NEET 2024 Re Q9: Finding dimensions of unknown constants (e.g., A² /B) — Full solution + Why they asked this

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1. TITLE

NEET 2024 Re Q9: Finding dimensions of unknown constants (e.g., A² /B) — Full solution + Why they asked this

2. THE ACTUAL NEET QUESTION

NEET 2024 Re — Question
The potential energy of a particle moving along x-direction varies as $V=\frac{Ax^{2}}{\sqrt{x}+B}$. The dimensions of $A^{2}/B$ are:
A) $[M^{3/2}L^{1/2}T^{-3}]$
B) $[M^{1/2}LT^{-3}]$
C) $[M^{2}L^{1/2}T^{-4}]$
D) $[ML^{2}T^{-4}]$

3. OFFICIAL ANSWER

NEET Answer Key: (C)

4. COMPLETE SOLUTION

Step-by-step calculation:
1. Find the dimension of B: According to the Principle of Homogeneity of Dimensions, physical quantities can only be added or subtracted if they have the exact same dimensions.
In the denominator ($\sqrt{x} + B$), $x$ represents distance (dimension $[L]$).
Therefore, $\sqrt{x}$ has the dimension $[L^{1/2}]$.
This means $B$ must also have the exact same dimension:
$[B] = [L^{1/2}]$.
2. Find the dimension of A: Now, look at the whole equation $V=\frac{Ax^{2}}{\sqrt{x}+B}$. The dimensions on the left side (Potential Energy) must equal the dimensions on the right side.
Potential energy ($V$) has the same dimensions as work/energy: $[ML^{2}T^{-2}]$.
Setting up the dimensional equation:
$[ML^{2}T^{-2}] = \frac{[A][L^{2}]}{[L^{1/2}]}$
Solving for $[A]$:
$[A] = \frac{[ML^{2}T^{-2}][L^{1/2}]}{[L^{2}]} = [ML^{1/2}T^{-2}]$.
3. Calculate $A^{2}/B$:
Substitute the dimensions of $A$ and $B$ into the requested expression:
$\frac{[A]^{2}}{[B]} = \frac{[ML^{1/2}T^{-2}]^{2}}{[L^{1/2}]}$
$= \frac{[M^{2}L^{1}T^{-4}]}{[L^{1/2}]}$
$= [M^{2}L^{1/2}T^{-4}]$.

Formula used:
Principle of Homogeneity: $[LHS] = [RHS]$ and terms added/subtracted must have identical dimensions.

Why this approach:
This is the fastest method. Always start with the denominator or any terms being directly added/subtracted to find the easiest unknown constant first (in this case, $B$). Then, use the overall equation to find the multiplier constant ($A$).

5. WHY NEET ASKED THIS

Explain the examiner's intent:
- Concept tested: The Principle of Homogeneity of Dimensions, specifically verifying if students know how to handle addition/subtraction of quantities with dimensions.
- Common mistake expected: Messing up the fractional exponents (roots) during the final division step, such as incorrectly dividing $L^1$ by $L^{1/2}$.
- The "trap": Examiners asked for $A^2/B$, not just $A$ or $A/B$. Students rushing through the exam often find $A$ and mistakenly look for it in the options, or forget to square $A$ in the final calculation.

6. TREND ANALYSIS

Prediction for NEET 2026:
Expect variations involving trigonometric or exponential functions (e.g., $y = A \sin(Bt + Cx)$), where you must remember that the argument of special functions must be entirely dimensionless to find the constants $B$ and $C$.

7. SIMILAR QUESTIONS TO PRACTICE

"If you understood this, you can solve:"
- NEET 2024 Q6: A force defined by $F=\alpha t^{2}+\beta t$ acts on a particle. Find the dimensionless factor.
- NEET 2006 Q43: The velocity $v$ of a particle at time $t$ is given by $v=at+\frac{b}{t+c}$. Find dimensions of $a, b, c$.
- NEET 1996 Q50: An equation is given as $(P+\frac{a}{V^{2}})=b\frac{\theta}{V}$. Find the dimensions of $a$.

8. QUICK REVISION FOR THIS TYPE

For finding dimensions of unknown constant questions, always:

• [First check] For terms being added or subtracted (like $\sqrt{x} + B$ or $t + c$). They MUST have the identical dimensions.
• [Then apply] The Principle of Homogeneity to the entire equation—meaning the overall LHS dimensions must equal the overall RHS dimensions—to find the remaining multipliers.
• [Watch out for] Fractional powers (like square roots) and always double-check exactly what the question asks you to find at the end (e.g., $A^2/B$ instead of just finding $A$).

9. NCERT BASE

This question tests: NCERT Class 11, Chapter 1 (Units and Measurement), Page 8 (Checking the Dimensional Consistency of Equations).

10. CTA

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