# NBHM (National Board for Higher Mathematics) MSc and MA Mathematics: Questions 65 - 73 of 101

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## Question 65

Appeared in Year: 2009

Essay▾

Evaluate

Let

Answer is zero

## Question 66

Appeared in Year: 2016

### Describe in Detail

Essay▾

Let be an arbitrary positive real number define

Which of the following statements are true?

a) For all , we have

b) The sequence is monotonic

c) The sequence convergent

### Explanation

… eq. (1)

That point (By Induction)

Result is true for

Assume that the result is true for

Result is true for

But result is true for

By method of Induction, result is true for all

From eq. (1)

(This is quadratic equation in )

is real

i.e.

This quadratic has a real solution

… eq. (2)

i.e. for all

(a) Option is true

( By (2) )

And

is monotonic de&#8230;

… (20 more words) …

## Question 67

Appeared in Year: 2006

### Question

MCQ▾

Let denote the set of all continuously differentiable real valued function defined on the real line. Define

where denotes the derivative of the function f. Pick out the true statements.

### Choices

Choice (4)Response

a.

A is an Infinite set.

b.

A is Empty set

c.

A is a finite and non-empty set

d.

All of the above

## Question 68

Appeared in Year: 2007

Essay▾

Evaluate

## Question 69

Appeared in Year: 2005

Short Answer▾

Evaluate:

## Question 70

Appeared in Year: 2017

Essay▾

Evaluate

### Explanation

Let us consider the sequence as follows

… eq. (1)

Using the result (from Cauchy՚s second theorem on limits)

If is a sequence such that and

Then

… eq. (2)

From eq. (1) & (2)

… (4 more words) …

## Question 71

Appeared in Year: 2005

Short Answer▾

Differentiate

## Question 72

Appeared in Year: 2009

### Describe in Detail

Essay▾

Write down the coefficient of in the Taylor series expansion of the function about the origin.

### Explanation

Taylor series expansion about the origin is given by

Now coefficient of is

(Neglecting higher terms)

Coefficient of

## Question 73

Appeared in Year: 2012

### Describe in Detail

Essay▾

Find all the square Roots of the complex number .

### Explanation

The square roots of are

And

Here let

Comparing

The square roots of are

Thus all the square roots of are