An unknown alcohol is treated with 'Lucas reagent' to distinguish amongst $1^o,2^o$ and $3^o$ alcohols. Which alcohol reacts faster and by which mechanism?

The correct option is B $3^o$ alcohol by $S_{N}1$
The reaction of an alcohol with Lucas reagent is mostly $S_{N}1$ reaction and the rate of reaction is directly proportional to the stability of carbocation formed in the reaction.
Since $3^o$ alcohol forms most stable carbocation, hence it will react faster.