When \(MnO_2\) is fused with \(KOH\), a coloured compound is formed. The product and its colour is:

\(K_2MnO_4\), purple green

The correct answer is option 1. \(K_2MnO_4\), purple green.

When \(MnO_2\) is fused with \(KOH\), a purple green coloured compound is formed. The compound is potassium manganate (\(K_2MnO_4\)). The formation of \(K_2MnO_4\) can be explained by the following reaction:

\(2MnO_2 + 4KOH + O_2 \longrightarrow 2K_2MnO_4 + 2H_2O\)

In this reaction, \(MnO_2\) is oxidized to \(K_2MnO_4\), and \(O_2\) is reduced to \(H_2O\). The purple green colour of \(K_2MnO_4\) is due to the presence of \(Mn(VII)\) in the compound.

The other options are incorrect.

Option (2), \(KMnO_4\), is incorrect because \(KMnO_4\) is a purple coloured compound.

Option (3), \(Mn_2O_3\), is incorrect because \(Mn_2O_3\) is a brown coloured compound.

Option (4), \(Mn_3O_4\), is incorrect because \(Mn_3O_4\) is a black coloured compound.