Q.Why does the solubility of alkaline earth metal hydroxides in water increase down the group?

Solubility and thermal stability of s-block salts are both governed by the balance between lattice enthalpy (how strongly the ions in the solid attract each other) and hydration enthalpy (how strongly the ions are stabilised by water) or, for thermal decomposition, by how strongly the cation polarises the anion. Alkaline-earth hydroxide solubility increases down the group (from almost insoluble $Be(OH)_2$ to soluble $Ba(OH)_2$) because lattice enthalpy — dominated by the small, size-fixed $OH^-$ ion — falls off faster than hydration enthalpy as the cation grows, favouring dissolution; but alkaline-earth carbonate and sulphate solubility DECREASES down the group because, for these bulkier anions, lattice enthalpy changes little with cation size while hydration enthalpy of the growing cation steadily weakens, making dissolution progressively less favourable. The same size/charge-density logic explains why alkali-metal salts (hydroxides, carbonates, nitrates, sulphates) are, group for group, both far more soluble and far more thermally stable than the corresponding alkaline-earth salts: a large, singly-charged, weakly polarising alkali cation neither locks the anion into a rigid lattice nor distorts it enough to promote early decomposition, e.g. $NaOH$/$Na_2CO_3$ dissolve freely and are thermally robust while $Mg(OH)_2$/$MgCO_3$ are only sparingly soluble and decompose at comparatively low temperature. Because polarising power falls steadily down Group 2 as the cation grows, thermal stability of the alkaline-earth carbonates rises in the order $MgCO_3 < CaCO_3 < SrCO_3 < BaCO_3$, so $BaCO_3$ is the most thermally stable of the four. The same competition explains why $BeO$ (very high lattice enthalpy between two small ions) is almost insoluble while $BeSO_4$, whose large sulphate ion keeps the lattice enthalpy comparatively low, dissolves; why, conversely, $BaO$ (large cation with the small oxide ion, weaker lattice enthalpy, favourable hydration) is soluble, while $BaSO_4$ (large cation matched with the large sulphate ion, and by now very low hydration enthalpy for $Ba^{2+}$) is insoluble; and why covalent $LiI$ dissolves in a low-polarity organic solvent like ethanol far better than the more ionic $KI$.