The most effective way to make your mum believe that you are responsible is by making her trust you. Trust is earned and keep in mind that you cannot manipulate people to trust you. For more on responsibility, visit http://www.wikihow.com/Be-Responsible.
Your not responisble if you have to prove it to her..but do your chores and your homework without her telling you to..take your own initiative...if you don't know what to do then your not responsible yet !!