PYQ + trends

1. TITLE

NEET 2025 Q2: Error Analysis in multi-variable expressions — Full solution + Why they asked this

2. THE ACTUAL NEET QUESTION

NEET 2025 — Question
A physical quantity P is related to four observations a, b, c and d as follows:
$P = a^{3}b^{2} / c\sqrt{d}$
The percentage errors of measurement in a, b, c and d are 1%, 3%, 2%, and 4% respectively. The percentage error in the quantity P is

A) 13%
B) 15%
C) 10%
D) 2%

3. OFFICIAL ANSWER

NEET Answer Key: (A)

4. COMPLETE SOLUTION

For Physics:
- Step-by-step calculation:
Given the expression for the physical quantity:
$P = \frac{a^{3}b^{2}}{c\sqrt{d}}$

Convert the square root to a fractional power:
$P = \frac{a^{3}b^{2}}{c^{1}d^{1/2}}$

Apply the percentage error formula:
Maximum % error in $P = 3(\frac{\Delta a}{a}\times100) + 2(\frac{\Delta b}{b}\times100) + 1(\frac{\Delta c}{c}\times100) + \frac{1}{2}(\frac{\Delta d}{d}\times100)$

Substitute the given percentage error values ($a=1\%$, $b=3\%$, $c=2\%$, $d=4\%$):
$\% P = 3(1\%) + 2(3\%) + 1(2\%) + \frac{1}{2}(4\%)$
$\% P = 3\% + 6\% + 2\% + 2\%$
$\% P = 13\%$

  • Formula used:
    When a quantity is a product or quotient, the maximum percentage error is found by adding the individual percentage errors. For powers, you multiply the percentage error of that variable by its power.

  • Why this approach:
    Errors combine in a specific way to reflect maximum uncertainty. Whether a variable is in the numerator or denominator, its absolute relative error always adds to the total error (errors never subtract or cancel out).

5. WHY NEET ASKED THIS

Explain the examiner's intent:
- What concept were they testing? They were testing the combination of errors rule (specifically the product/quotient rule) and evaluating a student's ability to handle exponents and fractional roots (like $\sqrt{d}$) in error analysis.
- What's the common mistake they expected students to make? The most common mistake is subtracting the percentage errors of the terms in the denominator ($c$ and $d$). Students often incorrectly calculate it as $+3\% + 6\% - 2\% - 2\% = 5\%$. Errors always add up, they never subtract. Another common mistake is forgetting to multiply the error of $d$ by $\frac{1}{2}$ for the square root.
- Which option was the "trap"? Options like 10% or 15% serve as traps for students who either miss the $\frac{1}{2}$ multiplier for the square root of $d$, or miscalculate the simple arithmetic.

6. TREND ANALYSIS

PATTERN ALERT:

Year How they asked Twist
2025 $P = a^{3}b^{2} / c\sqrt{d}$ with errors 1%, 3%, 2%, 4% Included a square root ($\sqrt{d}$) in the denominator, requiring a fractional multiplier of $\frac{1}{2}$.
2019 $X = \frac{A^{2}B^{1/2}}{C^{1/3}D^{3}}$ with errors 1%, 2%, 3%, 4% Multiple fractional powers in both the numerator ($1/2$) and denominator ($1/3$).
2013 $P = \frac{a^{3}b^{2}}{cd}$ with errors 1%, 2%, 3%, 4% Simpler integer powers in both the numerator and denominator.

Prediction for NEET 2026:
Expect a variation where the multi-variable expression is not given as a generic $P = a^x b^y$, but instead hidden within a known physical formula. For example, calculating the percentage error in the estimation of $g$ using $g = \frac{4\pi^{2}L}{T^{2}}$ or Kinetic Energy $K = \frac{1}{2}mv^2$, where you must remember to ignore dimensionless constants like $4\pi^2$ or $\frac{1}{2}$.

7. SIMILAR QUESTIONS TO PRACTICE

"If you understood this, you can solve:"
- NEET 2019 Q25: Percentage error in $X = \frac{A^{2}B^{1/2}}{C^{1/3}D^{3}}$
- NEET 2022 (Re-exam) Q17: Percentage error in $g$ using the formula $g = \frac{4\pi^{2}L}{T^{2}}$
- NEET 1995 Q56: Error estimation in Kinetic Energy ($K = \frac{1}{2}mv^2$)

8. QUICK REVISION FOR THIS TYPE

For Error Analysis in multi-variable expressions, always:

  • [First check] the formula and identify the powers (exponents) of each variable. Convert roots into fractional powers (e.g., $\sqrt{x} = x^{1/2}$). Ignore any exact numerical constants (like 2, $\pi$) as they have zero error.
  • [Then apply] the relative error sum rule. Multiply the percentage error of each variable by its respective power.
  • [Watch out for] the denominator! Errors in measurements always combine cumulatively. Never subtract an error simply because the variable is in the denominator. Add them all together!

9. NCERT BASE

This question tests: NCERT Class 11, Chapter 1 (Units and Measurement), Page 27 (Section 1.3.3 Rules for Determining the Uncertainty in the Results of Arithmetic Calculations).

10. CTA

All PYQs from this chapter → Physics and Measurement Guide
Practice similar questions → Mock exams