There are numerous courses for which you can apply without giving any competitive examinations.
For engineering:
There are plenty of engineering courses (private engineering colleges)where students can get admission through management quota.
You do not have to give any exams for that.
So even for engineering you can not sit for competitive exams but can get admission.
For Basic science:
Apart from IISc, IISER and NISER for  most of the courses you do not have to give any competitive exams.
Colleges take students on the basis of 10+2 only for B.Sc .
The popular courses are
B.Sc. in
Physics
Chemistry
Mathematics
Economics
Statistics
Zoology
Botany
Microbiology

hi,

see it is depends on what is your stream and which courses you are interested for. suppose you are a science student. you have the following career path to choose as engineering, mbbs/dental/other health science courses/ agriculture studies/pharmacy/ bsc courses/ bs-ms dual degree from iiser etc. now all of the courses except bsc has some entrance test to clear for applying in any good college. even for b.sc courses if you want to take admission in reputed college you need to go through the entrance exam, like if you want to pursue bachelors courses from delhi university/ banaras hindu university you need to appear in entrance test. however still there is some colleges left which offers b.sc admission through 10+2 score. hope you may understand.