The sentence 'He's looking after his (sick, ill) mother' is asking you to choose between the words 'sick' and 'ill' to describe the mother's condition. Both words can be used to indicate that someone is not in good health, but there are slight differences in their usage:
Sick : This word is often used in casual conversations and is commonly understood in many English-speaking regions. It generally refers to any kind of illness.
Ill : This term can be considered more formal than 'sick.' It is often used in medical contexts or more serious situations.
In everyday speech in regions like the United States, 'sick' might be more commonly used, while 'ill' might be used more in UK English or in formal situations. Either choice could work, but it depends on the context you want to convey. If the situation is informal, 'sick' might be the appropriate choice. If you intend to convey a more serious or formal context, 'ill' could be a better fit.
In summary, your choice between 'sick' and 'ill' may depend on the formality of the context and the regional variation of English you are aiming for.