Step 1: Match each named test with its purpose. Brady's reagent (2,4-dinitrophenylhydrazine) detects aldehydes and ketones; Tollen's reagent detects aldehydes and reducing sugars; Lucas reagent distinguishes primary, secondary and tertiary alcohols.
Step 2: None of those act on amines, so eliminate options (i), (ii) and (iv).
Step 3: The remaining choice, the Hinsberg test, treats the amine with benzenesulphonyl chloride. Primary amines yield a base-soluble sulphonamide, secondary amines a base-insoluble sulphonamide, and tertiary amines give no reaction.
Step 4: This three-way discrimination is exactly what is needed, so the answer is \[ \boxed{\text{Hinsberg test}} \] option (iii).