Is there any exam conducted to get admission into iist
No!
You will be required to fill admission form for IIST which is released in the
month of June.
The criteria for admission is JEE Main rank. Thus you will have to
sit for JEE Mains.
Also you have to qualify for JEE Advanced and fullfil the minimum cutoff
criteria in JEE Advanced.
A detailed information can be found on official IIST website.