(i) $\mathrm{A} + \mathrm{Na}_{2}\mathrm{CO}_{3} \longrightarrow \mathrm{B} + \mathrm{C}$ (ii) A $\xrightarrow{\mathrm{CO}_2}$ (Milky) C The chemical formula of A and B are respectively

From statement $\mathrm{A} \xrightarrow{\mathrm{CO}_2} \mathrm{C}$ with $\mathrm{C}$ described as milky, $\mathrm{A}$ must be lime water and $\mathrm{C}$ must be the white precipitate formed when carbon dioxide is passed through it. $$\mathrm{Ca(OH)_2 + CO_2 \rightarrow CaCO_3 + H_2O}$$ So $\mathrm{A = Ca(OH)_2}$ and $\mathrm{C = CaCO_3}$. Now use equation $(i)$. $$\mathrm{Ca(OH)_2 + Na_2CO_3 \rightarrow 2NaOH + CaCO_3}$$ Hence $\mathrm{B = NaOH}$. So the pair is $\mathrm{A = Ca(OH)_2}$ and $\mathrm{B = NaOH}$. Rechecking the options, this matches option $\mathrm{(B)}$.