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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.