# Indian Statistical Institute, ISI MMATH 2021 PMA Solutions & Discussions

## Problem 1

Suppose is a continuous function such that
for all . Then the value of is

(a) ;

(b) ;

(c) ;

(d) .

## Solution:

By continuity,

where the relation was used.

## Problem 2

A person throws a pair of fair dice. If the sum of the numbers on the dice is a perfect square, then the probability that the number 3 appeared on at least one of the dice is

(a)

(b)

(c)

(d)

## Solution:

The sum of the numbers on the dice takes values from to The only perfect squares that can appear are and The sum appears in cases:

The sum appears in cases:

So, the perfect square appears in cases. Among them in cases the number appears on one of the dice. The conditional probability of interest is

## Problem 3

Consider the system of linear equations: . Then

(a) the system is inconsistent

(b) the system has a unique solution

(c) the system has infinitely many solutions

(d) none of the above is true

## Solution:

We find that

Then the equations take the form

These two equations are identical. So, for any we have a solution

## Problem 4

If then is given by

(a)

(b)

(c)

(d)

## Solution:

We observe that

Hence,

integrate by parts

If
then is equal to

(a)

(b)

(c)

(d)

## Solution:

For we have the identity

Hence,

On the other hand,

Hence,

## Problem 6

Let be a sequence of functions defined as follows:

Then exists if and only if belongs to the interval

(a)

(b)

(c)

(d)

## Solution:

If then

If then

If then

and this does not converge. The limit of exists if and only if

## Problem 7

Let be a set of elements. The number of ways in which distinct non-empty subsets of can be chosen such that
, is

(a)

(b)

(c)

(d)

## Solution:

Denote Each difference contains at least one element. Denote this element by If then so All elements are distinct. Hence, is an ordering of the set . There are ways to order elements of the set and every ordering defines a sequence of non-empty distinct subsets by the rule

## Problem 8

Let be a matrix such that both and are non-null.
If , then the rank of is

(a) ;

(b) ;

(c) ;

(d) .

## Solution:

Elements of the adjugate matrix are equal (up to signs) to cofactors of elements of the matrix (i.e. minors of ). Since the rank of is Since there exists at least one non-zero minor of and the rank of is We deduce that the rank of is equal to .

## Problem 9

The set of all satisfying the inequality

is equal to the interval

(a)

(b)

(c)

(d)

## Solution:

At first we observe that Evaluate the integral:

The inequality transforms to

## Problem 10

Let be the set of all continuous functions and be the set of all differentiable functions such that the derivative is continuous. (Here, differentiability at 0 means right differentiability and differentiability at 1 means left differentiability.) If is defined by , then

(a) is one-to-one and onto

(b) is one-to-one but not onto

(c) is onto but not one-to-one

(d) is neither one-to-one nor onto.

## Solution:

Consider a constant function Then Since are distinct for different , the mapping is not one-to-one.

Given a continuous functions consider

By the Fundamental Theorem of Calculus, and The mapping is onto.

## Problem 11

Suppose are in A.P. and are in G.P. If and , then the value of is

(a)

(b)

(c)

(d)

## Solution:

Let be the difference of the arithmetic progression. Then and we find

Further,

and

Taking square roots we find

i.e. either or The case is impossible, as Also, the condition implies that and Correspondingly,

## Problem 12

The number of distinct even divisors of

is

(a) ;

(b) ;

(c) ;

(d) .

## Solution:

We have

Every even divisor of is of the type with There are even divisors.

## Problem 13

Let be the triangular region in the -plane with vertices at and . Then the value of

is

(a)

(b)

(c) ;

(d)

## Solution:

The region is defined by inequalities Hence

## Problem 14

Given a real number , define a sequence by the following recurrence relation:

If then the value of is

(a)

(b)

(c)

(d)

## Solution:

Consider the characteristic equation of the recurrence:

Roots are and Hence,

Since we find that To find we use first two values:

Then So,

## Problem 15

A straight line passes through the intersection of the lines given by and and makes equal intercepts of the same sign on the coordinate axes. The equation of the straight line is

(a)

(b)

(c)

(d)

## Solution:

We find the intersection of the lines:

The line we are looking for passes through the point and makes equal (positive) intercepts with the coordinate axes. The slope of such line is and its equation is of the form

We find

So, the equation of the line is

Equivalently,

## Problem 16

The series

(a) converges for and diverges for

(b) converges for all

(c) converges for and diverges for

(d) converges for and diverges for .

## Solution:

Denote Apply the ratio test:

Hence, the radius of convergence of the power series is equal to 1. The series is convergent for and is divergent for

We study convergence at Observe that

Hence, there exists such that for all we have

Equivalently,

Taking the sum from to we find

So,

The sequence of partial sums is bounded and the series is convergent.

## Problem 17

Suppose and are two square matrices such that the largest eigenvalue of is positive. Then the smallest eigen value of

(a) must be positive

(b) must be negative

(c) must be

(d) is none of the above

## Solution:

We compute the trace of the matrix:

Since the trace is the sum of eigenvalues and the largest eigenvalue of is positive, the smallest eigenvalue of is negative.

## Problem 18

The number of saddle points of the function is

(a) ;

(b) ;

(c) ;

(d) none of the above.

## Solution:

To find stationary points we compute the gradient of

Stationary point is found from the system

There are three stationary points: The matrix of second derivatives of is

Eigenvalues are and The stationary point is a saddle point if the first eigenvalue is negative. It happens only for the point

## Problem 19

Suppose is a cyclic group and . There does not exist any such that Also, there does not exist an such that . Then,

(a) there exists an element such that .

(b) there exists an element such that .

(c) the smallest exponent such that for some is

(d) none of the above is true.

## Solution:

Let be the generator of Then for some integer Observe that is odd. Similarly, for some odd . But then where is even. Define Then

## Problem 20

The number of real roots of the polynomial is

(a) ;

(b) ;

(c) ;

(d) .

## Solution:

Denote The derivative of is

The function is increasing on decreasing on , increasing on

The only real zero of is located in

## Problem 21

Suppose that a matrix has an eigen value . If the matrix
is equal to

then the eigen vectors of corresponding to the eigenvalue are in
the form,

(a) ;

(b) ;

(c) ;

(d)

## Solution:

Let be the eigenvector of corresponding to the eigenvalue Then
Denote

So, Let The vector is of the form

## Problem 22

Consider the function defined by . For and , the value of the limit

is

(a)

(b)

(c)

(d) none of the above

Hence

## Problem 23

Suppose is a solution of the differential equation such that and . Then

(a) as

(b) as

(c) as

(d) as

## Solution:

The characteristic equation is Its roots are and The solution is of the form

To find and we use initial conditions:

Hence, and the solution is In particular, as

## Problem 24

A fair die is rolled five times. What is the probability that the largest
number rolled is   ?

(a)

(b)

(c)

(d)

## Solution:

Denote by the event that all numbers rolled are Then The event of interest is Its probability is

## Problem 25

Two rows of chairs, facing each other, are laid out. The number of different ways that couples can sit on these chairs such that each person sits directly opposite to his/her partner is

(a)

(b)

(c)

(d)

## Solution:

There are choices how each couple can choose its pair of opposite chairs. When a pair of chairs is chosen, there are ways for each couple they can sit on these chairs, i.e. choices for all couples. Totally there are ways the couples can sit on the chairs.

## Problem 26

Consider the function defined on the complex plane by . For a real number , let and . Then

(a) both and are straight line segments

(b) is a circle and is a straight line segment

(c) is a straight line segment and is a circle

(d) both and are circles

## Solution:

Let Then and the set consists of points of the form

is a cricle of center nd radius

Let Then and the set consists of point of the form

is a straight line connecting and

## Problem 27

Consider two real valued functions and given by
for , and for
Which of the following statements about inverse functions is true?

(a) Neither nor exists

(b) exists, but not

(c) does not exist, but does

(d) Both and exist.

## Solution:

The function is strictly decreasing:

So, is a bijection of the set exists.

The function is strictly decreasing:

So, is a bijection of exists.

## Problem 28

A circle is drawn with centre at touching externally. Then the circle touches

(a) both the axes

(b) only the -axis

(c) none of the two axes

(d) only the -axis

## Solution:

Let be the radius of the circle centered at The equation of the second circle is

It is centered at and of radius The distance between centers of circles is

Hence, and the first circle touches both axes (as and belong to the first circle).

## Problem 29

Let for all and . Then the value of is

(a)

(b)

(c)

(d)

## Solution:

Observe that By assumption,

Differentiate this equation in

Differentiate in :

Then

Take

## Problem 30

Let

The number of elements in the set

is
(a)

(b)

(c)

(d)

## Solution:

The determinant of must be zero:

Hence, and the matrix is of the form

The rank of is if and only if So, there are possible values for