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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 \( {m_1 \text{and} m_2 }\) 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 $n_0$ 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 \( {m_1 \text{and} m_2 }\) 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 $n_0$ 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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