This post contains Indian Statistical Institute, ISI Entrance 2020 Problems and Solutions. Try them out.
This is a work in progress.
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:
(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
(Created by students). Please suggest changes in the comment section.
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.
This is a work in progress.
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:
(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
(Created by students). Please suggest changes in the comment section.
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
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.