To determine the correct answer for the metals $x$, $y$, and $z$ described in the question, let's analyze each metal chloride based on their solubility properties:
- $x$ is water soluble at room temperature:
- Among the options provided, $\mathrm{CuCl_2}$ and $\mathrm{AlCl_3}$ are known to be soluble in water at room temperature.
- $y$ is sparingly soluble in water at room temperature:
- $\mathrm{AgCl}$ and $\mathrm{PbCl_2}$ are examples of chlorides that are sparingly soluble in water at room temperature.
- Of the provided chlorides, $\mathrm{AgCl}$ has the least solubility, making it the likely candidate for $y$.
- $z$ is soluble in hot water:
- $\mathrm{PbCl_2}$ is more soluble in hot water than at room temperature.
- This is a characteristic property of lead(II) chloride, which fits $z$.
Based on the analysis of solubility:
- $x$ should be $\mathrm{CuCl_2}$ because it is water-soluble at room temperature.
- $y$ is likely $\mathrm{AgCl}$ as it is sparingly soluble in water at room temperature.
- $z$ is $\mathrm{PbCl_2}$, soluble in hot water.
Therefore, the correct answer is: $\mathrm{CuCl_2, AgCl\ and\ PbCl_2}$.