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Explore the Back-StoryHere, you will find all the questions of ISI Entrance Paper 2009 from Indian Statistical Institute's B.Stat Entrance. You will also get the solutions soon of all the previous year problems.

**Problem 1:**

Two train lines intersect each other at a junction at an acute angle . A train is passing along one of the two lines. When the front of the train is at the junction, the train subtends an angle at a station on the other line. It subtends an angle at the same station, when its rear is at the junction. Show that

**Problem 2: **

Let be a continuous function, whose first and second derivatives are continuous on and for all in . Show that

**Problem 3:**

** **Let be a right-angled triangle with . Let be any point on . Draw perpendiculars and on and respectively from . Define to be the maximum of the areas of , and . Find the minimum possible value of .

**Problem 4: **

A sequence is called an arithmetic progression of the first order if the differences of the successive terms are constant. It is called an arithmetic progression of the second order if the differences of the successive terms form an arithmetic progression of the first order. In general, for , a sequence is called an arithmetic progression of the -th order if the differences of the successive terms form an arithmetic progression of the -th order.

The numbers are the first six terms of an arithmetic progression of some order. What is its least possible order? Find a formula for the -th term of this progression.

**Problem 5:**

** **A cardboard box in the shape of a rectangular parallelopiped is to be enclosed in a cylindrical container with a hemispherical lid. If the total height of the container from the base to the top of the lid is centimetres and its base has radius centimetres, find the volume of the largest box that can be completely enclosed inside the container with the lid on.

**Problem 6:**

** **Let be a function satisfying for

Show that where denotes the -th derivative evaluated at .

**Problem 7: **

Show that the vertices of a regular pentagon are concyclic. If the length of each side of the pentagon is , show that the radius of the circumcircle is .

**Problem 8: **

Find the number of ways in which three numbers can be selected from the set , such that the sum of the three selected numbers is divisible by .

**Problem 9:**

** **Consider points located at . Let be the region consisting of all points in the plane whose distance from is smaller than that from any other . Find the perimeter of the region .

**Problem 10: **

Let be the -th non-square positive integer. Thus , etc. For a positive real number , denotes the integer closest to it by . If , where is an integer, then define . For example,

Here, you will find all the questions of ISI Entrance Paper 2009 from Indian Statistical Institute's B.Stat Entrance. You will also get the solutions soon of all the previous year problems.

**Problem 1:**

Two train lines intersect each other at a junction at an acute angle . A train is passing along one of the two lines. When the front of the train is at the junction, the train subtends an angle at a station on the other line. It subtends an angle at the same station, when its rear is at the junction. Show that

**Problem 2: **

Let be a continuous function, whose first and second derivatives are continuous on and for all in . Show that

**Problem 3:**

** **Let be a right-angled triangle with . Let be any point on . Draw perpendiculars and on and respectively from . Define to be the maximum of the areas of , and . Find the minimum possible value of .

**Problem 4: **

A sequence is called an arithmetic progression of the first order if the differences of the successive terms are constant. It is called an arithmetic progression of the second order if the differences of the successive terms form an arithmetic progression of the first order. In general, for , a sequence is called an arithmetic progression of the -th order if the differences of the successive terms form an arithmetic progression of the -th order.

The numbers are the first six terms of an arithmetic progression of some order. What is its least possible order? Find a formula for the -th term of this progression.

**Problem 5:**

** **A cardboard box in the shape of a rectangular parallelopiped is to be enclosed in a cylindrical container with a hemispherical lid. If the total height of the container from the base to the top of the lid is centimetres and its base has radius centimetres, find the volume of the largest box that can be completely enclosed inside the container with the lid on.

**Problem 6:**

** **Let be a function satisfying for

Show that where denotes the -th derivative evaluated at .

**Problem 7: **

Show that the vertices of a regular pentagon are concyclic. If the length of each side of the pentagon is , show that the radius of the circumcircle is .

**Problem 8: **

Find the number of ways in which three numbers can be selected from the set , such that the sum of the three selected numbers is divisible by .

**Problem 9:**

** **Consider points located at . Let be the region consisting of all points in the plane whose distance from is smaller than that from any other . Find the perimeter of the region .

**Problem 10: **

Let be the -th non-square positive integer. Thus , etc. For a positive real number , denotes the integer closest to it by . If , where is an integer, then define . For example,

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can someone explain to me the solution to the last problem.

Please discuss problem no.1 and forward video