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MATHEMATICS I & II
DIPLOMA COURSE IN ENGINEERING
FIRST SEMESTER
A Publication under
Government of Tamilnadu
Distribution of Free Textbook Programme
( NOT FOR SALE )
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DIRECTORATE OF TECHNICAL EDUCATION
GOVERNMENT OF TAMIL NADU
Government of Tamilnadu
First Edition – 2011
Chairperson
Thiru Kumar Jayanth I.A.S
Commissioner of Technical Education
Directorate of Technical Education, Chennai –25
Co-ordinator
Convener
Er. S. Govindarajan
P.L. Sankar
Principal
Dr. Dharmambal Government
Polytechnic College
Tharamani, Chennai – 113
Lecturer (Selection Grade)
Rajagopal Polytechnic College
Gudiyatham-632602
Reviewer
Dr. S. Paul Raj
Associate Professor, Dept of Mathematics
Anna University, MIT Campus, Chennai - 42.
Authors
R. Ramadoss
Lecturer, (Selection Grade)
TPEVR Govt. Polytechnic College
Vellore-632002
B.R. Narasimhan
Lecturer (Selection Grade)
Arulmigu Palaniandavar Polytechnic
College
Palani-624601
M. Devarajan
Lecturer (Selection Grade)
Dr. Dharmambal Government
Polytechnic College for Women
Taramani, Chennai-600113
Dr. L. Ramuppillai
Lecturer (Selection Grade)
Thiagarajar Polytechnic College
Selam-636005
K. Shanmugam
Lecturer (Selection Grade)
Government Polytechnic College
Purasawalkam, Chennai-600 012
Dr.A. Shanmugasundaram
Lecturer (Selection Grade)
Valivalam Desikar Polytechnic College
Nagappattinam-611 001
M. Ramalingam
Lecturer (Selection Grade)
Government Polytechnic College
Tuticorin-628008
R. Subramanian
Lecturer (Selection Grade)
Arasan Ganesan Polytechnic College
Sivakasi-626130
Y. Antony Leo
Lecturer
Mothilal Nehru Government
Polytechnic College
Pondicherry-605008
C. Saravanan
Lecturer (Senior Scale)
Annamalai Polytechnic College
Chettinad-630102
This book has been prepared by the Directorate of Technical Education
This book has been printed on 60 G.S.M Paper
Through the Tamil Nadu Text Book Corporation
FOREWORD
We take great pleasure in presenting this book of mathematics to
the students of Polytechnic Colleges. This book is prepared in
accordance with the new syllabus framed by the Directorate of
Technical Education, Chennai.
This book has been prepared keeping in mind, the aptitude and
attitude of the students and modern methods of education. The lucid
manner in which the concepts are explained, make the teaching
learning process more easy and effective. Each chapter in this book is
prepared with strenuous efforts to present the principles of the subject
in the most easy-to-understand and the most easy-to-workout
manner.
Each chapter is presented with an introduction, definition,
theorems, explanation, worked examples and exercises given are for
better understanding of concepts and in the exercises, problems have
been given in view of enough practice for mastering the concept.
We hope that this book serves the purpose i.e., the curriculum
which is revised by DTE, keeping in mind the changing needs of the
society, to make it lively and vibrating. The language used is very clear
and simple which is up to the level of comprehension of students.
List of reference books provided will be of much helpful for further
reference and enrichment of the various topics.
We extend our deep sense of gratitude to Thiru.S.Govindarajan,
Co-ordinator and Principal, Dr. Dharmambal Government polytechnic
College for women, Chennai and Thiru. P.L. Sankar, convener,
Rajagopal polytechnic College, Gudiyatham who took sincere efforts
in preparing and reviewing this book.
Valuable suggestions and constructive criticisms for
improvement of this book will be thankfully acknowledged.
Wishing you all success.
Authors
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SYLLABUS
FIRST SEMESTER MATHEMATICS - I
UNIT - I
DETERMINANTS
1.1 Definition and expansion of determinants of order 2 and 3 .Properties
of determinants .Cramer's rule to solve simultaneous equations in 2
and 3 unknowns-simple problems.
1.2 Problems involving properties of determinants
1.3 Matrices :Definition of matrix .Types of matrices .Algebra of matrices
such as equality, addition, subtraction, scalar multiplication and
multiplication of matrices. Transpose of a matrix, adjoint matrix and
inverse matrix-simple problems.
UNIT - II
BINOMIAL THEOREM
2.1 Definition of factorial notation, definition of Permutation and
Combinations with formula. Binomial theorem for positive integral
index (statement only), finding of general and middle terms. Simple
problems.
2.2 Problems finding co-efficient of xn, independent terms. Simple
problems. Binomial Theorem for rational index, expansions, only
upto – 3 for negative integers. Simple Expansions
2.3 Partial Fractions :Definition of Polynomial fraction, proper and
improper fractions and definition of partial fractions.
To resolve proper fraction into partial fraction with denominator
containing non repeated linear factors, repeated linear factors and
irreducible non repeated quadratic factors. Simple problems.
iv
UNIT - III
STRAIGHT LINES
3.1 Length of perpendicular distance from a point to the line and
perpendicular distance between parallel lines. Simple problems.
Angle between two straight lines and condition for parallel and
perpendicular lines. Simple problems
3.2 Pair of straight lines Through origin :Pair of lines passing through the
origin ax2+2hxy+by2=0 expressed in the form (y-m1x)(y-m2x) =0.
Derivation of angle between pair of straight lines. Condition for
parallel and perpendicular lines. Simple problems
3.3 Pair of straight lines not through origin: Condition for general
equation of the second degree ax2+2hxy+by2+2gx+2fy+c=0 to
represent pair of lines.(Statement only) Angle between them,
condition for parallel and perpendicular lines. Simple problems..
UNIT - IV
TRIGONOMETRY
4.1 Trigonometrical ratio of allied angles-Expansion of Sin(A+B) and
cos(A+B)- problems using above expansion
4.2 Expansion of tan(A+B) and Problems using this expansion
4.3 Trigonometrical ratios of multiple angles (2A only) and sub-multiple
angles. Simple problems.
UNIT - V
TRIGONOMETRY
5.1 Trigonometrical ratios of multiple angels (3A only) Simple
problems.
5.2 Sum and Product formulae-Simple Problems.
5.3 Definition of inverse trigonometric ratios, relation between inverse
trigonometric ratios-Simple Problems
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FIRST SEMESTER MATHEMATICS II
UNIT - I
CIRCLES
1.1 Equation of circle – given centre and radius. General Equation of
circle – finding center and radius. Simple problems.
1.2 Equation of circle through three non collinear points – concyclic
points. Equation of circle on the line joining the points (x1,y1) and
(x2,y2) as diameter. Simple problems.
1.3 Length of the tangent-Position of a point with respect to a circle.
Equation of tangent (Derivation not required). Simple problems.
UNIT-II
FAMILY OF CIRCLES:
2.1 Concentric circles – contact of circles (internal and external circles) –
orthogonal circles – condition for orthogonal circles.(Result only).
Simple Problems
2.2 Limits:Definition of limits x n - an
= na n-1
x ® a x-a
Lt
sin q
= 1,
q®0 q
Lt
Lt
q®0
tan q
=1
q
(q in radian)
[Results only] – Problems using the above results.
2.3 Differentiation:Definition – Differentiation of u xn, sinx, cosx, tanx,
v
cotx, secx, cosecx, logx, ex, u v, uv, uvw,
(Results only).
Simple problems using the above results.
UNIT- III
3.1 Differentiation of function of functions and Implicit functions. Simple
Problems.
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3.2 Differentiation of inverse trigonometric functions and parametric
functions. Simple problems.
3.3 Successive differentiation up to second order (parametric form not
included) Definition of differential equation, formation of differential
equation. Simple Problems
UNIT- IV
APPLICATION OF DIFFERENTIATION–I
4.1 Derivative as a rate measure-simple Problems.
4.2 Velocity and Acceleration-simple Problems
4.3 Tangents and Normals-simple Problems
UNIT-V
APPLICATION OF DIFFERENTIATION –II
5.1 Definition of Increasing function, Decreasing function and turning
points. Maxima and Minima (for single variable only) – Simple
Problems.
5.2 Partial Differentiation: Partial differentiation of two variable up to
second order only. Simple problems.
5.3 Definition of Homogeneous functions-Eulers theorem-Simple
Problems.
vii
FIRST SEMESTER
MATHEMATICS - I
Contents
Page No
Unit – 1
DETERMINANTS .......................................................1
1.1 Introduction ................................................................ 1
1.2 Problems Involving Properties of Determinants ......... 12
1.3 Matrices ...................................................................... 19
Unit – 2
BINOMIAL THEOREM............................................... 44
2.1 Introduction ................................................................ 45
2.2 Binomial Theorem ...................................................... 47
2.3 Partial Fractions .......................................................... 55
Unit – 3
STRAIGHT LINES ...................................................... 69
3.1 Introduction ................................................................ 69
3.2 Pair of straight lines through origin ............................. 81
3.3 Pair of straight lines not through origin ....................... 89
Unit – 4
TRIGONOMETRY ...................................................... 101
4.1 Trigonometrical Ratios of
Related or Allied Angles ............................................. 104
4.2 Compound Angles (Continued) ................................. 107
4.3 Multiple Angles of 2A Only and
Sub – Multiple Angles................................................. 119
Unit - 5
TRIGONOMETRY ...................................................... 129
5.1 Trigonometrical Ratios of Multiple Angle of 3A ........... 129
5.2 Sum and Product Formulae ....................................... 135
5.3 Inverse Trigonometric Function ................................. 142
MODEL QUESTION PAPER ............................................ 161
viii
MATHEMATICS – II
Contents
Page No
Unit – 1
CIRCLES ................................................................ 167
1.1 Circles .................................................................... 167
1.2 Concyclic Points...................................................... 171
1.3 Length of the Tangent to a circle
from a point(x1,y1) .................................................... 176
Unit – 2
FAMILY OF CIRCLES ............................................ 185
2.1 Family of circles ...................................................... 185
2.2 Definition of limits .................................................... 191
2.3 Differentiation.......................................................... 196
Unit – 3
Differentiation Methods ........................................ 206
3.1 Differentiation of function of functions ...................... 206
3.2 Differentiation of Inverse Trigonometric Functions ....214
3.3 successive differentiation ........................................ 223
Unit – 4
APPLICATION OF DIFFERENTIATION ..................232
4.1 Derivative as a Rate Measure .................................. 232
4.2 Velocity and Acceleration ........................................ 238
4.3 Tangents and Normals ............................................ 242
Unit – 5
APPLICATION OF DIFFERENTIATION-II ............. 254
5.1 Introduction ............................................................ 254
5.2. Partial derivatives.................................................... 271
5.3 Homogeneous Functions ........................................ 277
MODEL QUESTION PAPER .......................................... 292
ix