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Explore the Back-StoryHere, you will find all the questions of ISI Entrance Paper 2013 from Indian Statistical Institute's B.Stat Entrance. You will also get the solutions soon of all the previous year problems.

**Multiple-Choice Test**

**Problem 1:**

Let and . The number of distinct real numbers in the set is

(A)

(B)

(C)

(D) infinite.

**Problem 2:**

From a square of unit length, pieces from the corners are removed to form a regular octagon. Then, the value of the area removed is

(A)

(B)

(C)

(D) .

**Problem 3:**

We define the dual of a line to be the point . Consider a set of non-vertical lines, , passing through the point . Then the duals of these lines will always

(A) be the same

(B) lie on a circle

(C) lie on a line

(D) form the vertices of a polygon with positive area.

**Problem 4:**

Suppose and are three real numbers satisfying

.

Then the value of is

(A)

(C)

(D)

**Problem 5:**

The value of is

(A)

(B)

(C)

(D) .

**Problem 6:**

The distance between the two foci of the rectangular hyperbola defined by is

(A)

(B)

(C)

(D) .

**Problem 7:**

Suppose is a differentiable and increasing function on such that

Let Then

(A) is an increasing function on

(B) is a decreasing function on

(C) has a unique maximum in the open interval

(D) has a unique minimum in the open interval .

**Problem 8:**

In an isosceles triangle the angle Then the ratio of two sides is

(A)

(B)

(C)

(D)

**Problem 9:**

Let be positive real numbers. If the equation

has a solution for then and must satisfy

(A)

(B)

(C)

(D)

**Problem 10:**

Suppose and where and Then

is

(A)

(B)

(C)

(D)

**Problem 11:**

Let and be a sequence of complex numbers defined by and for . Then is

(A)

(B)

(C)

(D) .

**Problem 12:**

The last digit of the number is

(A)

(B)

(C)

(D) .

**Problem 13:**

The maximum value of subject to the condition is

(A)

(B)

(C)

(D) .

**Problem 14:**

Which of the following is correct?

(A) for all .

(B) for and for

(C) for all

(D) for and for .

**Problem 15:**

The area of a regular polygon of sides that can be inscribed in the circle is

(A) units

(B) units

(C) units

(D) units.

**Problem 16:**

Let . The set of all real values of for which the function is defined and is

(A)

(B)

(C)

(D)

**Problem 17:**

Let be the largest integer strictly smaller than where is also an integer. Consider the following inequalities:

(1)

(2)

and find which of the following is correct.

(A) Only (1) is correct.

(B) Only (2) is correct.

(C) Both (1) and (2) are correct.

(D) None of them is correct.

**Problem 18:**

The value of is

(A)

(B)

(C)

(D) .

**Problem 19:**

For integers and , let denote the function from the set of integers to itself, defined by

Let be the set of all such functions,

Call an element invertible if there exists an element such that for all integers . Then which of the following is true?

(A) Every element of is invertible.

(B) has infinitely many invertible and infinitely many non-invertible elements.

(C) has finitely many invertible elements.

(D) No element of is invertible.

**Problem 20:**

Consider six players and . A team consists of two players. (Thus, there are 15 distinct teams.) Two teams play a match exactly once if there is no common player. For example, team can not play with but will play with . Then the total number of possible matches is

(A)

(B)

(C)

(D)

**Problem 21:**

The minimum value of is

(A)

(B)

(C)

(D) .

**Problem 22:**

The number of s at the end of the integer is

(A)

(B)

(C)

(D)

**Problem 23:**

We denote the largest integer less than or equal to by . Consider the identity

Then

(A)

(B) and

(C)

(D) and .

**Problem 24:**

The number of four tuples of positive integers satisfying all three equations

is

(A)

(B)

(C)

(D) .

**Problem 25:**

The number of real roots of is

(A)

(B)

(C)

(D) .

**Problem 26:**

Suppose and are the roots of the equation . Then the value of is

(A)

(B)

(C)

(D) .

**Problem 27:**

Among the four time instances given in the options below, when is the angle between the minute hand and the hour hand the smallest?

(A) .

(B)

(C)

(D) .

**Problem 28:**

Suppose all roots of the polynomial are real and smaller than Then, for any such polynomial, the function

(A) is increasing

(B) is either increasing or decreasing

(C) is decreasing

(D) is neither increasing nor decreasing.

**Problem 29:**

Consider a quadrilateral in the XY-plane with all of its angles less than Let be an arbitrary point in the plane and consider the six triangles each of which is formed by the point and two of the points Then the total area of these six triangles is minimum when the point is

(A) outside the quadrilateral

(B) one of the vertices of the quadrilateral

(C) intersection of the diagonals of the quadrilateral

(D) none of the points given in (A), (B) or (C).

**Problem 30:**

The graph of the equation comprises

(A) one point

(B) union of a line and a parabola

(C) one line

(D) union of a line and a hyperbola.

**Short-Answer Type Test**

**Problem 1:**

Let be real numbers greater than . Let denote the sum

. Find the smallest possible value of .

**Problem 2:**

For define . Determine the set

**Problem 3:**

Let be a function satisfying for all . Show that , where is a constant.

**Problem 4:**

In a badminton singles tournament, each player played against all the others exactly once and each game had a winner. After all the games, each player listed the names of all the players she defeated as well as the names of all the players defeated by the players defeated by her. For instance, if defeats and defeats . then in the list of both and are included. Prove that at least one player listed the names of all other players.

**Problem 5:**

Let be a diameter of a circle of radius . Let be points on the semicircle (with distinct from ) so that . Determine the ratio of the length of the chord to the radius.

**Problem 6:**

Let be distinct polynomials with real coefficients such that the sum of the coefficients of each of the polynomials equals s. If then prove the following:

(1) for some integer and a polynomial with .

(2) where is as given in .

**Problem 7:**

Let be a positive integer such that is the square of a positive integer. Then determine all possible values of . (Note that is a prime number).

**Problem 8:**

Let be a square with the side lying on the line . Suppose lie on the parabola . Find the possible values of the length of the side of the square.

- ISI BStat and B.Math 2014 Problems and Solutions
- Our ISI and CMI Entrance Program
- ISI BSTAT 2020 Problem 2 | Polynomials and Double Roots- Watch & Learn

Download Pdf: ISI Entrance 2013 Paper

Here, you will find all the questions of ISI Entrance Paper 2013 from Indian Statistical Institute's B.Stat Entrance. You will also get the solutions soon of all the previous year problems.

**Multiple-Choice Test**

**Problem 1:**

Let and . The number of distinct real numbers in the set is

(A)

(B)

(C)

(D) infinite.

**Problem 2:**

From a square of unit length, pieces from the corners are removed to form a regular octagon. Then, the value of the area removed is

(A)

(B)

(C)

(D) .

**Problem 3:**

We define the dual of a line to be the point . Consider a set of non-vertical lines, , passing through the point . Then the duals of these lines will always

(A) be the same

(B) lie on a circle

(C) lie on a line

(D) form the vertices of a polygon with positive area.

**Problem 4:**

Suppose and are three real numbers satisfying

.

Then the value of is

(A)

(C)

(D)

**Problem 5:**

The value of is

(A)

(B)

(C)

(D) .

**Problem 6:**

The distance between the two foci of the rectangular hyperbola defined by is

(A)

(B)

(C)

(D) .

**Problem 7:**

Suppose is a differentiable and increasing function on such that

Let Then

(A) is an increasing function on

(B) is a decreasing function on

(C) has a unique maximum in the open interval

(D) has a unique minimum in the open interval .

**Problem 8:**

In an isosceles triangle the angle Then the ratio of two sides is

(A)

(B)

(C)

(D)

**Problem 9:**

Let be positive real numbers. If the equation

has a solution for then and must satisfy

(A)

(B)

(C)

(D)

**Problem 10:**

Suppose and where and Then

is

(A)

(B)

(C)

(D)

**Problem 11:**

Let and be a sequence of complex numbers defined by and for . Then is

(A)

(B)

(C)

(D) .

**Problem 12:**

The last digit of the number is

(A)

(B)

(C)

(D) .

**Problem 13:**

The maximum value of subject to the condition is

(A)

(B)

(C)

(D) .

**Problem 14:**

Which of the following is correct?

(A) for all .

(B) for and for

(C) for all

(D) for and for .

**Problem 15:**

The area of a regular polygon of sides that can be inscribed in the circle is

(A) units

(B) units

(C) units

(D) units.

**Problem 16:**

Let . The set of all real values of for which the function is defined and is

(A)

(B)

(C)

(D)

**Problem 17:**

Let be the largest integer strictly smaller than where is also an integer. Consider the following inequalities:

(1)

(2)

and find which of the following is correct.

(A) Only (1) is correct.

(B) Only (2) is correct.

(C) Both (1) and (2) are correct.

(D) None of them is correct.

**Problem 18:**

The value of is

(A)

(B)

(C)

(D) .

**Problem 19:**

For integers and , let denote the function from the set of integers to itself, defined by

Let be the set of all such functions,

Call an element invertible if there exists an element such that for all integers . Then which of the following is true?

(A) Every element of is invertible.

(B) has infinitely many invertible and infinitely many non-invertible elements.

(C) has finitely many invertible elements.

(D) No element of is invertible.

**Problem 20:**

Consider six players and . A team consists of two players. (Thus, there are 15 distinct teams.) Two teams play a match exactly once if there is no common player. For example, team can not play with but will play with . Then the total number of possible matches is

(A)

(B)

(C)

(D)

**Problem 21:**

The minimum value of is

(A)

(B)

(C)

(D) .

**Problem 22:**

The number of s at the end of the integer is

(A)

(B)

(C)

(D)

**Problem 23:**

We denote the largest integer less than or equal to by . Consider the identity

Then

(A)

(B) and

(C)

(D) and .

**Problem 24:**

The number of four tuples of positive integers satisfying all three equations

is

(A)

(B)

(C)

(D) .

**Problem 25:**

The number of real roots of is

(A)

(B)

(C)

(D) .

**Problem 26:**

Suppose and are the roots of the equation . Then the value of is

(A)

(B)

(C)

(D) .

**Problem 27:**

Among the four time instances given in the options below, when is the angle between the minute hand and the hour hand the smallest?

(A) .

(B)

(C)

(D) .

**Problem 28:**

Suppose all roots of the polynomial are real and smaller than Then, for any such polynomial, the function

(A) is increasing

(B) is either increasing or decreasing

(C) is decreasing

(D) is neither increasing nor decreasing.

**Problem 29:**

Consider a quadrilateral in the XY-plane with all of its angles less than Let be an arbitrary point in the plane and consider the six triangles each of which is formed by the point and two of the points Then the total area of these six triangles is minimum when the point is

(A) outside the quadrilateral

(B) one of the vertices of the quadrilateral

(C) intersection of the diagonals of the quadrilateral

(D) none of the points given in (A), (B) or (C).

**Problem 30:**

The graph of the equation comprises

(A) one point

(B) union of a line and a parabola

(C) one line

(D) union of a line and a hyperbola.

**Short-Answer Type Test**

**Problem 1:**

Let be real numbers greater than . Let denote the sum

. Find the smallest possible value of .

**Problem 2:**

For define . Determine the set

**Problem 3:**

Let be a function satisfying for all . Show that , where is a constant.

**Problem 4:**

In a badminton singles tournament, each player played against all the others exactly once and each game had a winner. After all the games, each player listed the names of all the players she defeated as well as the names of all the players defeated by the players defeated by her. For instance, if defeats and defeats . then in the list of both and are included. Prove that at least one player listed the names of all other players.

**Problem 5:**

Let be a diameter of a circle of radius . Let be points on the semicircle (with distinct from ) so that . Determine the ratio of the length of the chord to the radius.

**Problem 6:**

Let be distinct polynomials with real coefficients such that the sum of the coefficients of each of the polynomials equals s. If then prove the following:

(1) for some integer and a polynomial with .

(2) where is as given in .

**Problem 7:**

Let be a positive integer such that is the square of a positive integer. Then determine all possible values of . (Note that is a prime number).

**Problem 8:**

Let be a square with the side lying on the line . Suppose lie on the parabola . Find the possible values of the length of the side of the square.

- ISI BStat and B.Math 2014 Problems and Solutions
- Our ISI and CMI Entrance Program
- ISI BSTAT 2020 Problem 2 | Polynomials and Double Roots- Watch & Learn

Download Pdf: ISI Entrance 2013 Paper

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