JEE PYQ: Units & Measurements - Question ID cf400a30a886 (JEE Main 2025)

Core Formula & Concept:

When a physical quantity \( Q \) is expressed as a product (or quotient) of other measured quantities \( X, Y, Z \) raised to powers, i.e., \[ Q = X^{a} Y^{b} Z^{c}, \] the fractional error (or relative error) in \( Q \) is determined by the propagation of errors rule. The key formula is: \[ \frac{\Delta Q}{Q} = \left| a \right| \frac{\Delta X}{X} + \left| b \right| \frac{\Delta Y}{Y} + \left| c \right| \frac{\Delta Z}{Z}. \] Here, \( \frac{\Delta X}{X} \), \( \frac{\Delta Y}{Y} \), and \( \frac{\Delta Z}{Z} \) are the fractional errors in \( X, Y, \) and \( Z \) respectively. The absolute values ensure that errors add up in the worst-case scenario (maximum possible error).

Step-by-Step Derivation:

Given: \[ Q = X^{-2} Y^{+\frac{3}{2}} Z^{-\frac{2}{5}}. \] The fractional errors in \( X, Y, \) and \( Z \) are: \[ \frac{\Delta X}{X} = 0.1, \quad \frac{\Delta Y}{Y} = 0.2, \quad \frac{\Delta Z}{Z} = 0.5. \]

Step 1: Identify the exponents
The exponents of \( X, Y, \) and \( Z \) in \( Q \) are: \[ a = -2, \quad b = +\frac{3}{2}, \quad c = -\frac{2}{5}. \]

Step 2: Apply the error propagation formula
The maximum fractional error in \( Q \) is: \[ \frac{\Delta Q}{Q} = \left| a \right| \frac{\Delta X}{X} + \left| b \right| \frac{\Delta Y}{Y} + \left| c \right| \frac{\Delta Z}{Z}. \] Substitute the absolute values of the exponents and the given fractional errors: \[ \frac{\Delta Q}{Q} = \left| -2 \right| \cdot 0.1 + \left| +\frac{3}{2} \right| \cdot 0.2 + \left| -\frac{2}{5} \right| \cdot 0.5. \] Simplify: \[ \frac{\Delta Q}{Q} = 2 \cdot 0.1 + \frac{3}{2} \cdot 0.2 + \frac{2}{5} \cdot 0.5. \]

Step 3: Compute each term
\[ 2 \cdot 0.1 = 0.2, \] \[ \frac{3}{2} \cdot 0.2 = 0.3, \] \[ \frac{2}{5} \cdot 0.5 = 0.2. \]

Step 4: Sum the contributions
\[ \frac{\Delta Q}{Q} = 0.2 + 0.3 + 0.2 = 0.7. \]

Thus, the maximum fractional error in \( Q \) is \( 0.7 \).

Common Traps & Exam Tip:

1. Sign of exponents: Students often forget to take the absolute value of the exponents. Errors always add up, so negative exponents must be treated as positive when calculating fractional error.
2. Fractional exponents: Misapplying the exponents (e.g., squaring or cubing the fractional error instead of multiplying by the exponent) leads to incorrect results. Always multiply the fractional error by the absolute value of the exponent.
3. Maximum error assumption: The question asks for the maximum fractional error. This means all errors are assumed to contribute in the same direction (worst-case scenario). Do not average or subtract errors.
4. Unit consistency: While not directly relevant here, ensure that all quantities are in consistent units when applying error propagation in other problems.

Final Answer: The maximum fractional error in \( Q \) is \( 0.7 \), which corresponds to option C.