# 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 ? Option 1) Option 2) Option 3) Option 4)

$A.D= C \ln \left ( B.D \right )$

Hence $\left [ B.D \right ]= are \: dimensionless$

$\left [ B \right ]= \left [ \frac{1}{D} \right ]...............(1)$

&   $\left [ A.D \right ]= \left [ C \right ]...............(2)$

Relevent Concept here is :

Dimension -

The power to which fundamental quantities must be raised in order to express the given physical quantities.

-

Option 1)

$A^{2}-B^{2}\, C^{2}$

$\left [ A \right ]= \left [ \frac{C}{D} \right ]= \left [ C.B \right ]$

$\therefore$ Dimensions of A & B.C are same.

Option 2)

$\frac{A-C}{D}$

Dimension of $\left [ A \right ]and \left [ C \right ]$ are not same hence its meaningless.

Option 3)

$\frac{\left [ A \right ]}{\left [ B \right ]}-\left [ C \right ]= \left [ AD \right ]-\left [ C \right ]$

Since A D & C are of same dimension hence its meaningful.

Option 4)

$\frac{C}{BD}-\frac{AD^{2}}{C}$

Its meaningful.

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