JEE MAIN - Physics (2016 - 10th April Morning Slot - No. 19)
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
$${{\left( {A - C} \right)} \over D}$$
$${A \over B} - C$$
$${C \over {BD}} - {{A{D^2}} \over C}$$
Explanation
Given,
AD = c ln (BD)
As log is dimensionless.
So, [BD] = 1 $$ \Rightarrow $$ [B] = $${1 \over {\left[ D \right]}}$$
And [AD] = [C]
Now checking options one by one
(a) [B2 C2] = [B2] [A2 D2] = [A2] [B2 D2] = [A2]
$$ \therefore $$ This is meaningful.
(b) $$\left( {{{A - C} \over D}} \right)$$ is not meaningful.
As dimension of A $$ \ne $$ dimension of C
Hence (A $$-$$ C) is not possible.
(c) $$\left[ {{A \over B}} \right]$$ = [AD] = [C]
$$ \therefore $$ $${{A \over B}}$$ $$-$$ C is meaningful.
(d) $$\left[ {{C \over {BD}}} \right]$$ = $${{\left[ C \right]} \over {\left[ {BD} \right]}}$$ = $${{\left[ C \right]} \over 1}$$ = [C]
$$\left[ {{{A{D^2}} \over C}} \right]$$ = $${{\left[ {AD} \right]\left[ D \right]} \over {\left[ C \right]}}$$ = $${{\left[ C \right]\left[ D \right]} \over {\left[ C \right]}}$$ = [D]
As dimension of C and D are not same so it is not meaning ful.
AD = c ln (BD)
As log is dimensionless.
So, [BD] = 1 $$ \Rightarrow $$ [B] = $${1 \over {\left[ D \right]}}$$
And [AD] = [C]
Now checking options one by one
(a) [B2 C2] = [B2] [A2 D2] = [A2] [B2 D2] = [A2]
$$ \therefore $$ This is meaningful.
(b) $$\left( {{{A - C} \over D}} \right)$$ is not meaningful.
As dimension of A $$ \ne $$ dimension of C
Hence (A $$-$$ C) is not possible.
(c) $$\left[ {{A \over B}} \right]$$ = [AD] = [C]
$$ \therefore $$ $${{A \over B}}$$ $$-$$ C is meaningful.
(d) $$\left[ {{C \over {BD}}} \right]$$ = $${{\left[ C \right]} \over {\left[ {BD} \right]}}$$ = $${{\left[ C \right]} \over 1}$$ = [C]
$$\left[ {{{A{D^2}} \over C}} \right]$$ = $${{\left[ {AD} \right]\left[ D \right]} \over {\left[ C \right]}}$$ = $${{\left[ C \right]\left[ D \right]} \over {\left[ C \right]}}$$ = [D]
As dimension of C and D are not same so it is not meaning ful.
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