Here, you will find all the questions of ISI Entrance Paper 2006 from Indian Statistical Institute's B. Stat Entrance. You will also get the solutions soon of all the previous year problems.
Problem1 :
If the normal to the curve at some point makes an angle
with the
-axis, show that the equation of the normal is
Problem 2 :
Suppose that is an irrational number.
(a) If there is a real number such that both
and
are rational numbers, show that
is a quadratic surd. (
is a quadratic surd if it is of the form
or
for some rationals
and
, where
is not the square of a rational number).
(b) Show that there are two real numbers and
such that
i) is rational but
is irrational.
ii) is irrational but
is rational.
(Hint: Consider the two cases, where is a quadratic surd and
is not a quadratic surd, separately).
Problem3 :
Prove that is composite for all values of
greater than
.
Discussion
Problem4 :
In the figure below, is the midpoint of the arc
and the segment
is perpendicular to the chord
at
. If the length of the chord
is
, and that of the segment
is
, determine the length of
in terms of
.
Discussion
Problem 5 :
Let and
be three points on a circle of radius
.
(a) Show that the area of the triangle equals
(b) Suppose that the magnitude of is fixed. Then show that the area of the triangle
is maximized when
(c) Hence or otherwise, show that the area of the triangle is maximum when the triangle is equilateral.
Problem 6 :
(a) Let . Show that f(x) is an increasing function on
, and
.
(b) Using part (a) or otherwise, draw graphs of , and
for
using the same
and
axes.
Problem 7 :
For any positive integer greater than
, show that
Problem 8:
Show that there exists a positive real number such that
. Hence obtain the set of real numbers
such that
has only one real solution.
Problem 9 :
Find a four digit number such that the number
has the following properties.
(a) is also a four digit number
(b) has the same digits as in
but in reverse order.
Problem 10 :
Consider a function on nonnegative integers such that
,
and
+
=
+
for
. Show that
=
Here, you will find all the questions of ISI Entrance Paper 2006 from Indian Statistical Institute's B. Stat Entrance. You will also get the solutions soon of all the previous year problems.
Problem1 :
If the normal to the curve at some point makes an angle
with the
-axis, show that the equation of the normal is
Problem 2 :
Suppose that is an irrational number.
(a) If there is a real number such that both
and
are rational numbers, show that
is a quadratic surd. (
is a quadratic surd if it is of the form
or
for some rationals
and
, where
is not the square of a rational number).
(b) Show that there are two real numbers and
such that
i) is rational but
is irrational.
ii) is irrational but
is rational.
(Hint: Consider the two cases, where is a quadratic surd and
is not a quadratic surd, separately).
Problem3 :
Prove that is composite for all values of
greater than
.
Discussion
Problem4 :
In the figure below, is the midpoint of the arc
and the segment
is perpendicular to the chord
at
. If the length of the chord
is
, and that of the segment
is
, determine the length of
in terms of
.
Discussion
Problem 5 :
Let and
be three points on a circle of radius
.
(a) Show that the area of the triangle equals
(b) Suppose that the magnitude of is fixed. Then show that the area of the triangle
is maximized when
(c) Hence or otherwise, show that the area of the triangle is maximum when the triangle is equilateral.
Problem 6 :
(a) Let . Show that f(x) is an increasing function on
, and
.
(b) Using part (a) or otherwise, draw graphs of , and
for
using the same
and
axes.
Problem 7 :
For any positive integer greater than
, show that
Problem 8:
Show that there exists a positive real number such that
. Hence obtain the set of real numbers
such that
has only one real solution.
Problem 9 :
Find a four digit number such that the number
has the following properties.
(a) is also a four digit number
(b) has the same digits as in
but in reverse order.
Problem 10 :
Consider a function on nonnegative integers such that
,
and
+
=
+
for
. Show that
=