# NBHM (National Board for Higher Mathematics) MSc and MA Mathematics: Questions 37 - 46 of 101

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## Question 37

Appeared in Year: *2011*

### Describe in Detail

Essay▾Evaluate

### Explanation

And

This is exponential form

Using the Result:

If

Then

Now

… eq. (1)

Fore (1)

Required answer is

… (1 more words) …

## Question 38

Appeared in Year: *2007*

### Describe in Detail

Essay▾Find the Minimum value of the function:

### Explanation

Here

**Step I**:

**Step II**: Let us solve

i.e.

and

we get

is the only critical point

**Step III**:

**Step IV**:

Also

is Minimum at

And minimum value

is the required solution.

## Question 39

Appeared in Year: *2005*

### Describe in Detail

Essay▾Let denote the seventh roots of unity. Evaluate

### Explanation

Let x be nth root of unity. Then

(Where k is an integer)

Let . Then nth roots of unity are , i.e.. nth roots of unity are

**Product of the roots**

… (A)

Here is the seventh roots of unity

Product of the roots

… (1 more words) …

## Question 40

Appeared in Year: *2006*

### Describe in Detail

Essay▾What is the relation between the height and radius of a right circular cylinder of fixed volume and Minimal Total surface area?

### Explanation

Here given that is the radius of the base of right circular cylinder and its height. is the volume and Let be the total surface area.

Also

Now

Now

When

S is Minimum when

i.e.

This is the required relation between the height h and radius r.

## Question 41

Appeared in Year: *2012*

### Describe in Detail

Essay▾Find the area of the pentagon whose vertices are the fifth root of unity in the complex plane.

### Explanation

To find fifth root of unity.

Let

Putting we get

Thus the fifth roots of unity are

i.e..

Where

Now area of pentagon whose vertices are the fifth roots of unity in the complex plane is given by

Where n is the number of sides

Therefore the required answer is

## Question 42

Appeared in Year: *2015*

### Describe in Detail

Essay▾Does the integral exist? If it exists give its value.

### Explanation

Yes, its value is …

Note that is not defined for . Here we can write;

(Integration by parts)

… eq. (A)

Now

(Using Hosptital rule)

Hence from eq. (A)

Yes, its value exists and its value is .

… (2 more words) …

## Question 43

Appeared in Year: *2014*

### Describe in Detail

Essay▾Which of the following series are convergent?

a)

b)

c) , Where

### Explanation

Option (a) The given series is where

Take

Which is finite and non – zero

By comparison test

behave alike

But is divergent by p – test

is also divergent

So option (a) is not convergent

Option b) Convergent (Already solved)

Option c)

The given series is where

The given series is convergent (By ratio test)

… (2 more words) …

## Question 44

Appeared in Year: *2016*

### Describe in Detail

Essay▾Let be a fined positive integer. Let

Evaluate

### Explanation

Consider the expansion … eq. (1)

Integrating both sides of (1) within limits , we get

… (1 more words) …

## Question 45

Appeared in Year: *2007*

### Write in Short

Short Answer▾Let be a seventh root of unity. Write down a polynomial equation of degree satisfied by w.

## Question 46

Appeared in Year: *2010*

### Describe in Detail

Essay▾Let C be the semicircle where varies from , in the complex plane. Evaluate:

### Explanation

Equation of given circle is