The salient feature of this question is that the dying person is an 11-year old child, rather than an adult. Our sympathy is aroused and we wish to protect the child from pain, especially the pain of knowing that s/he is about to die. I think it is important to first ask the question: if the dying person was an adult, would it be appropriate to lie about the prognosis? And I think the answer is "No. The adult has a right to be told the truth about their prognosis if they ask." So, if an adult has that right, why not a child? Some would say that a child "can't handle the information." But I think that depends on the child, and many 11-year olds might be able to handle the information as well as adults, and might use it to e.g. make final statements or requests. The work of anthropologist Myra Bluebond-Langner on dying children shows that children often handle the topic of their own death as maturely (or more maturely) as adults do. The beloved family doctor could probably have made a better decision.
Read another response by Miriam Solomon
An 11-year-old child lies on the operating table, dying from an accident. He asks his doctor if he is going to die. The doctor says "no", knowing the child will be dead in minutes. I say the doctor (it is not me) did nothing wrong. What good does it do for a child to be told he will be dead in a few minutes. This beloved family doctor has been conflicted over this 40-year-old problem and teared up when telling someone (me) for the first time.