The shyness can be worked on, repeatedly every time you go out together you can ask your daughter to do little jobs, give her 50p at the supermarket and make her buy something, go through the til a few along from yours and be audibly polite . Don't ever let her off the hook by answering for her when she is asked questions by adults in a social setting, she will simply stay silent until you do the work. Don't force her to speak but make her realise that until she does she will be squirming, even an acknowledging nod counts as communicating to begin with. Make her tell you the eye colour of the person who she is talking too, looking at the floor, or away from the face is classic. She will probably always be shy, but she needs help to get along.
Remember children who say "I can't, I am shy." To an adult are far removed from shy, they are actually confronting an adult with refusal, shy kids are often complicit as they can't abide drawing attention.
The worrying part of your story is the blank sheet of sums, this suggests her shyness has gone one step further, where she will not engage with the teacher in any way but is blocking out the teachers existence in her space. My son briefly did this, he would sit with empty paper, invisible to all, particularly if he missed an instruction, no matter how insignificant, it would throw him off course and he quite literally could not ask for help. He had one great teacher who pulled him out of himself, gently gently.
He is at grammar now, finally this year at parents evening a teacher said, "I guess if he hasn't been keen to put his hand up, or involve himself in debates for the last ten years we are not going to change him now, but when he does,talk it's always interesting and intelligent." Or words v similar. We all breathed a sigh of relief as "should be more involved in class" is repeated on report slips year after year. He is happy to chat to people now, can ask for help after lesson (not in lesson with a big hand in the air gesture) and will raise his hand to answer questions occasionally and can now answer when picked without falling to pieces. However this did not come easily, naturally or without upset, you absolutely have to persevere.
At school ask if she can hand out or collect books, pens etc. She will have to pass something to the teacher so has silent but essential communication with her.
Do not force her to speak though and make sure her teachers don't, the more she is forced, the more often a question is repeated at her the smaller inside she will become and the more likely she is to avoid all contact.
Good luck, it's hard to watch your child being last in every queue, have questions answered for them by other often arrogant kids and get overlooked by teachers, but you can help her. Academic assistance alone is not the answer but will help her results of course, but lots of communication anxiety homework should help her hugely in class.