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

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Question number: 10

» Definite Integral as Limit of a Sum

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Appeared in Year: 2006

Essay Question▾

Describe in Detail

Evaluate

Explanation

It is of the form where

Replace

Lower limit

Lower limit

Required limit is

Putting

At

At

… (47 more words) …

Question number: 11

» Indeterminate Forms

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Appeared in Year: 2006

Short Answer Question▾

Write in Short

Evaluate

Question number: 12

» Convergence of Positive Series Test

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Appeared in Year: 2013

MCQ▾

Question

Which of the following series are convergent?

Choices

Choice (4)Response

a.

b.

c.

d.

Both a. and b. are correct

Question number: 13

» Differentiability in Complex Plane

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Appeared in Year: 2012

Essay Question▾

Describe in Detail

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.

… (446 more words) …

Question number: 14

» Points of Continuity

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Appeared in Year: 2012

Essay Question▾

Describe in Detail

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:

… (254 more words) …

Question number: 15

» Binomial Series

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Appeared in Year: 2013

Essay Question▾

Describe in Detail

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

… (195 more words) …

Question number: 16

» Geometry in Complex Plane

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Appeared in Year: 2014

Essay Question▾

Describe in Detail

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

Figure of Equilateral Triangle

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Question number: 17

» Maxima and Minima

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Appeared in Year: 2014

Essay Question▾

Describe in Detail

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

A Right Circular Cylinder

In

……………eq. (1)

Volume of cylinder, V

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