# 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*

### Describe in Detail

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*

### Question

True-False▾### Statements

If … eq. (1)

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

If is continuous and differentiable in , where and if

Then there exists such that

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. |