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

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## Question 1

Appeared in Year: *2006*

### Write in Short

Short Answer▾For what value of p does the following series converge?

## Question 2

Appeared in Year: *2005*

### Describe in Detail

Essay▾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. one-to-one

i.e..

so is we want f to be one-to-one

… (1 more words) …

## Question 3

Appeared in Year: *2009*

### Describe in Detail

Essay▾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

## Question 4

Appeared in Year: *2006*

### Question

MCQ▾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 5

Appeared in Year: *2008*

### Write in Short

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

## Question 6

Appeared in Year: *2008*

### Describe in Detail

Essay▾Evaluate:

### Explanation

This is exponential form form

Using the Result

If

Then

[using the result]

## Question 7

Appeared in Year: *2014*

### Describe in Detail

Essay▾Find the sum of the following infinite series.

### Explanation

## Question 8

Appeared in Year: *2013*

### Describe in Detail

Essay▾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. (1) & (2) and dividing by 2

From eq. (A)

… (3 more words) …

## Question 9

Appeared in Year: *2005*

### Write in Short

Short Answer▾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 .