First of all Jee main is the national level engineering entrance exam conducted by cbse in online and offline modes every year for aspiring students for engineering colleges. If you have written wrong father's name don't worry it will not hamper anything. You can correct it going on the official site or give an application to correct it. 

It really does. A mistake in your name, your father's name, or any initial in the JEE form, exam paper, or anything related to it can create great problems. Therefore, you should always take utmost care to feed in the right information. Rectification of these mistakes can be a costly affair, or an impossible affair with disappointing consequences.