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Monday, September 1, 2014

Tensor's Test

In the 1990s there used to be a tradition of newly joined IITians from Hyderabad, to conduct a pre-IIT JEE examination for the next batch, called the Tensor's test (is it still going on?). IIT-JEE =  joint entrance examination for admission to the IITs.

I had the opportunity to conduct the test (with Amit, Rajasekhar and Ramana).

Here are a few math questions from that test.

1) f:RR is a continuous function such that f(f(x))=x, xR. Show that, for some cR,f(c)=c.

2) Find all natural numbers x,y,x>y, such that xy=yx. Hint (given as part of the test): What is the maximum value of f(x)=x1/x?

3) fn is a sequence such that f1>0 and 3fn+1=2fn+Af2n for some constant A>0 and all n1. Show that fn+1fnn>1

Problem 3 was a particular favourite of mine (at that time) as I had discovered a way to compute nth-roots (the above sequence converges to 3A), with a nice (in my opinion :-)) elementary proof that the sequence is bounded and monotonic.

Problem 3 basically asks for the monotonicity part of that proof. Only later did I realize that this was just Netwon-Raphson's method.

[If you are interested, the solutions to the above three problems are here: http://ruffnsluff.blogspot.com/p/tensors-test-solutions.html]

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