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NCERT Class 12 Mathematics Part 1


NCERT Class 12 Mathematics Part 1

289 Pages · 2014 · 3.16 MB · 54,084 Downloads· English



Foreword v
Preface 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


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
In this chapter, we will study different types of relations and functions, composition
of functions, invertible functions and binary operations.

Chapter 2 


Mathematics, in general, is fundamentally the science of
self-evident things. — FELIX KLEIN

2.1 Introduction
In Chapter 1, we have studied that the inverse of a function
f, denoted by f

–1, exists if f is one-one and onto. There are
many functions which are not one-one, onto or both and
hence we can not talk of their inverses. In Class XI, we
studied that trigonometric functions are not one-one and
onto over their natural domains and ranges and hence their
inverses do not exist. In this chapter, we shall study about
the restrictions on domains and ranges of trigonometric
functions which ensure the existence of their inverses and
observe their behaviour through graphical representations.
Besides, some elementary properties will also be discussed.
The inverse trigonometric functions play an important
role 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 Concepts
In 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} ✂ R
cotangent function, i.e., cot : R – { x : x = n☎, n ✥ Z} ✂ R
secant 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


The essence of Mathematics lies in its freedom. — CANTOR
3.1 Introduction
The knowledge of matrices is necessary in various branches of mathematics. Matrices
are one of the most powerful tools in mathematics. This mathematical tool simplifies
our work to a great extent when compared with other straight forward methods. The
evolution of concept of matrices is the result of an attempt to obtain compact and
simple methods of solving system of linear equations. Matrices are not only used as a
representation of the coefficients in system of linear equations, but utility of matrices
far exceeds that use. Matrix notation and operations are used in electronic spreadsheet
programs for personal computer, which in turn is used in different areas of business
and science like budgeting, sales projection, cost estimation, analysing the results of an
experiment etc. Also, many physical operations such as magnification, rotation and
reflection through a plane can be represented mathematically by matrices. Matrices
are also used in cryptography. This mathematical tool is not only used in certain branches
of sciences, but also in genetics, economics, sociology, modern psychology and industrial
In this chapter, we shall find it interesting to become acquainted with the
fundamentals of matrix and matrix algebra.
3.2 Matrix
Suppose we wish to express the information that Radha has 15 notebooks. We may
express it as [15] with the understanding that the number inside [ ] is the number of
notebooks that Radha has. Now, if we have to express that Radha has 15 notebooks
and 6 pens. We may express it as [15 6] with the understanding that first number
inside [ ] is the number of notebooks while the other one is the number of pens possessed
by Radha. Let us now suppose that we wish to express the information of possession

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