Dependence on clearing an entrance exam for admission varies from course to course and institution to institution. Popular and reputable colleges require exams like JEE main for technical courses like BTech or Aerospace Engineering. Law subjects such as BSc LLB require the CLAT, while journalism programs like BJMC may offer CUET or specific institute tests. Management programs like BBA FIA which offer other respective courses sometimes require DU JAT or its equivalent. Merit based or exam based admission criteria can be adopted for hospitality management and aeronautical engineering.
Postgraduate courses such as MSc Maths or MSc Agriculture will have certain colleges that will require entrance tests while others will admit students based on undergraduate performance. These ones in particular are typically viewed as furthering ones education so private institutions do exist. Like the rest of the country, the admission process is rarely uniform and often includes interviews or case assignments. Make sure to verify exact requirements if you are looking at colleges from Thanjavur or Vadodara as they may differ significantly. Always check the admission policies of the institution you selected in order to not miss any necessary information.