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(i)
is small in size so it has high polarising power and 
is also small in size. Compatibility of both the cation and anion are very high. So their lattice energy is also very high. When BeO is dissolved in water it's hydration energy is not sufficient to overcome its lattice energy. So, therefore, it is insoluble in water.
On the other hand,
ions are large in size. Hence 
ion can easily polarise 
ions and making it unstable and because of that lattice energy of 
is not very high and so it is soluble in water.
(ii)
is soluble because 
cation is large in size as compare to 
anion. Size compatibility between them is not good. Therefore 
is unstable. Hence lattice energy during the formation of their lattice is not high So it can be easily overcome by hydration energy. Therefore 
is soluble in water. In case of 
, we know that down the group hydration enthalpy decreases and both the anion and cation have very good size compatibility. So, lattice energy cannot be overcome by hydration energy. As a result, 
is not soluble in water.
(iii) The
ion has high polarising power. It is very small in size as compare to 
ion. So, it has a high tendency to distort the electron cloud around the negative iodide (
) ion. As a result of high polarizability, it has high covalent character than.
Hence it is more soluble in methanol.