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

Also see: ISI and CMI Entrance Course at Cheenta

- 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 - 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.
- 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, \( {\int_{1}^{x} [u]([u]+1)f(u)du = 2\sum_{i=1}^{[x]} i \int_{i}^{x} f(u)du }\)
- Show that it is not possible to have a triangle with sides a,b, and c whose medians have length \( {\frac{2}{3}a, \frac{2}{3}b \text{and} \frac{4}{5}c}\).
- Show that \( {-2 \le \cos \theta\left(\sin \theta + \sqrt{\sin ^2 \theta +3}\right) \le 2 }\) for all values of \( {theta}\).
- 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
- \( {f(s,r)=f(r,s) \text{for all} r,s \in S}\)
- \( {{f(r,s): s\in S}=S \text{for all} r\in S}\)Show that \( {{f(r,r): r\in S}=S}\)

- 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.
- 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. - Let X \( {\subset \mathbb{R}^2}\) be a set satisfying the following properties:
- if \( {(x_1,y_1)}\) and \( {(x_2,y_2)}\) are any two distinct elements in X, then

\( {\text{ either, } x_1 > x_2 \text{ and } y_1 > y_2 \text{ or, } x_1 < x_2 \text{ and } y_1 < y_2}\) - there are two elements \( {(a_1,b_1)}\) and \( {(a_2,b_2)}\) in X such that for any \( {(x,y) in X}\),

\( {a_1\le x \le a_2 \text{ and } b_1\le y \le b_2 }\) - 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 \( {\lambda in [0,1], x=\lambda a_1 +(1-\lambda)a_2, y=\lambda b_1 +(1-\lambda)b_2 }\)

- if \( {(x_1,y_1)}\) and \( {(x_2,y_2)}\) are any two distinct elements in X, then
- Let A be a set of positive integers satisfying the following properties:
- if m and n belong to A, then m+n belong to A;
- 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.

How to solve the 3rd problem?