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Explore the Back-StoryHere, you will find all the questions of ISI Entrance Paper 2007 from Indian Statistical Institute's B.Stat Entrance. You will also get the solutions soon of all the previous year problems.

**Problem 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

**Problem 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.

**Problem 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 \( x>1\), \( {\int_{1}^{x} [u]([u]+1)f(u)du = 2\sum_{i=1}^{[x]} i \int_{i}^{x} f(u)du }\)

**Problem 4:**

Show that it is not possible to have a triangle with sides \( a,b \), and \(c\) whose medians have length \(\frac{2}{3}a, \frac{2}{3}b\) and \(\frac{4}{5}c\).

**Problem 5:**

Show that \( {-2 \leq \cos \theta\left(\sin \theta + \sqrt{\sin ^2 \theta +3}\right) \leq 2 }\) for all values of \( {\theta}\).

**Problem 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

(a) \(f(s,r)=f(r,s)\) for all \(r,s \in S\)

(b) \(f(r,s): s \in S=S\) for all \(r\in S\) Show that \( {{f(r,r): r\in S}=S}\)

**Problem 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.

**Problem 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.

**Problem 9:**

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

\( {\text{ either, } x_1 > x_2 \text{ and } y_1 > y_2 \text{ or, } x_1 < x_2 \text{ and } y_1 < y_2}\)

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

**Problem 10: **

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 \( {m_1 and m_2 }\) 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 \(n_0\)and \(n_0+1\) be two consecutive integers belonging to \(A\). Show that if \( {n\geq n_0^2 }\) then \(n\) belongs to \(A\).

Here, you will find all the questions of ISI Entrance Paper 2007 from Indian Statistical Institute's B.Stat Entrance. You will also get the solutions soon of all the previous year problems.

**Problem 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

**Problem 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.

**Problem 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 \( x>1\), \( {\int_{1}^{x} [u]([u]+1)f(u)du = 2\sum_{i=1}^{[x]} i \int_{i}^{x} f(u)du }\)

**Problem 4:**

Show that it is not possible to have a triangle with sides \( a,b \), and \(c\) whose medians have length \(\frac{2}{3}a, \frac{2}{3}b\) and \(\frac{4}{5}c\).

**Problem 5:**

Show that \( {-2 \leq \cos \theta\left(\sin \theta + \sqrt{\sin ^2 \theta +3}\right) \leq 2 }\) for all values of \( {\theta}\).

**Problem 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

(a) \(f(s,r)=f(r,s)\) for all \(r,s \in S\)

(b) \(f(r,s): s \in S=S\) for all \(r\in S\) Show that \( {{f(r,r): r\in S}=S}\)

**Problem 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.

**Problem 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.

**Problem 9:**

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

\( {\text{ either, } x_1 > x_2 \text{ and } y_1 > y_2 \text{ or, } x_1 < x_2 \text{ and } y_1 < y_2}\)

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

**Problem 10: **

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 \( {m_1 and m_2 }\) 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 \(n_0\)and \(n_0+1\) be two consecutive integers belonging to \(A\). Show that if \( {n\geq n_0^2 }\) then \(n\) belongs to \(A\).

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