Here, you will find all the questions of ISI Entrance Paper 2015 from Indian Statistical Institute's B.Stat Entrance. You will also get the solutions soon of all the previous year problems.
Problem 1:
Let denote the set of complex numbers and
where
denotes the complex conjugate of
Then
has:
(A) two elements
(B) three elements
(C) four elements
(D) six elements.
Problem 2:
The number of one-to-one functions from a set with elements to a set with
elements is
(A)
(B)
(C)
(D) .
Problem 3:
Two sides of a triangle are of length and
Then, the maximum possible area (in
) of the triangle is:
(A)
(B)
(C)
(D) .
Problem 4:
The number of factors of which are either perfect squares or perfect cubes (or both) is:
(A)
(B)
(C)
(D) .
Problem 5:
The minimum value of the function where
, is
(A)
(B)
(C)
(D) .
Problem 6:
The minimum area of the triangle formed by any tangent to the ellipse 1 and the coordinate axes is
(A)
(B)
(C)
(D) .
Problem 7:
The angle between the hyperbolas and
(at their point of intersection) is
(A)
(B)
(C)
(D) .
Problem 8:
The population of a city doubles in years. In how many years will it triple, under the assumption that the rate of increase is proportional to the number of inhabitants?
(A) years
(B) years
(C) years
(D) years.
Problem 9:
We define a set of polynomials to be a linearly dependent set if there exist real numbers
not all zero, such that
for all real numbers
. Which of the following is a linearly dependent set?
(A)
(B)
(C)
(D) .
Problem 10:
A set of numbers is said to be multiplicatively closed if
whenever both
and
Let
and
be a non-real cube root of unity. Let
Problem 11:
When the product of four consecutive odd positive integers is divided by the set of remainder(s) is
(A)
(B)
(C)
(D) .
Problem 12:
Consider the equation where
and
. How many solutions
exist such that both
and
are non-negative integers?
(A) None
(B) Exactly one
(C) Exactly two
(D) Greater than two.
Problem 13:
Let be a point on the circle
above the
-axis, and
be a point on the circle
below the
-axis such that the line joining
and
is tangent to both these circles. Then the length of
is
(A) units
(B) units
(C) units
(D) units.
Problem 14:
Let viewed as a subset of the plane. For every point
in
let
denote the sum of the distances from
to the point
and the point
. The number of points
in
such that
is the least among all elements in the set
, is
(A)
(B)
(C)
(D)
Problem 15:
Let and
be the angles of a triangle. Suppose that
and
are the roots of the equation
. Then
equals
(A)
(B)
(C)
(D) .
Problem 16:
Let and
. The number of functions
for which the sum
is an even number, is
(A)
(B)
(C)
(D) .
Problem 17:
Define
Let be the function given by
where
is the set of real numbers. Then the number of discontinuities of
is:
(A)
(B)
(C)
(D) .
Problem 18:
Suppose is a subset of real numbers and
is a bijection (that is, one-to-one and onto) satisfying
for all
Then
cannot be:
(A) the set of integers
(B) the set of positive integers
(C) the set of positive real numbers
(D) the set of real numbers.
Problem 19:
The set of real numbers satisfying the inequality
Problem 20:
Let be a 10 -digit number, where all the digits are distinct. Further,
are consecutive odd numbers and
are consecutive even numbers. Then
is
(A)
(B)
(C)
(D) .
Problem 21:
Let are prime numbers,
The number of elements in
is
(A)
(B)
(C)
(D) .
Problem 22:
Let
Then is equal to
(A)
(B)
(C)
(D) .
Problem 23:
Let be a differentiable function. Suppose also that
for all
Which of the following is ALWAYS true?
(A) is increasing
(B) is one-to-one
(C) for all
(D)
Problem 24:
Consider 50 evenly placed points on a circle with centre at the origin and radius such that the arc length between any two consecutive points is the same. The complex numbers represented by these points form
(A) an arithmetic progression with common difference
(B) an arithmetic progression with common difference
(C) a geometric progression with common ratio
(D) a geometric progression with common ratio
Problem 25:
Given two complex numbers with unit modulus (i.e.,
), which of the following statements will ALWAYS be correct?
(A) and
(B) and
(C) or
(D) or
Problem 26:
The number of points in the region satisfying
is
(A)
(B)
(C)
(D) .
Problem 27:
If all the roots of the equation are positive, then
(A) must be
(B) can be any number strictly between and
(C) must be
(D) can be any number strictly between and
Problem 28:
Let denote the origin and
denote respectively the points (-10,0) and (7,0) on the
-axis. For how many points
on the
-axis will the lengths of all the line segments
and
be positive integers?
(A)
(B)
(C)
(D) infinite.
Problem 29:
Let where
is any real number and
is a continuous function such that
for all real
Then,
(A) and
has a local maximum or minimum at
.
(B) For any real number , the equation
has a unique solution.
(C) There exists a real number such that
has no solution.
(D) There exists a real number such that
has more than one solution.
Problem 30:
There are real numbers having the property that the sum of any
of them is less than the sum of the remaining
Then,
(A) all the numbers must be positive
(B) all the numbers must be negative
(C) all the numbers must be equal
(D) such a system of real numbers cannot exist.
Here, you will find all the questions of ISI Entrance Paper 2015 from Indian Statistical Institute's B.Stat Entrance. You will also get the solutions soon of all the previous year problems.
Problem 1:
Let denote the set of complex numbers and
where
denotes the complex conjugate of
Then
has:
(A) two elements
(B) three elements
(C) four elements
(D) six elements.
Problem 2:
The number of one-to-one functions from a set with elements to a set with
elements is
(A)
(B)
(C)
(D) .
Problem 3:
Two sides of a triangle are of length and
Then, the maximum possible area (in
) of the triangle is:
(A)
(B)
(C)
(D) .
Problem 4:
The number of factors of which are either perfect squares or perfect cubes (or both) is:
(A)
(B)
(C)
(D) .
Problem 5:
The minimum value of the function where
, is
(A)
(B)
(C)
(D) .
Problem 6:
The minimum area of the triangle formed by any tangent to the ellipse 1 and the coordinate axes is
(A)
(B)
(C)
(D) .
Problem 7:
The angle between the hyperbolas and
(at their point of intersection) is
(A)
(B)
(C)
(D) .
Problem 8:
The population of a city doubles in years. In how many years will it triple, under the assumption that the rate of increase is proportional to the number of inhabitants?
(A) years
(B) years
(C) years
(D) years.
Problem 9:
We define a set of polynomials to be a linearly dependent set if there exist real numbers
not all zero, such that
for all real numbers
. Which of the following is a linearly dependent set?
(A)
(B)
(C)
(D) .
Problem 10:
A set of numbers is said to be multiplicatively closed if
whenever both
and
Let
and
be a non-real cube root of unity. Let
Problem 11:
When the product of four consecutive odd positive integers is divided by the set of remainder(s) is
(A)
(B)
(C)
(D) .
Problem 12:
Consider the equation where
and
. How many solutions
exist such that both
and
are non-negative integers?
(A) None
(B) Exactly one
(C) Exactly two
(D) Greater than two.
Problem 13:
Let be a point on the circle
above the
-axis, and
be a point on the circle
below the
-axis such that the line joining
and
is tangent to both these circles. Then the length of
is
(A) units
(B) units
(C) units
(D) units.
Problem 14:
Let viewed as a subset of the plane. For every point
in
let
denote the sum of the distances from
to the point
and the point
. The number of points
in
such that
is the least among all elements in the set
, is
(A)
(B)
(C)
(D)
Problem 15:
Let and
be the angles of a triangle. Suppose that
and
are the roots of the equation
. Then
equals
(A)
(B)
(C)
(D) .
Problem 16:
Let and
. The number of functions
for which the sum
is an even number, is
(A)
(B)
(C)
(D) .
Problem 17:
Define
Let be the function given by
where
is the set of real numbers. Then the number of discontinuities of
is:
(A)
(B)
(C)
(D) .
Problem 18:
Suppose is a subset of real numbers and
is a bijection (that is, one-to-one and onto) satisfying
for all
Then
cannot be:
(A) the set of integers
(B) the set of positive integers
(C) the set of positive real numbers
(D) the set of real numbers.
Problem 19:
The set of real numbers satisfying the inequality
Problem 20:
Let be a 10 -digit number, where all the digits are distinct. Further,
are consecutive odd numbers and
are consecutive even numbers. Then
is
(A)
(B)
(C)
(D) .
Problem 21:
Let are prime numbers,
The number of elements in
is
(A)
(B)
(C)
(D) .
Problem 22:
Let
Then is equal to
(A)
(B)
(C)
(D) .
Problem 23:
Let be a differentiable function. Suppose also that
for all
Which of the following is ALWAYS true?
(A) is increasing
(B) is one-to-one
(C) for all
(D)
Problem 24:
Consider 50 evenly placed points on a circle with centre at the origin and radius such that the arc length between any two consecutive points is the same. The complex numbers represented by these points form
(A) an arithmetic progression with common difference
(B) an arithmetic progression with common difference
(C) a geometric progression with common ratio
(D) a geometric progression with common ratio
Problem 25:
Given two complex numbers with unit modulus (i.e.,
), which of the following statements will ALWAYS be correct?
(A) and
(B) and
(C) or
(D) or
Problem 26:
The number of points in the region satisfying
is
(A)
(B)
(C)
(D) .
Problem 27:
If all the roots of the equation are positive, then
(A) must be
(B) can be any number strictly between and
(C) must be
(D) can be any number strictly between and
Problem 28:
Let denote the origin and
denote respectively the points (-10,0) and (7,0) on the
-axis. For how many points
on the
-axis will the lengths of all the line segments
and
be positive integers?
(A)
(B)
(C)
(D) infinite.
Problem 29:
Let where
is any real number and
is a continuous function such that
for all real
Then,
(A) and
has a local maximum or minimum at
.
(B) For any real number , the equation
has a unique solution.
(C) There exists a real number such that
has no solution.
(D) There exists a real number such that
has more than one solution.
Problem 30:
There are real numbers having the property that the sum of any
of them is less than the sum of the remaining
Then,
(A) all the numbers must be positive
(B) all the numbers must be negative
(C) all the numbers must be equal
(D) such a system of real numbers cannot exist.
where to find detailed solutions?
Where are the solutions?