b
(b) \(L \propto {v^x}{A^y}{F^z}\) \( \Rightarrow \) \(L = k{v^x}{A^y}{F^z}\)

Putting the dimensions in the above relation

\([M{L^2}{T^{ - 1}}] = k{[L{T^{ - 1}}]^x}{[L{T^{ - 2}}]^y}{[ML{T^{ - 2}}]^z}\)

\(⇒\) \([M{L^2}{T^{ - 1}}] = k[{M^z}{L^{x + y + z}}{T^{ - x - 2y - 2z}}]\)

Comparing the powers of \(M,\,L\) and \(T\)

\(z = 1\) …\((i)\)

\(x + y + z = 2\) …\((ii)\)

\( - x - 2y - 2z = - 1\) …\((iii)\)

On solving \((i)\), \((ii)\) and \((iii)\) \(x = 3,\,y = - 2,\,z = 1\)

So dimension of \(L\) in terms of \(v,\,A\) and \(f\)
\([L] = [F{v^3}{A^{ - 2}}]\)