Insulinoma produces excess insulin from pancreatic beta cells, leading to fasting hypoglycaemia with symptoms relieved by glucose (Whipple triad). The diagnostic strategy is to catch the tumour secreting insulin while the patient is hypoglycaemic. A monitored prolonged fast lowers glucose, and the failure of insulin to switch off is the cornerstone finding, so the fasting glucose test is central. To confirm the insulin is the body's own and not injected, C-peptide is measured; since it is released molecule-for-molecule with native insulin, a raised C-peptide level locates the source as the pancreas. The insulin to glucose ratio is then derived from these paired values, and an inappropriately high ratio during low glucose points to the tumour. Among the listed choices, only the xylulose test is unrelated to insulin biology. Xylulose handling is assessed in pentose metabolism and pentosuria, with nothing to say about beta-cell function, so it cannot diagnose insulinoma. Eliminating the three genuine endocrine assays leaves the xylulose test as the odd one out.
\[\boxed{\text{Xylulose test}}\]