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Explore the Back-StoryThis post contains Indian Statistical Institute, ISI Entrance 2020 Problems and Solutions. Try them out.

Subjective Paper - ISI Entrance 2020 Problems and Solutions

Objective Paper - ISI Entrance 2020 Problems and Solutions

**Problem 1**

The number of subsets of having an odd number of elements is

(A) (B) (C) (D)

**Problem 2**

For the function on the real line given by , which of the following is true?

(A) It is differentiable everywhere.

(B) It is differentiable everywhere except at and

(C) It is differentiable everywhere except at

(D) It is differentiable everywhere except at

**Problem 3**

If are real-valued differentiable functions on the real line such that and then equals

(A) (B) (C) (D)

**Problem 4**

The number of real solutions of is

(A) (B) (C) (D) infinite.

**Problem 5**

What is the limit of as tends to

(A) The limit does not exist. (B) (C) (D)

**Problem 6**

A group of 64 players in a chess tournament needs to be divided into 32 groups of 2 players each. In how many ways can this be done?

(A) (B) (C) (D)

**Problem 7**

The integral part of equals

(A) (B) (C) (D)

**Problem 8**

Let be the number of subsets of that do not contain any two consecutive numbers. Then

(A) (B)

(C) (D)

**Discussion and Solution**

**Problem 9**

There are numbers which are arranged in a circular pattern in clockwise order. We start deleting numbers from this set in a clockwise fashion as follows. First delete the number then skip the next available number (which is 3) and delete 4. Continue in this manner, that is, after deleting a number, skip the next available number clockwise and delete the number available after that, till only one number What is the last number left?

(A) (B) (C) (D) None of the above.

**Problem 10**

Let and be complex numbers lying on the circles of radii 2 and 3 respectively, with centre If the angle between the corresponding vectors is 60 degrees, then the value of is:

(A) (B) (C) (D)

**Problem 11**

Two vertices of a square lie on a circle of radius and the other two vertices lie on a tangent to this circle. Then the length of the side of the square is

(A) (B) (C) (D)

**Problem 12**

For a real number let denote the greatest integer less than or equal to Then the number of real solutions of is

(A) (B) (C) (D)

**Problem 13**

Let be differentiable functions on the real line with Assume that the set is non-empty and that for all Then which of the following is necessarily true?

(A) If then . (B) For any

(C) For any

(D) None of the above.

**Problem 14**

Consider the sequence obtained by writing one two 's, three 's and so on. What is the term in the sequence?

(A) (B) (C) (D)

**Problem 15**

Let and be two sets of real numbers. What is the total number of functions such that is onto and

(A) (B) (C) (A)

**Problem 16**

The number of complex roots of the polynomial which have modulus is

(A) (B) (C) (D) more than

**Problem 17**

The number of real roots of the polynomial

(A) (B) (C) (D)

**Problem 18**

. Which of the following is the sum of an infinite geometric sequence whose terms come from the set

(A) (B) (C) (D)

**Problem 19**

If are distinct odd natural numbers, then the number of rational roots of the polynomial

(A) must be .

(B) must be .

(C) must be .

(D) cannot be determined from the given data.

**Problem 20**

Let be finite subsets of the plane such that and are all empty. Let . Assume that no three points of are collinear and also assume that each of and has at least 3 points. Which of the following statements is always true?

(A) There exists a triangle having a vertex from each of that does not contain any point of in its interior.

(B) Any triangle having a vertex from each of must contain a point of in its interior.

(C) There exists a triangle having a vertex from each of that contains all the remaining points of in its interior.

(D) There exist 2 triangles, both having a vertex from each of such that the two triangles do not intersect.

**Problem 21**

Shubhaangi thinks she may be allergic to Bengal gram and takes a test that is known to give the following results:

- (i)For people who really do have the allergy, the test says "Yes" of the time.
- (ii)For people who do not have the allergy, the test says "Yes" of the time.

- If of the population has the allergy and Shubhangi's test says "Yes" then the chances that Shubhaangi does really have the allergy are

(A)

(B)

(C)

(D) cannot be determined from the given data.

**Problem 22**

If then is

(A)

(B)

(C)

(D)

**Problem 23**

If the word PERMUTE is permuted in all possible ways and the different resulting words are written down in alphabetical order (also known as dictionary order , irrespective of whether the word has meaning or not, then the word would be:

(A) EEMPRTU (B) EUTRPME (C) UTRPMEE (D) MEET-PUR.

**Problem 24**

The points (4,7,-1),(1,2,-1),(-1,-2,-1) and (2,3,-1) in are the vertices of a

(A) rectangle which is not a square.

(B) rhombus.

(C) parallelogram which is not a rectangle.

(D) trapezium which is not a parallelogram.

**Problem 25**

Let be functions on the real line such that both and are differentiable. Which of the following is FALSE?

(A) is necessarily differentiable.

(B) is differentiable if and only if is differentiable.

(C) and are necessarily continuous.

(D) If for all then is differentiable.

**Problem 26**

Let be the set consisting of all those real numbers that can be written as where and are the perimeter and area of a right-angled triangle having base length 1 . Then is

(A)

(B)

(C)

(D) the real line .

**Problem 27**

Let For any non-empty subset of , let l(a) denote the largest number in If that is, is the sum of the numbers while ranges over all the nonempty subsets of , then is

( A )

(B)

( C)

(D)

**Problem 28**

The area of the region in the plane given by points satisfying and is

(A)

(B)

(G)

(D)

**Problem 29**

Let be a positive integer and Then equals

(A)

(B)

(C)

(D)

**Problem 30**

For any real number let be the greatest integer such that Then the number of points of discontinuity of the function on the interval is

(A)

(B)

(C)

(D)

Objective Paper - Answer Key

1. B | 2. B | 3. A | 4. D | 5. C |

6. A | 7. B | 8. A | 9. A | 10. A |

11. D | 12. A | 13. C | 14. C | 15. A |

16. C | 17. B | 18. B | 19. A | 20. A |

21. B | 22. B | 23. B | 24. C | 25. D |

26. A | 27. D | 28. C | 29. A | 30. D |

Some useful links

This post contains Indian Statistical Institute, ISI Entrance 2020 Problems and Solutions. Try them out.

Subjective Paper - ISI Entrance 2020 Problems and Solutions

Objective Paper - ISI Entrance 2020 Problems and Solutions

**Problem 1**

The number of subsets of having an odd number of elements is

(A) (B) (C) (D)

**Problem 2**

For the function on the real line given by , which of the following is true?

(A) It is differentiable everywhere.

(B) It is differentiable everywhere except at and

(C) It is differentiable everywhere except at

(D) It is differentiable everywhere except at

**Problem 3**

If are real-valued differentiable functions on the real line such that and then equals

(A) (B) (C) (D)

**Problem 4**

The number of real solutions of is

(A) (B) (C) (D) infinite.

**Problem 5**

What is the limit of as tends to

(A) The limit does not exist. (B) (C) (D)

**Problem 6**

A group of 64 players in a chess tournament needs to be divided into 32 groups of 2 players each. In how many ways can this be done?

(A) (B) (C) (D)

**Problem 7**

The integral part of equals

(A) (B) (C) (D)

**Problem 8**

Let be the number of subsets of that do not contain any two consecutive numbers. Then

(A) (B)

(C) (D)

**Discussion and Solution**

**Problem 9**

There are numbers which are arranged in a circular pattern in clockwise order. We start deleting numbers from this set in a clockwise fashion as follows. First delete the number then skip the next available number (which is 3) and delete 4. Continue in this manner, that is, after deleting a number, skip the next available number clockwise and delete the number available after that, till only one number What is the last number left?

(A) (B) (C) (D) None of the above.

**Problem 10**

Let and be complex numbers lying on the circles of radii 2 and 3 respectively, with centre If the angle between the corresponding vectors is 60 degrees, then the value of is:

(A) (B) (C) (D)

**Problem 11**

Two vertices of a square lie on a circle of radius and the other two vertices lie on a tangent to this circle. Then the length of the side of the square is

(A) (B) (C) (D)

**Problem 12**

For a real number let denote the greatest integer less than or equal to Then the number of real solutions of is

(A) (B) (C) (D)

**Problem 13**

Let be differentiable functions on the real line with Assume that the set is non-empty and that for all Then which of the following is necessarily true?

(A) If then . (B) For any

(C) For any

(D) None of the above.

**Problem 14**

Consider the sequence obtained by writing one two 's, three 's and so on. What is the term in the sequence?

(A) (B) (C) (D)

**Problem 15**

Let and be two sets of real numbers. What is the total number of functions such that is onto and

(A) (B) (C) (A)

**Problem 16**

The number of complex roots of the polynomial which have modulus is

(A) (B) (C) (D) more than

**Problem 17**

The number of real roots of the polynomial

(A) (B) (C) (D)

**Problem 18**

. Which of the following is the sum of an infinite geometric sequence whose terms come from the set

(A) (B) (C) (D)

**Problem 19**

If are distinct odd natural numbers, then the number of rational roots of the polynomial

(A) must be .

(B) must be .

(C) must be .

(D) cannot be determined from the given data.

**Problem 20**

Let be finite subsets of the plane such that and are all empty. Let . Assume that no three points of are collinear and also assume that each of and has at least 3 points. Which of the following statements is always true?

(A) There exists a triangle having a vertex from each of that does not contain any point of in its interior.

(B) Any triangle having a vertex from each of must contain a point of in its interior.

(C) There exists a triangle having a vertex from each of that contains all the remaining points of in its interior.

(D) There exist 2 triangles, both having a vertex from each of such that the two triangles do not intersect.

**Problem 21**

Shubhaangi thinks she may be allergic to Bengal gram and takes a test that is known to give the following results:

- (i)For people who really do have the allergy, the test says "Yes" of the time.
- (ii)For people who do not have the allergy, the test says "Yes" of the time.

- If of the population has the allergy and Shubhangi's test says "Yes" then the chances that Shubhaangi does really have the allergy are

(A)

(B)

(C)

(D) cannot be determined from the given data.

**Problem 22**

If then is

(A)

(B)

(C)

(D)

**Problem 23**

If the word PERMUTE is permuted in all possible ways and the different resulting words are written down in alphabetical order (also known as dictionary order , irrespective of whether the word has meaning or not, then the word would be:

(A) EEMPRTU (B) EUTRPME (C) UTRPMEE (D) MEET-PUR.

**Problem 24**

The points (4,7,-1),(1,2,-1),(-1,-2,-1) and (2,3,-1) in are the vertices of a

(A) rectangle which is not a square.

(B) rhombus.

(C) parallelogram which is not a rectangle.

(D) trapezium which is not a parallelogram.

**Problem 25**

Let be functions on the real line such that both and are differentiable. Which of the following is FALSE?

(A) is necessarily differentiable.

(B) is differentiable if and only if is differentiable.

(C) and are necessarily continuous.

(D) If for all then is differentiable.

**Problem 26**

Let be the set consisting of all those real numbers that can be written as where and are the perimeter and area of a right-angled triangle having base length 1 . Then is

(A)

(B)

(C)

(D) the real line .

**Problem 27**

Let For any non-empty subset of , let l(a) denote the largest number in If that is, is the sum of the numbers while ranges over all the nonempty subsets of , then is

( A )

(B)

( C)

(D)

**Problem 28**

The area of the region in the plane given by points satisfying and is

(A)

(B)

(G)

(D)

**Problem 29**

Let be a positive integer and Then equals

(A)

(B)

(C)

(D)

**Problem 30**

For any real number let be the greatest integer such that Then the number of points of discontinuity of the function on the interval is

(A)

(B)

(C)

(D)

Objective Paper - Answer Key

1. B | 2. B | 3. A | 4. D | 5. C |

6. A | 7. B | 8. A | 9. A | 10. A |

11. D | 12. A | 13. C | 14. C | 15. A |

16. C | 17. B | 18. B | 19. A | 20. A |

21. B | 22. B | 23. B | 24. C | 25. D |

26. A | 27. D | 28. C | 29. A | 30. D |

Some useful links

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HELLO SIR I THINK 13 SHOULD BE D.NONE.

Take f(x)=x+1 and g(x)= x^2

Pls check that for ur assumed function the condition of f'(t)>=g'(t) is not satisfied for all t belonging to m

What will be the cut off for B.math 2020

25 -C.........take a case f(X)={1 for x>=0, and -1 for x<0 g(x)=-f(x)

I think you are right. I wrongly chose option D.