Q.48 Physics · Section B ⭐ 360° Option Elimination Active

NEET 2024 Physics Question 48 Solution

Topic: Units and Measurements · Official NTA Booklet Code T1

Question

A force defined by F = αt² + βt acts on a particle at a given time t. The factor which is dimensionless, if α and β are constants, is:

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Verified Against NTA Final Answer Key

Option (4) : αt / β

📚 Units and Measurements Easy

📋 Given Parameters & Problem Setup

Force equation F = αt² + βt

Constants α and β are dimensional constants

Time variable t

⚡

The 30-Second Exam Hall Shortcut: ⚡ Dimensional Ratio Inspection

⏱️ Save ~35 seconds

Both terms have dimension of Force: [αt²] = [βt] -> [αt²] / [βt] = [αt / β] = 1 (Dimensionless)!

1 Principle of Dimensional Homogeneity: In F = αt² + βt, every term added has identical dimensions of Force: [αt²] = [βt] = [F].

2 Divide the first term by the second term: (αt²) / (βt) = αt / β.

3 The ratio of two quantities having identical dimensions is strictly DIMENSIONLESS: [αt / β] = [F] / [F] = 1 = [M⁰L⁰T⁰].

4 Option (4) in 3 seconds!

🛡️

Why You Can 100% Trust This Shortcut: Principle of Dimensional Homogeneity (Terms added in a physical equation have identical dimensions).

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Step-by-Step Rigorous Workflow & Mathematical Proof

Step 1

Principle of Dimensional Homogeneity

According to Fourier’s principle of dimensional homogeneity, physical quantities can be added or equated only if they have the same dimensions. In the force equation F = αt² + βt, both terms on the right must independently have dimensions of force [M L T⁻²]:

[\alpha t^2] = [F] = [M L T^{-2}], \quad [\beta t] = [F] = [M L T^{-2}]

Step 2

Determination of Dimensions of α and β

Solving for the dimensional formulas of α and β:

[\alpha] = \frac{[F]}{[t^2]} = \frac{[M L T^{-2}]}{[T^2]} = [M L T^{-4}]

Step 3

Dimensions of β

Solving for the dimensions of β:

[\beta] = \frac{[F]}{[t]} = \frac{[M L T^{-2}]}{[T]} = [M L T^{-3}]

Step 4

Testing the Ratio αt / β

Evaluating the dimensions of the expression αt / β:

\left[\frac{\alpha t}{\beta}\right] = \frac{[\alpha] [t]}{[\beta]} = \frac{[M L T^{-4}] [T]}{[M L T^{-3}]} = \frac{[M L T^{-3}]}{[M L T^{-3}]} = [M^0 L^0 T^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:

[αβt] = [M² L² T⁻⁶], not dimensionless.

⚠️

Student Fallacy (Why candidates pick this trap): Multiplying dimensions instead of taking ratio.

2 (2)

✗ Incorrect

Verdict & Scientific Reason:

[αβ / t] = [M² L² T⁻⁸], not dimensionless.

⚠️

Student Fallacy (Why candidates pick this trap): Dimensional mismatch.

3 (3)

✗ Incorrect

Verdict & Scientific Reason:

[βt / α] = [T²], has dimensions of time squared.

⚠️

Student Fallacy (Why candidates pick this trap): Inverting the ratio.

4 (4)

✓ 100% Correct

Verdict & Scientific Reason:

Rigorous dimensional cancellation yields [M⁰L⁰T⁰].

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Why this is airtight: None.

⚠️

NTA Trap Alert: Detailed Algebraic Derivation vs Direct Ratio Inspection

Students waste 2 minutes calculating [α] and [β] separately and then testing all four options one by one.

🛡️ How to avoid the -1 mark: Direct shortcut: Since αt² and βt both equal Force, their ratio (αt²)/(βt) = αt/β MUST be dimensionless!

📖

Core NEET Syllabus Standard

Physics Core General Principles · Units and Measurements Standard Benchmark

Core Scientific Principle Tested:

"Terms added together in an equation of physics must have identical dimensions; their ratio is always dimensionless."

Syllabus Focus: Dimensional Analysis and Its Applications

✓ Core NEET UG Syllabus Anchor Syllabus & Academic Standards ↗

Both parts equal Force in size: Divide them out and dimensions die!

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NEET 2024 Physics Question 48 Solution

Question

The 30-Second Exam Hall Shortcut: ⚡ Dimensional Ratio Inspection

Step-by-Step Rigorous Workflow & Mathematical Proof

Principle of Dimensional Homogeneity

Determination of Dimensions of α and β

Dimensions of β

Testing the Ratio αt / β

360° Option Elimination Engine Key Feature

NTA Trap Alert: Detailed Algebraic Derivation vs Direct Ratio Inspection

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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