# NBHM (National Board for Higher Mathematics) MSc and MA Mathematics: Questions 27 - 36 of 101

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## Question 27

## Question 28

Appeared in Year: *2011*

### Describe in Detail

Essay▾Find the points z in the complex plane where exists and evaluate it at those points

a)

b) , Where denotes the imaginary part of z.

### Explanation

(a) Let

Clearly all the four partial derivatives exist and continuous functions of

The points z in the complex plane where exist are those for which Cauchy – Riemann conditions are satisfied

i.e.

i.e.

Thus the points where exist are

(b) Let

Now all these partial derivatives exist and continuous functions of

The points where exists are those wher…

… (4 more words) …

## Question 29

Appeared in Year: *2017*

### Describe in Detail

Essay▾Let , be a fixed positive integer. Let for

Evaluate:

### Explanation

Numerical value of last term of

i.e.

And

Here

The given series is

Differentiating both sides with respect to , we get

Putting , then we get

Thus the answer is zero

Alternative Method

L. H. S

in bracket put , then

## Question 30

Appeared in Year: *2015*

### Describe in Detail

Essay▾Write down the power series expansion of the function in a neighbourhood of .

### Explanation

So that

This is the Required power series Expansion of the function in the neighbourhood of .

## Question 31

Appeared in Year: *2005*

### Describe in Detail

Essay▾What is the Radius of Convergence of the power series

?

### Explanation

The given power series is

Now

## Question 32

Appeared in Year: *2006*

### Describe in Detail

Essay▾Let . Define Evaluate as a function of x.

### Explanation

Leibnitz Rule is given by

If the function are defined on and are differentiation at a point is continuous, then

[Using Leibnitz Rule]

Which is a function of x.

## Question 33

Appeared in Year: *2010*

### Describe in Detail

Essay▾Let C denote the circle in the complex plane, described in the positive i.e. (anti-clockwise) sense. Evaluate

### Explanation

Here let

Here is a simple pole and lies within the circle

By Cauchy residue theorem

## Question 34

## Question 35

Appeared in Year: *2012*

### Question

MCQ▾Let be a sequence of functions defined on . Determine . For each the following.

### Choices

Choice (4) | Response | |
---|---|---|

a. | ||

b. | ||

c. | ||

d. | All a., b. and c. are correct |

## Question 36

Appeared in Year: *2013*

### Describe in Detail

Essay▾Find the area of the polygon whose vertices are represented in the complex plane by the eighth roots of unity.

### Explanation

In triangle OAB

area of triangle

Area of polygon whose vertices are represented in the complex plane by the eighth roots of unity