# Indian Statistical Institute, ISI BStat & BMath 2020 UGA Solutions & Discussions

## Problem 1

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) .

## Solution:

is differentiable everywhere except at

is differentiable everywhere except at

is differentiable everywhere.

is differentiable everywhere except at and

## Problem 2

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

(a) ;

(b) ;

(c) ;

(d) .

## Solution:

Differentiate equality

## Problem 3

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

(a) ;

(b) ;

(c) ;

(d) .

## Solution:

Let be the number of subsets of having odd number of elements. Let be a subset having odd number of elements. If then There are such subsets. If then has even number of elements. There are such subsets.

## Problem 4

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

(a) ;

(b)

(c)

(d)

## Solution:

First, we permute all the players in ways and then group the first two, second two, and so on and make groups, but inside each group the two players can get permuted in ways and so we divide by 2^{32} and since the groups can get interchanged we divide it by .

## Problem 5

The number of real solution of is

(a) ;

(b) ;

(c) ;

(d) infinity.

## Solution:

for all On each segment the function increases from to , hence it intersects

## Problem 6

What is the limit of as tends to

(a)The limit dose not exist;

(b) ;

(c) ;

(d) .

## Solution:

Consider partition of the segment On each segment of the partition choose the value of at the right end-point. Corresponding Riemann sum is

## Problem 7

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.

## Solution:

Let Then The set

is not empty.

and satisfy the conditions of the problem. However, neither of (A),(B),(C) is true.

## Problem 8

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

(a) ;

(b) ;

(c) ;

(d) .

## Solution:

The last position where the integer appears is If then The number appears on positions

## Problem 9

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

(a) ;

(b) ;

(c) ;

(d) .

## Solution:

To define an increasing mapping of onto we must split elements of into nonempty groups of consecutive integers (first group will be mapped to second — to and so on), i.e. to place barriers on available places between s.

## Problem 10

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

(a) ;

(b) ;

(c) ;

(d) more that

## Solution:

Let be a solution to such that Denote Then

Write

Only two possibilities for

## Problem 11

The number of real roots of the polynomial

is

(a) ;

(b) ;

(c) ;

(d) .

## Solution:

Equation has no real roots.

Equation has one real root

Equation has two real roots

## Problem 12

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

(a) ;

(b) ;

(c) ;

(d) .

## Solution:

Let be the first term, and be the second term of the progression, Then the common ratio of the progression is The sum is equal to

The difference in the denominator is odd, so we take Let Then

This is the sum of the progression

## Problem 13

The integral part of equals

(a) ;

(b) ;

(c) ;

(d) .

## Solution:

Compare the sum with the integral of

Hence, the integer part of the sum is

## Problem 14

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

(a) ;

(b) ;

(c)

(d) .

## Solution:

Consider a subset that does not contain any two consecutive numbers. If then (there are such subsets). If then and (there are such subsets).

## Problem 15

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 ) and delete . 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 remains. What is the last number left

(a) ;

(b) ;

(c) ;

(d) None of the above.

## Solution:

Consider the same algorithm but with numbers arrange in the clockwise order. If then we will delete and the only number remained will be By induction we prove that the last number always be Assume the statement is proved for and consider numbers arranged circularly in clockwise order. After the first steps we will delete all even numbers and will get numbers with the first available element By the inductive assumption, the last number left will be

## Problem 16

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

(a) ;

(b) ;

(c) ;

(d) .

## Problem 17

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) .

## Solution:

Let be the side of the square. The sides of the right-angled triangle

are equal

Hence,

## Problem 18

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) .

## Solution:

Let Then and

Either and or or and or

If then is

(a) ;

(b) ;

(c) ;

(d)

Let Then

## Problem 20

If the word PERMUTE is permuted is 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) ;

(b) ;

(c) ;

(d) .

## Solution:

Put the letters of the word PERMUTE in the dictionary order: EEMPRTU. Since the last six letters are distinct, the first permutations do not change the position of the first letter and include all possible permutations of the last six letters. Then the 720th word is defined by the last (in the dictionary order) permutation of EMPRTU, i.e. it is EUTRPME

## Problem 21

The points and 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.

## Solution:

All four points belong to the plane Denote and Observe equality of vectors

Vectors and are not orthogonal and of different lengths.

## Problem 22

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

(a) is necessarily differentiable.

(b) f(x) is differentiable if and only if is differentiable.

(c) f(x) and are necessarily continuous.

(d) If for all , then is differentiable.

## Solution:

Let

Then are differentiable, while both and are discontinuous.

## Problem 23

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 . Then is

(a) ;

(b) ;

(c) ;

(d) the real line .

## Solution:

Let be the height of a triangle. Then is the set of all real numbers of the form

## Problem 24

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

(a) ;

(b) ;

(c) ;

(d) .

## Solution:

If then all other elements of can be arbitrary elements of It means that there are subsets of with

We compute

At we get

## Problem 25

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.

## Solution:

There are rational roots of the polynomial if is rational. It is possible only when the discriminant is a perfect square:

Then Observe that is odd, hence and are even.

and both and are odd:

But then is even. This is impossible.

## Problem 26

Let be finite subsets of the plan 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 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 triangles, both having a vertex from each of such that two triangles do not intersect.

## Solution:

Let be the minimal number of points of that can be in the interior of a triangle that has a vertex from each Consider such triangle where Let and be inside the interior of Without loss of generality assume that Then has points of in its interior, which is impossible. So,

## Problem 27

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

• For people who really do have the allergy, the test says “Yes” of the time.
•  For people who do not have the allergy, the test says “Yes” of the time

If of the population has the allergy and Shubhaangi’s test says “Yes”, then the chance that Shubhaangi does really have the allergy are
(a) ;

(b) ;

(c) ;

(d) cannot be determined from the given data

## Solution:

Let denote the event that the test says “Yes”. Consider two hypotheses:

— Shubhaangi has the allergy;

— Shubhaangi does not have the allergy.

Then

By the Bayes rule

## Problem 28

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) .

## Solution:

We observe that is even and for It is enough to find positive points of discontinuity and multiply the result by 2. The function is increasing on from to when is a point of discontinuity of if and only if is integer. There are integers in the interval

## Problem 29

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

(a) ;

(b) ;

(c) ;

(d) .

## Solution:

Parametrize the region as follows Then the area is

## Problem 30

Let be a positive integer and . Then
equals

(a) ;

(b) ;

(c) ;

(d)

## Solution:

• fayaque ali

For problem 22, option B is also a correct answer right? That is to say that the statement is false. Both f(x) and g(x) are independent of each other. The differentiability of g does not effect the differentiability of f.

• “Both f(x) and g(x) are independent of each other” How do you prove your claim? Do you have an example where is differentiable but is not or vice versa?

• Soumil

for q22, option B should also be the answer right?
Example : f(x)= 100 + |x-1|
g(x) = 100 – |x-1|

• Soumil

Sorry, I meant in q22, option *d* should also be the answer

• Soumil

Yes for q22, option B should be the answer.
Example : f(x)= 100 + |x-1|
g(x) = 100 – |x-1|