Question

Sol. [Key Concept: Express each SI base unit dimensionally in terms of $h,G,c$ (e.g. mass $= \sqrt{\frac{hc}{G}}$) and match the lists.]

$${\lbrack h\rbrack = \left\lbrack ML^{2}T^{- 1} \right\rbrack }{\lbrack c\rbrack = \left\lbrack {LT}^{- 1} \right\rbrack }$$and $\lbrack G\rbrack = \left\lbrack M^{- 1}{\text{ }L}^{3}{\text{ }T}^{- 2} \right\rbrack$

then $\left\lbrack \sqrt{\frac{hc}{G}} \right\rbrack = \left\lbrack \frac{{ML}^{2}{\text{ }T}^{- 1}{LT}^{- 1}}{M^{- 1}{\text{ }L}^{3}{\text{ }T}^{- 2}} \right\rbrack^{1/2} = \lbrack M\rbrack$

$${\left\lbrack \sqrt{\frac{Gh}{c^{5}}} \right\rbrack = \left\lbrack \frac{M^{- 1}{\text{ }L}^{3}{\text{ }T}^{- 2}{ML}^{2}{\text{ }T}^{- 1}}{{\text{ }L}^{5}{\text{ }T}^{- 5}} \right\rbrack^{1/2} = \lbrack T\rbrack }{\left\lbrack \sqrt{\frac{k^{2}{\text{ }L}^{2}c^{3}}{Gh}} \right\rbrack = \left\lbrack \frac{k^{2}{\text{ }L}^{2}{\text{ }L}^{3}{\text{ }T}^{- 3}}{M^{- 1}{\text{ }L}^{3}{\text{ }T}^{- 2}{ML}^{2}{\text{ }T}^{- 1}} \right\rbrack^{1/2} = \lbrack k\rbrack }{\left\lbrack \sqrt{\frac{Gh}{c^{3}}} \right\rbrack = \left\lbrack \frac{M^{- 1}{\text{ }L}^{3}{\text{ }T}^{- 2}{ML}^{2}{\text{ }T}^{- 1}}{{\text{ }L}^{3}{\text{ }T}^{- 3}} \right\rbrack^{1/2} = \lbrack L\rbrack }{A \rightarrow IV,B \rightarrow II,C \rightarrow I,D \rightarrow III }$$