ENGINEERING MATHEMATICS - I

1 ENGINEERING MATHEMATICS - I DIPLOMA COURSE IN ENGINEERING & TECHNOLOGY FIRST SEMESTER A Publication under Government of Tamilnadu Distribution of Free Textbook Programme (NOT FOR SALE) Untouchability is a sin Untouchability is a crime Untouchability is an inhuman DIRECTORATE OF TECHNICAL EDUCATION GOVERNMENT OF TAMILNADU ii Government of Tamilnadu First Edition 2020 CHAIRPERSON Thiru. K. VIVEKANANDAN, , Director of Technical Education Directorate of Technical Education Guindy, Chennai - 600 025. Coordinator: Dr. Padmanabhan, Principal (i/c), Central Polytechnic College, Chennai 600 113. Conveners: 1. Tmt. M. Narayanavadivoo, Lecturer/ MATHEMATICS , Murugappa Polytechnic College, Sathyamurthy Nagar, Avadi, Chennai 62. 2. Thiru. J. Krishnan, Senior Lecturer/ MATHEMATICS , Ramakrishna Mission Polytechnic College, Mylapore, Chennai 4.

2 Authors: 1. Thiru. I. Nagarajan, HOS/Maths, Thiagarajar Polytechnic College, Salem 636005. 5. Tmt. V. Kavithamani, Lecturer(SS)/Maths Arasan Ganesan Polytechnic College, Anaikuttam ( ), Viruthunagar Main Road, Sivakasi 626 103. 2. Thiru. R. Saravanakumar, Lecturer/ Maths, GRG Polytechnic College, Kuppepalayam, SS kulam ( ), Coimbatore 641 107. 6. Thiru. N. Eswaran, Lecturer/ Maths, Thiru. K. Sekar, Lecturer/ Maths, Ramasamy, Lecturer/ Maths, Sri Ramakrishna Mission Vidyalaya Polytechnic College, Coimbatore 20. 3. Tmt. R. Valarmathi, Lecturer/ Maths PAC Ramasamy Raja Polytechnic College, Kumarasamy Raja Nagar, Rajapalayam 626 108. 7. Tmt. Suganthi, Lecturer (SS)/ Maths, TPEVR Government Polytechnic College, Vellore.

3 4. Tmt. M. Sasikala, Lecturer (SG)/Maths, Tmt. M. Sivapriya, Lecturer (SG)/Maths, PSG Polytechnic College, Coimbatore 641 004. 8. R. Muthu Bhavani, Lecturer/ Maths, Rajagopal Polytechnic College, Gandhi Nagar, Gudiyatham, Vellore 632 602. This book has been prepared by the Directorate of Technical Education This book has been printed on 60 Paper Through the Tamil Nadu Text book and Educational Services Corporation Printed by Web Offset at: iii THE NATIONAL ANTHEM FULL VERSION Jana-gana-mana-adhinayaka jaya he Bharata-bhagya-vidhata Punjaba-Sindhu-Gujarata-Maratha- Dravida-Utkala-Banga Vindhya-Himachala-Yamuna-Ganga Uchchhala-jaladhi-taranga Tava Subha name jage,TavaSubhaasisa mage, Gahe tava jaya-gatha. Jana-gana-mangala-dayaka jaya he Bharata-bhagya-vidhata. Jaya he, jaya he, jaya he, Jaya jaya jaya jaya he.

4 - Rabindranath Tagore SHORT VERSION Jana-gana-mana-adhinayaka jaya he Bharata-bhagya-vidhata. Jaya he, jaya he, jaya he, Jaya jaya jaya jaya he. AUTHENTIC ENGLISH TRANSLATION OF THE NATIONAL ANTHEM Thou art the ruler of the minds of all people, Thou dispenser of India's destiny. Thy name rouses the hearts of the Punjab, Sind, Gujarat and Maratha, of Dravida, Orissa and Bengal It echoes in the hills of the Vindhyas and Himalayas, mingles in the music of the Yamuna and Ganges and is chanted by the waves of the Indian Sea. They pray for Thy blessings and sing Thy praise The saving of all people waits in Thy hand, Thou dispenser of India's destiny. Victory, Victory, Victory to Thee iv THE NATIONAL INTEGRATION PLEDGE I solemnly pledge to work with dedication to preserve and strengthen the freedom and integrity of the nation.

5 I further affirm that I shall never resort to violence and that all differences and disputes relating to religion, language, region or other political or economic grievances should be settled by peaceful and constitutional means. INVOCATION TO GODDESS TAMIL Bharat is like the face beauteous of Earth clad in wavy seas; Deccan is her brow crescent-like on which the fragrant 'Tilak' is the blessed Dravidian land. Like the fragrance of that 'Tilak' plunging the world in joy supreme reigns Goddess Tamil with renown spread far and wide. Praise unto You, Goddess Tamil, whose majestic youthfulness, inspires awe and ecstasy . v PREFACE 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 under N scheme framed by the Directorate of Technical Education, Chennai.

6 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 and learning process more easy and effective. Each chapter in this book is prepared with strenuous effort to present the principles of the subject in the most easy to understand and the most easy to workout manner. Each chapter and section is presented with QR code, an introduction, learning objective, definitions, theorems , explanation, worked examples, summary and exercises with answer 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 the book serve the purpose keeping in mind the changing needs of the society to make it lively and vibrating.

7 The language used is very clear and simple which is up to the level of comprehension of students. We extend our deep sense of gratitude to Thiru. , the Chairperson for giving valuable inputs and suggestions in bringing out this text book for the benefit of the student community. We also thank the Co-ordinator Dr. Padmanabhan, Principal(i/c), Central Polytechnic College, Chennai and Conveners who took sincere efforts in preparing and reviewing this book. Valuable suggestions and corrections for the improvement of this book is most welcome and will be acknowledge most gratefully. Mail your suggestions to - AUTHORS vi ANNEXURE- I STATE BOARD OF TECHNICAL EDUCATION &TRAINING, TAMILNADU DIPLOMA IN ENGINEERING / TECHNOLOGY SYLLABUS N-SCHEME (Implemented from the Academic year 2020 - 2021 onwards) Course Name : All branches of Diploma in ENGINEERING and Technology and Special Programs except DMOP, HMCT and Film &TV.

8 Subject Code : 40012 Semester : I Subject Title : ENGINEERING MATHEMATICS - I TEACHING AND SCHEME OF EXAMINATION No. of weeks per semester: 16 weeks Subject Instructions Examination Hours / Week Hours / Semester Marks Duration Internal Assessment Board Examinations Total ENGINEERING MATHEMATICS I 5 80 25 100* 100 3 Hrs. * Examinations will be conducted for 100 marks will be reduced to 75 marks. TOPICS AND ALLOCATION OF HOURS: Sl. No. Topics Time (Hrs) 1 Algebra 15 2 Complex Number 15 3 Trigonometry 14 4 Differential Calculus I 15 5 Differential Calculus II 14 Test & Model Exam 7 TOTAL 80 vii 40012 ENGINEERING MATHEMATICS I DETAILED SYLLABUS Contents: Theory UNIT NAME OF THE TOPICS HOURS I ALGEBRA MATRICES AND DETERMINANTS: MATRICES: Definition, Concept and Types of Matrices.

9 OPERATIONS ON MATRICES: Multiplication of a Matrix by a scalar, Addition/Subtraction of two Matrices. Multiplication of two Matrices - properties. Reducing a Matrix into triangular and echelon form. Transpose of a Matrix and its properties. DETERMINANTS: Definition and Evaluation of 2nd and 3rd order Determinants. Properties of determinants, product of Determinants. Determinant of a square Matrix singular and non singular Matrices - simple problems. APPLICATIONS OF MATRICES AND DETERMINANTS: Co-factor, Adjoint of Matrix, Inverse of Matrix and Rank of a matrix Simple problems. Solution of simultaneous equations using Cramer s rule - Matrix Inversion method - Gaussian Elimination method simple problems. Characteristic Equation Eigen Values and Eigen Vectors of a real matrix consistency and inconsistency of system of linear equations.

10 BINOMIAL THEOREM: Introduction Factorial, Permutation and Combinations Values of nPr and nCr. Statement of Binomial theorem for positive integral index. Expansion of Binomial - Finding general term Middle term Coefficient of xn and Term independent of x Binomial Theorem for rational index up to -3. Applications of binomial theorem Finding the remainder, digits of a number and greatest term simple problems. 3 7 5 II COMPLEX NUMBERS ALGEBRA OF COMPLEX NUMBERS Introduction Complex Numbers Conjugates Algebra of complex numbers (without geometrical proof), Properties of complex conjugates - Modulus and Amplitude - Polar and Euler form of a complex number Simple problems. Argand Diagram Collinear points, four points forming square, rectangle, rhombus and parallelogram only - Simple problems.