This question tests your understanding of verb patterns after certain verbs, specifically the verb "hate".

Concept: The verb "hate" can be followed by either a gerund (verb + ing) or an infinitive (to + verb), but the meaning can change slightly.

Gerund (verb + ing): We use a gerund after "hate" to talk about an action that we find unpleasant or that we dislike in a general sense. It often refers to a real, experienced, or habitual action.

Infinitive (to + verb): We use an infinitive after "hate" to talk about a specific occasion or to be more emphatic about the unpleasantness. It can sometimes refer to a potential or future action.

Analysis of the Sentence: The sentence is "He hates _____ when he is having his after-lunch nap." The clause "when he is having his after-lunch nap" describes a specific, recurring situation. The feeling of hatred is directed at the experience of the action happening to him (being disturbed), not at his own action of disturbing someone else.

Step-by-Step Evaluation of Options:

Step 1: "being disturbed" - This is a gerund in the passive voice. It correctly means "the act of someone disturbing him." This fits perfectly because he dislikes the experience of this action during his nap.

Step 2: "having been disturbed" - This is a perfect gerund. It refers to an action that was completed in the past before the main verb ("hates"). The sentence is about hating the disturbance during the nap, not after it was already over.

Step 3: "to have disturbed" - This is a perfect infinitive. It implies he performed the action of disturbing someone else and now hates that he did it. This does not match the context of the sentence.

Step 4: "to disturb" - This is a simple infinitive. It implies that he hates to perform the action of disturbing someone else. Again, the context shows he is the one being disturbed, not the one doing the disturbing.

Final Answer: The most appropriate word is being disturbed.

Related Topic: Gerunds and Infinitives

Certain verbs in English are followed strictly by a gerund, others by an infinitive, and some by either. The meaning can remain the same (e.g., "I started to read" / "I started reading") or can change significantly (e.g., "I forgot to buy milk" vs. "I forgot buying milk"). Verbs that often take a gerund include dislike, enjoy, avoid, and suggest. Verbs that often take an infinitive include want, need, plan, and promise.

Related Topic: Passive Voice

The passive voice is used when the focus is on the action itself or the recipient of the action, not on who is performing the action. The structure is [be + past participle]. In this case, "being disturbed" uses the passive voice because "he" is receiving the action of the verb "disturb".

Formulae:

Verb + Gerund: $\mathrm{hate}+\mathrm{verb}+\mathrm{ing}$ (e.g., hate running)

Verb + Infinitive: $\mathrm{hate}+\mathrm{to}+\mathrm{verb}$ (e.g., hate to run)

Passive Gerund: $\mathrm{being}+\mathrm{past}\mathrm{participle}$ (e.g., being told)