# NBHM (National Board for Higher Mathematics) MSc and MA Mathematics: Questions 94 - 100 of 101

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## Question 94

Appeared in Year: 2008

### Write in Short

Short Answer▾

Let be a simple closed contour in the complex plane described in the positive sense. Evaluate.

When

a) lies inside and

b) lies outside

## Question 95

Appeared in Year: 2013

### Describe in Detail

Essay▾

In each of the following find the points of continuity of f.

a)

b)

### Explanation

Option (a) is discontinuous for all values of x on real line

Points of continuity

Option (b) is continous when

Points of continuity are

Points of continuity are

Case a)

b)

Case (a) (Explanation) Two case arise

Case I: is Rational

Now, we will check left hand limit and right hand limit at

Here is either rational or irrational

can be either , so R. …

… (61 more words) …

## Question 96

Appeared in Year: 2012

### Describe in Detail

Essay▾

Let . Fill in the blanks with the correct sign :

a) ________

b) ________

### Explanation

(a) Let

Then satisfies the conditions of LMVT in consequently there exists some satisfying such that

… eq. (A)

(from eq. (A) )

Correct sign is “

Option (b) Let

So that

Since is continous in and derivable in .

So by LMVT, there exists some such that

… eq. (B)

Now and

(Using B)

Correct sign is “

… (6 more words) …

## Question 97

Appeared in Year: 2007

### Question

MCQ▾

Which of the following series are convergent?

### Choices

Choice (4)Response

a.

b.

c.

d.

Both b. and c. are correct

## Question 98

Appeared in Year: 2007

### Describe in Detail

Essay▾

Find the interval of convergence of the series

### Explanation

The given series is

Put

So the given series is

(By Cauchy՚s second theorem on limits)

The given series converges for

At , the series is

Which is convergent by Leibnitz test.

At , the series is

Which is not convergent by comparison test

Hence the interval of convergence is

… (2 more words) …

## Question 99

Appeared in Year: 2011

Essay▾

Evaluate

### Explanation

Let … eq. (1)

… eq. (2)

Subtracting eq. (2) from (1)

If [ First term, Common Ratio]

… (2 more words) …

## Question 100

Appeared in Year: 2012

True-False▾

### Statements

1. If … eq. (1)

Where , then the polynomial has a root in the interval .

2. If is continuous and differentiable in , where and if

Then there exists such that

3. The polynomial has only one real root.

### Choices

Choice (4)Response

a.

None of the statements is correct.

b.

Both statement Ⅰ & statement Ⅱ are true.

c.

All the statements are correct.

d.

Only statement Ⅱ is true.