Question

Match the following:

Choose the correct option:

• A. (1)→(i), (2)→(iii), (3)→(i), (4)→(iv)
• B. (1)→(iii), (2)→(i), (3)→(ii), (4)→(iv)
• C. (1)→(i), (2)→(ii), (3)→(iii), (4)→(iv)
• D. (1)→(iii), (2)→(ii), (3)→(i), (4)→(iv)

Hint:

Always derive dimensions from fundamental definitions rather than memorizing them. This avoids confusion in matching-type questions.

Answer 1

The Correct Option is D

Step 1: Spring constant From Hooke’s law: \[ F = kx \Rightarrow k = \frac{F}{x} \] \[ [k] = \frac{MLT^{-2}}{L} = MT^{-2} \] \[ \Rightarrow (1) \to (iii) \]
Step 2: Thermal conductivity Heat conduction equation: \[ Q = kA\frac{(T_1-T_2)}{\ell}t \] \[ [k] = \frac{ML^2T^{-2}}{LTK \cdot T} = MLT^{-3}K^{-1} \] \[ \Rightarrow (2) \to (ii) \]
Step 3: Boltzmann constant Definition: \[ k_B = \frac{\text{Energy}}{\text{Temperature}} \] \[ [k_B] = \frac{ML^2T^{-2}}{K} = ML^2T^{-2}K^{-1} \] \[ \Rightarrow (3) \to (i) \]
Step 4: Inductance From energy stored in an inductor: \[ U = \frac{1}{2}LI^2 \] \[ [L] = \frac{ML^2T^{-2}}{A^2} = ML^2T^{-2}A^{-2} \] \[ \Rightarrow (4) \to (iv) \] Final Matching: \[ (1)\to(iii),\quad (2)\to(ii),\quad (3)\to(i),\quad (4)\to(iv) \] \[ \boxed{\text{Correct option is (4)}} \]