BASIC MATHEMATICSFOR POLYTECHNIC

BASICMATHEMATICS

FOR POLYTECHNIC

By

Dr. N.R. PandyaPrincipal (I/C)

Government Polytechnic, Kheda (Kapadwanj)Gujarat

An ISO 9001:2008 Company

BENGALURU � CHENNAI � COCHIN � GUWAHATI � HYDERABAD

JALANDHAR � KOLKATA � LUCKNOW � MUMBAI � RANCHI � NEW DELHI

BOSTON (USA) � ACCRA (GHANA) � NAIROBI (KENYA)

BASIC MATHEMATICS FOR POLYTECHNICS

© by Laxmi Publications Pvt. Ltd. All rights reserved including those of translation into other languages. In accordance with the Copyright (Amendment) Act, 2012, no part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise. Any such act or scanning, uploading, and or electronic sharing of any part of this book without the permission of the publisher constitutes unlawful piracy and theft of the copyright holder’s intellectual property. If you would like to use material from the book (other than for review purposes), prior written permission must be obtained from the publishers.

Printed and bound in India Typeset at Goswami Associates, Delhi

First Edition : 2016ISBN 978-93-85750-33-5

Limits of Liability/Disclaimer of Warranty: The publisher and the author make no representation or warranties with respect to the accuracy or completeness of the contents of this work and specifically disclaim all warranties. The advice, strategies, and activities contained herein may not be suitable for every situation. In performing activities adult supervision must be sought. Likewise, common sense and care are essential to the conduct of any and all activities, whether described in this book or otherwise. Neither the publisher nor the author shall be liable or assumes any responsibility for any injuries or damages arising here from. The fact that an organization or Website if referred to in this work as a citation and/or a potential source of further information does not mean that the author or the publisher endorses the information the organization or Website may provide or recommendations it may make. Further, readers must be aware that the Internet Websites listed in this work may have changed or disappeared between when this work was written and when it is read.

All trademarks, logos or any other mark such as Vibgyor, USP, Amanda, Golden Bells, Firewall Media, Mercury, Trinity, Laxmi appearing in this work are trademarks and intellectual property owned by or licensed to Laxmi Publications, its subsidiaries or affiliates. Notwithstanding this disclaimer, all other names and marks mentioned in this work are the trade names, trademarks or service marks of their respective owners.

Published in india by

An ISO 9001:2008 Company 113, GOLDEN HOUSE, DARYAGANJ,NEW DELHI - 110002, INDIA Telephone : 91-11-4353 2500, 4353 2501 Fax : 91-11-2325 2572, 4353 2528 C—www.laxmipublications.com info@laxmipublications.com Printed at:

& Bengaluru 080-26 75 69 30

& Chennai 044-24 34 47 26, 24 35 95 07

& Cochin 0484-237 70 04, 405 13 03

& Guwahati 0361-254 36 69, 251 38 81

& Hyderabad 040-27 55 53 83, 27 55 53 93

& Jalandhar 0181-222 12 72

& Kolkata 033-22 27 43 84

& Lucknow 0522-220 99 16

& Mumbai 022-24 91 54 15, 24 92 78 69

& Ranchi 0651-220 44 64

Bran

ches

1. BASIC ALGEBRA ... 1–27

1.1 Introduction 11.2 Logarithm ... 51.3 Types of Logarithms ... 71.4 Theorems ... 71.5 Use of Graphical Software for Logarithms ... 23

2. MATRIX ALGEBRA ... 28–57

2.1 Introduction ... 282.2 Determinant ... 282.3 Matrix: A Basic Idea ... 302.4 Types of Matrix ... 312.5 Basic Operations ... 332.6 Transpose of a Matrix ... 382.7 Co-factor ... 392.8 Adjoint of a Matrix: Adj(M) ... 402.9 Inverse of a Matrix ... 402.10 Simultaneous Equations ... 432.11 Kick Start ... 51

3. TRIGONOMETRY (CIRCULAR FUNCTION) ... 58–106

3.1 Introduction ... 583.2 Measurements of Angles ... 593.3 Relation between Degree and Radian ... 593.4 Trigonometric Ratios ... 603.5 Identities ... 613.6 Standard Angles ... 653.7 Allied Angles ... 673.8 Compound Angles ... 723.9 Multiple and Submultiple Angles ... 753.10 Product Formulae ... 793.11 Periodic Functions ... 823.12 Graphs ... 833.13 Inverse Trigonometric Function ... 86

Contents

Chapter Pages

( v )

( vi )

4. VECTOR ALGEBRA ... 107–131

4.1 Introduction ... 1074.2 Scalars and Vectors ... 1074.3 Position Vector ... 1084.4 Addition and Subtraction ... 1094.5 Direction Cosine ... 1114.6 Product of Two Vectors ... 1134.7 Applications ... 118

5. MENSURATION ... 132–156

5.1 Introduction ... 1325.2 Area of Regular Figure ... 1325.3 Area of Circle ... 1405.4 Surface Area and Volume ... 143

Preface

Friends, time has come to know something more about Mathematics. Mathematics hasits own inevitable part as oxygen for a human. It is absolutely important to balance ourtechnical ecology in the form of Mathematics. It requires the supporting device as computingenvironment.

In this book, I have tried to incorporate the graphical dimension with MATLAB muse.Graph is very important to any rudimentary engineer both to be guided and glided. It cansolve the practical problem without performing the actual participations that may involvehuge money or time.

I have made the concepts of MATLAB to the lowest platform. Just go and grab it, mydear friends. That will be elevated to the highest pick in future.

And I don’t want to talk about Mathematics; you just experience and enlighten your soul.Here, at this juncture I want to thank Laxmi Publication Pvt. Ltd., New Delhi to

provide me the meeting with my beloved students and learned faculties. I am also thankfulto Dr Sonal Mehta (Lecturer in English, Government Girls Polytechnic, Ahmedabad) toassist me in linguistic part of this book and my past work. Lastly, I can not forget my wifeBhamini for consuming her precious time and Indirect support.

—Author

( vii )

1

Chapter1 BASIC ALGEBRA

1.1 INTRODUCTION

Dear friends, you have studied indices and related topics in your school studies. Also, youknow the simple concept of function. Logarithm is very important function in Engineeringand Science. With the help of logarithmic functions or expressions, many concepts and practicalproblems become simple to explain. The graph of huge data can be converted into logarithmicgraph without loosing the original characteristics.

In this section, Basic Algebra includes Indices, Surds and Logarithms. There are certaintools required as basic techniques to the various forthcoming concepts. Though you may havestudied these topics in previous years, it is repeated for the sake of revision and betterunderstanding. So, let us start with Indices.

Indices

As we know, x2 = x × x, 1/2x x . Here, observe that x is raised to the power of 2 and ½. Indicerefers to the power to which a number is raised. Look at these,

x3 × x2 = (x × x × x) × (x × x) = x3+2 = x5

Or

x3/x2 = (x × x × x)/(x × x) = x3–2 = x

And likewise we can conclude many results. These results can be considered as Formulae.

2 BASIC MATHEMATICS

Laws of Indices

Observe the following laws of Indices, where x and y are any real numbers and m, n arerational numbers.

1. m n m nx x x 2. /m n m nx x x

3. ( )m n mnx x 4. ( )m m nx y x y

5.⎛ ⎞

⎜ ⎟⎝ ⎠

m m

m

x xy y

Note:

(i) If m = n, for 2nd law, x0 = 1. (ii) ( ) andn n mm n m m nx x x x

(iii)

00, 0 and .

0a

a

Example 1.1. Simplify 82/3 × 2–1/2.

Solution. Let us apply the rule 82/3 × 2–1/2 = 2 1

3 3 2(2 ) 2

=

1 1

22 2 22 2 2 = 322

Example 1.2. Simplify {[(a2)2/3]–2}–3/8.

Solution. Our expression is {[(a2)3/2]–2}–3/8 = {[a4/3]–2}–3/8

= {a–8/3}–3/8

= 8 3

( ) ( )3 8a = a

Example 1.3. If x y z3 = 9 = 27 , show that 1 1 2+ =

x z y.

Solution. 3x = 9y = 32y x = 2y

1x

= 12y

...(1)

Also, 9y = 27z

32y = 33z 2y = 3z

32y

= 1z

...(2)

Adding (1) and (2),

1 1 1 32 2x z y y

1 1 42x z y

1 1x z

= 2y

, which is proved.

BASIC ALGEBRA 3

Example 1.4. Solve 2x + 4 = 2x + 3 + 4.

Solution. 2x + 4 = 2x + 3 + 4

2x · 24 = 2x · 23 + 4

16 · 2x = 8 · 2x + 4

8 · 2x = 4 2x = 12

2x = 2–1 x = – 1

So, x = – 1 is the solution.

EXERCISE 1.11. Find the value of the following expressions:

(i) 163/2 (ii) 23( 27) (iii) 2 33 22 2

2. Simplify

22 1

2 2

.( ) ( ).

a b aba b ba

× b6

3. Prove that

⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠

2 2 2 2 2 2

. .a ab b b bc c c ca aa b c

b c ax x xx x x

= 1

4. If 2x = 5y = (100)p and 1 1x y

= 2, find value of p.

5. If (xm)n = nmx , then show that mn – 1 · nm – 1 = nm.

Answers

1. (i) 64 (ii) 9 (iii) 256

2. 1 4. 1/6

Surd: A surd is a number or quantity that cannot be expressed as a/b, where a and b are integers.In other words, SURD are the irrational root of a Rational Number.

32,2 3, 2 and 2 5 are surds.

The value of 2 6 2 ( 2 3) = ( 2 2) 3 2 3

Conjugate of 5 2 is 5 3

Rationalize the Surd:

12

is simple expression. Rationalization of surd means remove the surd from

denominator.

i.e.,12

= 22 2

= 2

2

4 BASIC MATHEMATICS

Example 1.5. Simplify 1

3 2.

Solution. 13 2

= 3 213 2 3 2

using the conjugate surd

=3 23 2

= 3 2

No surd in denominator.

Example 1.6. Simplify

1 2+

3 + 1 5 3.

Solution. 1 23 1 5 3

= 3 1 5 31 23 1 3 1 5 3 5 3

using the conjugate surds.

= 3 1 2( 5 3)

2 2 =

2 5 3 3 12

.

Example 1.7. Simplify 8 2 15 .

Solution. We have the surd 8 2 15 , we have to find square root of 8 2 15 . We will use

(a + b)2 formula.

We want to numbers whose sum of its squre is 8 and product is 15. That is 5 and 3.

8 2 15 = 2 2. .( 5) 2 5 3 ( 3)

= ( 5 3) = ( 5 3)

So, square root of ( 5 3) .

Example 1.8. Prove that 3 2 1

5 2 3 28 + 60 = 0.

Solution. 8 60 = 2 2 . .( 5) ( 3) 2 5 3

= 2( 5 3) ( 5 3)

LHS =

3 2 15 2 3 28 60

= 3 2 15 2 5 3 3 2

=5 2 5 3 3 23 2 1

5 2 5 2 5 3 5 3 3 2 3 2

=3( 5 2) 2( 5 3) 3 2

3 2 1

BASIC ALGEBRA 5

= ( 5 2) – ( 5 3 ) – ( 3 2) = 0

Now, friends you know how to use the surds. Also, you will be able to apply thetechnique in future for various expressions. It’s time to solve some examples.

EXERCISE 1.21. Simplify

(i) 2 3 6 2 (ii) 75 108

2. Simplify 3

3 23. Simplify 5 24

4. Prove that1 3 4

6 5 5 2 6 2

= 0

5. Prove that 4 7 4 7 = 2

6. Simplify

2 3 1

4 15 7 40 5 6

Answers

1. (i) 6 (ii) 11 3 2. 3 6

3. 3 2 6. 0

1.2 LOGARITHM

The notion of logarithm, first, introduced by John Napier in his book Miritici LogarithmorumCanonis, Description, in Scotland (1614). His concept is useful in Navigation, Surveying,Astronomy and even in solving the real-time data problems of Numerical Methods.

John Napier observed that any number may be represented as the power of some numberto a base number.

i.e., 8 = 23

Here 8 is a number, 3 is exponent and 2 is base.

i.e., If N = ax, the logarithm is given by loga N = x.

Logarithm of some number (N) to some base (a) is nothing but the exponent.If x, y, a R and a > 0, a 1, y > 0 and y = ax, then the logarithm of y to the base a is x for

example, loga y = x.Here, we can say that a number y which is expressed as the expression ax, then the

logarithm of the number y to the base a is the index x.

6 BASIC MATHEMATICS

As we know 18

= 0.125 or 2–3 = 18

i.e., 0.125 = 2–3. So, the logarithm of 0.125 can be

expressed as

log2(0.125) = –3 or ⎛ ⎞⎜ ⎟⎝ ⎠

21

log8

= –3.

Let us see some simple examples.Example 1.9. Evaluate:

(i) log2 8 (ii) log10 (0.01)

(iii) log27 3 (iv) 21

log8

⎛ ⎞⎜ ⎟⎝ ⎠

.

Solution. (i) Let log2 8 = x, using the definition,2x = 8 2x = 23 x = 3

(ii) Let log10 (0.01) = x 10x = 0.01 10x = 10–2

x = –2(iii) Let log27 3 = x 27x = 3 (33)

x = 3

33x = 31 3x = 1 (Using index)

x = 13

(iv) Let ⎛ ⎞⎜ ⎟⎝ ⎠

21

log8

= x

2x =18

= 2–3

x = –3So, we have evaluated simple logarithmic expressions.Notes 1. We know that a = a, and a0 = 1. So,

a1 = a loga a = 1 and a0 = 1 loga1 = 0.

2. We know that y = ax loga y = x. Replace the result in y = ax

y = loga ya

i.e., 2log 32 = 3.

Example 1.10. Simplify 2log 34 .

Solution. 2log 34 = 24 log 3( 2)

=4

2log 3( 2 ) = 2log 81( 2) = 81 (Using the result y = loga ya )

Basic Mathematics for Polytechnics

Publisher : Laxmi Publications ISBN : 9789385750335 Author : Dr N.R.Pandya

Type the URL : http://www.kopykitab.com/product/11785

Get this eBook

40%OFF