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Question:

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 ln(BD) holds true. Then which of the combination is not a meaningful quantity: A²−B²C², (A−C)D, CBD−AD²/C, AB−C

A²−B²C²

(A−C)D

CBD−AD²/C

AB−C

Solution:

The correct option is (A−C)D
Since the product BD is inside the logarithmic function, it must be dimensionless. Hence [B][D] = 1.
Also, from the equality of two quantities in the expression, [C] = [AD]. This expression clearly means that the dimensions of A and C are not the same since D is not dimensionless. Hence A and C cannot be subtracted and hence option (A−C)D is not a meaningful quantity.