Here, you will find all the questions of ISI Entrance Paper 2009 from Indian Statistical Institute's B.Stat Entrance. You will also get the solutions soon of all the previous year problems.
Problem 1:
If times the sum of the first
natural numbers is equal to the sum of the squares of the first
natural numbers, then
is
(A) ;
(B) ;
(C) ;
(D) .
Problem 2:
Two circles touch each other at . The two common tangents to the circles, none of which pass through
meet at
. They touch the larger circle at
and
The larger circle has radius
units and
has length
units. Then the radius of the smaller circle is
(A) ;
(B) ;
(C) ;
(D) .
Problem 3:
Suppose is a ten-digit number, where the digits are all distinct. Moreover,
satisfy
are consecutive even digits and
are consecutive odd digits. Then
is
(A) ;
(B) ;
(C) ;
(D) .
Problem 4:
Let be a right angled triangle with
. Construct three equilateral triangles
and
so that
and
are on opposite sides of
and
are on opposite sides of
and
are on opposite sides of
. Then
(A)
(B)
(C)
(D)
Problem 5:
The value of is
(A) ;
(B) a multiple of
(C) not an integer;
(D) a multiple of .
Problem 6:
Let where
Problem 7:
The number of solutions of in the interval
satisfying
(A) ;
(B) ;
(C) ;
(D) .
Problem 8:
A building with ten storeys, each storey of height metres, stands on one side of a wide street. From a point on the other side of the street directly opposite to the building, it is observed that the three uppermost storeys together subtend an angle equal to that subtended by the two lowest storeys. The width of the street is
(A) metres;
(B) metres;
(C) metres;
(D) metres.
Problem 9:
A collection of black and white balls are to be arranged on a straight line, such that each ball has at least one neighbour of different colour. If there are black balls, then the maximum number of white balls that allows such an arrangement is
(A) ;
(B) ;
(C) ;
(D) .
Problem 10:
Let be a real-valued function satisfying
where
and
are distinct real numbers and
and
are non-zero real numbers. Then
will have real solution when
(A) ;
(B) ;
(C) ;
(D)
Problem 11:
A circle is inscribed in a square of side , then a square is inscribed in that circle, a circle is inscribed in the latter square, and so on. If
is the sum of the areas of the first
circles so inscribed, then,
is
(A) ;
(B) ;
(C) ;
(D) .
Problem 12:
Let and
be two arithmetic progressions. Then the number of distinct integers in the collection of first
terms of each of the progressions is
(A) ;
(B) ;
(C) ;
(D) .
Problem 13:
Consider all the -letter words that can be formed by arranging the letters in
in all possible ways. Any two such words are called equivalent if those two words maintain the same relative order of the letters
and
For example,
and
are equivalent. How many words are there which are equivalent to
?
(A)
(B)
(C)
(D)
Problem 14:
The limit
(A) ;
(B) ;
(C) ;
(D) .
Problem 15:
Let and
be real numbers satisfying
Then the set of real numbers
such that the equations
and
have real solutions for
and
is
(A) ;
(B)
(C)
(D) .
Problem 16:
Let be an onto and differentiable function defined on
to
such that
Which of the following statements is necessarily true?
(A) is greater than or equal to
for all
;
(B) is smaller than
for all
;
(C) is greater than or equal to
for some
;
(D) is smaller than
for some
.
Problem 17:
The area of the region bounded by is
(A) ;
(B) ;
(C) ;
(D) .
Problem 18:
Let and
be two positive valued functions defined on
such that
and
is an even function with
. Then
satisfies
(A) ;
(B) ;
(C)
(D) .
Problem 19:
How many possible values of with
real, are there such that
and
(A) ;
(B) ;
(C) ;
(D) .
Problem 20:
What is the maximum possible value of a positive integer , such that for any choice of seven distinct elements from
there will exist two numbers
and
satisfying
(A)
(B)
(C)
(D) .
Problem 21:
Which of the following are roots of the equation
(A) ;
(B) ;
(C) ;
(D) .
Problem 22:
The equation has
(A) two positive roots;
(B) two real roots;
(C) three real roots;
(D) none of the above.
Problem 23:
If then
(A)
(B)
(C)
(D) .
Problem 24:
Suppose is a quadrilateral such that the coordinates of
and
are
and
respectively. For which choices of coordinates of
will
be a trapezium?
(A) ;
(B) ;
(C) ;
(D) .
Problem 25:
Let and
be two real numbers such that
holds. Which of the following are possible values of
(A) ;
(B) ;
(C) ;
(D) .
Problem 26:
Let be a differentiable function satisfying
for all
. Then
(A) is an odd function;
(B) for all
;
(C) for all
(D) If then
.
Problem 27:
Consider the function
Then
(A) ;
(B) ;
(C) is continuous for all
;
(B) is differentiable for all
.
Problem 28:
Which of the following graphs represent functions whose derivatives have a maximum in the interval ?
Problem 29:
A collection of geometric figures is said to satisfy Helly property if the following condition holds:
for any choice of three figures from the collection satisfying
and
one must have
Which of the following collections satisfy Helly property?
(A) A set of circles;
(B) A set of hexagons;
(C) A set of squares with sides parallel to the axes;
(D) A set of horizontal line segments.
Problem 30:
Consider an arrav of rows and
columns obtained by arranging the first
integers in some order. Let
be the maximum of the numbers in the i-th row and
be the minimum of the numbers in the
-th column. If
Here, you will find all the questions of ISI Entrance Paper 2009 from Indian Statistical Institute's B.Stat Entrance. You will also get the solutions soon of all the previous year problems.
Problem 1:
If times the sum of the first
natural numbers is equal to the sum of the squares of the first
natural numbers, then
is
(A) ;
(B) ;
(C) ;
(D) .
Problem 2:
Two circles touch each other at . The two common tangents to the circles, none of which pass through
meet at
. They touch the larger circle at
and
The larger circle has radius
units and
has length
units. Then the radius of the smaller circle is
(A) ;
(B) ;
(C) ;
(D) .
Problem 3:
Suppose is a ten-digit number, where the digits are all distinct. Moreover,
satisfy
are consecutive even digits and
are consecutive odd digits. Then
is
(A) ;
(B) ;
(C) ;
(D) .
Problem 4:
Let be a right angled triangle with
. Construct three equilateral triangles
and
so that
and
are on opposite sides of
and
are on opposite sides of
and
are on opposite sides of
. Then
(A)
(B)
(C)
(D)
Problem 5:
The value of is
(A) ;
(B) a multiple of
(C) not an integer;
(D) a multiple of .
Problem 6:
Let where
Problem 7:
The number of solutions of in the interval
satisfying
(A) ;
(B) ;
(C) ;
(D) .
Problem 8:
A building with ten storeys, each storey of height metres, stands on one side of a wide street. From a point on the other side of the street directly opposite to the building, it is observed that the three uppermost storeys together subtend an angle equal to that subtended by the two lowest storeys. The width of the street is
(A) metres;
(B) metres;
(C) metres;
(D) metres.
Problem 9:
A collection of black and white balls are to be arranged on a straight line, such that each ball has at least one neighbour of different colour. If there are black balls, then the maximum number of white balls that allows such an arrangement is
(A) ;
(B) ;
(C) ;
(D) .
Problem 10:
Let be a real-valued function satisfying
where
and
are distinct real numbers and
and
are non-zero real numbers. Then
will have real solution when
(A) ;
(B) ;
(C) ;
(D)
Problem 11:
A circle is inscribed in a square of side , then a square is inscribed in that circle, a circle is inscribed in the latter square, and so on. If
is the sum of the areas of the first
circles so inscribed, then,
is
(A) ;
(B) ;
(C) ;
(D) .
Problem 12:
Let and
be two arithmetic progressions. Then the number of distinct integers in the collection of first
terms of each of the progressions is
(A) ;
(B) ;
(C) ;
(D) .
Problem 13:
Consider all the -letter words that can be formed by arranging the letters in
in all possible ways. Any two such words are called equivalent if those two words maintain the same relative order of the letters
and
For example,
and
are equivalent. How many words are there which are equivalent to
?
(A)
(B)
(C)
(D)
Problem 14:
The limit
(A) ;
(B) ;
(C) ;
(D) .
Problem 15:
Let and
be real numbers satisfying
Then the set of real numbers
such that the equations
and
have real solutions for
and
is
(A) ;
(B)
(C)
(D) .
Problem 16:
Let be an onto and differentiable function defined on
to
such that
Which of the following statements is necessarily true?
(A) is greater than or equal to
for all
;
(B) is smaller than
for all
;
(C) is greater than or equal to
for some
;
(D) is smaller than
for some
.
Problem 17:
The area of the region bounded by is
(A) ;
(B) ;
(C) ;
(D) .
Problem 18:
Let and
be two positive valued functions defined on
such that
and
is an even function with
. Then
satisfies
(A) ;
(B) ;
(C)
(D) .
Problem 19:
How many possible values of with
real, are there such that
and
(A) ;
(B) ;
(C) ;
(D) .
Problem 20:
What is the maximum possible value of a positive integer , such that for any choice of seven distinct elements from
there will exist two numbers
and
satisfying
(A)
(B)
(C)
(D) .
Problem 21:
Which of the following are roots of the equation
(A) ;
(B) ;
(C) ;
(D) .
Problem 22:
The equation has
(A) two positive roots;
(B) two real roots;
(C) three real roots;
(D) none of the above.
Problem 23:
If then
(A)
(B)
(C)
(D) .
Problem 24:
Suppose is a quadrilateral such that the coordinates of
and
are
and
respectively. For which choices of coordinates of
will
be a trapezium?
(A) ;
(B) ;
(C) ;
(D) .
Problem 25:
Let and
be two real numbers such that
holds. Which of the following are possible values of
(A) ;
(B) ;
(C) ;
(D) .
Problem 26:
Let be a differentiable function satisfying
for all
. Then
(A) is an odd function;
(B) for all
;
(C) for all
(D) If then
.
Problem 27:
Consider the function
Then
(A) ;
(B) ;
(C) is continuous for all
;
(B) is differentiable for all
.
Problem 28:
Which of the following graphs represent functions whose derivatives have a maximum in the interval ?
Problem 29:
A collection of geometric figures is said to satisfy Helly property if the following condition holds:
for any choice of three figures from the collection satisfying
and
one must have
Which of the following collections satisfy Helly property?
(A) A set of circles;
(B) A set of hexagons;
(C) A set of squares with sides parallel to the axes;
(D) A set of horizontal line segments.
Problem 30:
Consider an arrav of rows and
columns obtained by arranging the first
integers in some order. Let
be the maximum of the numbers in the i-th row and
be the minimum of the numbers in the
-th column. If