JEE Main 2023 — Units and Dimensions Question with Solution

The dimensional formula of force is

$F=ML{T}^{-2}$

The dimensional formula of velocity is

$V=L{T}^{-1}$

The formula for density is 

 Density $=\frac{M}{L^3}=M{L}^{-3}$

Comparing the density dimensions with the assumed fundamental quantities in the question,
$$\begin{gathered} M{L}^{-3}={\left[F\right]}^x{\left[V\right]}^y{\left[T\right]}^z \\ \Rightarrow M{L}^{-3}={\left[ML{T}^{-2}\right]}^x{\left[L{T}^{-1}\right]}^y[T{]}^z \\ \Rightarrow M{L}^{-3}=M^x{L}^{(x+y)}{T}^{-2x-y+z} \end{gathered}$$

Comparing the exponents,
$$\begin{gathered} x=1 ...\left(i\right) \\ x+y=-3 ...\left(ii\right) \\ -2x-y+z=0 ...\left(iii\right) \end{gathered}$$

Solving equations (i), (ii) & (iii), we get

$y=-4, z=-2$.

The dimensions of density $=F{V}^{-4}{T}^{-2}$.