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1. Relations and Functions 1 1.1 Introduction 1 1.2 Types of Relations 2 1.3 Types of Functions 7 1.4 Composition of Functions and Invertible Function 12 1.5 Binary Operations 19 2. Inverse Trigonometric Functions 33 2.1 Introduction 33 2.2 Basic Concepts 33 2.3 Properties of Inverse Trigonometric Functions 42 3. Matrices 56 3.1 Introduction 56 3.2 Matrix 56 3.3 Types of Matrices 61 3.4 Operations on Matrices 65 3.5 Transpose of a Matrix 83 3.6 Symmetric and Skew Symmetric Matrices 85 3.7 Elementary Operation (Transformation) of a Matrix 90 3.8 Invertible Matrices 91 4. Determinants 103 4.1 Introduction 103 4.2 Determinant 103 4.3 Properties of Determinants 109 4.4 Area of a Triangle 121 4.5 Minors and Cofactors 123 4.6 Adjoint and Inverse of a Matrix 126 4.7 Applications of Determinants and Matrices 133 5. Continuity and Differentiability 147 5.1 Introduction 147 5.2 Continuity 147 5.3 Differentiability 161 5.4 Exponential and Logarithmic Functions 170 5.5 Logarithmic Differentiation 174 5.6 Derivatives of Functions in Parametric Forms 179 5.7 Second Order Derivative 181 5.8 Mean Value Theorem 184 6. Application of Derivatives 194 6.1 Introduction 194 6.2 Rate of Change of Quantities 194 6.3 Increasing and Decreasing Functions 199 6.4 Tangents and Normals 206 6.5 Approximations 213 6.6 Maxima and Minima 216 Appendix 1: Proofs in Mathematics 247 A.1.1 Introduction 247 A.1.2 What is a Proof? 247 Appendix 2: Mathematical Modelling 256 A.2.1 Introduction 256 A.2.2 Why Mathematical Modelling? 256 A.2.3 Principles of Mathematical Modelling 257 Answers 268 7. Integrals 287 7.1 Introduction 288 7.2 Integration as an Inverse Process of Differentiation 288 7.3 Methods of Integration 300 7.4 Integrals of some Particular Functions 307 7.5 Integration by Partial Fractions 316 7.6 Integration by Parts 323 7.7 Definite Integral 331 7.8 Fundamental Theorem of Calculus 334 7.9 Evaluation of Definite Integrals by Substitution 338 7.10 Some Properties of Definite Integrals 341 8. Application of Integrals 359 8.1 Introduction 359 8.2 Area under Simple Curves 359 8.3 Area between Two Curves 366 9. Differential Equations 379 9.1 Introduction 379 9.2 Basic Concepts 379 9.3 General and Particular Solutions of a 383 Differential Equation 9.4 Formation of a Differential Equation whose 385 General Solution is given 9.5 Methods of Solving First order, First Degree 391 Differential Equations 10. Vector Algebra 424 10.1 Introduction 424 10.2 Some Basic Concepts 424 10.3 Types of Vectors 427 10.4 Addition of Vectors 429 10.5 Multiplication of a Vector by a Scalar 432 10.6 Product of Two Vectors 441 11. Three Dimensional Geometry 463 11.1 Introduction 463 11.2 Direction Cosines and Direction Ratios of a Line 463 11.3 Equation of a Line in Space 468 11.4 Angle between Two Lines 471 11.5 Shortest Distance between Two Lines 473 11.6 Plane 479 11.7 Coplanarity of Two Lines 487 11.8 Angle between Two Planes 488 11.9 Distance of a Point from a Plane 490 11.10 Angle between a Line and a Plane 492 12. Linear Programming 504 12.1 Introduction 504 12.2 Linear Programming Problem and its Mathematical Formulation 505 12.3 Different Types of Linear Programming Problems 514 13. Probability 531 13.1 Introduction 531 13.2 Conditional Probability 531 13.3 Multiplication Theorem on Probability 540 13.4 Independent Events 542 13.5 Bayes' Theorem 548 13.6 Random Variables and its Probability Distributions 557 13.7 Bernoulli Trials and Binomial Distribution 572 Answers 588