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## Monday, November 16, 2020

# NCERT Class 12 Mathematics Part 1

289 Pages · 2014 · 3.16 MB · English

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

Answers

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.

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)

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

management.

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

**About AMAN KUMAR**

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Very nice book bhai

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