Salt ($\text{X}$) is soluble in water. Salt ($\text{Y}$) is sparingly soluble in water. Salt ($\text{Z}$) is soluble only in hot water. $\text{X}, \text{Y}, \text{Z}$ respectively are.
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Solubility rules for Group 1 metal ions ($\text{Ag}^+, \text{Pb}^{2+}, \text{Hg}_2^{2+}$) are essential. $\text{PbCl}_2$ is uniquely distinguished by its significant solubility increase upon heating.
$\text{X}$: Soluble in cold water. $\text{AlCl}_3$ is highly soluble.
$\text{Y}$: Sparingly soluble in cold water. $\text{AgCl}$ is insoluble/sparingly soluble.
$\text{Z}$: Soluble in hot water. $\text{PbCl}_2$ is characteristic of this behavior.
The correct sequence is $\text{AlCl}_3, \text{AgCl}, \text{PbCl}_2$.