Here it is
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Fajan's rule is based on more covalent character due to following two reasons

High polarizability of the big anions

High polarizing power of small cations with high oxidation numbers
More is the covalent character lesser is the solubility of the compound.
Now in your question  a ,b and  d can be explained on the basis of charges, oxidation numbers and sizes with Fajan's rule.
However c cannot be explained on the basis of vasant role because barium is bigger in size than magnesium.
This can be explained only on the basis of stable lattice bigger cations can stabilize bigger polyatomic ions
This results in more stable lattice which makes them less soluble in water