# NBHM (National Board for Higher Mathematics) MSc and MA Mathematics: Questions 1 - 9 of 101

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## Question number: 1

» Sequences and Series (Convergence)

Appeared in Year: 2006

### Write in Short

For what value of p does the following series converge?

## Question number: 2

» Continuity and Differentiability

Appeared in Year: 2005

### Describe in Detail

Let is a differentiable function such that for all . For what value of will be the function be necessarily one-to-one?

### Explanation

Let

Then

if ………. . eq. (A)

If

Then [By LMVT]

[From (A) ]

then the function is not injective i. e o

… (34 more words) …

## Question number: 3

» Line Integral in Complex Plane

Appeared in Year: 2009

### Describe in Detail

Let and let f be defined by

If C is the straight-line segment joining ; compute

### Explanation

Let P be the point in which represents the complex number .

is the line from

On so that

… (84 more words) …

## Question number: 4

Appeared in Year: 2006

### Question

Pick out the function which are continous atleast at one point in the real line.

### Choices

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

a. |
| |

b. |
| |

c. |
| |

d. | Both b. and c. are correct |

## Question number: 5

Appeared in Year: 2008

### Write in Short

Write down an equation of degree four satisfied by all the complex fifth roots of unity.

## Question number: 6

Appeared in Year: 2008

### Describe in Detail

Evaluate:

### Explanation

This is exponential form form

Using the Result

If

Then

… (70 more words) …

## Question number: 7

Appeared in Year: 2014

### Describe in Detail

Find the sum of the following infinite series.

### Explanation

… (76 more words) …

## Question number: 8

Appeared in Year: 2013

### Describe in Detail

Let . Find the sum of the infinite series.

### Explanation

Let

Put

…………… eq. (A)

Now we know the exponential series

…………. . eq. (1)

Replacing by in (1)

…………. . eq. (2)

Adding eq

… (92 more words) …

## Question number: 9

Appeared in Year: 2005

### Write in Short

Let be two twice continuously differentiable functions on satisfying the Cauchy –Riemann equation. Let . Define . Express the complex derivative of i. e. in terms of the partial derivatives of .