An expression for a dimensionless quantity P is given by $$P = {\alpha \over \beta }{\log _e}\left( {{{kt} \over {\beta x}}} \right)$$; where $$\alpha$$ and $$\beta$$ are constants, x is distance; k is Boltzmann constant and t is the temperature. Then the dimensions of $$\alpha$$ will be :

If $$Z = {{{A^2}{B^3}} \over {{C^4}}}$$, then the relative error in Z will be :

Identify the pair of physical quantities that have same dimensions:

Identify the pair of physical quantities which have different dimensions: