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