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) \(f:\mathbb{R}\to\mathbb{R}\) is a continuous function such that \(f(f(x)) = x,\ \forall x \in \mathbb{R}\). Show that, for some \(c \in \mathbb{R}, f(c) = c\).
2) Find all natural numbers \(x, y, x \gt y\), such that \(x^y = y^x\). Hint (given as part of the test): What is the maximum value of \(f(x) = x^{1/x}\)?
3) \(f_n\) is a sequence such that \(f_1 \gt 0\) and $$3f_{n+1} = 2f_n + \frac{A}{f_n^2}$$ for some constant \(A \gt 0\) and all \(n \ge 1\). Show that \(f_{n+1} \le f_n \forall n \gt 1\)
Problem 3 was a particular favourite of mine (at that time) as I had discovered a way to compute \(n^{th}\)-roots (the above sequence converges to \(\sqrt[3]{A}\)), 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]
I had the opportunity to conduct the test (with Amit, Rajasekhar and Ramana).
Here are a few math questions from that test.
1) \(f:\mathbb{R}\to\mathbb{R}\) is a continuous function such that \(f(f(x)) = x,\ \forall x \in \mathbb{R}\). Show that, for some \(c \in \mathbb{R}, f(c) = c\).
2) Find all natural numbers \(x, y, x \gt y\), such that \(x^y = y^x\). Hint (given as part of the test): What is the maximum value of \(f(x) = x^{1/x}\)?
3) \(f_n\) is a sequence such that \(f_1 \gt 0\) and $$3f_{n+1} = 2f_n + \frac{A}{f_n^2}$$ for some constant \(A \gt 0\) and all \(n \ge 1\). Show that \(f_{n+1} \le f_n \forall n \gt 1\)
Problem 3 was a particular favourite of mine (at that time) as I had discovered a way to compute \(n^{th}\)-roots (the above sequence converges to \(\sqrt[3]{A}\)), 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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