That's pretty normal for teen/preteen girls! You gotta use discipline & stick to it! If you take her computer, keep it until she earns it back etc... It'll work out, trust me, I've been there! My smart mouth teen daughter is now a 30 yr old Mom of 3, working for Dept. of Homeland Security! She recently apologized for her teen behavior & thanked me for not giving up on her! She's a very successful woman! Believe me, when I say she was a handful, there were days when I wanted to strangle her! If it gets too bad, try counseling!
I find it hard to believe a mother wants her 11 year old gone. Maybe she senses the "unwanted" feeling and therefore acting out. How much time do you spend with your daughter. What does her dad think? Have you tried family counseling? Getting rid of your daughter is a horrible thing. Maybe look at yourself and what you might be doing wrong.
Try to actually discipline her. Take away her things and ground her when she does something you don't like. Parents don't seem to do that anymore! The minute she lies, talks back, brats out, take away stuff. Don't be her friend, be her parent. Don't give her everything she wants; tell her no, and no means no. :)
Let's see, I bet she likes to fix her hair and wear makeup. If she does, take them all away, curling iron too. Tell her when she can act right, she can have one thing back at a time. If she doesn't fix her hair or wear makeup, take away the things she uses, TV, phone, take everything out of her room but the bed. You'll get her attention trust me, works everytime.
Some kids need a good ol' fashioned time out. No hanging with friends, No electronics. To school and back. To make sure she comes straight home if you can pick her up at the end of the day. It will show her she can't do whatever she wants. And then if she gets to out of hand, send her to a counselor.