Sort by receptor family: Group the options by signalling type and the odd one out is the answer.
TSH, LH and MSH are all anterior-pituitary-axis peptide hormones that work through G-protein coupled receptors. They activate Gs, raise cAMP, and stimulate protein kinase A. So all three share the GPCR-cAMP route.
Insulin is different. Its receptor is a receptor tyrosine kinase, a heterotetramer of 2 alpha (extracellular, ligand-binding) and 2 beta (transmembrane, with kinase domains) subunits held by disulfide bonds. Insulin binding switches on the beta-subunit tyrosine kinase, causing autophosphorylation and recruitment of IRS proteins, driving metabolic effects.
So insulin is the drug/hormone acting via a tyrosine kinase receptor.
Pearl: growth factors, insulin and IGF-1 use RTKs; most peptide hormones (TSH, LH, MSH, ACTH, glucagon) use GPCRs.
Ref: K. D. Tripathi, 7th Edition, receptor pharmacology.