The structure of ‘A’ and type of isomerism in the above reaction are respectively: 

1. Prop-1-en-2-ol, metamerism
2. Prop-1-en-1-ol, tautomerism
3. Prop-2-en-2-ol, geometrical isomerism
4. Prop-1-en-2-ol, tautomerism

Hint: Oxymercuration-Demercuration reaction

The chemical reaction is as follows:

 CH3-CCH 1%HgSO440%H2SO4CH3-C OH
  |=CH
 |H Prop-1-en-2-olketo-enol tautomerismCH3-CO
||-CH3
  Propyne                                                                              Acetone

[A] is prop-1-en-2-ol, which undergo tautomerism to form acetone.