Force F and density d are related as `F=(alpha)/(beta+sqrtd)`, Then find the dimensions of `alpha` and `beta`

Options

A. `[M^(3//2)L^(-1//2)T^(-2)],[M^(1//2)L^(-3//2)]` respectively
B. `[M^(-3//2)L^(1//2)T^(-2)],[M^(-1//2)L^(3//2)]` respectively
C. `[M^(3//2)L^(-1//2)T^(2)],[M^(-1//2)L^(3//2)]` respectively
D. `[M^(3//2)L^(1//2)T^(-2)],[M^(1//2)L^(3//2)]` respectively

Answer

A

Explanation

The dimensions of left hand side expression should be same as right hand sides expression
`[(alpha)/(beta + sqrt(d))] = [F] = [MLT^(-2)]`
The dimensional for `beta` should be same as the dimenasional formula for `sqrt(b)`
`[beta]= [sqrt(d)]= [ML^(-3)]^(1//2) = [M^(1//2)L^(-3//2)]`
Hence `[alpha] = [beta] [F] = [M^(1//2)L^(-3//2)] [MLT^(-2)]`
`rArr [alpha] = [M^(1//2)L^(-1//2)T^(-2)]`