详解
Choice B is the best answer. The last sentence of the passage states that Mr. Peters's mischaracterization of himself makes him act in ways that are not "becoming" for a man of his age. In this context, "becoming" suggests behavior that is appropriate or fitting.
Choices A, C, and D are incorrect because in the context of describing one's behavior, "becoming" means appropriate or fitting, not becoming known (choice A), becoming more advanced (choice C), or simply occurring (choice D).