Partners authority and liability

Hello

I want to ask that whether I am right in the following sentences? If not then please correct me in cases where am I wrong

1) If partner have actual authority then firm (i.e all individual partners) is liable , otherwise only individual partner is liable
2) If transaction relate to business carried on by firm, then firm is liable, otherwise only individual partner is liable
3) If other party know that the partner had no authority then firm is NOT liable, only individual partner is liable
4) If other party know that partner was a partner then firm is liable, otherwise only individual partner

Thanks