Step 1: Understanding the reactions.
- The organic compound C$_8$H$_9$NO, when reacted with KOH/BBr$_2$, forms 'P'. This suggests that the compound contains a reactive group such as amide or amine. KOH/BBr$_2$ is known to react with amides to form an acylated derivative.
- The diazotisation of 'P' forms 'Q', indicating that an amine group is present. Diazotisation is a reaction typically involving amines.
- The reaction with CuCN forms 'R', suggesting the formation of an aryl nitrile group.
- The hydrolysis of 'R' to form 'S' suggests that 'R' is an aryl nitrile, which when hydrolyzed (acidic conditions) forms a carboxylic acid.
- 'S' reacts with KMnO$_4$/H$^+$ to form 'T', which contains two types of hydrogen atoms. This reaction indicates oxidation, possibly of a methyl group to a carboxyl group.
Step 2: Identifying compound X.
Based on the given reactions, X must contain a structure that allows these transformations: starting from an amide, undergoing diazotisation, forming a nitrile, and being hydrolyzed to a carboxylic acid. The structure of X fits with CH$_3$C$=$NH, which is an imidic acid derivative.
Step 3: Conclusion.
The correct structure of X is CH$_3$C$=$NH, as it explains all the reactions described in the question. Hence, the correct answer is (3).