Is it permissible to lie to an Alzheimer’s patient by answering him that he has prayed, even though he has not, in order to avoid his repetitive questions that agitate those around him?

There is no harm in using tawriya (ambiguous speech) with a patient who frequently asks about prayer. For example, one might say to him, "You have prayed," intending that the patient has prayed in the past. This avoids lying, alleviates hardship, and brings comfort to the patient.

If the illness reaches a stage of continuous dementia, then prayer and other acts of worship are not obligatory for him, as accountability is lifted from him, and there is no need to remind him.

However, if his mind sometimes wanders and sometimes he is lucid, then he must perform prayer when he is lucid, and you must remind him if he asks you, and tawriya is forbidden in such a case.