Question

MY TEACHER TOLD US THAT NOW THE WORD "SINCERELY " IS NOT USED IN LETTERS NOW ...IS THAT RIGHT ?

Answer 1

I don't know why your teacher said like that, maybe there could be any reason of her saying this.
But formally sincerely is used in letters. Sincerely is used when you begin with, e.g. Dear Mr. Sidhant, in this case you can use "yours sincerely". But when you begin a letter with, Dear Sidhant, then you will end with "your faithfully".
 he main thing is MR., so don't forget this.

Thus if you write Dear Mr. Sidhant, then you'll use "yours sincerely", and if you write Dear sidhant, then you'll use "yours faithfully".