NBHM (National Board for Higher Mathematics) MSc and MA Mathematics: Questions 10 - 17 of 101

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Question 10

Appeared in Year: 2006

Describe in Detail

Essay▾

Evaluate

Explanation

It is of the form where

Replace

Lower limit

Lower limit

Required limit is

Putting

At

At

Question 11

Appeared in Year: 2006

Write in Short

Short Answer▾

Evaluate

Question 12

Appeared in Year: 2013

Question

MCQ▾

Which of the following series are convergent?

Choices

Choice (4)Response

a.

b.

c.

d.

Both a. and b. are correct

Question 13

Appeared in Year: 2012

Describe in Detail

Essay▾

Define

Which of the following statements are true?

a) is not differentiable at

b) is differentiable at

c) is differentiable only at

Explanation

Using the Result

If

Then is continous at i.e. if is root of , then is continous at and these are the only points at which function will be continuous.

will be continuous at points where

is continuous at only.

is discontinuous at

is not discontinuous at

Differentiability at

is Rational

Now if is Rational

Then if is irrational

(Both the case)

I…

… (18 more words) …

Question 14

Appeared in Year: 2012

Describe in Detail

Essay▾

List the let of points of discontinuity, if any for the following function

a) Defined by

b) Defined by

c) Defined by

Where is the largest integer less than, or equal to .

Explanation

(a) Two cases arise

Case I: is irrational

Left hand limit

Here can be rational or rational

can be

L. H. L does not exist

Similarly R. H. L does not exist

So is discontinuous at all irrational points of

Case II: is Rational

Figure of the Rational

i.e. can be either

R. H. L. dose not exist

Similarly L. H. L does not exist

is discontinuous at rational …

… (66 more words) …

Question 15

Appeared in Year: 2013

Describe in Detail

Essay▾

Find the sum of the infinite series

Explanation

… eq. (A)

Comparing eq. (A) with the standard form i.e..

Therefore … eq. (1)

… eq. (2)

And … eq. (3)

Rewrite eq. (2)

(Using (1) )

Hence sum of the series S is given by

Answer

… (4 more words) …

Question 16

Appeared in Year: 2014

Describe in Detail

Essay▾

Find the area of the triangle in the complex plane whose vertices are the points representing the number , the cube roots of unity.

Explanation

Let then

Or

Or

So the cube roots of unity are

Also note that cube roots of unity represents vertices of equilateral triangle on the Argand plane

Figure of Equilateral Triangle

Sq. units

Question 17

Appeared in Year: 2014

Describe in Detail

Essay▾

A right circular cylinder is inscribed in a sphere of radius . What is the height of the cylinder when its volume is maximal?

Explanation

Let be the radius and height of the inscribed cylinder respectively.

A Right Circular Cylinder

In

… eq. (1)

Volume of cylinder, V

Also

is Maximum when

… (1 more words) …