The major product of the following reaction is:

The given compound contains a phenolic$$\ \ -OH$$ and an aliphatic$$\ \ -OH$$

$$K_2CO_3$$ is a weak base. It will selectively deprotonate the phenolic$$\ \ -OH$$ as the resulting phenoxide ion $$Ar\ -\ O^{-\ }$$ is stabilized by resonance, while the aliphatic$$\ \ -OH$$ remains mostly undissociated.

The phenoxide ion then acts as a nucleophile and performs an $$S_N2$$ reaction with $$CH_3I$$ to form $$Ar-OCH_3$$, which is the compound given in option A.