Match List I with List II. The force F acting on a particle in a medium is given by the equation F = A e^ -Bt + C D + x^2 , where t is time and x is distance. List I List II (A) B (I) [ M L ^3 ~T ⁻²] (B) D (II) [ M L T ⁻¹] (C) A B (III) [ L ^2] (D) C (IV) [ T ⁻¹] Choose the correct answer from the options given below:

Step-by-step solution

According to the principle of homogeneity of dimensions: 1. The argument of an exponential function must be dimensionless. [Bt] = [ M ^0 L ^0 T ^0] [B] = 1 [ T ] = [ T ⁻¹] Thus, (A) matches with (IV). 2. Quantities added together must have the same dimensions. In the denominator (D + x^2) : [D] = [x^2] = [ L ^2] Thus, (B) matches with (III). 3. Every term in an equation must have the same dimensions as the whole. For the first term: [A e^ -Bt ] = [F] [A] = [ M L T ⁻²] Then, [ A B ] = [ M L T ⁻²] [ T ⁻¹] = [ M L T ⁻