Is it true that Carey Mulligan wrote to her headteacher to help her get cast in her Pride and Prejudice?

It may well be that her headmistress knew Julian Fellowes and had either spoken about him, or perhaps he had visited the school she went to, so knew that the headmistress would have a contact address for him. I doubt very much that the headmistress could get Carey seen if there had been no connection at all between them. Showbusiness is full of 'who you know'. I think Carey was a bit cheeky, but it worked for her.