Manager is trying to extend my notice period.

Niehter the manager nor the company can force you to work beyond the notice period even if there is a clause mentioned in the employment offer letter.

It is contrary to the section 27 of Indian contract act.

If the management is still insisting to work beyond the notice period you may plan to take it up legally.

Because of no strict country level policies and lack integrity in most Indian companies. You will find employees being exploited this way. If you don't agree to them then they might not release your experience letter, also give bad feedback in the future.

Now if your resignation was accepted in the system count 90 days from that and your term ends there.

I handed over my resignation to my employer over an email as I would like to go for higher studies. The standard notice period is 3 months and the managaer is asking me to work extended notice. There is following clause in my letter of appointment: 

The Notice period for the employment would be three months and may be extended at the discretion of the Company. Employee has to receive clearance from the human resource department before leaving the organization. The Company reserves the right to decide the relieving date of the employee based on the ongoing activities and satisfactory handing over of work.

During my appointment they had taken cheque bearing 3 months of my CTC without any date. It will not be given back to me along with the holded salary if my manager doesn't approve of my relieving.

What should I do as I want to be relieved from my duties within standard notice period of 3 months and receive my f&f and relieving letter along with the cheque?

As already suggested such condition is not legally enforceable.

well advised.