What is the chemical name of – (a) Lime water (b) The white insoluble solid formed on reaction of carbon dioxide with lime water.

The qualitative analysis of carbon dioxide gas uses the interaction of lime water solution with carbon dioxide. This reaction results in the development of a white precipitate, which dissolves when too much gas is passed over it.
(A) Calcium hydroxide is the chemical term for lime water. Calcium hydroxide is represented chemically by the symbol Ca(OH)₂.
b)We add diluted hydrochloric acid to the provided salt for the qualitative analysis of carbon dioxide, and the development of the briskness of effervescence is then visible.
The developed gas is then passed through lime water (Ca(OH)2) for gas confirmation, and a white precipitate forms, turning the lime water milky.
The following response will support the assertion:

$\mathrm{Ca}{\left(\mathrm{OH}\right)}_2+{\mathrm{CO}}_2\to {\mathrm{CaCO}}_3+{\mathrm{H}}_2\mathrm{O}$

Calcium carbonate is the name for the precipitate that is generated and is insoluble. And CaCO₃ can be used to denote the formula.
However, when too much carbon dioxide is added to the resultant solution, the milkiness goes away and calcium bicarbonate forms, giving rise to a clear solution.

The following reaction will justify:

${\mathrm{CaCO}}_3+{\mathrm{H}}_2\mathrm{O}+{\mathrm{CO}}_2\to \mathrm{Ca}{\left({\mathrm{HCO}}_3\right)}_{2\left(aq\right)}$

Due to the fact that calcium bicarbonate is a soluble calcium salt, the precipitate progressively dissolves in the solution. It also develops when the water has to have its temporary hardness removed.
As a result, calcium bicarbonate is often used to temporarily soften water.
Hence option A is correct.