Step 1: The key to this question is matching each sulphonamide with how long it persists in plasma, since duration of action tracks the elimination half-life.
Step 2: Sulfamethiazole and sulfamethoxazole are cleared relatively quickly, so they fall into the short-to-intermediate band and need repeated dosing. Sulfadiazine sits in the intermediate range.
Step 3: Sulphadoxine stands apart because its half-life stretches to several days, placing it in the ultra-long-acting category alongside sulfamethopyrazine. This is why a single dose can cover a malaria treatment course.
Step 4: Comparing all four, sulphadoxine clearly has the slowest clearance and therefore the longest action.
\[\boxed{\text{Sulphadoxine}}\]