# I.S.I. M.Math Subjective 2014

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)

## 4 Replies to “I.S.I. M.Math Subjective 2014”

1. 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. 🙂