Solubility and thermal stability both increase drown the group for (A.M.`=` Alkaline metal`,`
`A.E.M.=`Alkaline earth metal) `:`

Options

A. Hydrixide of A.E.M. [`M(OH_(2)`]
B. Fluorides of A.M. (MF)
C. Perchlorates of A.M. (`MClO_(4)`)
D. Hydrides of N-family (`MH_(3)`)

Answer

To solve the question regarding the solubility and thermal stability of compounds as we move down the groups of alkaline metals (AM) and alkaline earth metals (AEM), we will analyze the options provided and determine which one correctly shows an increase in both solubility and thermal stability.

### Step-by-Step Solution:

1. **Understanding the Groups:**
- Alkaline metals (Group 1) include elements like Lithium (Li), Sodium (Na), Potassium (K), etc.
- Alkaline earth metals (Group 2) include elements like Magnesium (Mg), Calcium (Ca), Strontium (Sr), etc.

2. **Analyzing Hydroxides of AEM:**
- Hydroxides of alkaline earth metals (e.g., Mg(OH)₂) become more soluble and thermally stable as we move down the group.
- For example, the solubility of hydroxides increases from Mg(OH)₂ to Ba(OH)₂.

3. **Analyzing Fluorides of AM:**
- Fluorides of alkaline metals (e.g., NaF) do not show a consistent increase in solubility or thermal stability down the group.
- Generally, the solubility of fluorides decreases down the group.

4. **Analyzing Chlorides of AM:**
- Chlorides of alkaline metals (e.g., NaCl) also do not show an increase in solubility or thermal stability down the group.
- The trend is similar to that of fluorides.

5. **Analyzing Hydrides of AM:**
- Hydrides of alkaline metals (e.g., LiH) do not exhibit an increase in solubility or thermal stability down the group.
- The stability of hydrides decreases as we move down the group.

6. **Conclusion:**
- The only option that shows an increase in both solubility and thermal stability down the group is the hydroxides of alkaline earth metals (AEM).

### Final Answer:
The correct answer is the hydroxides of alkaline earth metals (AEM), as both solubility and thermal stability increase down the group.

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