# NBHM (National Board for Higher Mathematics) MSc and MA Mathematics: Questions 84 - 93 of 101

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## Question 84

Appeared in Year: 2011

### Describe in Detail

Essay▾

Test the following series for convergence.

a)

b)

### Explanation

Option (a) The given series is

Take

which is non – zero and finite

converge or diverge together.

But is convergent by p – test

Given series is convergent

option (a) is correct

Option (b) The given series is

Where

Let

which is finite & non – zero.

By comparison test, converge or diverge together.

But is convergent

Given series is convergent.

… (3 more words) …

## Question 85

Appeared in Year: 2006

### Describe in Detail

Essay▾

Find the interval of convergence of the series

### Explanation

The given series is

Put

The given power series

Let

By Cauchy second theorem on limits, we have

The given series converges for

This is the interval of convergence

At the series converges by Leibnitz test

At , the series converges by comparison test

So answer is

## Question 86

Appeared in Year: 2013

Essay▾

Evaluate

### Explanation Graph of the Evaluate

## Question 87

Appeared in Year: 2008

Essay▾

Evaluate

### Explanation

Replacing by the sign of

Upper limit

Lower limit

## Question 88

Appeared in Year: 2016

### Question

MCQ▾

Let Which of the following statements are true?

### Choices

Choice (4)Response

a.

is rational for all

b.

is Irrational for all

c.

is rational if

d.

All of the above

## Question 89

Appeared in Year: 2010

### Describe in Detail

Essay▾

Let be continuously differentiable.

Evaluate

### Explanation

It is of the form

So using

Function is continuously differentiable)

## Question 90

Appeared in Year: 2012

### Question

MCQ▾

Pick out the true statements:

### Choices

Choice (4)Response

a.

for all

b.

for all

c.

for all

d.

All of the above

## Question 91

Appeared in Year: 2016

Essay▾

Evaluate

### Explanation Graph of the Points of Intersection

Drow the graphs of .

Their points of intersection are given by

The region in the dark is the

## Question 92

Appeared in Year: 2012

### Describe in Detail

Essay▾

Write down explicitly the Expression for the nth derivative of the function.

### Explanation

If are function of x possessing nth derivative then

[Leibnitz Theorem]

Let

.

.

By Leibnitz Theorem

(all other terms are zero)

This is the Required solution of nth derivative of

## Question 93

Appeared in Year: 2012

### Describe in Detail

Essay▾

Let be the Boundary of the square in the complex plane with vertices at the points which is described in the anticlockwise direction. Evaluate .

### Explanation The Boundary of the Square

Along OA

… eq. (1)

Along AB

… eq. (2)

Along BC

… eq. (3)

Along CO

… eq. (4)

Now

… (4 more words) …