# 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

Appeared in Year: 2006

### Describe in Detail

Evaluate

### Explanation

It is of the form where

Replace

Lower limit

Lower limit

Required limit is

Putting

At

At

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## Question number: 11

Appeared in Year: 2006

### Write in Short

Evaluate

## Question number: 12

» Convergence of Positive Series Test

Appeared in Year: 2013

### 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

Appeared in Year: 2012

### 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.

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## Question number: 14

Appeared in Year: 2012

### 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:**

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## Question number: 15

Appeared in Year: 2013

### 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

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## Question number: 16

Appeared in Year: 2014

### 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

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## Question number: 17

Appeared in Year: 2014

### 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.

In

……………eq. (1)

Volume of cylinder, V

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