Here, you will find all the questions of ISI Entrance Paper 2011 from Indian Statistical Institute's B.Stat Entrance. You will also get the solutions soon of all the previous year problems.
Problem 1:
Let be positive reals with
. Then show that
Problem 2:
Consider three positive real numbers and
. Show that there cannot exist two distinct positive integers
and
such that both
and
hold.
Problem 3:
Let denote the set of real numbers. Suppose a function
satisfies
for all
. Show that
(i) is one-one,
(ii) cannot be strictly decreasing, and
(iii) if is strictly increasing, then
for all
.
Problem 4:
Let be a twice differentiable function on the open interval
such that
. Suppose
also satisfies
and
, for all
. Show that
.
Problem 5:
is a trapezium such that
and
>1. Suppose
and
are points on
and
respectively, such that
Prove that is a parallelogram.
Problem 6:
Let be a complex number such that both
and
have modulus
. If for a positive integer
,
is an
-th root of unity, then show that
is also an
-th root of unity and
is a multiple of
.
Problem 7:
(i) Show that there cannot exists three prime numbers, each greater than , which are in arithmetic progression with a common difference less than
.
(ii) Let be an integer. Show that it is not possible for
prime numbers, each greater than
, to be in an arithmetic progression with a common difference less than or equal to
.
Problem 8:
Let . Arrange
in increasing order of magnitude. Justify your answer.
Problem 9:
Consider all non-empty subsets of the set . For every such subset, we find the product of the reciprocals of each of its elements. Denote the sum of all these products as
. For example,
1. Show that .
2. Hence or otherwise, deduce that .
Problem 10:
Show that the triangle whose angles satisfy the equality is right angled.
Here, you will find all the questions of ISI Entrance Paper 2011 from Indian Statistical Institute's B.Stat Entrance. You will also get the solutions soon of all the previous year problems.
Problem 1:
Let be positive reals with
. Then show that
Problem 2:
Consider three positive real numbers and
. Show that there cannot exist two distinct positive integers
and
such that both
and
hold.
Problem 3:
Let denote the set of real numbers. Suppose a function
satisfies
for all
. Show that
(i) is one-one,
(ii) cannot be strictly decreasing, and
(iii) if is strictly increasing, then
for all
.
Problem 4:
Let be a twice differentiable function on the open interval
such that
. Suppose
also satisfies
and
, for all
. Show that
.
Problem 5:
is a trapezium such that
and
>1. Suppose
and
are points on
and
respectively, such that
Prove that is a parallelogram.
Problem 6:
Let be a complex number such that both
and
have modulus
. If for a positive integer
,
is an
-th root of unity, then show that
is also an
-th root of unity and
is a multiple of
.
Problem 7:
(i) Show that there cannot exists three prime numbers, each greater than , which are in arithmetic progression with a common difference less than
.
(ii) Let be an integer. Show that it is not possible for
prime numbers, each greater than
, to be in an arithmetic progression with a common difference less than or equal to
.
Problem 8:
Let . Arrange
in increasing order of magnitude. Justify your answer.
Problem 9:
Consider all non-empty subsets of the set . For every such subset, we find the product of the reciprocals of each of its elements. Denote the sum of all these products as
. For example,
1. Show that .
2. Hence or otherwise, deduce that .
Problem 10:
Show that the triangle whose angles satisfy the equality is right angled.
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problem 8 is based on basic inequalities
Let f be a twice differentiable function on the open interval (-1,1) such that f(0)=1. Suppose f also satisfies f(x)≥0,f′(x)≤0 and f”(x)≤f(x), for all x\ge 0. Show that f′(0)≥−2–√.
How to proceed with no. 4?
Can you help me solving the 19 number question ??...
Consider the L-shaped diagram below ..that's one