First, a couple of fairly uncontroversial observations.
There will always be stiff competition for entry into elite colleges -- even if India creates 50 IIts, there will still be a scramble for a seat at, say, IIT-B (going by current trends). With stiff competition, there will always be a need for some filtering mechanism.
A mechanism that considers many aspects of a student's academic (and even non-academic) record, skills, and preparation is better than one that relies on his/her performance in one high-stakes exam.
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JEE is a bad exam in many ways -- this post from six years ago for some of the details. The most important badness is in its insistence on ranking students -- an exercise with little or no statistical basis that produces noisy results with poor reproducibility.
Unfortunately, the new mechanism proposed by the government does not allow us to move away from the statistically bankrupt idea of ranking students. Even if the two proposed national exams (JEE-Main and JEE-Advanced) are standardized, can their percentile scores be expressed with a precision of one in a million?
And, the proposed mechanism is rigid in fixing the relative weights for the three components, and using this rigid formula for ranking students. From this point on, there's no difference at all between the new mechanism and AIEEE (or JEE).
We can do better.
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Here's one possibility:
Let's say there are N exams in all -- and students get a percentile score in each of them. Now, let the individual institutions choose their weights for each of the N exams.
An institution need not stop there -- it can even insist on ensuring a diverse student body by creating state-wise quotas, and quotas for students from poor families, and for women (in IITs) or men (at St. Stephen's).
Each institution can design its policy so that truly exceptional students (top rankers in National Olympiads, top athletes and sportspersons, accomplished artists, ...) are admitted (as long as they fulfill a certain minimum threshold, and as long as these special talents are backed by objective measures).
This sort of stuff would be horrendously complicated if it were to be implemented manually; with computers, it's just a matter of coding an institution's admissions criteria and let them do the hard work.
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Articulating a vision for your institution's student body and devising an admissions policy that helps you get there are at the core of academic freedom and autonomy.
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All our top institutions have a duty to participate in these policy debates to steer them to a broadly acceptable outcome which gives institutions a viable way to exercise their autonomy.
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Ah, yes, one-handed typing is a bloody pain!