C}{C}$ This is incorrect because errors are always additive in magnitude. The negative sign in the function does not translate to a negative sign in the error expression.

Trap 2: Forgetting to Multiply by Exponents
Some students add the relative errors directly: $\frac{\Delta A}{A} + \frac{\Delta B}{B} + \frac{\Delta C}{C}$ This ignores the fact that the exponents $2, 3, 4$ amplify the errors. Always multiply each relative error by the absolute value of its exponent.

Exam Tip:
Whenever you see a function involving powers, use logarithmic differentiation. It simplifies the process and reduces the chance of sign errors. Remember: Errors add in magnitude, not direction.

Thus, the correct answer is Option C.