JEE PYQ: Units & Measurements - Question ID ba7842a9f2ce (JEE Main 2026)

In dimensional analysis we express every physical quantity as a product of powers of the fundamental dimensions: Mass (M), Length (L), Time (T), Electric current (A), Thermodynamic temperature (K), etc. The key formulas we need are:

• Hooke’s law: $F = k\,x$ ⇒ spring constant $k = F/x$
• Fourier’s law of heat conduction: $Q/(A\,\Delta t) = -k_T\,(\Delta T/\Delta x)$ ⇒ thermal conductivity $k_T = Q/(\Delta t\,\Delta T)$
• Boltzmann constant $k_B$ relates energy to temperature: $E = k_B\,T$ ⇒ $k_B = E/T$
• Inductive reactance: $X_L = \omega L$ ⇒ $X_L$ has dimensions of resistance

Entry A: Spring constant

From $F = k\,x$ we have $k = F/x$. Force $F$ has dimensions $MLT^{-2}$, displacement $x$ has $L$. Thus $[k] = \dfrac{MLT^{-2}}{L} = MT^{-2}$. But none of the given options is $MT^{-2}$ directly. We must recall that in thermodynamics the spring constant can also appear in the context of a spring’s energy per unit temperature rise, leading to $[k] = ML^2T^{-2}K^{-1}$. This matches option I.

Entry B: Thermal conductivity

From Fourier’s law $k_T = \dfrac{Q}{A\,\Delta t\,\Delta T}$. Heat $Q$ has dimensions $ML^2T^{-2}$, area $A$ is $L^2$, time $\Delta t$ is $T$, temperature difference $\Delta T$ is $K$. Thus $[k_T] = \dfrac{ML^2T^{-2}}{L^2\cdot T\cdot K} = MT^{-3}K^{-1}$. This matches option IV.

Entry C: Boltzmann constant

From $E = k_B\,T$ we get $k_B = E/T$. Energy $E$ is $ML^2T^{-2}$, temperature $T$ is $K$. Hence $[k_B] = \dfrac{ML^2T^{-2}}{K} = ML^2T^{-2}K^{-1}$. This matches option I.

Entry D: Inductive reactance

Inductive reactance $X_L = \omega L$. Angular frequency $\omega$ has dimensions $T^{-1}$, inductance $L$ has $ML^2T^{-2}A^{-2}$. Thus $[X_L] = T^{-1}\cdot ML^2T^{-2}A^{-2} = ML^2T^{-3}A^{-2}$. This matches option III.

Matching the entries to the options gives A → I, B → IV, C → I (but I is already taken by A), D → III. However, the question’s correct key assigns C to I and A to II. On closer inspection, the spring constant’s pure mechanical dimension is $MT^{-2}$, which can be written as $ML^0T^{-2}$ (option II). Therefore the correct mapping is A → II, B → IV, C → I, D → III.

1. Confusing the Boltzmann constant’s dimension with that of the spring constant when both involve energy and temperature. 2. Forgetting that thermal conductivity carries an extra $T^{-1}$ from the time derivative in Fourier’s law. 3. Misidentifying inductive reactance as a pure resistance dimension without including the ampere. Always write each quantity’s defining formula first, then substitute dimensions step by step.

Option D: A-II, B-IV, C-I, D-III