Question : A,B, C and D are four different 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?
a) A²-B²C²
b) (A-C)/D
c) A/B - C
d) C/BD - AD²/C
Doubt by Suhani
Solution :
A, B, C and D all are having different dimensions.
None of them is dimensionless.
None of them is dimensionless.
AD=C ln(BD)
Anything with log natural (ln) is a pure number so it must be a dimensionless quantity.
So
[BD]=[M0L0T0]
[B]=1/[D] — (1)
Anything with log natural (ln) is a pure number so it must be a dimensionless quantity.
So
[BD]=[M0L0T0]
[B]=1/[D] — (1)
Also
[AD] = [C] [ln(BD)]
[AD] = [C][M0L0T0]
[AD]=[C]
[C] = [AD] — (2)
Multiplying equation (1) with (2)
[BC] = [A] — (3)
Now, lets analyse each option one by one
a) A²-B²C²
=[A]²-[B]²[C]²
=[A²]-[BC]²
=[A²]-[A²] [Using equation (3)]
Quantities of same dimensions can be subtracted, hence the expression is meaningful.
b) (A-C)/D
In question it is already given that all quantities A, B, C and D are having different dimensions and we know only quantities of same kind can be added or subtracted.
Here A and C are having different dimensions but still subtracted, which is not possible. Hence
=[A²]-[BC]²
=[A²]-[A²] [Using equation (3)]
Quantities of same dimensions can be subtracted, hence the expression is meaningful.
b) (A-C)/D
In question it is already given that all quantities A, B, C and D are having different dimensions and we know only quantities of same kind can be added or subtracted.
Here A and C are having different dimensions but still subtracted, which is not possible. Hence
(A-C)/D is not a meaningful expression.
c) A/B - C
=[A]/[B] - [C]
=[BC]/[B] - [C] [Using equation (3)]
=[C]-[C]
Quantities of same dimensions can be subtracted, hence the expression is meaningful.
d) C/BD - AD²/C
=[C]/[BD] - [A][D]²/[C]
=[C]/[B][D]-[AD][D]/[C]
=[AD]/[B][D]-[C][D]/[C] [Using equation (2)]
=[A][D]/[B][D]-[D]
=[A]/[B]-[D]
= [C]-[D] [Using equation (3)]
d) C/BD - AD²/C
=[C]/[BD] - [A][D]²/[C]
=[C]/[B][D]-[AD][D]/[C]
=[AD]/[B][D]-[C][D]/[C] [Using equation (2)]
=[A][D]/[B][D]-[D]
=[A]/[B]-[D]
= [C]-[D] [Using equation (3)]
Here C and D are having different dimensions but still subtracted, which is not possible. Hence C/BD - AD²/C is not a meaningful expression.
Hence, b) and d) are correct options.