JEE PYQ: Units & Measurements - Question ID e6d150125da3 (JEE Main 2024)

Core Formula & Concept:

In experimental physics and error analysis, when a physical quantity \( Q \) is expressed as a product or quotient of other measured quantities \( a, b, c \), the relative error (or percentage error) in \( Q \) is determined by combining the relative errors of the individual quantities. The key formula is:

$\frac{\Delta Q}{Q} = \left| n \frac{\Delta a}{a} \right| + \left| m \frac{\Delta b}{b} \right| + \left| p \frac{\Delta c}{c} \right|$

where \( Q = k \cdot a^n \cdot b^m \cdot c^p \) (with \( k \) being a dimensionless constant), and \( \frac{\Delta a}{a}, \frac{\Delta b}{b}, \frac{\Delta c}{c} \) are the relative errors in \( a, b, c \) respectively. The exponents \( n, m, p \) can be positive or negative, and their absolute values are used to ensure the error contributions add up.

In this problem, \( Q = \frac{a^4 b^3}{c^2} \), so the exponents are \( n = 4 \), \( m = 3 \), and \( p = -2 \). The percentage errors in \( a, b, c \) are given as \( 3\% \), \( 4\% \), and \( 5\% \) respectively.

Step-by-Step Derivation:

Step 1: Express the relative error in \( Q \)

Given \( Q = \frac{a^4 b^3}{c^2} \), take the natural logarithm on both sides: $\ln Q = 4 \ln a + 3 \ln b - 2 \ln c$

Step 2: Differentiate both sides

Differentiating with respect to each variable: $\frac{dQ}{Q} = 4 \frac{da}{a} + 3 \frac{db}{b} - 2 \frac{dc}{c}$

Step 3: Convert differentials to errors

Replace differentials with errors (assuming small errors, so \( \Delta \approx d \)): $\frac{\Delta Q}{Q} = 4 \frac{\Delta a}{a} + 3 \frac{\Delta b}{b} + 2 \frac{\Delta c}{c}$ Note: The negative sign for \( c \) becomes positive because errors are always added in magnitude.

Step 4: Substitute percentage errors

Given: $\frac{\Delta a}{a} = 3\% = 0.03$ $\frac{\Delta b}{b} = 4\% = 0.04$ $\frac{\Delta c}{c} = 5\% = 0.05$

Substitute into the error formula: $\frac{\Delta Q}{Q} = 4(0.03) + 3(0.04) + 2(0.05)$ $= 0.12 + 0.12 + 0.10$ $= 0.34$

Step 5: Convert to percentage

$\frac{\Delta Q}{Q} \times 100\% = 0.34 \times 100\% = 34\%$

Conclusion: The percentage error in \( Q \) is 34%, which corresponds to option B.

Common Traps & Exam Tip:

1. Sign Confusion: Students often forget that the exponent of \( c \) is negative in the formula for \( Q \), but the error contribution is always added in magnitude. The negative sign in the exponent does not carry over to the error calculation.

2. Forgetting to Multiply by Exponents: A common mistake is to simply add the percentage errors (3% + 4% + 5% = 12%) without multiplying by the respective exponents. This leads to incorrect results.

3. Misapplying the Formula: Some students confuse this with the formula for errors in sums/differences (where absolute errors add) rather than products/quotients (where relative errors add). Always remember: for products/quotients, relative errors are combined.

Exam Tip: When dealing with percentage errors in products or quotients, always:

1. Identify the exponents of each variable in the formula.
2. Multiply each percentage error by the absolute value of its exponent.
3. Add all contributions to get the total percentage error.