# NCERT Class 12 Mathematics Part 1

## ContentsPART I

### Foreword vPreface vii

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
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

# Chapter 1

## RELATIONS AND FUNCTIONS

There is no permanent place in the world for ugly mathematics ... . It may
be very hard to define mathematical beauty but that is just as true of
beauty of any kind, we may not know quite what we mean by a
beautiful poem, but that does not prevent us from recognising
one when we read it. — G. H. HARDY

1.1 Introduction
Recall that the notion of relations and functions, domain,
co-domain and range have been introduced in Class XI
along with different types of specific real valued functions
and their graphs. The concept of the term ‘relation’ in
mathematics has been drawn from the meaning of relation
in English language, according to which two objects or
quantities are related if there is a recognisable connection
or link between the two objects or quantities. Let A be
the set of students of Class XII of a school and B be the
set of students of Class XI of the same school. Then some
of the examples of relations from A to B are
(i) {(a, b) ✂ A × B: a is brother of b},
(ii) {(a, b) ✂ A × B: a is sister of b},
(iii) {(a, b) ✂ A × B: age of a is greater than age of b},
(iv) {(a, b) ✂ A × B: total marks obtained by a in the final examination is less than
the total marks obtained by b in the final examination},
(v) {(a, b) ✂ A × B: a lives in the same locality as b}. However, abstracting from
this, we define mathematically a relation R from A to B as an arbitrary subset
of A × B.
If (a, b) ✂ R, we say that a is related to b under the relation R and we write as
a R b. In general, (a, b) ✂ R, we do not bother whether there is a recognisable
connection or link between a and b. As seen in Class XI, functions are special kind of
relations.
In this chapter, we will study different types of relations and functions, composition
of functions, invertible functions and binary operations.

# Chapter 2

## INVERSE TRIGONOMETRICFUNCTIONSMathematics, in general, is fundamentally the science ofself-evident things. — FELIX KLEIN2.1 IntroductionIn Chapter 1, we have studied that the inverse of a functionf, denoted by f–1, exists if f is one-one and onto. There aremany functions which are not one-one, onto or both andhence we can not talk of their inverses. In Class XI, westudied that trigonometric functions are not one-one andonto over their natural domains and ranges and hence theirinverses do not exist. In this chapter, we shall study aboutthe restrictions on domains and ranges of trigonometricfunctions which ensure the existence of their inverses andobserve their behaviour through graphical representations.Besides, some elementary properties will also be discussed.The inverse trigonometric functions play an importantrole in calculus for they serve to define many integrals.The concepts of inverse trigonometric functions is also used in science and engineering.2.2 Basic ConceptsIn Class XI, we have studied trigonometric functions, which are defined as follows:sine function, i.e., sine : R ✂ [– 1, 1]cosine function, i.e., cos : R ✂ [– 1, 1]tangent function, i.e., tan : R – { x : x = (2n + 1) 2✁, n ✥ Z} ✂ Rcotangent function, i.e., cot : R – { x : x = n☎, n ✥ Z} ✂ Rsecant function, i.e., sec : R – { x : x = (2n + 1) 2✁, n ✥ Z} ✂ R – (– 1, 1)cosecant function, i.e., cosec : R – { x : x = n☎, n ✥ Z} ✂ R – (– 1, 1)# Chapter 3 MATRICESThe essence of Mathematics lies in its freedom. — CANTOR3.1 IntroductionThe knowledge of matrices is necessary in various branches of mathematics. Matricesare one of the most powerful tools in mathematics. This mathematical tool simplifiesour work to a great extent when compared with other straight forward methods. Theevolution of concept of matrices is the result of an attempt to obtain compact andsimple methods of solving system of linear equations. Matrices are not only used as arepresentation of the coefficients in system of linear equations, but utility of matricesfar exceeds that use. Matrix notation and operations are used in electronic spreadsheetprograms for personal computer, which in turn is used in different areas of businessand science like budgeting, sales projection, cost estimation, analysing the results of anexperiment etc. Also, many physical operations such as magnification, rotation andreflection through a plane can be represented mathematically by matrices. Matricesare also used in cryptography. This mathematical tool is not only used in certain branchesof sciences, but also in genetics, economics, sociology, modern psychology and industrialmanagement.In this chapter, we shall find it interesting to become acquainted with thefundamentals of matrix and matrix algebra.3.2 MatrixSuppose we wish to express the information that Radha has 15 notebooks. We mayexpress it as  with the understanding that the number inside [ ] is the number ofnotebooks that Radha has. Now, if we have to express that Radha has 15 notebooksand 6 pens. We may express it as [15 6] with the understanding that first numberinside [ ] is the number of notebooks while the other one is the number of pens possessedby Radha. Let us now suppose that we wish to express the information of possession

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