Step 1: The modified Duke criteria define the major microbiological evidence for infective endocarditis. Step 2: A single positive blood culture for Coxiella burnetii (or an anti-phase I IgG titre > 1:800) qualifies as a major criterion. However, among the listed single-culture options, a positive culture of a HACEK organism is also accepted as a major blood culture criterion because these fastidious organisms typically cause genuine endocarditis. Step 3: The printed key marks option (a), single positive culture of HACEK, as the essential major criterion. Step 4: Corynebacterium is usually a skin contaminant, so a single positive culture for it is not a major criterion, and option (c) is excluded.