Q.34 Physics ⭐ 360° Option Elimination Active

NEET 2023 Physics Question 34 Solution

Topic: Units and Measurements · Official NTA Booklet Code G5

Question

A metal wire has mass (0.4 ± 0.002) g, radius (0.3 ± 0.001) mm and length (5 ± 0.02) cm. The maximum possible percentage error in the measurement of density will nearly be:

✓

Verified Against NTA Final Answer Key

Option (4) : 1.6%

📚 Units and Measurements Easy

📋 Given Parameters & Problem Setup

Mass (m) (0.4 ± 0.002) g

Radius (r) (0.3 ± 0.001) mm

Length (L) (5 ± 0.02) cm

Formula for density (ρ) ρ = m / (π r² L)

⚡

The 30-Second Exam Hall Shortcut: ⚡ Error Propagation Sum: %Δρ = %Δm + 2(%Δr) + %ΔL

⏱️ Save ~45 seconds

%Δm = (0.002/0.4)×100 = 0.5%. 2(%Δr) = 2×(0.001/0.3)×100 = 0.67%. %ΔL = (0.02/5)×100 = 0.4%. Total = 0.5 + 0.67 + 0.4 = 1.57% ≈ 1.6% in 20 seconds!

1 Density of cylindrical wire: ρ = m / V = m / (π r² L).

2 Take fractional errors: Δρ/ρ = Δm/m + 2(Δr/r) + ΔL/L.

3 Calculate % error in mass: (0.002 / 0.4) × 100 = 0.5%.

4 Calculate % error in radius (multiplied by power 2): 2 × (0.001 / 0.3) × 100 = 2 × 0.333% = 0.667%.

5 Calculate % error in length: (0.02 / 5) × 100 = 0.4%.

6 Add all maximum errors: 0.5% + 0.667% + 0.4% = 1.567% ≈ 1.6%. Option (4) locked!

🛡️

Why You Can 100% Trust This Shortcut: Experimental Physics · Error Analysis (In a product/quotient of powers, maximum fractional error is the sum of fractional errors weighted by their respective exponents).

📐

Step-by-Step Rigorous Workflow & Mathematical Proof

Step 1

Mathematical Formulation of Density of a Cylindrical Wire

A wire is modeled as a uniform cylinder of mass m, radius r, and length L. Its volume is V = π r² L. The density ρ is defined as mass per unit volume:

\rho = \frac{m}{V} = \frac{m}{\pi r^2 L}

Step 2

Error Propagation Formula by Logarithmic Differentiation

Taking the natural logarithm and differentiating to obtain the maximum relative (fractional) error, where errors always add up to give the upper bound:

\frac{\Delta \rho}{\rho} = \frac{\Delta m}{m} + 2\,\frac{\Delta r}{r} + \frac{\Delta L}{L}

Step 3

Individual Percentage Error Computations

Convert each measurement uncertainty into percentage error (% = Δx / x × 100):

\begin{aligned} \text{\% Error in } m &= \frac{0.002}{0.4} \times 100\% = 0.5\% \\[6pt] \text{\% Error in } r &= \frac{0.001}{0.3} \times 100\% = \frac{1}{3}\% \approx 0.333\% \\[6pt] \text{\% Error in } L &= \frac{0.02}{5} \times 100\% = 0.4\% \end{aligned}

Step 4

Summation to Determine Maximum Percentage Error in Density

Substitute the percentage errors into the total error formula (noting the power of 2 for radius):

\frac{\Delta \rho}{\rho} \times 100\% = 0.5\% + 2\left(\frac{1}{3}\%\right) + 0.4\% = 0.5\% + 0.667\% + 0.4\% = 1.567\% \approx 1.6\%

⭐

360° Option Elimination Engine Key Feature

Diagnose every choice: Know exactly why the right option works and why distractors fail.

4-Option Deep Scan

1 (1)

✗ Incorrect

Verdict & Scientific Reason:

Underestimates error by omitting the multiplier 2 on radius.

⚠️

Student Fallacy (Why candidates pick this trap): Candidate forgot the power of 2 on radius: 0.5 + 0.33 + 0.4 = 1.23% or rounded down incorrectly.

2 (2)

✗ Incorrect

Verdict & Scientific Reason:

Omitted the factor of 2 in radius error completely (0.5 + 0.33 + 0.4 = 1.23% ≈ 1.2%).

⚠️

Student Fallacy (Why candidates pick this trap): Candidate treated radius as power 1 instead of power 2.

3 (3)

✗ Incorrect

Verdict & Scientific Reason:

Arithmetic approximation error.

⚠️

Student Fallacy (Why candidates pick this trap): Candidate rounded intermediate steps prematurely.

4 (4)

✓ 100% Correct

Verdict & Scientific Reason:

Accurate maximum percentage error: 1.567% rounds to 1.6%.

💡

Why this is airtight: None. Candidate correctly doubled the percentage error of radius.

⚠️

NTA Trap Alert: Forgetting to Double the Radius Error (r² in Volume Formula)

Because volume is V = π r² L, radius appears as r². The power rule states Δ(r²)/r² = 2(Δr/r). Forgetting the factor of 2 yields ~1.2% (Option 2), which is an intentional NTA trap.

🛡️ How to avoid the -1 mark: Always inspect powers: If a variable is squared, its fractional error is multiplied by 2.

📖

Core NEET Syllabus Standard

National Eligibility cum Entrance Test (NEET UG) · Units and Measurements Standard Benchmark

Core Scientific Principle Tested:

"In formula X = A^a B^b / C^c, %ΔX = a(%ΔA) + b(%ΔB) + c(%ΔC)."

Syllabus Focus: Combination and Propagation of Errors

Preserved in Rationalized Curriculum Syllabus & Academic Standards ↗

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NEET 2023 Physics Question 34 Solution

Question

The 30-Second Exam Hall Shortcut: ⚡ Error Propagation Sum: %Δρ = %Δm + 2(%Δr) + %ΔL

Step-by-Step Rigorous Workflow & Mathematical Proof

Mathematical Formulation of Density of a Cylindrical Wire

Error Propagation Formula by Logarithmic Differentiation

Individual Percentage Error Computations

Summation to Determine Maximum Percentage Error in Density

360° Option Elimination Engine Key Feature

NTA Trap Alert: Forgetting to Double the Radius Error (r² in Volume Formula)

Core NEET Syllabus Standard

Ready to attempt under real exam pressure?

🔗 Related Questions — Units and Measurements

Vernier Calliper Negative Zero Error & Full Marks Bonus Audit

Dimensional Analysis of Combined Thermal Radiation Constants (sigma_s * k_B^-1 * b)

Vernier Callipers Least Count, Positive Zero Error & Corrected Reading

Percentage Error in Physical Quantity P = a³ b² / (c √d)

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