Categories

# ISI Entrance Paper BMath 2007 – Subjective

ISI Entrance Paper BMath 2007 – from Indian Statistical Institute’s Entrance

1. Let n be a positive integer . If n has odd number of divisors ( other than 1 and n ) , then show that n is a perfect square .
2. Let a and b be two non-zero rational numbers such that the equation $\mathbf{ax^2+by^2=0}$ has a non-zero solution in rational numbers . Prove that for any rational number t , there is a solution of the equation $\mathbf{ax^2+by^2=t}$.
3. For a natural number n>1 , consider the n-1 points on the unit circle $\mathbf{e^{\frac{2\pi ik}{n}} (k=1,2,...,n-1)}$ . Show that the product of the distances of these points from 1 is n.
4. Let ABC be an isosceles triangle with AB=AC=20 . Let P be a point inside the triangle ABC such that the sum of the distances of P to AB and AC is 1 . Describe the locus of all such points inside triangle ABC.
5. Let P(X) be a polynomial with integer coefficients of degree d>0.(a) If $\mathbf{\alpha}$ and $\mathbf{\beta}$ are two integers such that $\mathbf{P(\alpha)=1}$ and $\mathbf{P(\beta)=-1}$ , then prove that $\mathbf{|\beta - \alpha|}$ divides 2.(b) Prove that the number of distinct integer roots of $\mathbf{P^2(x)-1}$ is at most d+2.
6. In ISI club each member is on two committees and any two committees have exactly one member in common . There are 5 committees . How many members does ISI club have?
7. Let $\mathbf{0\leq \theta\leq \frac{\pi}{2}}$ . Prove that $mathbf{sin \theta \geq \frac{2\theta}{\pi}}$.
Solution
8. Let $\mathbf{P:\mathbb{R} to \mathbb{R}}$ be a continuous function such that P(X)=X has no real solution. Prove that P(P(X))=X has no real solution.
Solution
9. In a group of five people any two are either friends or enemies , no three of them are friends of each other and no three of them are enemies of each other . Prove that every person in this group has exactly two friends .
10. The eleven members of a cricket team are numbered 1,2,…,11. In how many ways can the entire cricket team sit on the eleven chairs arranged around a circular table so that the numbers of any two adjacent players differ by one or two ?

## By Dr. Ashani Dasgupta

Ph.D. in Mathematics, University of Wisconsin, Milwaukee, United States.

Research Interest: Geometric Group Theory, Relatively Hyperbolic Groups.

Founder, Cheenta

## 3 replies on “ISI Entrance Paper BMath 2007 – Subjective”

please send me the solution of the forth no question.

eshasays: