Lime reacts exothermally with water to give ‘ A ’ which...

$\mathop{CaO}_{Lime}\overset{Limitedwater}{\to}Ca{\left(OH\right)}_{2}$

The process is known as slaking of lime and the product is slaked lime. It is an exothermic process.

$Ca{\left(OH\right)}_{2}aq$ solution is known as lime water and used for test of ${CO}_{2}$.

$Ca{\left(OH\right)}_{2}\left(aq\right)\overset{{CO}_{2}}{\to}{CaCO}_{3}{CaCO}_{3} + {CO}_{2}\left(excess\right) + {H}_{2}O\overset{}{\to}Ca{\left({HCO}_{3}\right)}_{2}$

Milkiness is due to the formation of calcium carbonate and disappearance of milkiness is due to the formation of calcium bicarbonate.