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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 is a complex number such that If is a positive integer, find the value of

Discussion

**Problem 2:**

Use calculus to find the behaviour of the function and sketch the graph of the function for . Show clearly the locations of the maxima, minima and points of inflection in your graph.

**Problem 3:**

Let be a continuous function and, for any real number , let denote the greatest integer less than or equal to . Show that for any ,

**Problem 4:**

Show that it is not possible to have a triangle with sides , and whose medians have length and .

**Problem 5:**

Show that for all values of .

**Problem 6:**

Let where is an odd integer. Let be a function defined on taking values in such that

Â (a) for all

(b) for all Show that

**Problem 7:**

Consider a prism with triangular base. The total area of the three faces containing a particular vertex is . Show that the maximum possible volume of the prism is and find the height of this largest prism.

**Problem 8:**

The following figure shows a grid divided into subgrids of size . This grid has cells, in each subgrid.

Now consider an grid divided into subgrids of size . Find the number of ways in which you can select 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 be a set satisfying the following properties:

- if and are any two distinct elements in , then

2. there are two elements and in such that for any ,

Â 3. if are two elements of , then for all

Show that if , then for some

**Problem 10: **

Let be a set of positive integers satisfying the following properties:

- if and belong to , then belong to ;
- there is no prime number that divides all elements of . (a) Suppose are two integers belonging to such that . Show that you can find two integers in such that

(b) Hence show that there are two consecutive integers belonging to .

(c) Let and be two consecutive integers belonging to . Show that if then belongs to .

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 is a complex number such that If is a positive integer, find the value of

Discussion

**Problem 2:**

Use calculus to find the behaviour of the function and sketch the graph of the function for . Show clearly the locations of the maxima, minima and points of inflection in your graph.

**Problem 3:**

Let be a continuous function and, for any real number , let denote the greatest integer less than or equal to . Show that for any ,

**Problem 4:**

Show that it is not possible to have a triangle with sides , and whose medians have length and .

**Problem 5:**

Show that for all values of .

**Problem 6:**

Let where is an odd integer. Let be a function defined on taking values in such that

Â (a) for all

(b) for all Show that

**Problem 7:**

Consider a prism with triangular base. The total area of the three faces containing a particular vertex is . Show that the maximum possible volume of the prism is and find the height of this largest prism.

**Problem 8:**

The following figure shows a grid divided into subgrids of size . This grid has cells, in each subgrid.

Now consider an grid divided into subgrids of size . Find the number of ways in which you can select 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 be a set satisfying the following properties:

- if and are any two distinct elements in , then

2. there are two elements and in such that for any ,

Â 3. if are two elements of , then for all

Show that if , then for some

**Problem 10: **

Let be a set of positive integers satisfying the following properties:

- if and belong to , then belong to ;
- there is no prime number that divides all elements of . (a) Suppose are two integers belonging to such that . Show that you can find two integers in such that

(b) Hence show that there are two consecutive integers belonging to .

(c) Let and be two consecutive integers belonging to . Show that if then belongs to .

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