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?
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?
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