This is a post that contains Group A and Group B Subjective problems from ISI MMath 2014. Try to solve them out.
- Let f(x) be a twice continuously differentiable function such that . f(0) = f(1) = 0. Prove that f(x) is uniformly continuous in the interval [0, 1].
- Suppose f(x) is a twice continuously differentiable function which satisfies the following conditions:
- f(0) = f(1) = 0
- f satisfies the following equation:
Prove that if f attains a maximum M in the interval (0, 1) then M = 0. Hence or otherwise show that f(x) = 0 in the interval [0, 1]
- Let f be a continuous function defined from . It is given that for all values of n > 1. Does their exist such a function.
- Prove that there exists a constant c > 0 such that , for all
- Let G and H be two nonzero subgroups of (Q, +). Show that the intersection of G and H is non empty.
- Find surjective homomorphisms from
- (Q, +) to (Z, +)
- (R, +) to (Z, +)
- Find the units (invertible elements) of this ring.
- Demonstrate a proper ideal of this ring
- Is this ideal a prime ideal?
- Construct a polynomial with integer coefficient which has as a root.
(problems are collected from student feed back)