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Explore the Back-StoryHere, 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

Which one of the following statements is true?

(A) Both and are multiplicatively closed.

(B) is multiplicatively closed but is not.

(C) is multiplicatively closed but is not.

(D) Neither nor is multiplicatively closed.

**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

is:

(A)

(B)

(C)

(D) .

**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

Which one of the following statements is true?

(A) Both and are multiplicatively closed.

(B) is multiplicatively closed but is not.

(C) is multiplicatively closed but is not.

(D) Neither nor is multiplicatively closed.

**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

is:

(A)

(B)

(C)

(D) .

**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.

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