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Explore the Back-StoryHere, you will find all the questions of ISI Entrance Paper 2005 from Indian Statistical Institute's B.Stat Entrance. You will also get the solutions soon of all the previous year problems.

**Problem 1:**

Let and be the sides of a right angled triangle. Let be the smallest angle of this triangle. If and are also the sides of a right angled triangle then show that

**Problem 2:**

Let for all real . Sketch the graph of . What is the minimum value of ?

**Problem 3:**

Let be a function defined on satisfying for all i = + - for all such that . Find all real solutions of the equation .

**Problem 4:**

Consider an acute angled triangle such that and are the circumcentre, incentre and orthocentre respectively. Suppose and , measured in degrees, are respectively. Show that >

**Problem 5:**

Let be a function defined on as follows: . Let be a function defined for all as . Suppose that for all . Show that is a strictly increasing function. Show that there exists a real number such that is strictly decreasing in the interval and strictly increasing in the interval .

**Problem 6:**

For integers , Let and denote the following sets:

a) = : given that for all

b) = : = given that and for all

c) = : given that for all

d) Define a one-one onto map from onto .

e) Find the number of elements of the sets and

**Problem 7:**

A function is defined on the set of positive integers is said to be multiplicative if whenever and have no common factors greater than . Are the following functions multiplicative? Justify your answer.

**Problem 8:**

where k is the number of distinct primes which divide . if is divisible by for some integer ...., 1 otherwise.

**Problem 9:**

Suppose that to every point of the plane a colour, either red or blue, is associated. Show that if there is no equilateral triangle with all vertices of the same colour then there must exist three points and of the same colour such that is the midpoint of . Show that there must be an equilateral triangle with all vertices of the same colour.

**Problem 10:**

Let be a triangle. Take n point lying on the side (different from and ) and connect all of them by straight lines to the vertex . Similarly, take n points on the side and connect them to . Into how many regions is the triangle partitioned by these lines? Further, take points on the side also and join them with . Assume that no three straight lines meet at a point other than and . Into how many regions is the triangle partitioned now?

Here, you will find all the questions of ISI Entrance Paper 2005 from Indian Statistical Institute's B.Stat Entrance. You will also get the solutions soon of all the previous year problems.

**Problem 1:**

Let and be the sides of a right angled triangle. Let be the smallest angle of this triangle. If and are also the sides of a right angled triangle then show that

**Problem 2:**

Let for all real . Sketch the graph of . What is the minimum value of ?

**Problem 3:**

Let be a function defined on satisfying for all i = + - for all such that . Find all real solutions of the equation .

**Problem 4:**

Consider an acute angled triangle such that and are the circumcentre, incentre and orthocentre respectively. Suppose and , measured in degrees, are respectively. Show that >

**Problem 5:**

Let be a function defined on as follows: . Let be a function defined for all as . Suppose that for all . Show that is a strictly increasing function. Show that there exists a real number such that is strictly decreasing in the interval and strictly increasing in the interval .

**Problem 6:**

For integers , Let and denote the following sets:

a) = : given that for all

b) = : = given that and for all

c) = : given that for all

d) Define a one-one onto map from onto .

e) Find the number of elements of the sets and

**Problem 7:**

A function is defined on the set of positive integers is said to be multiplicative if whenever and have no common factors greater than . Are the following functions multiplicative? Justify your answer.

**Problem 8:**

where k is the number of distinct primes which divide . if is divisible by for some integer ...., 1 otherwise.

**Problem 9:**

Suppose that to every point of the plane a colour, either red or blue, is associated. Show that if there is no equilateral triangle with all vertices of the same colour then there must exist three points and of the same colour such that is the midpoint of . Show that there must be an equilateral triangle with all vertices of the same colour.

**Problem 10:**

Let be a triangle. Take n point lying on the side (different from and ) and connect all of them by straight lines to the vertex . Similarly, take n points on the side and connect them to . Into how many regions is the triangle partitioned by these lines? Further, take points on the side also and join them with . Assume that no three straight lines meet at a point other than and . Into how many regions is the triangle partitioned now?

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doubt question no. 4and 5

We have posted the requested discussions. Find them here: http://cheenta.com/2015/05/07/solutions-to-an-equation-b-stat-2005-subjective-problem-4-solution/

http://cheenta.com/2015/05/07/geometric-inequality-i-s-i-b-stat-2005-problem-5-solution/

I want to all the solutions of above questions

Do you please help me giving solutions of I.S.I. B.STAT ENTRANCE SUBJECTIVE PAPERS of 2

005, 2006,2007,2008,2009,2010,2011,2012