**Group A**

- Let f(x) be a twice continuously differentiable function such that \(\mathbf {|f”(x)| \le M }\) . 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: \(\mathbf { x^2 f”(x) + x^4 f'(x) – f(x) = 0 }\)

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 \(\mathbf { [0, 1] to [0, \infty] }\) . It is given that \(\mathbf { \int_0^1 x^n f(x) dx = 1}\) for all values of n > 1. Does their exist such a function.
- Prove that there exists a constant c > 0 such that \(\mathbf { \sum_{\nge x} \frac{1}{n^2}\le\frac{c}{x}}\), for all \(x {\in [1, \infty] }\)

**Group B**

- 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, +)

- Define \(\mathbf { R = { \frac {2^k m } {n} \text{m and n are odd, k is non negative}} }\)
- 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 \(\sqrt{2 – i} \) as a root.

(problems are collected from student feed back)

*Related*

question 1 of group A seems incomplete. Simply |f”(x)|<M does not imply uniform cont. Please check it.

we have revised that problem

Thanks. But still the given information are not useful.(Continuous functions on a closed and bounded interval are uniformly cont.??) Or may be they want the given hypothesis to used somehow to arrive at the conclusion. 🙂

Indeed cont functions on closed bounded intervals are uniformly cont. This can be proved by Heine Borel theorem