Q.17 Physics · Section A ⭐ 360° Option Elimination Active

NEET 2024 Physics Question 17 Solution

Topic: Dual Nature of Radiation and Matter · Official NTA Booklet Code T1

Question

The graph which shows the variation of (1/λ²) and its kinetic energy, E is (where λ is de Broglie wavelength of a free particle):

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Verified Against NTA Final Answer Key

Option (2) : Straight line passing through origin with positive slope

📚 Dual Nature of Radiation and Matter Easy

📋 Given Parameters & Problem Setup

de Broglie wavelength λ = h / p

Kinetic energy of free particle E = p² / (2m)

Y-axis variable 1 / λ²

X-axis variable E

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The 30-Second Exam Hall Shortcut: ⚡ Linear Equation Identification

⏱️ Save ~40 seconds

λ = h / √(2mE) -> λ² = h² / (2mE) -> 1/λ² = (2m/h²) · E => Form y = kx -> Straight line through origin!

1 Formula: λ = h / √(2mE).

2 Square both sides: λ² = h² / (2mE).

3 Invert both sides: 1/λ² = (2m / h²) × E.

4 Compare to standard line equation y = mx: y = 1/λ², x = E, slope = 2m/h² = constant > 0.

5 This represents a straight line passing through the origin (0, 0) with constant positive slope.

6 Graph (2) is a straight line through the origin -> Pick Option (2) in under 5 seconds!

🛡️

Why You Can 100% Trust This Shortcut: de Broglie Hypothesis & Matter Wave Relations (λ = h/p = h/√(2mE)).

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Step-by-Step Rigorous Workflow & Mathematical Proof

Step 1

Relation Between de Broglie Wavelength and Kinetic Energy

For a non-relativistic free particle of mass m moving with momentum p and kinetic energy E, the momentum is expressed in terms of kinetic energy as p = √(2mE). The de Broglie wavelength is:

\lambda = \frac{h}{p} = \frac{h}{\sqrt{2mE}}

Step 2

Squaring and Reciprocal Manipulation

Squaring both sides of the relation:

\lambda^2 = \frac{h^2}{2mE}

Step 3

Formulation of Linear Graph Function

Taking the reciprocal gives the function plotted on the vertical axis (1/λ²) in terms of the horizontal axis variable E:

\left(\frac{1}{\lambda^2}\right) = \left(\frac{2m}{h^2}\right) E

Step 4

Geometric Interpretation

Since 2m/h² is a positive constant, the relation has the mathematical form y = k · x with zero intercept (c = 0). The corresponding graph is a straight line passing through the origin with positive slope.

y = kx \quad \text{where } y = \frac{1}{\lambda^2}, \quad x = E, \quad k = \frac{2m}{h^2} > 0

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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:

Incorrect; describes y ∝ √x (which would represent 1/λ vs E, not 1/λ² vs E).

⚠️

Student Fallacy (Why candidates pick this trap): Confusing 1/λ with 1/λ².

2 (2)

✓ 100% Correct

Verdict & Scientific Reason:

Exact linear dependence: 1/λ² = (2m/h²) E.

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Why this is airtight: None.

3 (3)

✗ Incorrect

Verdict & Scientific Reason:

Represents y ∝ x².

⚠️

Student Fallacy (Why candidates pick this trap): Squaring the energy variable mistakenly.

4 (4)

✗ Incorrect

Verdict & Scientific Reason:

Represents y ∝ 1/x (which would describe λ² vs E, not 1/λ² vs E).

⚠️

Student Fallacy (Why candidates pick this trap): Plotting λ² vs E instead of its reciprocal.

⚠️

NTA Trap Alert: Plotting Direct Variable vs Reciprocal Square

Students frequently remember λ ∝ 1/√E and sketch a hyperbolic or curved graph, forgetting that the y-axis is explicitly 1/λ².

🛡️ How to avoid the -1 mark: Always substitute y for the entire y-axis expression: Let y = 1/λ² and x = E -> y = kx is a straight line!

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Core NEET Syllabus Standard

Physics Core Modern Physics · Dual Nature of Radiation and Matter Standard Benchmark

Core Scientific Principle Tested:

"The reciprocal of the square of de Broglie wavelength is directly proportional to kinetic energy: 1/λ² = (2m/h²) E."

Syllabus Focus: Wave Nature of Matter and de Broglie Relation

✓ Core NEET UG Syllabus Anchor Syllabus & Academic Standards ↗

Inverted and Squared makes it Linear and Cleared (1/λ² vs E is a straight line)!

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NEET 2024 Physics Question 17 Solution

Question

The 30-Second Exam Hall Shortcut: ⚡ Linear Equation Identification

Step-by-Step Rigorous Workflow & Mathematical Proof

Relation Between de Broglie Wavelength and Kinetic Energy

Squaring and Reciprocal Manipulation

Formulation of Linear Graph Function

Geometric Interpretation

360° Option Elimination Engine Key Feature

NTA Trap Alert: Plotting Direct Variable vs Reciprocal Square

Core NEET Syllabus Standard

Ready to attempt under real exam pressure?

🔗 Related Questions — Dual Nature of Radiation and Matter

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