A, B, C and D are four different physical quantities having different dimensions. None of them is dimensionless. But we know that the equation `AD= C 1n(BD)` holds true. Then which of the combination is not a meaningful quantity :-

Options

A. `A^(2)-B^(2)C`
B. `((A-C))/(D)`
C. `(A)/(B)-C`
D. `(C )/(BD)-(AC^2)/(C )`

Answer

B, D

Explanation

The product of B and D is dimensionless and the product of A and D has same dimensions as C. Dimensionally we can also write :
`B propto (1)/(D)`
`AD= C`
(i) `A^(2)-(BC)^(2) implies ((C )/(D))^(2) - ((C )/(D))^(2)` meaningful
(ii) `((A-C))/(D)` The dimensions of A and C are not same, hence this expression in meaningless
(iii) `(A)/(B)-C implies (AD)/(BD)-C" or "(C )/(BD)-C`
BD is dimensionless thus both the terms `C"/"BD` and C have the same dimensions
(iv) `(C )/(BD)-(AD^2)/(C )`
`implies (C )/(BD)-(AD^2)/(AD)`
`implies (C )/(BD)-D`
The dimensions of C and D are not same, thus this combination does not give a meaningful quantity.