The second sentence in the question already contains a verb. Thus, the answer must be a non finite verb. Further, the first question suggests that the answer must describe the state in which the subject of the first question was found in the armchair. Thus, the answer must also be an adjective that means 'asleep'.
Option A is incorrect. 'sleep' is a verb. Having two verbs in one sentence will make it grammatically incorrect. Thus, it is the incorrect answer.
Option B is incorrect. 'to sleep' is an infinitive. Thus, it is a non finite verb, but, infinitives are generally used as a noun or an adverb and not an adjective. Thus, it does not describe the subject of the first question, and is not the correct answer.
Option C is incorrect. 'sleeping' here, means 'asleep'. Further, it is also a non finite verb. Here, 'sleeping' describes the state of the subject of the first question, that is, it shows that the person was 'asleep' or was in the state of 'sleeping'.
Option D is incorrect. 'slept' is a verb. Having two verbs in one sentence will make it grammatically incorrect. Thus, it is the incorrect answer..