ISI Entrance Paper 2007 – from Indian Statistical Institute’s B.Stat Entrance

1. Suppose a is a complex number such that $${ a^2+a+\frac{1}{a}+\frac{1}{a^2}+1=0 }$$ If m is a positive integer, find the value of $${a^{2m}+a^m+\frac{1}{a^m}+\frac{1}{a^{2m}}}$$
Discussion
2. Use calculus to find the behaviour of the function $${ y=e^x\sin{x} -\infty <x< +\infty}$$ and sketch the graph of the function for $${-2\pi \le x \le 2\pi}$$. Show clearly the locations of the maxima, minima and points of inflection in your graph.
3. Let f(u) be a continuous function and, for any real number u, let [u] denote the greatest integer less than or equal to u. Show that for any x>1, $${\int_{1}^{x} [u]([u]+1)f(u)du = 2\sum_{i=1}^{[x]} i \int_{i}^{x} f(u)du }$$
4. Show that it is not possible to have a triangle with sides a,b, and c whose medians have length $${\frac{2}{3}a, \frac{2}{3}b \text{and} \frac{4}{5}c}$$.
5. Show that $${-2 \le \cos \theta\left(\sin \theta + \sqrt{\sin ^2 \theta +3}\right) \le 2 }$$ for all values of $${theta}$$.
6. Let $${S={1,2,\cdots ,n}}$$ where n is an odd integer. Let f be a function defined on $${{(i,j): i\in S, j \in S}}$$ taking values in S such that
1. $${f(s,r)=f(r,s) \text{for all} r,s \in S}$$
2. $${{f(r,s): s\in S}=S \text{for all} r\in S}$$Show that $${{f(r,r): r\in S}=S}$$
7. Consider a prism with triangular base. The total area of the three faces containing a particular vertex A is K. Show that the maximum possible volume of the prism is $${\sqrt{\frac{K^3}{54}}}$$ and find the height of this largest prism.
8. The following figure shows a $${3^2 \times 3^2 }$$ grid divided into $${3^2}$$ subgrids of size $${3 \times 3}$$. This grid has 81 cells, 9 in each subgrid. Now consider an $${n^2 \times n^2}$$ grid divided into $${n^2}$$ subgrids of size $${n \times n}$$. Find the number of ways in which you can select n^2 cells from this grid such that there is exactly one cell coming from each subgrid, one from each row and one from each column.
9. Let X $${\subset \mathbb{R}^2}$$ be a set satisfying the following properties:
1. if $${(x_1,y_1)}$$ and $${(x_2,y_2)}$$ are any two distinct elements in X, then
$${\text{ either, } x_1 > x_2 \text{ and } y_1 > y_2 \text{ or, } x_1 < x_2 \text{ and } y_1 < y_2}$$
2. there are two elements $${(a_1,b_1)}$$ and $${(a_2,b_2)}$$ in X such that for any $${(x,y) in X}$$,
$${a_1\le x \le a_2 \text{ and } b_1\le y \le b_2 }$$
3. if $${(x_1,y_1) \text{and} (x_2,y_2)}$$ are two elements of X, then for all $${\lambda \in [0,1], \left(\lambda x_1+(1-\lambda)x_2, \lambda y_1 + (1-\lambda)y_2\right) \in X }$$
Show that if $${(x,y) \in X}$$, then for some $${\lambda in [0,1], x=\lambda a_1 +(1-\lambda)a_2, y=\lambda b_1 +(1-\lambda)b_2 }$$
10. Let A be a set of positive integers satisfying the following properties:
1. if m and n belong to A, then m+n belong to A;
2. there is no prime number that divides all elements of A.(a) Suppose $${ n_1 \text{and} n_2 }$$ are two integers belonging to A such that $${n_2-n_1 > 1}$$. Show that you can find two integers $${m_1 \text{and} m_2 }$$ in A such that $${0 < m_2-m_1 < n_2-n_1}$$
(b) Hence show that there are two consecutive integers belonging to A.
(c) Let $${n_0 \text{and} n_0+1 }$$ be two consecutive integers belonging to A. Show that if $${n\geq n_0^2 }$$ then n belongs to A.