s. chand's

S. Chand'sMATHEMATICS

FOR CLASS XIIVOLUME – I

AS PER NEW SYLLABUS

H. K. DASSM.Sc., Diploma in Specialist Studies (Maths.)

University of Hull (England)

Dr. RAMA VERMAM.Sc. (Gold Medalist), Ph.D.

HOD (Mathematics)Mata Sundri College(Delhi University)

This book has been written according to the new NCERT syllabus prescribedby the Central Board of Secondary Education (CBSE) for Class XII

With

Value Based

Questions

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© 2005, H.K. DassAll rights reserved. No part of this publication may be reproduced or copied in any material form (includingphotocopying or storing it in any medium in form of graphics, electronic or mechanical means and whetheror not transient or incidental to some other use of this publication) without written permission of the copyrightowner. Any breach of this will entail legal action and prosecution without further notice.Jurisdiction : All disputes with respect to this publication shall be subject to the jurisdiction of the Courts,Tribunals and Forums of New Delhi, India only.First Edition 2005Subsequent Editions and Reprints 2007, 2008, 2009, 2010 (Twice), 2011, 2012, 2013Reprint with Correction 2014ISBN : 81-219-2900-8 Code : 14B 549PRINTED IN INDIA

By Rajendra Ravindra Printers Pvt. Ltd., 7361, Ram Nagar, New Delhi -110 055and published by S. Chand & Company Pvt. Ltd., 7361, Ram Nagar, New Delhi -110 055.

PREFACE TO THE FIFTH REVISED EDITIONI feel happy to bring out this thoroughly revised edition of XII class Vol.–I, strictly in accordancewith new NCERT, CBSE syllabus.In this revised edition we have included :

(i) Solved and Unsolved Value Based Problems (ii) Hints to the selected questions of the exercises (iii) Very short answer type questions after each chapter

(iv) Multiple Choice questions after each chapter (v) Solved Question papers 2012, 2013.

We are thankful to the Management Team and the Editorial Department of S. Chand & CompanyPvt. Ltd. for all help and support in the publication of this book.Misprints which came to my knowledge have been corrected.Suggestions and healthy criticism from students and teachers to improve the book shall bepersonally acknowledged and deeply appreciate to help me to make it an ideal book for all.

AuthorsD-1/87, JanakpuriNew Delhi-110 058Tel. 011-32985078, 28525078Mobile : 09350055078e-mail : [email protected] : www.hkdass.com

Disclaimer : While the authors of this book have made every effort to avoid any mistake or omission and have used their skill,expertise and knowledge to the best of their capacity to provide accurate and updated information. The authors and S. Chand donot give any representation or warranty with respect to the accuracy or completeness of the contents of this publication and areselling this publication on the condition and understanding that they shall not be made liable in any manner whatsoever. S.Chandand the authors expressly disclaim all and any liability/responsibility to any person, whether a purchaser or reader of this publicationor not, in respect of anything and everything forming part of the contents of this publication. S. Chand shall not be responsible forany errors, omissions or damages arising out of the use of the information contained in this publication.Further, the appearance of the personal name, location, place and incidence, if any; in the illustrations used herein is purelycoincidental and work of imagination. Thus the same should in no manner be termed as defamatory to any individual.

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CLASS XII

SYLLABUSOne Paper Three Hours Marks : 100

Units Marks

I Relations and Functions 10II Algebra 13

III Calculus 44IV Vectors and Three Dimensional Geometry 17V Linear Programming 06

VI Probability 10 Total 100

UNIT-I: RELATIONS AND FUNCTIONS1. Relations and Functions: (10) Periods

Types of relations: reflexive, symmetric, transitive and equivalence relations. One to one andonto functions, composite functions, inverse of a function. Binary operations.

2. Inverse Trigonometric Functions:Definition, range, domain, principal value branches. Graphs of inverse trigonometric func-tions. Elementary properties of inverse trigonometric functions.

UNIT-II: ALGEBRA1. Matrices: (18) Periods

Concept, notation, order, equality, types of matrices, zero matrix, transpose of a matrix, sym-metric and skew symmetric matrices. Addition, multiplication and scalar multiplication ofmatrices, simple properties of addition, multiplication and scalar multiplication. Non-commu-tativity of multiplication of matrices and existence of non-zero matrices whose product is thezero matrix (restrict to square matrices of order 2). Concept of elementary row and columnoperations. Invertible matrices and proof of the uniqueness of inverse, if it exists; (Here allmatrices will have real entries).

2. Determinants: (20) PeriodsDeterminant of a square matrix (up to 3 × 3 matrices), properties of determinants, minors,cofactors and applications of determinants in finding the area of a triangle. Adjoint and ‘inverse of a square matrix. Consistency, inconsistency and number of solutions of system oflinear equations by examples, solving system of linear equations in two or three variables(having unique solution) using inverse of a matrix.

UNIT-III: CALCULUS1. Continuity and Differentiability: (18) Periods

Continuity and differentiability, derivative of composite functions, chain rule, derivatives ofinverse trigonometric functions, derivative of implicit function. Concept of exponential andlogarithmic functions and their derivative. Logarithmic differentiation. Derivative of functionsexpressed in parametric forms. Second order derivatives. Rolle’s andLagrange’s Mean Value Theorems (without proof) and their geometric interpretations.

2. Applications of Derivatives: (10) PeriodsApplications of derivatives: rate of change, increasing/decreasing functions, tangents & normals,approximation, maxima and minima (first derivative test motivated geometrically and secondderivative test given as a provable tool). Simple problems (that illustrate basic principles andunderstanding of the subject as well as real-life situations).

(iv)


3. Integrals: (20) PeriodsIntegration as inverse process of differentiation. Integration of a variety of functions by substitu-tion, by partial fractions and by parts, only simple integrals of the type

2 2 22 2 2 2 2, , , , ,

–

dx dx dx dx dxx a ax bx cx a a x ax bx c

2 2 2 22 2

( ) ( ), , and –px q px qdx dx a x dx x a dxax bx c ax bx c

to be evaluated.Definite integrals as a limit of a sum. Fundamental Theorem of Calculus (without proof). Basicproperties of definite integrals and evaluation of definite integrals

4. Applications of the Integrals: (10) PeriodsApplications in finding the area under simple curves, especially lines, arcs of circles/parabolas/ellipses (in standard form only), area between the two above said curves (the region should beclearly identifiable).

5. Differential Equations: (10) PeriodsDefinition, order and degree, general and particular solutions of a differential equation.Formation of differential equation whose general solution is given. Solution of differentialequations by method of separation of variables, homogeneous differential equations of firstorder and first degree. Solutions of linear differential equation of the type:

( ) ( ),dy p x y q xdx

where p(x) and q(x) are functions of x.

UNIT- IV: VECTORS AND THREE - DIMENSIONAL GEOMETRY1. Vectors: (12) Periods

Vectors and scalars, magnitude and direction of a vector. Direction cosines/ratios of vectors.Types of vectors (equal, unit, zero, parallel and collinear vectors), position vector of a point,negative of a vector, components of a vector, addition of vectors, multiplication of a vector by ascalar, position vector of a point dividing a line segment in a given ratio. Scalar (dot) product ofvectors, projection of a vector on a line. Vector (cross) product of vectors.

2. Three -dimensional Geometry: (12) PeriodsDirection cosines/ratios of a line joining two points. Cartesian and vector equation of a line,coplanar and skew lines, shortest distance between two lines. Cartesian and vector equation of aplane. Angle between (i) two lines, (ii) two planes, (iii) a line and a plane. Distance of a pointfrom a plane.

UNIT-V : LINEAR PROGRAMMING1. Linear Programming: (12) Periods

Introduction, definition of related terminology such as constraints, objective function,optimization, different types of linear programming (L.P.) problems, mathematical formulationof L.P. problems, graphical method of solution for problems in two variables, feasible andinfeasible regions, feasible and infeasible solutions, optimal feasible solutions (up to threenon-trivial constraints).

UNIT-VI: PROBABILITY1. Probability: (18) Periods

Multiplication theorem on probability. Conditional probability, independent events, totalprobability, Baye’s theorem. Random variable and its probability distribution, mean and varianceof haphazard variable. Repeated independent (Bernoulli) trials and Binomial distribution.


CONTENTSChapter Description Page

No.UNIT — I: RELATIONS AND FUNCTIONS

1. Relations 1-272. Functions 28-783. Binary Operations 79-1074. Inverse Trigonometric Functions 108-158

UNIT — II: ALGEBRA

5. Determinants 159-2196. Matrices 220-2657. Multiplication of Matrices 266-3058. Adjoint and Inverse of a Matrix 306-3499. Solutions of Simultaneous Linear Equations 350-398

UNIT — III: CALCULUS

10. Continuity 399-44411. Differentiability 445-46512. Differentiation 466-57813. Second Order Derivatives 579-60714. Rolle’s Theorem and Mean Value Theorem 608-64615. Rate of Change of Quantitives 647-67316. Tangents and Normals 674-72417. Differentials, Errors and Approximations 725-74618. Increasing And Decreasing Functions 747-78019. Maxima and Minima 781-886

Question Paper 2012 887-893Question Paper 2013 894-903Value Based Questions 904-927Table of Important solutions in the text book at a glance. 929-936

(vi)

UNIT — III20. Indefinite Integrals21. Integration By Substitution22. Integration by using Partial Fractions23. Integration of Rational and Irrational

Functions24. Integration by Parts25. Definite Integrals26. Sketching of Simple Curves27. Area of Bounded Regions28. Differential Equations29. Differential Equations with Variables

Separable30. Homogeneous Differential Equations31. Linear Differential Equations32. Applications of Differential Equations

UNIT — IV33. Algebra of Vectors34. Scalar Product of two Vectors35. Vector Product of Two Vectors36. Three Dimensional Co-ordinate System37. Straight Lines in Space38. Plane

UNIT — V39. Linear Programming

UNIT — VI40. Probability41. Probability Distribution42. Binomial Distribution

Rest of the topics are given in Mathematics for Class XII, Volume II as details given below:


CHAPTER 1

RELATIONS

SYLLABUSTypes of relations : reflexive, symmetric, transitive and equivalence relations.

IMPORTANT FACTS1. Ordered pair. If a pair of elements is listed in a specific order, then such a pair is called an

ordered pair. It is denoted by (a, b), where a being first element and b being second element.2. Relation. If A and B be two non-empty sets. Then, a relation R from A to B is a subset of

A × B.i.e. R A B

3. Total number of relations. Total number of relations from A to B is 2mn.4. Empty Relation. If there is no relation between the elements of a set or sets, then the

relation is called empty relation.R

5. Universal Relation. A relation R in a set A is called universal relation, if each elements ofA is related to every element of A, i.e., R = A × A.

6. Identity Relation.The relation ( , ) :AI a a a A is called the identity relation on A.7. Reflexive Relation. A relation R on a set A is said to be reflexive if every element of A is

related to itself. In symbolic form :R is reflexive (a, a) R for all a A

Note : Identity relation is always subset of Reflexive relation.8. Symmetric Relation. A relation R on a set A is said to be a symmetric relation.

If ( , ) ( , )a b R b a R 9. Transitive Relation. A relation R on a set A is said to be transitive relation.

If ( , )a b R and ( , ) ( , )b c R a c R 10. Antisymmetric Relation

A relation R on a set A is said to be an antisymmetric relation.If (i) ( , )a b R but ( , )b a R (ii) ( , )a b R and ( , )b a R for a = b.

11. Equivalance RelationA relation which is reflexive, symmetric and transitive, is called an equivalence relation.

12. Congruence Modulo m. Let m be an arbitrary but fixed integer. Two integers a and b aresaid to be congruence modulo m if a – b is divisible by m and we write (mod ).a b m

Thus, (mod )a b m a b is divisible by m.13. Equivalence classes. Let R be an equivalence relation on a set A. Let .a A Then the set

of all those elements of A which are related to A is called the equivalence class determineby [a] or a .

14. Partition of a Set. By the partition of a set A, we mean that A is union of some mutuallydisjoint subsets of A.i.e., A = A0 + A1 + A2

UNIT – I: RELATIONS AND FUNCTIONS

1


The Book is Dedicated toINDIAN MATHEMATICIAN

BHASKARA I(A.D. 600 – 680)

Presumably Bhaskara’s I was born near Saurashtra in Gujarat and died in Ashmaka.His astronomical education was given by his father. Bhaskara is considered the mostimportant scholar of Aryabhata’s astronomical school.

Bhaskara I (A.D. 600 - 680) was a 7th century Indian mathematician, who wasapparently the first to write numbers in the Hindu-Arabic decimal system with a circle forthe zero, and who gave a unique and remarkable rational approximation of the sine func-tion in his commentary on Aryabhata’s work.

Bhaskara’s probably most important mathematical contribution concerns the repre-sentation of numbers in a positional system. The first positional representations were knownto Indian astronomers about 500. However, the numbers were not written in figures, but inwords or allegories, and were organized in verses. For instance, the number 1 was givenas moon, since it exists only once; the number 2 was represented by wings, twins, or eyes,since they always occur in pairs; the number 5 was given by the (5) senses. Similar to ourcurrent decimal system, these words were aligned such that each number assigns thefactor of the power of ten corresponding to its position, only in reverse order: the higherpowers were right from the lower ones. For example,

1052 = wings senses void moon.

About 510, Aryabhata used a different method (“Aryabhata cipher”) assigning syl-lables to the numbers. His number system has the basis 100, and not 10 (Ifrah 2000, p.449). In his commentary to Aryabhata’s Aryabhatiya in 629, Bhaskara modified this sys-tem to a true positional system with the base 10, containing a zero. He used properlydefined words for the numbers, began with the ones, then writes the tens, etc. For in-stance, he wrote the number 4,320,000 as

viyat ambara akasha sunya yama rama vedasky atmosphere ether void primordial couple (Yama & Yami) Rama Veda

0 0 0 0 2 3 4


1.1. INTRODUCTIONIn English language the meaning of relation is two people are related if there is a link betweenthem.For example, the statement ‘‘x is the mother of y’’ describes a relation on the set P of allliving people. The statement is true for a certain pair (x, y) of P and for all other pairs it isfalse.

Mother and daughter Son and Father Brother and sisters

Similarly, in mathematics there is a variety of relations whose knowledge is crucial.1.2. ORDERED PAIR

If a pair of elements is listed in a specific order, then such a pair is called an ordered pair. Theordered pair of two elements a and b is denoted by (a, b); a being first element and b issecond element.The set of all ordered pairs (a, b) of elements ,a A b B is called the cartesian product ofset A and set B and is denoted by A × B.

{( , ); , }A B a b a A b B For example; If A = {1, 2, 3} and B = {6, 10}, then

A × B = {(1, 6), (1, 10), (2, 6), (2, l0), (3, 6), (3, 10)}Remember: A B B A

1.3. RELATIONIf A and B be two non empty sets. Then, a relation R from A to B is a subset of A × B.

R A B

If ( , )a b R , then we say that a is related to b and denote this by a R b.If ( )a b R then we write a R b and we say that a is not related to b by the relation R.

1

RELATIONS


4 Mathematics for 12th Class (Vol – I)

Remark : Let A and B be any two non-empty finite sets containing m and n elementsrespectively.

Number of ordered pairs in A × B is mn. Total number of subsets of A × B is 2mn. Total number of relations from A to B is 2mn.

SOLVED EXAMPLESExample 1. Let A = {1, 2, 3, 4, 5, 6}, B = {1, 2, 3, 4, 5, 6, 7, 8, 9}.If R is a relation from set A to set B and is defined as

{( , ) : 10},R a b A B a b find R.Solution. Here, 1 + 9 = 10, 2 + 8 = 10, 3 + 7 = 10,

4 + 6 = 10, 5 + 5 = 10, 6 + 4 = 10 R = {(1, 9), (2, 8) (3, 7) (4, 6), (5, 5), (6, 4)} Ans.

Example 2. Let A = {3, 5, 7, 12}, B = {6, 8, 10}.If R is a relation from set A to set B is defined by ‘‘is lessthan’’. Find R.Solution. Here, 3 < 6, 3 < 8, 3 < 10,

5 < 6, 5 < 8, 5 < 10,7 < 8, 7 < 10

R = {(3, 6), (3, 8) (3, 10), (5, 6), (5, 8), (5, 10), (7, 8), (7, l0)} Ans.Example 3. Let A = {2, 3, 4, 6}. Let R be the relation on A defined by

{( , ) : , , }.a b a A b A a divides b

Solution. 2 divides 2, 2 divides 4, 2 divides 6,3 divides 3, 3 divides 6, 4 divides 4, 6 divides 6; R = {(2, 2), (2, 4), (2, 6), (3, 3), (3, 6) (4, 4), (6, 6)} Ans.

Example 4. Let A = {5, 7, 9, l0)} and B = (4, 6, 9}. There is arelation R from A to B,

R = (x, y) : –

,,x y is oddx A y B

Find R.

Solution. Here, 5 – 4 = 1, 7 – 4 = 3, 7 – 6 = 1, 9 – 4 = 5, 9 – 6 = 3 , 10 – 9 = 1 R = {(5, 4), (7, 4), (7, 6) (9, 4) (9, 6) (10, 9)} Ans.Example 5. Let A = {1, 2, 3, ..... 9} there is a relation R from

A to A by

R = {(x, y); 3x – y = 0} where , .x y A Find R.Solution. Here, A = {1, 2, 3, 4, 5, 6, 7, 8, 9}

R = {(x, y) : 3x – y = 0, where ,x y A }

R = {(1, 3), (2, 6), (3, 9)} [By Roaster method]

R ={(x, y) : 3x – y = 0, where ,x y A } [By Set Builder method] Ans.

2

346

4

A B

2

3

6

123456789

123456789

A A

123

6

47

5

A B

98

654

32

1

3

57

1210

A B

6

8

5

79

10 9

A B

4

6


Relations 5

EXERCISE 1.11 Mark Questions

1. Let A = {1, 2} and B = {3, 4}. Find the number of relations from A into B.

2. Let A = {1, 2, 3, 4, 5}, and B = {2, 4, 6, 8, 10}. Let {( , ) : , , divides }R a b a A b B a b be arelation from A to B, find R.

3. Given A = {2, 3, 4, 5, 6}. List the elements of each of the following relations :

(i) {( , ) : , }xx y A A x y Wy

(ii) {( , ) :x y A A x is divisor of y and }x y .4 Marks Questions

4. Let A = {1, 2, 3, 4}, B = {5, 6, 7, 8}. If R is a relation from set A to B is defined as(i) R = {(a, b) : (a, b) A × B, a, b are odd }(ii) R = {(a, b) : (a, b) A × B; a, b are even}(iii) R = {(a, b) : (a, b) A × B; a, b are primes}

(iv) R = {(a, b) : (a, b) A × B, ba is odd}

ANSWERS1. 162. R = {(1, 2), (1, 4), (1, 6), (1, 8), (1, 10), (2, 2), (2, 4), (2, 6), (2, 8), (2, 10), (3, 6), (4, 4), (4, 8), (5, 10)}.3. (i) {(3, 2), (4, 3), (5, 2), (5, 3), (5, 4), (6, 4), (6, 5)}

(ii) {(2, 4), (2, 6), (3, 6)}. 4. (i) {(1, 5), (1, 7), (3, 5), (3, 7)} (ii) {(2, 6), (2, 8), (4, 6), (4, 8)}

(iii) {(2, 5), (2, 7), (3, 5), (3, 7)} (iv) {(1, 5), (1, 7), (2, 6)}

1.4 TYPES OF TRIVIAL RELATIONSThere are two types of relations.1. Empty Relation. If there is no relation between the elements of a set or sets, then therelation is called empty relation.

R set A A For example;Consider the relation R in the set A = {1, 2, 3, 5} given by

2{( , ) : , , }R a b a b a A b A

This is empty relation, as no pair (a, b) satisfies the condition a2 = b.Definition. A relation R in a set A is called empty relation, if no element of A is related to anyelement of A, i.e., R A A

SOLVED EXAMPLESExample 1. Let A = {4, 6, 8, 20}, R = {(a, b) : a + b = 25 ; , }a b A

Show that the relation R is empty.Solution. This is an empty set as no pair (a, b) satisfies the condition a + b = 25.Hence, ,R showing that R is the empty relation.



2. Universal RelationA relation R in a set A is called universal relation, if every element of A is related to each ofthe elements of A, i.e., R = A × A.Note :(1) Both the empty relation and the univeral relation are sometimes called trivial relations.(2) The empty and universal relations on a set A are respectively the smallest and the largest

relations on A.Example 2. Let the set A = {3, 5, 7}. Show that the relation R in A is given by

{( , ) : 5; , }R a b a b a b A is the universal relation.Solution. We have,

A = {3, 5, 7} A × A = {(3, 3), (3, 5), (3, 7), (5, 3), (5, 5), (5, 7), (7, 3), (7, 5), (7, 7)}

Here, A × A = RAll the ordered pairs satisfy the given relation a – b < 5.Hence, R is universal relation.Example 3. Let the set A of males = {Mohan Singh, Sohan Singh, Pratap Singh, KartarSingh}. Show that the relation R in A is given by R = {(a, b): b is the mother of a} is emptyrelation.Solution. Since in set A all are males and there is no lady belonging to A. Male cannot be themother of any child. Hence ,R showing that R is the empty relation.

1.5. SOME IMPORTANT RELATIONS

1. Identity Relation :

The relation, {( , ) : }AI a a a A is called the identity relation on A.

For example : If A = {2, 3, 4}, then the identity relation on A is given byIA = {(2, 2,) (3, 3,) (4, 4)}.

2. Reflexive Relation :A relation R on a set A is said to be reflexive if every element of A is related to itself.In symbolic form it can be written as :

R is reflexive if ( , )a a R for every .a A

Note : Identity relation is always subset of Reflexive relation.For example : If A = {2, 4, 7} then the relation R1 = {2, 2) (2, 4) (4, 4) (7, 7)} is reflexive. Butthe relation R2 = {(2, 4), (4, 4), (7, 7), (4, 7)} is not reflexive because 2 A but 2(2, 2) .R

3. Symmetric Relation :A relation R on a set A is said to be a symmetric relation,

If ( , )a b R ( , )b a R , a, b Ai.e., If a R b b R aFor example : If A = {2, 4, 7}, then the relation {(2, 4), (4, 2), (7, 7)} is a symmetric.

4. Transitive Relation :A relation R on a set A is said to be transitive relation.


Relations 7

If ( , )a b R and ( , )b c R ( , )a c R i.e., If a R b and b R c a R cFor example : If A = {2, 4, 7}, then the relation {(2, 4), (4, 7), (2, 7), (4, 4)} is transitive.

5. Antisymmetric Relation :A relation R on a set A is said to be an antisymmetric relation

iff (i) ( , )a b R but ( , )b a R for all .a b

(ii) ( , )a b R and ( , ) ,b a R a = b for all , .a b A

For example 1. The identity relation on a set A is an antisymmetric relation.For example 2. If A = {2, 4, 7}, then {(2, 4), (2, 7), (4, 7), (2, 2), (4, 4), (7, 7)} is antisymmetricrelation.Note: If a relation satisfies all the conditions of symmetry with some extra relations. Thisrelation is still symmetric inspite of extra relations.Similarly, this rule also apply on reflexive relation and transitive relation.For example; Ashok satisfies all the conditions of honesty but he is duffer. It means Ashokis still honest inspite of his dufferness.

SOLVED EXAMPLESExample 1. Show that the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} issymmetric but neither reflexive nor transitive. (CBSE 2011, TEXTBOOK)Solution. R = {(1, 2), (2, 1)}Reflexivity : As (1, 1), (2, 2) RSo R is not reflexive.Symmetry. As (1, 2) and (2, 1) RSo R is symmetric.

Transitivity. As (1, 2) , but (2, 3), (1, 3)R R . So R is not transitive. Ans.Note : For transitivity there should be atleast three elements in R.Example 2. The relation R is defined on set A = {1, 2, 3} as follows

R = {(1, 2) (2, 3), (3, 1)}Find whether each of R is reflexive, symmetric and transitive.Solution. Reflexive : R is not reflexive relation on A, because none of (1, 1), (2, 2) and (3, 3)are elements of R.

Symmetric : R is not symmetric on A, because (2, 3) R but (3, 2) R .

Transitive : R is not transitive on A, because (1, 2) R and (2,3) R but (1,3) .R Ans.Example 3. The relation R is defined on set A = {1, 2, 3} as follows

R = {(1, 2), (2, 1), (1, 3), (3, 1)}.Find whether R is reflexive, symmetric and transitive.

Solution. Reflexive: R is not reflexive on A, because (1, 1), (2, 2) and (3,3) R

Symmetric : R is symmetric relation on A, because the ordered pair obtained byinterchanging the components of ordered pairs in R are also in R.



Transitive : R is not transitive relation on A, because (1, 2) R and (2,1) R but (1,1) R .Ans.

Example 4. The relation R is defined on set A = {1, 2, 3} as follows :R = {(1, 1), (1, 2), (1, 3), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)}

Find whether R is reflexive symmetric and transitive ?Solution.(i) Reflexive : R is reflexive on A, because {(1, 1), (2, 2) (3, 3)} i.e., identity relation is a subset

of R.

(ii) Symmetric : R is not symmetric on A because (1, 2) R but (2, 1) .R

(iii) Transitive : R is not transitive, because we observe that (2, 3) R and (3, 1) R but(2, 1) .R Ans.

Example 5. Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} asR = {(a, b): b = a + 1} is reflexive, symmetric or transitive. [TEXTBOOK]Solution. Let A = {1, 2, 3, 4, 5, 6} and R = {(a, b) : b = a + 1}By Roaster method,

R = {(1, 2), (2, 3), (3, 4), (4, 5), (5, 6)} ... (1)

Reflexive : Here, (1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6) R [From (1)]So, R is not reflexive.

Symmetric : (1, 2) R but (2, 1) R [From (1)]So, R is not symmetric.

Transitive : (1, 2), (2, 3) ,R but (1, 3) R [From (1)]So R is not transitive.Hence, R is neither reflexive nor symmetric nor transitive. Ans.Example 6. Check the following relation R for reflexivity, symmetry and transitivity : a R bif b is divisible by a, , .a b N

Solution. We have, |aRb a b for all , .a b N

Reflexivity : For any ,a N we have | .a a aRa Thus, aRa for all .a N So, R isreflexive on N.Symmetry : R is not symmetric because if a | b, then b may not divide a. For example, 2 | 6

but 6 2.

Transitivity : Let , ,a b c N such that a R b and b R c. Then a R b and |b R c a b and

| | .b c a c a R c So, R is a transitive relation on N. Ans.Example 7. ‘‘Show that the relation R ‘‘is perpendicular to’’ on the set S of all straight linesin a plane is symmetric, but it is neither reflexive nor transitive.Solution.Reflexive : It is not reflexive, because any line is not perpendicular to it self i.e., ( , ) .l l R

Symmetric : It is symmetric, because l m m l . (l, m) and ( , ) .m l R


S.Chand’s Mathematics Class XIIVolume 1

Publisher : SChand Publications ISBN : 9788121929004Author : H K Dass AndRama Verma

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