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. (a) `M prop sqrtc`
B. (b) `M prop sqrtG`
C. (c) `L prop sqrth`
D. (d) `L prop sqrtg`

Answer

A, C, D

Explanation

(a,c,d)
`L prop h^(x)c^(y)G^(z)`
Ddimensionally
`[M^(0)L^(1)T^(0)] = [ML^(2)T^(-1)]^(x)[LT^(-1)^(y)[M^(-1)L^(3)T^(-2)]^(z)`
`M^(0)L^(1)T(0) = M^(x - z) L^(2x + y + 3z) T^(-x - y - 2z)`
:. `x - z = 0`
:. `2x + y + 3z = 1 and - x- y - 2z =0`
On solving we get
`x = (1)/(2), y = -(3)/(2), z = (1)/(2)`
:. `L prop sqrth`
`L prop sqrtG`
`C,D` are correct options
`M prop h^(x)c^(y)G^(z)`
`[M'L^(0)T^(0)] = [ML^(2)T^(-1)]^(x)[LT^(-1)^(y)[M^(-1)L^(3)T^(-2)]^(z)`
:. `M'L^(1)T(0) = M^(x - z) L^(2x + y + 3z) T^(-x - y - 2z)`
`x - z = 1`
`2x + y + 3z = 0 and - x- y - 2z =0`
On solving we get
`x = (1)/(2), y = (1)/(2), z = -(1)/(2)`
:. `M prop sqrtc`
`A` is the correct option.