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Group A

1. 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].
2. Suppose f(x) is a twice continuously differentiable function which satisfies the following conditions:
1. f(0) = f(1) = 0
2. 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]
3. 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.
4. 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

1. Let G and H be two nonzero subgroups of (Q, +). Show that the intersection of G and H is non empty.
2. Find surjective homomorphisms from
1. (Q, +) to (Z, +)
2. (R, +) to (Z, +)
3. Define $\mathbf { R = { \frac {2^k m } {n} \text{m and n are odd, k is non negative}} }$
1. Find the units (invertible elements) of this ring.
2. Demonstrate a proper ideal of this ring
3. Is this ideal a prime ideal?
4. Construct a polynomial with integer coefficient which has $\sqrt{2 - i}$ as a root.

(problems are collected from student feed back)