Here, you will find all the questions of ISI Entrance Paper 2008 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 continuous function. Suppose
for all and all
. Then show that there exists a constant
such that
for all
.
Problem 2 :
Suppose that is a polynomial with real coefficients such that for some positive real numbers
and for all natural numbers
we have
Prove that has a real zero.
Problem 3 :
Let be a complex number such that
are collinear in the complex plane. Show that
is a real number.
Problem 4 :
Let be integers. Show that there exists integers
and
such that the sum
is divisible by .
Problem 5 :
If a polynomial with integer coefficients has three distinct integer zeroes, then show that
for any integer
.
Problem 6 :
Let denote the binomial coefficient
and
be the
th Fibonacci number given by
and
for all
. Show that
Here, the above sum is over all pairs of integers with
.
Problem 7 :
Let,
How many squares can be formed in the plane all of whose vertices are in and whose sides are parallel to the
-axis and
-axis?
Problem 8 :
Let,
Prove that,
Problem 9 :
For determine all real solutions of the system of
equations:
Problem 10 :
If is a prime number and
is a natural number, then show that the greatest common divisor of the two numbers
and
is either
or
.
Here, you will find all the questions of ISI Entrance Paper 2008 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 continuous function. Suppose
for all and all
. Then show that there exists a constant
such that
for all
.
Problem 2 :
Suppose that is a polynomial with real coefficients such that for some positive real numbers
and for all natural numbers
we have
Prove that has a real zero.
Problem 3 :
Let be a complex number such that
are collinear in the complex plane. Show that
is a real number.
Problem 4 :
Let be integers. Show that there exists integers
and
such that the sum
is divisible by .
Problem 5 :
If a polynomial with integer coefficients has three distinct integer zeroes, then show that
for any integer
.
Problem 6 :
Let denote the binomial coefficient
and
be the
th Fibonacci number given by
and
for all
. Show that
Here, the above sum is over all pairs of integers with
.
Problem 7 :
Let,
How many squares can be formed in the plane all of whose vertices are in and whose sides are parallel to the
-axis and
-axis?
Problem 8 :
Let,
Prove that,
Problem 9 :
For determine all real solutions of the system of
equations:
Problem 10 :
If is a prime number and
is a natural number, then show that the greatest common divisor of the two numbers
and
is either
or
.