Here, you will find all the questions of ISI Entrance Paper 2010 from Indian Statistical Institute's B.Stat Entrance. You will also get the solutions soon of all the previous year problems.
Problem 1:
There are 8 balls numbered and 8 boxes numbered
. The number of ways one can put these balls in the boxes so that each box gets one ball and exactly 4 balls go in their corresponding numbered boxes is
(A) ;
(B) ;
(C) ;
(D) .
Problem 2:
Let and
be two positive real numbers. For every integer
, define
Then is equal to
(A)
(B)
(C)
(D) .
Problem 3:
Let be a function given by
where
and
are fixed positive integers. Suppose that
is one-one. Then
(A) both and
must be odd;
(B) at least one of and
must be odd;
(C) exactly one of and
must be odd;
(D) neither nor
can be odd.
Problem 4:
equals
(A) ;
(B) ;
(C) ;
(D)
Problem 5:
A circle is inscribed in a triangle with sides and
centimetres. The radius of the circle in centimetres is
(A) ;
(B) ;
(C) ;
(D) none of the above.
Problem 6:
Let and
be the angles of an acute angled triangle. Then the quantity
(A) can have any real value;
(B) is ;
(C) is ;
(D) none of the above.
Problem 7:
Let for
. Then
(A) is differentiable at
and
(B) is not differentiable at
and
;
(C) is differentiable at
but not differentiable at
(D) is not differentiable at
but differentiable at
.
Problem 8:
Consider a rectangular cardboard box of height breadth 4 and length 10 units. There is a lizard in one corner
of the box and an insect in the corner
which is farthest from
. The length of the shortest path between the lizard and the insect along the surface of the box is
(A) units;
(B) units;
(C) units;
(D) units.
Problem 9:
Recall that, for any non-zero complex number which does not lie on the negative real axis, arg
denotes the unique real number
in
such that
(A) any value in the set but none from outside;
(B) any value in the interval (-1,0) but none from outside;
(C) any value in the interval (0,1) but none from outside;
(D) any value in the set (-1,0) but none from outside.
Problem 10:
An aeroplane is moving in the air along a straight line path which passes through the points
and
, and makes an angle
with the ground. Let
be the position of an observer as shown in the figure below. When the plane is at the position
its angle of elevation is
and when it is at
its angle of elevation is
from the position of the observer. Moreover, the distances of the observer from the points
and
respectively are 1000 metres and
metres.
Then is equal to
(A) ;
(B) ;
(C) ;
(D) .
Problem 11:
The sum of all even positive divisors of is
(A) ;
(B) ;
(C) ;
(D) .
Problem 12:
The equation has two real roots
and
If
then the area under the curve
between
and
is
(A) ;
(B)
(C)
(D) .
Problem 13:
The minimum value of subject to
and
is
(A)
(B) ;
(C) ;
(D) .
Problem 14:
The value of
(A)
(B) ;
(C) ;
(D) .
Problem 15:
For any real number , let
denote the unique real number
in
such that
. Then
(A) is equal to ;
(B) is equal to ;
(C) does not exist;
(D) none of the above
Problem 16:
Let be an integer. The number of primes which divide both
and
is
(A) at most one;
(B) exactly one;
(C) exactly two;
(D) none of the above.
Problem 17:
The value of
(A) ;
(B) ;
(C) ;
(D) .
Problem 18:
A person standing at a point
on a flat plane starts walking. At each step, he walks exactly 1 foot in one of the directions North, South, East or West. Suppose that after 6 steps
comes back to the original position
. Then the number of distinct paths that
can take is
(A) ;
(B) ;
(C) ;
(D) .
Problem 19:
Consider the branch of the rectangular hyperbola in the first quadrant. Let
be a fixed point on this curve. The locus of the mid-point of the line segment joining
and an arbitrary point
on the curve is part of
(A) a hyperbolas;
(B) a parabola;
(C) an ellipse;
(D) none of the above.
Problem 20:
The digit at the unit place of is
(A) ;
(B)
(C) ;
(D) .
Problem 21:
Let be the vertices of a regular polygon and
be its
sides. If
then the value of is
(A) ;
(B) ;
(C) ;
(D) .
Problem 22:
Suppose that and
are two distinct numbers in the interval
. If
then the value of is
(A) ;
(B) ;
(C)
(D)
Problem 23:
Consider the function Then
has
(A) no solution;
(B) only one solution;
(C) exactly two solutions;
(D) exactly three solutions.
Problem 24:
Consider the quadratic equation The number of pairs
for which the equation has solutions of the form
and
for some
is
(A)
(B) ;
(C) ;
(D) infinite.
Problem 25:
Let Then
(A)
(B)
(C)
(D) .
Problem 26:
Consider the following two curves on the interval (0,1) :
Then on
(A) lies above
;
(B) lies above
(C) and
intersect at exactly one point;
(D) none of the above.
Problem 27:
Let be a real valued function on
such that
is twice differentiable. Suppose that
vanishes only at 0 and
is everywhere negative. Define a function
by
where
Then
(A) has a local minima in
;
(B) has a local maxima in
;
(C) is monotonically increasing in
;
(D) is monotonically decreasing in
.
Problem 28:
Consider the triangle with vertices and
. Let
denote the region enclosed by the above triangle. Consider the function
defined by
Then the range of
is the interval
(A) ;
(B) ;
(C) ;
(D) .
Problem 29:
For every positive integer , let
denote the integer closest to
. Let
The number of elements in
is
(A) ;
(B)
(C) ;
(D) .
Problem 30:
Consider a square inscribed in a circle of radius
Let
and
be two points on the (smaller) arcs
and
respectively, such that
is a pentagon in which
. If
denotes the area of the pentagon
then
(A) can not be equal to
;
(B) lies in the interval
;
(C) is greater than or equal to
;
(D) none of the above.
Here, you will find all the questions of ISI Entrance Paper 2010 from Indian Statistical Institute's B.Stat Entrance. You will also get the solutions soon of all the previous year problems.
Problem 1:
There are 8 balls numbered and 8 boxes numbered
. The number of ways one can put these balls in the boxes so that each box gets one ball and exactly 4 balls go in their corresponding numbered boxes is
(A) ;
(B) ;
(C) ;
(D) .
Problem 2:
Let and
be two positive real numbers. For every integer
, define
Then is equal to
(A)
(B)
(C)
(D) .
Problem 3:
Let be a function given by
where
and
are fixed positive integers. Suppose that
is one-one. Then
(A) both and
must be odd;
(B) at least one of and
must be odd;
(C) exactly one of and
must be odd;
(D) neither nor
can be odd.
Problem 4:
equals
(A) ;
(B) ;
(C) ;
(D)
Problem 5:
A circle is inscribed in a triangle with sides and
centimetres. The radius of the circle in centimetres is
(A) ;
(B) ;
(C) ;
(D) none of the above.
Problem 6:
Let and
be the angles of an acute angled triangle. Then the quantity
(A) can have any real value;
(B) is ;
(C) is ;
(D) none of the above.
Problem 7:
Let for
. Then
(A) is differentiable at
and
(B) is not differentiable at
and
;
(C) is differentiable at
but not differentiable at
(D) is not differentiable at
but differentiable at
.
Problem 8:
Consider a rectangular cardboard box of height breadth 4 and length 10 units. There is a lizard in one corner
of the box and an insect in the corner
which is farthest from
. The length of the shortest path between the lizard and the insect along the surface of the box is
(A) units;
(B) units;
(C) units;
(D) units.
Problem 9:
Recall that, for any non-zero complex number which does not lie on the negative real axis, arg
denotes the unique real number
in
such that
(A) any value in the set but none from outside;
(B) any value in the interval (-1,0) but none from outside;
(C) any value in the interval (0,1) but none from outside;
(D) any value in the set (-1,0) but none from outside.
Problem 10:
An aeroplane is moving in the air along a straight line path which passes through the points
and
, and makes an angle
with the ground. Let
be the position of an observer as shown in the figure below. When the plane is at the position
its angle of elevation is
and when it is at
its angle of elevation is
from the position of the observer. Moreover, the distances of the observer from the points
and
respectively are 1000 metres and
metres.
Then is equal to
(A) ;
(B) ;
(C) ;
(D) .
Problem 11:
The sum of all even positive divisors of is
(A) ;
(B) ;
(C) ;
(D) .
Problem 12:
The equation has two real roots
and
If
then the area under the curve
between
and
is
(A) ;
(B)
(C)
(D) .
Problem 13:
The minimum value of subject to
and
is
(A)
(B) ;
(C) ;
(D) .
Problem 14:
The value of
(A)
(B) ;
(C) ;
(D) .
Problem 15:
For any real number , let
denote the unique real number
in
such that
. Then
(A) is equal to ;
(B) is equal to ;
(C) does not exist;
(D) none of the above
Problem 16:
Let be an integer. The number of primes which divide both
and
is
(A) at most one;
(B) exactly one;
(C) exactly two;
(D) none of the above.
Problem 17:
The value of
(A) ;
(B) ;
(C) ;
(D) .
Problem 18:
A person standing at a point
on a flat plane starts walking. At each step, he walks exactly 1 foot in one of the directions North, South, East or West. Suppose that after 6 steps
comes back to the original position
. Then the number of distinct paths that
can take is
(A) ;
(B) ;
(C) ;
(D) .
Problem 19:
Consider the branch of the rectangular hyperbola in the first quadrant. Let
be a fixed point on this curve. The locus of the mid-point of the line segment joining
and an arbitrary point
on the curve is part of
(A) a hyperbolas;
(B) a parabola;
(C) an ellipse;
(D) none of the above.
Problem 20:
The digit at the unit place of is
(A) ;
(B)
(C) ;
(D) .
Problem 21:
Let be the vertices of a regular polygon and
be its
sides. If
then the value of is
(A) ;
(B) ;
(C) ;
(D) .
Problem 22:
Suppose that and
are two distinct numbers in the interval
. If
then the value of is
(A) ;
(B) ;
(C)
(D)
Problem 23:
Consider the function Then
has
(A) no solution;
(B) only one solution;
(C) exactly two solutions;
(D) exactly three solutions.
Problem 24:
Consider the quadratic equation The number of pairs
for which the equation has solutions of the form
and
for some
is
(A)
(B) ;
(C) ;
(D) infinite.
Problem 25:
Let Then
(A)
(B)
(C)
(D) .
Problem 26:
Consider the following two curves on the interval (0,1) :
Then on
(A) lies above
;
(B) lies above
(C) and
intersect at exactly one point;
(D) none of the above.
Problem 27:
Let be a real valued function on
such that
is twice differentiable. Suppose that
vanishes only at 0 and
is everywhere negative. Define a function
by
where
Then
(A) has a local minima in
;
(B) has a local maxima in
;
(C) is monotonically increasing in
;
(D) is monotonically decreasing in
.
Problem 28:
Consider the triangle with vertices and
. Let
denote the region enclosed by the above triangle. Consider the function
defined by
Then the range of
is the interval
(A) ;
(B) ;
(C) ;
(D) .
Problem 29:
For every positive integer , let
denote the integer closest to
. Let
The number of elements in
is
(A) ;
(B)
(C) ;
(D) .
Problem 30:
Consider a square inscribed in a circle of radius
Let
and
be two points on the (smaller) arcs
and
respectively, such that
is a pentagon in which
. If
denotes the area of the pentagon
then
(A) can not be equal to
;
(B) lies in the interval
;
(C) is greater than or equal to
;
(D) none of the above.