(A) - (III) eloquently → Speaking eloquently (fluently and persuasively) mesmerizes an audience.
(B) - (IV) reluctantly → "After much persuasion" indicates he agreed unwillingly/reluctantly.
(C) - (I) vaguely → Remembering something from "years ago" would be unclear/vague.
(D) - (II) briskly → Walking briskly (quickly and efficiently) shows urgency to reach the crime scene without compromising clues.
Hence, Option 4: (A) - (III), (B) - (IV), (C) - (I), (D) - (II) → Each adverb perfectly matches the context: eloquent speech mesmerizes, reluctant agreement after persuasion, vague memories from long ago, and brisk movement to preserve evidence → correct