# NBHM (National Board for Higher Mathematics) MSc and MA Mathematics: Questions 55 - 64 of 101

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## Question 55

Appeared in Year: 2010

### Describe in Detail

Essay▾

Find the order of the pole and its residue at of the function

### Explanation

Order of pole

At , order of pole is n if

And , is the smallest such integer.

But

Order of pole at is

Formula for a pole of order n.

## Question 56

Appeared in Year: 2017

### Describe in Detail

Essay▾

Let C be the contour in the complex plane consisting of two straight line segments, one from and the other from . Evaluate

Where

### Explanation Graph of the Two Straight Line Segments

Along OM

The imaginary axis from is the line segment

Now on so that

Also y varies from

… eq. (1)

Along MP

MP is the line parallel to the x-axis (real axis) from on the line segment

Also x varies from

… eq. (2)

[From 1 and 2]

… (2 more words) …

## Question 57

Appeared in Year: 2007

Essay▾

Evaluate

### Explanation

Let … eq. (1)

… eq. (2)

Adding (1) and (2) , we get

… eq. (3)

Let

Put

When

When

From eq. (3)

… (3 more words) …

## Question 58

Appeared in Year: 2008

### Describe in Detail

Essay▾

Which of the following maps are differentiable everywhere?

a)

b) Such that for all

c) Where

### Explanation

Option (a)

Clearly f is continuous for all

Clearly f is differentiable for all except we can՚t say at

At

Since

So f is differentiable at

is differentiable everywhere

Option (b)

Let i.e. take limit as

is a constant

is differentiable everywhere [constant function is differentiable everywhere]

Option (c)

Clearly function is differentiable at all poi…

… (26 more words) …

## Question 59

Appeared in Year: 2010

### Describe in Detail

Essay▾

In each of the following verify whether the series is absolutely convergent conditionally or divergent:

a)

b)

c)

### Explanation

Option (a) The given series is

Where

Now

… eq. (1)

does not converge

Given series is not absolutely convergent

Also it is not convergent by Leibnitz test (by eq. (1) )

The given series diverge

Option (b)

The given series is . Where

Take

Both converge or diverge together

Converges by p test

(By comparison test)

is absolutely convergent

Option (c) …

… (51 more words) …

## Question 60

Appeared in Year: 2005

### Describe in Detail

Essay▾

Find the sum of the series

## Question 61

Appeared in Year: 2009

### Describe in Detail

Essay▾

Test the following series for convergence.

a)

b)

Where is a convergent series of positive terms.

### Explanation

Option (a) The given series is

Here

By Cauchy՚s Root test the given series is convergent.

option (a) is correct

Option (b) Using Result are convergent, then is convergent.

Now the given series is

Let

And is convergent (given)

Take

is convergent (By p – test)

By result (stated above)

is also convergent

i.e.. is convergent

is convergent

Here (b) is…

… (4 more words) …

## Question 62

Appeared in Year: 2013

Short Answer▾

Evaluate

## Question 63

Appeared in Year: 2014

### Describe in Detail

Essay▾

Find the sum of the following infinite series

### Explanation

… eq. (A) [ Rearranging the terms]

Now we know that

Put

… eq. (1)

Also … eq. (2)

Using eq. (1) & (2) in (A)

Required sum of the infinite series is

… (3 more words) …

## Question 64

Appeared in Year: 2012

### Question

MCQ▾

Determine if the following claim is true or false:

### Choices

Choice (4)Response

a.

b.

c.

d.

Question does not provide sufficient data or is vague