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)