Deadlock among Directors

As per provisions of Section 179 of Companies Act, 2013 read with Rule 8 of Companies (Meetings of Board & its Powers) Rules, 2014, the shareholders are authorised to exercise all the powers of the Board in case of a deadlock in Board.

So in your case, a general meeting should be called and another director should be appointed first after which the provisions of section 169 should be followed for removal of Director A.

Can the Director holding 40% shares of the company call an AGM to appoint new Director?
would the provisiona of Quorum not contravened by this?

Well first you in position of shareholder should send notice to company for conducting EGM. The company will be bound to conduct EGM in that case.
Then company will have to send notice to all the members which includes your director A for meeting. If he doesn't show up in EGM then obviously quorum will not be present & then as per Section 103 the meeting will be adjourned to same place same time same day next week and again you will have to send notice of adjourned AGM again atleast 3 days before adjourned EGM.
If Mr. A again doesn't appear again then the members present, i.e., you will constitute the quorum.

Thanks for your prompt response sir. But I am concerned about Quorum. The Co. Act states that there should be atleast 2 members must be present even at adjourned EGM of Pvt Co.
In this case, there are only 2 members in the company who are also directors.

Can Director-B transfer his share to new person to increase number of members, to successfully convene EGM?

Yes its a better idea. if you transfer some of your shares then quorum requirements of 2 members will be met in first meeting itself and thereafter you can appoint bew directors. But remember you will have to send notice to Mr. A about the meeting along with auditors of company and other required person atleast 21 days before meeting (short notice is not possible in your case) and if he does come to meeting then he have majority voting power and this plan might not work.
And remember to issue new shares just after appointment of bew director because Mr. A is still majority shareholder after appointment of new director. Once Mr. A is not a majority shareholder then you can do whatever you want in accordance with provisions of Companies Act 2013.

And do consult with some professional say your CA or CS for validity of such planning and how to execute the same.

You need to tranfer your share to someone to increase the shareholder quorum than only you can appoint other director. rest you can call me for further details.
Pritesh 9812955551