Here, you will find all the questions of ISI Entrance Paper 2011 from Indian Statistical Institute's B. Math Entrance. You will also get the solutions soon of all the previous year problems.
Problem 1 :
Let be a constant such that
for all
What can you say about
? Justify your answer.
Problem 2 :
Let for
. Define a function
on the nonnegative real numbers as follows: for each integer
, the graph of the function
on the interval
is the straight line segment connecting the points
and
. Find the total area of the region which lies between the curves of
and
.
Problem 3 :
For any positive integer , show that
.
Problem 4 :
If are not necessarily distinct real numbers such that
for all
, then show that we can choose three of them such that they are the lengths of the sides of a triangle.
Problem 5 :
For any real number let
denote the largest integer which is less than or equal to
. Let
be the sequence of non-square positive integers. If the
th non-square positive integer satisfies
then show that
Problem 6 :
Let and
be two cubes with sides of lengths
and
respectively, where
and
are positive integers. Show that the difference of their volumes equals the difference of their surface areas, if and only if
.
Problem 7 :
Let be any triangle and let
be a point on the line segment
Show that there exists a line parallel to
which divides the triangle
into two equal parts of equal area.
Problem 8 :
Let be real numbers, and consider the function
be given by
Show that
Here, you will find all the questions of ISI Entrance Paper 2011 from Indian Statistical Institute's B. Math Entrance. You will also get the solutions soon of all the previous year problems.
Problem 1 :
Let be a constant such that
for all
What can you say about
? Justify your answer.
Problem 2 :
Let for
. Define a function
on the nonnegative real numbers as follows: for each integer
, the graph of the function
on the interval
is the straight line segment connecting the points
and
. Find the total area of the region which lies between the curves of
and
.
Problem 3 :
For any positive integer , show that
.
Problem 4 :
If are not necessarily distinct real numbers such that
for all
, then show that we can choose three of them such that they are the lengths of the sides of a triangle.
Problem 5 :
For any real number let
denote the largest integer which is less than or equal to
. Let
be the sequence of non-square positive integers. If the
th non-square positive integer satisfies
then show that
Problem 6 :
Let and
be two cubes with sides of lengths
and
respectively, where
and
are positive integers. Show that the difference of their volumes equals the difference of their surface areas, if and only if
.
Problem 7 :
Let be any triangle and let
be a point on the line segment
Show that there exists a line parallel to
which divides the triangle
into two equal parts of equal area.
Problem 8 :
Let be real numbers, and consider the function
be given by
Show that