If we consider that $\frac{1}{6}$, in place of $\frac{1}{12}$, mass of carbon atom is taken to be the relative atomic mass unit, the mass of one mole of a substance will

The correct option is C Remain unchanged
Let's call the new unit amu'. Taking this to be the standard, the mass of 1 atom would be 1/2 times the old amu - hence, 1 amu = 2 amu', but hold on, the mass of 1 mole of carbon will be the mass of carbon in amu' * mass of amu' * avagadro number
= 2 amu' * 1/2 (mass of amu in g) * N = the old mass itself!
So mass remains unchainged.