An aqueous compound of an inorganic compound `(X)` shows the following reactions:
a. it decolourises and acidified `KMnO_(4)` solution accompanied by the evolution of oxygen.
b. it liberates `I_(2)` from an acidified `KI` solution.
c. It gives a brown precipitate with alkaline `KMnO_(4)` solution with evolution of oxygen.
d. It removes black stains from old oil paintings. Identify `(X)` and give chemical equtions for the reaction at steps (a) to (d).

Explanation

In reactions (a) and (c), (X) acts as a reducing agent, while in (b), it acts as an oxidising agent. Thus these reactions indicate that `(X)` is `H_(2)O_(2)` which is further confirmed from reactions `(d)`.
a. `underset("coloured")underset("Purple")(2KMnO_(4))+3H_(2)SO_(4)+5H_(2)O_(2)toK_(2)SO_(4)+underset("Colourless")(2MnSO_(4))`
b. `2KI+H_(2)SO_(4)+H_(2)O_(2)toK_(2)SO_(4)+I_(2)+2H_(2)O`
c. `2KMnO_(4)+3H_(2)O_(2)to2KOH+underset("Brown")(2MnO_(2)+2H_(2)O_(2)+3O_(2)`
d. `underset("black")(PbS)+4H_(2)O_(2)tounderset("White")(PbSO_(4))+4H_(2)O`