JEE PYQ: Units & Measurements - Question ID 16f84086c81c (JEE Main 2026)

Core Formula & Concept:

In the study of electrical circuits, the three fundamental passive elements are:

• Resistance ($R$),
• Inductance ($L$), and
• Capacitance ($C$).

Each of these has well-defined dimensional formulas in terms of mass ($M$), length ($L$), time ($T$), and electric current ($A$):

• Resistance: $[R] = \mathrm{ML}^2 \mathrm{T}^{-3} \mathrm{A}^{-2}$
• Inductance: $[L] = \mathrm{ML}^2 \mathrm{T}^{-2} \mathrm{A}^{-2}$
• Capacitance: $[C] = \mathrm{M}^{-1} \mathrm{L}^{-2} \mathrm{T}^4 \mathrm{A}^2$

The question asks us to identify which combination of $R$, $L$, and $C$ yields the dimensional formula $\mathrm{ML}^2 \mathrm{T}^{-4} \mathrm{A}^{-2}$. To solve this, we will compute the dimensions of each option and compare them to the target.

Step-by-Step Derivation:

Step 1: Write down the dimensional formulas of $R$, $L$, and $C$.

$[R] = \mathrm{ML}^2 \mathrm{T}^{-3} \mathrm{A}^{-2}$ $[L] = \mathrm{ML}^2 \mathrm{T}^{-2} \mathrm{A}^{-2}$ $[C] = \mathrm{M}^{-1} \mathrm{L}^{-2} \mathrm{T}^4 \mathrm{A}^2$

Step 2: Compute the dimensions of the product $LC$.

$[LC] = [L] \cdot [C] = \left(\mathrm{ML}^2 \mathrm{T}^{-2} \mathrm{A}^{-2}\right) \cdot \left(\mathrm{M}^{-1} \mathrm{L}^{-2} \mathrm{T}^4 \mathrm{A}^2\right)$ $= \mathrm{M}^{1-1} \mathrm{L}^{2-2} \mathrm{T}^{-2+4} \mathrm{A}^{-2+2} = \mathrm{M}^0 \mathrm{L}^0 \mathrm{T}^2 \mathrm{A}^0 = \mathrm{T}^2$

Step 3: Compute the dimensions of $\sqrt{LC}$.

Since $[LC] = \mathrm{T}^2$, taking the square root gives: $[\sqrt{LC}] = \mathrm{T}$

Step 4: Compute the dimensions of $\frac{R}{\sqrt{LC}}$ (Option A).

$\left[\frac{R}{\sqrt{LC}}\right] = \frac{[R]}{[\sqrt{LC}]} = \frac{\mathrm{ML}^2 \mathrm{T}^{-3} \mathrm{A}^{-2}}{\mathrm{T}} = \mathrm{ML}^2 \mathrm{T}^{-4} \mathrm{A}^{-2}$

Step 5: Verify that this matches the target dimensional formula.

The computed dimensions $\mathrm{ML}^2 \mathrm{T}^{-4} \mathrm{A}^{-2}$ exactly match the given formula.

Step 6: Check other options for completeness.

Option B: $\frac{R}{LC}$

$\left[\frac{R}{LC}\right] = \frac{[R]}{[LC]} = \frac{\mathrm{ML}^2 \mathrm{T}^{-3} \mathrm{A}^{-2}}{\mathrm{T}^2} = \mathrm{ML}^2 \mathrm{T}^{-5} \mathrm{A}^{-2} \quad (\text{Does not match})$

Option C: $\frac{C}{\sqrt{LR}}$

First compute $[LR]$: $$[LR] = [L] \cdot [R] = \left(\mathrm{ML}^2 \mathrm{T}^{-2} \mathrm{A}^{-2}\right) \cdot \left(\mathrm{ML}^2 \mathrm{T}^{-3} \mathrm{A}^{-2}\right) = \mathrm{M}^2 \mathrm{L}^4 \mathrm{T}^{-5} \mathrm{A}^{-4}$$ Then $[\sqrt{LR}] = \mathrm{M} \mathrm{L}^2 \mathrm{T}^{-2.5} \mathrm{A}^{-2}$ Now: $$\left[\frac{C}{\sqrt{LR}}\right] = \frac{[C]}{[\sqrt{LR}]} = \frac{\mathrm{M}^{-1} \mathrm{L}^{-2} \mathrm{T}^4 \mathrm{A}^2}{\mathrm{M} \mathrm{L}^2 \mathrm{T}^{-2.5} \mathrm{A}^{-2}} = \mathrm{M}^{-2} \mathrm{L}^{-4} \mathrm{T}^{6.5} \mathrm{A}^4 \quad (\text{Does not match})$$

Option D: $\frac{1}{R} \sqrt{\frac{L}{C}}$

First compute $\left[\frac{L}{C}\right]$: $$\left[\frac{L}{C}\right] = \frac{[L]}{[C]} = \frac{\mathrm{ML}^2 \mathrm{T}^{-2} \mathrm{A}^{-2}}{\mathrm{M}^{-1} \mathrm{L}^{-2} \mathrm{T}^4 \mathrm{A}^2} = \mathrm{M}^2 \mathrm{L}^4 \mathrm{T}^{-6} \mathrm{A}^{-4}$$ Then $\left[\sqrt{\frac{L}{C}}\right] = \mathrm{M} \mathrm{L}^2 \mathrm{T}^{-3} \mathrm{A}^{-2}$ Now: $$\left[\frac{1}{R} \sqrt{\frac{L}{C}}\right] = \frac{\left[\sqrt{\frac{L}{C}}\right]}{[R]} = \frac{\mathrm{M} \mathrm{L}^2 \mathrm{T}^{-3} \mathrm{A}^{-2}}{\mathrm{ML}^2 \mathrm{T}^{-3} \mathrm{A}^{-2}} = \mathrm{M}^0 \mathrm{L}^0 \mathrm{T}^0 \mathrm{A}^0 = 1 \quad (\text{Dimensionless, does not match})$$ Common Traps & Exam Tip:

Students often make the following mistakes:

• Incorrectly computing dimensions of $LC$: Forgetting that $[LC] = \mathrm{T}^2$ and not $\mathrm{T}^4$ or another power.
• Miscounting exponents in multiplication/division: Especially when dealing with negative exponents in capacitance ($[C] = \mathrm{M}^{-1} \mathrm{L}^{-2} \mathrm{T}^4 \mathrm{A}^2$).
• Confusing $\sqrt{LC}$ with $LC$: Not taking the square root leads to an incorrect time dimension.
• Overlooking units in complex expressions: For example, in Option D, students may not realize the entire expression becomes dimensionless.

Exam Tip: Always write down the dimensional formula of each quantity involved before combining them. Double-check each arithmetic operation on exponents, especially when multiplying or dividing dimensions.