Planck's constant `h`, speed of light `c` and gravitational constant `G`are used to form a unit of length `L` and a unit of mass `M`. Then the correct option `(s)` is `(are)`

Options

A. `M prop sqrt(c )`
B. `M prop sqrt(G )`
C. `L prop sqrt(h )`
D. `L prop sqrt(G)`

Answer

A, C, D

Explanation

`[M]= [h]^(p) [C ]^(q) [ G]^( r )`
`[M] = [ M^(1) L^(2) LT^(-1)]^(p) [ L^(1) T^(-1)]^(p) [L^(1) T^(-1)]^(q) [m^(-1) L^(3) T^(-2)]^( r )`
`[M] = [M]^(p)-r [L ]^( 2p + q + 3r) [T]^( -p -q - 2 r)`
:. `p - r = 1` ……(i)
`2p + q + 3r = 0` ......(ii)
`-p -q -2r = 0` .....(iii)
On solving (i) , (ii) and (iii) , we get
`P = (1)/(2) , r = (-1)/(2) and q = (1)/(2) rArr [M] prop sqrt(h)`
`[M] prop sqrt( C)`
`[M] prop (1) sqrt( G)`
Similarly for `[L]`
`p - r = 0` ....(iv)
`2 p + q + 3r = 1` .... (v)
`-p -q - 2r = 0 ` ....(vi) On solving (iv) ,(v) and( vi)
`P = (1)/(2) , q = (-3)/(2) , r = (1)/(2) rAr [L] prop sqrt( h)`
`[L] prop C^(3//2)`
`[L] prop sqrt(G)`