Preface. This book covers topics including algebra, trigonometry, vectors & scalars, matrices & determinants and mensuration.

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1 Preface We are thankful to Almighty ALLAH Who gave us an opportunity to write the book, named Applied Mathematics-I as Textbook of Diploma of Associate Engineer (DAE), intending to cover the new syllabus for the first year students Throughout the book, emphasis is on correct methods of computation, transposition of formulae, logical layout of solutions, neatness and clarity of arrangement of material, systematic use of all the normal mathematical and other tables. This book covers topics including algebra, trigonometry, vectors & scalars, matrices & determinants and mensuration. Normally students face difficulty in solving complicated problems because they do not make a systematic attempt. We have attempted to help the students to overcome the difficulty by providing detailed instructions for an orderly approach. Difficult procedures and types of problems appearing in the exercise are illustrated by carefully explained examples. In the presentation of these illustrated examples, we have avoided unnecessary explanations. It is hoped that this book will help to give students a good foundation in old and new techniques. Students are reminded that in order to acquire a proper understanding of the subject and its application, it is necessary to learn a number of sound basic rules and methods.no scientific or engineering subject can be fully comprehended and satisfactorily studied without a sound mathematical background. We would like to express, sincere and thanks to Mr.Jawad Ahmed Qureshi Chief Operating Officer TEVTA,Engr. Mr.Azhar Iqbal Shad G.M Academic,Engr.Mazhar Abbas Naqvi Manager (Curriculum) and Engr. Syed Muhammad Waqar ud- Din Deputy Director(technical ) Curriculum Section Acade mics Wing, who took keen interest and inspired us for the completion of this task. We made every effort to make the book valuable both for students and teachers, however we shall gratefully welcome to receive any suggestion for the further improvement of the book. Authors.

2 APPLIED A Textbook of MATHEMATICS-I for Math 113 First year Diploma of Associate Engineer (DAE) Including: Objective Type and Authors/MDS Sanaullah Khan Tahir Hameed Nasir-ud-Din Mahmood Associate Professor Assistant Professor Assistant Professor Govt. College of Tech. Govt. College of Tech. Govt. College of Tech. RailwayRoad Lahore Raiwind Road, Lahore for Printing&Graphic Arts Reviewed by(mec) Muhammad Javaid Iqbal Associate Professor Govt. College of Tech. RailwayRoad Lahore Syed Tanvir Haider Assistant Professor Govt. College of Tech. Raiwind Road Lahore

3 CONTENTS MATH 113 PAPER A From Chapter 1 to 7 Chapter 1 Quadratic Equation Equation: 1.2 Degree of an Equation 1.3 Polynomial Equation of Degree n 1.4 Linear and Cubic Equation 1.5 Quadratic Equation 1.6 Roots of the Equation 1.7 Methods of Solving Quadratic Equation Exercise Classification of Numbers 1.9 Nature of the roots of the Equation 1.10 Sum and Product of the Roots Exercise Formation of Quadratic Equation from the given roots Exercise 1.4 Chapter 2 Sequences and Series Introduction 2.2 Sequences 2.3 Progression 2.4 Arithmetic Sequence 2.5 nth term or General term(or, last term)of an Arithmetic Progression Exercise Arithmetic Mean 2.7 n Arithmetic Mean s between a and b Exercise Arithmetic Series Exercise Geometric Sequence or Progression (G.P) 2.10 nth term or General term(or, last term) of a Geometric Progression (G.P) Exercise Geometric Mean 2.12 n G.Ms Between a and b Exercise Geometric Series

4 Exercise Infinite Geometric Sequence 2.14 Recurring Decimals Exercise 2.7 Short questions Chapter 3 Binomial Theorem Introduction 3.2 Factorial of a Positive Integer 3.3 Combination 3.4 The Binomial Theorem 3.5 General Term 3.6 Middle Term in the Expansion (a + b) n Exercise Binomial Series 3.8 Application of the Binomial Series; Approximations: Exercise 3.2 Chapter 4 Partial Fractions Introduction 4.2 Partial fractions 4.3 Polynomial 4.4 Rational fraction 4.5 Proper Fraction 4.6 Improper Fraction 4.7 Process of Finding Partial Fraction 4.8 Type I Exercise Type II Exercise Type III Exercise Type IV Exercise 4.4 Chapter 5 Fundamentals of Trigonometry Introduction

5 5.2 Types of Trigonometry 5.3 Angle 5.4 Quadrants 5.5 Measurement of Angles 5.6 Relation between Degree and Radian Measure 5.7 Relation between Length of a Circular Arc and the Radian Measure of its Central Angle Exercise Trigonometric Function and Ratios 5.9 Reciprocal Functions 5.10 Rectangular Co-ordinates and Sign Convention 5.11 Signs of Trigonometric Functions 5.12 Trigonometric Ratios of Particular Angles Exercise Fundamental Identities Exercise Graph of Trigonometric Functions Chapter 6 General Identities Introduction 6.2 Distance formula 6.3 Fundamental law of trigonometry 6.4 Deductions from fundamental law Exercise Double Angle Identities 6.6 Half Angle identities 6.7 Triple angle identities Exercise Conversion of sum or difference to products 6.9 Converting Products to Sum or Difference Exercise 6.3 Chapter 7 Solution of Triangles Solution of Triangles Exercise Application of Right Angled Triangles 7.3 Angle of Elevation and Depression Exercise Law of Sines Exercise7.3

6 7.5 The Law of Cosines Exercise Solution of Oblique Triangles Exercise 7.5 PAPER B From Chapter 8 to 19 Chapter 8 Vectors and Scalars Introduction 8.2 Scalars and Vectors 8.3 Vector Representations 8.4 Types of Vectors 8.5 Addition and Subtraction of Vectors 8.6 Multiplication of a Vector by a Scalar 8.7 The Unit Vectors i, j, k (orthogonal system of unit Vectors) 8.8 Representation of a Vector in the form of Unit Vectors i, j and k. 8.9 Components of a Vector when the Tail is not at the Origin 8.10 Magnitude or Modulus of a Vector 8.11 Direction Cosines Exercise Product of Vectors 8.13 Rectangular form of a x b (Analytical expression of ) Exercise 8.2 Objective Type Question Chapter 9 Matrices and Determinants Introduction 9.2 Matrix 9.3 Some types of matrices 9.4 Operations on matrices Exercise Determinants 9.6 Minor and Cofactor of Element 9.7 Properties of the Determinant 9.8 Solution of Linear Equations by Determinants Exercise Special Matrices 9.11 Solution of Linear Equations by Matrices Exercise 9.3

7 Objective type Exercise Chapter 10 Area of triangles Introduction to Mensuration 10.2 Plane Figures 10.3 Triangle 10.4 Area of Triangles Exercise 10 Objective type Questions Chapter 11 Area of Quadrilateral Quadrilateral 11.2 Area of Quadrilateral Exercise 11 Objective Type Exercise Chapter 12 Area of Polygons Polygon 12.2 Method for Finding Area of Regular Polygon Exercise 12 Chapter 13 Area of Circle Circle 13.2 Area of the Circle 13.3 Concentric Circles 13.4 Sector of a Circle 13.5 Area of Segment 13.6 Ellipse Exercise 13 Chapter 14 Area of Irregular Plane Figures Mid Ordinate Rule 14.2 Trapezoidal Rule 14.3 Simpson s Rule Exercise 14

8 Chapter 15 Mensuration of Solid Solid 15.2 Mensuration of Prisms 15.3 Surface Area of a Prims 15.4 Volume of a Prism 15.5 Types of a Prism 15.6 Frustum of a Prism Exercise 15 Chapter 16 Mensuration of Cylinder Cylinder 16.2 Volume and Surface area of Different Kinds of Cylinders 16.3 Frustum of a Right Circular Cylinder Exercise 16 Chapter 17 Mensuration of Pyramid Pyramid 17.2 Right or Regular Pyramid 17.3 The surface area and Volume of a Regular Pyramid Exercise 17(A) 17.4 Frustum of a Pyramid 17.5 Volume and surface area of Frustum of a Regular Pyramid Exercise 17(B) Chapter 18 Mensuration of Cone Cone 18.2 Volume and Surface area of a Right Circular Cone Exercise 18(A) 18.3 Frustum of Cone 18.4 Volume and Surface area of the Frustum of a Cone Exercise 18(B) Chapter 19 Mensuration of Sphere

9 19.1 Sphere 19.2 Surface Area and Volume of a Sphere Exercise 19(A) 19.3 Zone (Frustum) of a Sphere 19.4 Volume and Surface Area of the Zone 19.5 Spherical Segment or Cap of a Sphere 19.6 Volume and Surface Area of Spherical Segment 19.7 Sector of Sphere Exercise 19(B)

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