$(a)$ $A$ solution of a substance $'X'$ is used for testing carbon dioxide. Write the equation of the reaction of $'X'$ with carbon dioxide.
$(b)$ How is $'X'$ obtained? Write the chemical equation.

(N/A) The substance $'X'$ is lime water,which is calcium hydroxide $\left[ Ca(OH)_{2} \right]$.
The chemical equation for the reaction of lime water with carbon dioxide is:
$Ca(OH)_{2}(aq) + CO_{2}(g) \longrightarrow CaCO_{3}(s) + H_{2}O(l)$
$(b)$ $'X'$ (calcium hydroxide) is obtained by the reaction of quick lime (calcium oxide) with water (slaking of lime).
The chemical equation is:
$CaO(s) + H_{2}O(l) \longrightarrow Ca(OH)_{2}(aq)$