Here, you will find all the questions of ISI Entrance Paper 2007 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 the circle
. The point
is
(A) inside but not the centre of
;
(B) outside ;
(C) on :
(D) the centre of .
Problem 2:
The number of distinct real roots of the equation
(A) ;
(B) ;
(C) ;
(D) .
Problem 3:
The set of complex numbers satisfying the equation
(A) a straight line;
(B) a pair of intersecting straight lines;
(C) a pair of distinct parallel straight lines;
(D) a point.
Problem 4:
Let be the set
and
the subset
The number of subsets
of
such that
is
(A) ;
(B)
(C)
(D) .
Problem 5:
The number of triplets of integers such that
and
are sides of a triangle with perimeter
is
(A) ;
(B) ;
(C) ;
(D) .
Problem 6:
Suppose and
are three numbers in G.P. If the equations
and
have a common root, then
and
are in
(A) A.P.
(B) G.P.;
(C) H.P.;
(D) none of the above.
Problem 7:
The number of solutions of the equation is
(A) ;
(B) ;
(C) ;
(D) .
Problem 8:
Suppose has two real roots
and
with
If
and
are the roots of
then the equation
has
(A) one positive and one negative root;
(B) two distinct positive roots;
(C) two distinct negative roots:
(D) no real roots.
Problem 9:
Suppose is a quadrilateral such that
and
If
is the point of intersection of
and
then the value of
is
(A) ;
(B) ;
(C) ;
(D) .
Problem 10:
Let be the set of all real numbers. The function
defined by
(A) one-to-one, but not onto:
(B) one-to-one and onto;
(C) onto, but not one-to-one;
(D) neither one-to-one nor onto.
Problem 11:
The angles of a convex pentagon are in A.P. Then, the minimum possible value of the smallest angle is
(A)
(B)
(C) ;
(D) .
Problem 12:
The number of points lying on the circle
such that the quadratic equation
has real roots, is
(A) infinite;
(B) ;
(C) ;
(D)
Problem 13:
Let be the point
and
be a point on the
-axis such that
has slope
Then the locus of the midpoint of
as
varies over all real values, is
(A)
(B)
(C)
(D)
Problem 14:
Suppose and
Which of the following statements is true?
(A) for all
(B) for all
(C) There exist such that
;
(D) None of the above.
Problem 15:
A triangle has a fixed base
. If
then the locus of the vertex
is
(A) a circle whose centre is the midpoint of ;
(B) a circle whose centre is on the line but not the midpoint of
;
(C) a straight line;
(D) none of the above.
Problem 16:
Suppose for some
. Then
(A) ;
(B) ;
(C) ;
(D) .
Problem 17:
Let The set of all values taken by
as
varies over the interval
is
(A) ;
(B) ;
(C) ;
(D) .
Problem 18:
is a
digit number. All the digits except the
th from the right are
If
is divisible by
then the unknown digit is
(A) ;
(B) ;
(C) ;
(D) .
Problem 19:
If , then the
-th derivative of
equals
(A) ;
(B) ;
(C)
(D) .
Problem 20:
Suppose . The maximum value of the integral
(D) .
Problem 21:
For any the value of
lies between
(A) and
;
(B) and
(C) and
;
(D) none of the above.
Problem 22:
Let denote a cube root of unity which is not equal to
Then the number of distinct elements in the set
Problem 23:
The graph of a function defined on (0,2) with the property
is as follows:
The graph of the derivative of the above function looks like:
Problem 24:
The value of the integral
Problem 25:
A hollow right circular cone rests on a sphere as shown in the figure. The height of the cone is metres and the radius of the base is
metre. The volume of the sphere is the same as that of the cone. Then, the distance between the centre of the sphere and the vertex of the cone is
(A) metres;
(B) metres;
(C) metres;
(D) metres.
Problem 26:
For each positive integer , define a function
on
as follows:
Then, the value of is
(A) :
(B) :
(C)
(D)
Problem 27:
The limit
Problem 28:
Let be the set of all points
such that
Given a point
in the plane, let
be the point in
which is closest to
. Then the points
for which
are
(A) all points with
;
(B) all points with
and
;
(C) all points with
and
;
(D) all points with
and
.
Problem 29:
In a win-or-lose game, the winner gets 2 points whereas the loser gets 0. Six players and
play each other in a preliminary round from which the top three players move to the final round. After each player has played four games, A has 6 points,
has 8 points and
has 4 points. It is also known that
won against
. In the next set of games
and
win their games against
and
respectively. If
and
move to the final round, the final scores of
and
are, respectively,
(A) and
;
(B) and
(C) and
;
(D) and
.
Problem 30:
The number of ways in which one can select six distinct integers from the set such that no two consecutive integers are selected, is
(A) ;
(B) ;
(C) ;
(D) .
Here, you will find all the questions of ISI Entrance Paper 2007 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 the circle
. The point
is
(A) inside but not the centre of
;
(B) outside ;
(C) on :
(D) the centre of .
Problem 2:
The number of distinct real roots of the equation
(A) ;
(B) ;
(C) ;
(D) .
Problem 3:
The set of complex numbers satisfying the equation
(A) a straight line;
(B) a pair of intersecting straight lines;
(C) a pair of distinct parallel straight lines;
(D) a point.
Problem 4:
Let be the set
and
the subset
The number of subsets
of
such that
is
(A) ;
(B)
(C)
(D) .
Problem 5:
The number of triplets of integers such that
and
are sides of a triangle with perimeter
is
(A) ;
(B) ;
(C) ;
(D) .
Problem 6:
Suppose and
are three numbers in G.P. If the equations
and
have a common root, then
and
are in
(A) A.P.
(B) G.P.;
(C) H.P.;
(D) none of the above.
Problem 7:
The number of solutions of the equation is
(A) ;
(B) ;
(C) ;
(D) .
Problem 8:
Suppose has two real roots
and
with
If
and
are the roots of
then the equation
has
(A) one positive and one negative root;
(B) two distinct positive roots;
(C) two distinct negative roots:
(D) no real roots.
Problem 9:
Suppose is a quadrilateral such that
and
If
is the point of intersection of
and
then the value of
is
(A) ;
(B) ;
(C) ;
(D) .
Problem 10:
Let be the set of all real numbers. The function
defined by
(A) one-to-one, but not onto:
(B) one-to-one and onto;
(C) onto, but not one-to-one;
(D) neither one-to-one nor onto.
Problem 11:
The angles of a convex pentagon are in A.P. Then, the minimum possible value of the smallest angle is
(A)
(B)
(C) ;
(D) .
Problem 12:
The number of points lying on the circle
such that the quadratic equation
has real roots, is
(A) infinite;
(B) ;
(C) ;
(D)
Problem 13:
Let be the point
and
be a point on the
-axis such that
has slope
Then the locus of the midpoint of
as
varies over all real values, is
(A)
(B)
(C)
(D)
Problem 14:
Suppose and
Which of the following statements is true?
(A) for all
(B) for all
(C) There exist such that
;
(D) None of the above.
Problem 15:
A triangle has a fixed base
. If
then the locus of the vertex
is
(A) a circle whose centre is the midpoint of ;
(B) a circle whose centre is on the line but not the midpoint of
;
(C) a straight line;
(D) none of the above.
Problem 16:
Suppose for some
. Then
(A) ;
(B) ;
(C) ;
(D) .
Problem 17:
Let The set of all values taken by
as
varies over the interval
is
(A) ;
(B) ;
(C) ;
(D) .
Problem 18:
is a
digit number. All the digits except the
th from the right are
If
is divisible by
then the unknown digit is
(A) ;
(B) ;
(C) ;
(D) .
Problem 19:
If , then the
-th derivative of
equals
(A) ;
(B) ;
(C)
(D) .
Problem 20:
Suppose . The maximum value of the integral
(D) .
Problem 21:
For any the value of
lies between
(A) and
;
(B) and
(C) and
;
(D) none of the above.
Problem 22:
Let denote a cube root of unity which is not equal to
Then the number of distinct elements in the set
Problem 23:
The graph of a function defined on (0,2) with the property
is as follows:
The graph of the derivative of the above function looks like:
Problem 24:
The value of the integral
Problem 25:
A hollow right circular cone rests on a sphere as shown in the figure. The height of the cone is metres and the radius of the base is
metre. The volume of the sphere is the same as that of the cone. Then, the distance between the centre of the sphere and the vertex of the cone is
(A) metres;
(B) metres;
(C) metres;
(D) metres.
Problem 26:
For each positive integer , define a function
on
as follows:
Then, the value of is
(A) :
(B) :
(C)
(D)
Problem 27:
The limit
Problem 28:
Let be the set of all points
such that
Given a point
in the plane, let
be the point in
which is closest to
. Then the points
for which
are
(A) all points with
;
(B) all points with
and
;
(C) all points with
and
;
(D) all points with
and
.
Problem 29:
In a win-or-lose game, the winner gets 2 points whereas the loser gets 0. Six players and
play each other in a preliminary round from which the top three players move to the final round. After each player has played four games, A has 6 points,
has 8 points and
has 4 points. It is also known that
won against
. In the next set of games
and
win their games against
and
respectively. If
and
move to the final round, the final scores of
and
are, respectively,
(A) and
;
(B) and
(C) and
;
(D) and
.
Problem 30:
The number of ways in which one can select six distinct integers from the set such that no two consecutive integers are selected, is
(A) ;
(B) ;
(C) ;
(D) .