NCERT solution

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CBSEClass–12Mathematics

NCERTsolution

Chapter-1

Relations&Functions-Exercise1.1

1.Determinewhethereachofthefollowingrelationsarereflexive,symmetricand

transitive:

(i)RelationRinthesetA={1,2,3,………..13,14}definedasR= .

(ii)RelationRinthesetNofnaturalnumbersdefinedasR={(x,y):y=x+5andx<4}

(iii)RelationRinthesetA={1,2,3,4,5,6}asR=

(iv)RelationRinthesetZofallintegersdefinedasR=

(v)RelationRinthesetAofhumanbeingsinatownataparticulartimegivenby:

(a)R=

(b)R=

(c)R=

(d)R=

(e)R=

Ans.(i)R= inA={1,2,3,4,5,6,……13,14}

ClearlyR={(1,3),(2,6),(3,9),(4,12)}

Since, R, Risnotreflexive.

Again Rbut R Risnotsymmetric.

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Also(1,3) Rand(3,9) Rbut(1,9) R, Risnottransitive.

Therefore,Risneitherreflexive,norsymmetricandnortransitive.

(ii)R= insetNofnaturalnumbers.

ClearlyR={(1,6),(2,7),(3,8)}

Now R, Risnotreflexive.

Again Rbut R Risnotsymmetric.

Also(1,6) Rand(2,7) Rbut(1,7) R, Risnottransitive.


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

ClearlyR=(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),(2,2),(2,4),(2,6),(3,3),(3,6),(4,4),(5,5),(6,6)

Now i.e.,(1,1),(2,2)and(3,3) R Risreflexive.

Again i.e.,(1,2) Rbut R Risnotsymmetric.

Also(1,4) Rand(4,4) Rand(1,4) R, Ristransitive.

Therefore,Risreflexiveandtransitivebutnotsymmetric.

(iv)R= insetZofallintegers.

Now i.e.,(1,1)=1–1=0 Z Risreflexive.

Again Rand R,i.e., and areaninteger Rissymmetric.

Also Zand Zand

R, Ristransitive.



Therefore,Risreflexive,symmetricandtransitive.

(v)RelationRinthesetAofhumanbeinginatownataparticulartime.

(a)R=

Since R,because and workatthesameplace. Risreflexive.

Now,if Rand R,since and workatthesameplaceand and

workatthesameplace. Rissymmetric.

Now,if Rand R R. Ristransitive


(b)R=

Since R,because and liveinthesamelocality. Risreflexive.

Also R Rbecause and liveinsamelocalityand and also

liveinsamelocality. Rissymmetric.

Again Rand R R Ristransitive.


(c)R=

isnotexactly7cmtallerthan ,so R Risnotreflexive.

Also isexactly7cmtallerthan but isnot7cmtallerthan ,so Rbut

R Risnotsymmetric.

Now isexactly7cmtallerthan and isexactly7cmtallerthan thenitdoesnot

implythat isexactly7cmtallerthan Risnottransitive.




(d)R=

isnotwifeof ,so R Risnotreflexive.

Also iswifeof but isnotwifeof ,so Rbut R Risnot

symmetric.

Also Rand Rthen R Risnottransitive.


(e)R=

isnotfatherof ,so R Risnotreflexive.

Also isfatherof but isnotfatherof ,

so Rbut R Risnotsymmetric.

Also Rand Rthen R Risnottransitive.


2.ShowthattherelationRinthesetRofrealnumbersdefinedasR=

isneitherreflexivenorsymmetricnortransitive.

Ans.R= ,RelationRisdefinedasthesetofrealnumbers.

(i)Whether R,then whichisfalse. Risnotreflexive.

(ii)Whether ,then and ,itisfalse. Risnotsymmetric.

(iii)Now , ,whichisfalse. Risnottransitive




3.CheckwhethertherelationRdefinedintheset{1,2,3,4,5,6}asR=

isreflexive,symmetricortransitive.

Ans.R= R

NowR={(1,2),(2,3),(3,4),(4,5),(5,6)}and

(i)When whichisfalse,so R, Risnotreflexive.

(ii)Whether ,then and ,false Risnotsymmetric.

(iii)Nowif R, R R

(iv)Then and whichisfalse. Risnottransitive.


4.ShowthattherelationRinRdefinedasR= ,isreflexiveand

transitivebutnotsymmetric.

Ans.(i) whichistrue,so R, Risreflexive.

(ii) but whichisfalse. Risnotsymmetric.

(iii) and whichistrue. Ristransitive.


5.CheckwhethertherelationRinRdefinedbyR= isreflexive,

symmetricortransitive.

Ans.(i)For whichisfalse. Risnotreflexive.

(ii)For and whichisfalse. Risnotsymmetric.

(iii)For and , whichisfalse. Risnottransitive.




6.ShowthattherelationinthesetA={1,2,3}givenbyR={(1,2),(2,1)}issymmetric

butneitherreflexivenortransitive.

Ans.R={(1,2),(2,1)},sofor ,(1,1) R. Risnotreflexive.

Alsoif then R Rissymmetric.

Now Rand thendoesnotimply R Risnottransitive..

Therefore,Rissymmetricbutneitherreflexivenortransitive.

7.ShowthattherelationRinthesetAofallthebooksinalibraryofacollege,givenby

R= isanequivalencerelation.

Ans.Books and havesamenumberofpages R Risreflexive.

If R R,so Rissymmetric.

Nowif R, R R Ristransitive.

SinceRisreflexive,symmetricandtransistive,therefore,Risanequivalencerelation.

8.ShowthattherelationRinthesetA={1,2,3,4,5}givenbyR=

isanequivalencerelation.Showthatalltheelementsof{1,3,

5}arerelatedtoeachotherandalltheelementsof{2,4}arerelatedtoeachother.But

noelementof{1,3,5}isrelatedtoanyelementof{2,4}.

Ans.A={1,2,3,4,5}andR= ,thenR={(1,3),(1,5),(3,5),(2,4)}

(a)For whichiseven. Risreflexive.

If iseven,then isalsoeven. Rissymmetric.



Now,if and iseventhen isalsoeven. Ris

transitive.

Therefore,Risanequivalencerelation.

(b)Elementsof{1,3,5}arerelatedtoeachother.

Since allareevennumbers

Elementsof{1,3,5}arerelatedtoeachother.

Similarlyelementsof(2,4)arerelatedtoeachother.

Since anevennumber,thennoelementoftheset{1,3,5}isrelatedtoany

elementof(2,4).

Hencenoelementof{1,3,5}isrelatedtoanyelementof{2,4}.

9.ShowthateachoftherelationRinthesetA= givenby:

(i)R=

(ii)R= isanequivalencerelation.Findthesetofallelementsrelatedto1in

eachcase.

Ans.A={x∈Z:0≤x≤12}={0,1,2,3,4,5,6,7,8,9,10,11,12}

(i)R={(a,b):|a-b|isamultipleof4}

Foranyelementa∈A,wehave(a,a)∈Ras|a-a=0|isamultipleof4.

∴Risreflexive.Now,let(a,b)∈R⇒|a-b|isamultipleof4.

⇒|-(a-b)|⇒|b-a|isamultipleof4.

⇒(b,a)∈R∴Rissymmetric.

Now,let(a,b),(b,c)∈R.⇒|(a-b)|isamultipleof4and|(b-c)|isamultipleof4.



⇒(a-b)isamultipleof4and(b-c)isamultipleof4.

⇒(a-c)=(a–b)+(b–c)isamultipleof4.

⇒|a-c|isamultipleof4.

⇒(a,c)∈R∴Ristransitive.Hence,Risanequivalencerelation.

Thesetofelementsrelatedto1is{1,5,9}since

|1-1|=0isamultipleof4,

|5-1|=4isamultipleof4,and

|9-1|=8isamultipleof4.

(ii)R={(a,b):a=b}

Foranyelementa∈A,wehave(a,a)∈R,sincea=a.∴Risreflexive.Now,let(a,b)∈R.⇒a=b⇒b=a⇒(b,a)∈R∴Rissymmetric.

Now,let(a,b)∈Rand(b,c)∈R.⇒a=bandb=c⇒a=c⇒(a,c)∈R∴Ristransitive.

Hence,Risanequivalencerelation.

TheelementsinRthatarerelatedto1willbethoseelementsfromsetAwhichareequalto1.

Hence,thesetofelementsrelatedto1is{1}.

10.Giveanexampleofarelation,whichis:

(i)Symmetricbutneitherreflexivenortransitive.

(ii)Transitivebutneitherreflexivenorsymmetric.



(iii)Reflexiveandsymmetricbutnottransitive.

(iv)Reflexiveandtransitivebutnotsymmetric.

(v)Symmetricandtransitivebutnotreflexive.

Ans.(i)Therelation“isperpendicularto” isnotperpendicularto

If then ,howeverif and then isnotperpendicularto

SoitisclearthatR“isperpendicularto”isasymmetricbutneitherreflexivenor

transitive.

(ii)RelationR=

Weknowthat isfalse.Also but isfalseandif , this

implies

Therefore,Ristransitive,butneitherreflexivenorsymmetric.

(iii)“isfriendof”R=

Itisclearthat isfriendof Risreflexive.

Also isfriendof and isfriendof Rissymmetric.

Alsoif isfriendof and isfriendof then

cannotbefriendof Risnottransitive.

Therefore,Risreflexiveandsymmetricbutnottransitive.

(iv)“isgreaterorequalto”R=

Itisclearthat Risreflexive.

And doesnotimply Risnotsymmetric.

But , Ristransitive.




(v)“isbrotherof”R=

Itisclearthat isnotthebrotherof Risnotreflexive.

Also isbrotherof and isbrotherof Rissymmetric.

Alsoif isbrotherof and isbrotherof then

canbebrotherof Ristransitive.

Therefore,Rissymmetricandtransitivebutnotreflexive.

11.ShowthattherelationRinthesetAofpointsinaplanegivenbyR={(P.Q):

distanceofthepointPfromtheoriginissameasthedistanceofthepointQfromthe

origin},isanequivalencerelation.Further,showthatthesetofallpointsrelatedtoa

pointP (0,0)isthecirclepassingthroughPwithoriginascentre.

Ans.PartI:R={(P,Q):distanceofthepointPfromtheoriginisthesameasthe

distanceofthepointQfromtheorigin}

LetP andQ andO(0,0).

OP=OQ = =

Now,For(P,P),OP=OP Risreflexive.

AlsoOP=OQandOQ=OP (P,Q)=(Q,P) R Rissymmetric.

AlsoOP=OQandOQ=OR OP=OQ Ristransitive.

Therefore,Risanequivalentrelation.

PartII:As = = (let) whichrepresentsacirclewith

centre(0,0)andradius



12.ShowthattherelationRdefinedinthesetAofalltrianglesasR={(T1,T2):T1is

similartoT2},isanequivalencerelation.ConsiderthreerightangletrianglesT1with

sides3,4,5,T2withsides5,12,13andT3withsides6,8,10.WhichtrianglesamongT1,

T2andT3arerelated?

Ans.PartI:R={(T1,T2):T1issimilartoT2}andT1,T2aretriangle.

Weknowthateachtrianglesimilartoitselfandthus(T1,T2) R Risreflexive.

Alsotwotrianglesaresimilar,thenT1 T2 T1 T2 Rissymmetric.

Again,ifthenT1 T2andthenT2 T3 thenT1 T3 Ristransitive.


PartII:ItisgiventhatT1,T2andT3arerightangledtriangles.

T1withsides3,4,5,T2withsides5,12,13and

T3withsides6,8,10

Since,twotrianglesaresimilarifcorrespondingsidesareproportional.

Therefore,

Therefore,T1andT3arerelated.

13.ShowthattherelationRdefinedinthesetAofallpolygonsasR={(P1,P2):P1and

P2havesamenumberofsides},isanequivalencerelation.Whatisthesetofall

elementsinArelatedtotherightangletriangleTwithsides3,4,and5?

Ans.PartI:R={(P1,P2):P1andP2havesamenumberofsides}

(i)Considertheelement(P1,P2),itshowsP1andP2havesamenumberofsides.

Therefore,Risreflexive.



(ii)If(P1,P2) Rthenalso(P2,P1) R

(P1,P2)=(P2,P1)asP1andP2havesamenumberofsides,therefore,Rissymmetric.

(iii)If(P1,P2) Rand(P2,P3) Rthenalso(P1,P3) RasP1,P2andP3havesame

numberofsides,therefore,Ristransitive.


PartII:weknowthatif3,4,5arethesidesofatriangle,thenthetriangleisrightangled

triangle.Therefore,thesetAisthesetofrightangledtriangle.

14.LetLbethesetofalllinesinXYplaneandRbetherelationinLdefinedasR={(L1,

L2):L1isparalleltoL2}.ShowthatRisanequivalencerelation.Findthesetofalllines

relatedtotheline

Ans.PartI:R={(L1,L2):L1isparalleltoL2}

(i)ItisclearthatL1 L1i.e.,(L1,L1) R Risreflexive.

(ii)IfL1 L2andL2 L1then(L1,L2) R Rissymmetric.

(iii)IfL1 L2andL2 L3 L1 L3 Ristransitive.


PartII:Allthelinesrelatedtotheline and where isareal

number.

15.LetRbetherelationintheset{1,2,3,4}givenbyR={(1,2),(2,2),(1,1),(4,4),(1,3),

(3,3),(3,2)}.Choosethecorrectanswer:

(A)Risreflexiveandsymmetricbutnottransitive.

(B)Risreflexiveandtransitivebutnotsymmetric.

(C)Rissymmetricandtransitivebutnotreflexive.



(D)Risanequivalencerelation.

Ans.LetRbetherelationintheset{1,2,3,4}isgivenby

R={(1,2),(2,2),(1,1),(4,4),(1,3),(3,3),(3,2)}

(a)(1,1),(2,2),(3,3),(4,4) R Risreflexive.

(b)(1,2) Rbut(2,1) R Risnotsymmetric.

(c)If(1,3) Rand(3,2) Rthen(1,2) R Ristransitive.

Therefore,option(B)iscorrect.

16.LetRbetherelationinthesetNgivenbyR= Choosethe

correctanswer:

(A)(2,4) R

(B)(3,8) R

(C)(6,8) R

(D)(8,7) R

Ans.Given:

(A) ,Here isnottrue,therefore,thisoptionisincorrect.

(B) and 3=6,whichisfalse.

Therefore,thisoptionisincorrect.

(C) and 6=6,whichistrue.

Therefore,option(C)iscorrect.

(D) and 8=5,whichisfalse.




NCERTsolution

Chapter-1


1. Show that the function defined by is one-one and onto,

where isthesetofallnon-zerorealnumbers.Istheresulttrue,ifthedomain is

replacedbyNwithco-domainbeingsameas ?

Ans.

PartI: and

If then

isone-one.

isonto.

Part II: When domain R is replaced by N, co-domain R remaining the same, then,



If

where N

isone-one.

Buteveryrealnumberbelongingtoco-domainmaynothaveapre-imageinN.

e.g. N isnotonto.

2.Checktheinjectivityandsurjectivityofthefollowingfunctions:

(i) givenby

(ii) givenby

(iii) givenby

(iv) givenby

(v) givenby

Ans.(i) givenby

If then

isinjective.

Therearesuchnumbersofco-domainwhichhavenoimageindomainN.



e.g.3 co-domainN,butthereisnopre-imageindomainof

therefore isnotonto. isnotsurjective.

(ii) givenby

Since,Z= therefore,

and1havesameimage. isnotinjective.

Therearesuchnumbersofco-domainwhichhavenoimageindomainZ.

e.g.3 co-domain,but domainof isnotsurjective.

(iii) givenby

As

and1havesameimage. isnotinjective.

e.g. co-domain,but domainRof isnotsurjective.

(iv) givenby

If then

i.e.,forevery N,hasauniqueimageinitsco-domain. isinjective.

There are many such members of co-domain of which do not have pre-image in its

domaine.g.,2,3,etc.

Therefore isnotonto. isnotsurjective.

(v) givenby

If then



i.e.,forevery Z,hasauniqueimageinitsco-domain. isinjective.

There are many such members of co-domain of which do not have pre-image in its

domain.

Therefore isnotonto. isnotsurjective.

3. Prove that the Greatest integer Function , given by is

neitherone-onenoronto,where denotesthegreatestintegerlessthanorequalto

Ans.Function ,givenby

and

isnotone-one.

Alltheimagesof Rbelongtoitsdomainhaveintegersastheimagesinco-domain.But

no fraction proper or improper belonging to co-domain of has any pre-image in its

domain.

Therefore, isnotonto.

4.Showthat theModulusFunction ,givenby isneitherone-

onenoronto,whereis if ispositiveor0and is if isnegative.

Ans.ModulusFunction ,givenby



Now

contains

Thusnegativeintegersarenotimagesofanyelement. isnotone-one.

AlsosecondsetRcontainssomenegativenumberswhicharenotimagesofanyrealnumber.

isnotonto.

5. Show that the Signum Function , given by is

neitherone-onenoronto.

Ans.SignumFunction ,givenby

for

isnotone-one.

Except noothermembersofco-domainof hasanypre-imageitsdomain.

isnotonto.

Therefore, isneitherone-onenoronto.

6.LetA={1,2,3},B={4,5,6,7}andlet ={(1,4),(2,5),(3,6)}beafunctionfromAto

B.Showthat isone-one.



Ans.A={1,2,3},B={4,5,6,7}and ={(1,4),(2,5),(3,6)}

Here, and

Here,alsodistinctelementsofAhavedistinctimagesinB.

Therefore, isaone-onefunction.

7. In each of the following cases, state whether the function is one-one, onto or

bijective.Justifyyouranswer.

(i) definedby

(ii) definedby

Ans.(i) definedby

Now,if R,then and

Andif ,then isone-one.

Again,ifeveryelementofY(–R)isimageofsomeelementofX(R)under i.e.,forevery

Y,thereexistsanelement inXsuchthat

Now

isontoorbijectivefunction.



(ii) definedby

Now,if R,then and

Andif ,then

isnotone-one.

Again,ifeveryelementofY(–R)isimageofsomeelementofX(R)under i.e.,forevery

Y,thereexistsanelement inXsuchthat

Now,

isnotonto.

Therefore, isnotbijective.

8.LetAandBbesets.Showthat :AxB BxAsuch that isa

bijectivefunction.

Ans.Injectivity:Let and AxBsuchthat

and

=

= forall AxB

So, isinjective.

Surjectivity:Let beanarbitraryelementofBxA.Then Band A.



AxB

Thus,forall BxA,theirexists AxBsuchthat

So, AxB BxAisanontofunction,therefore isbijective.

9.Let bedefinedby forall N.

Statewhetherthefunction isbijective.Justifyyouranswer.

Ans. bedefinedby

(a) and

Theelements1,2,belongingtodomainof havethesameimage1initsco-domain.

So, isnotone-one,therefore, isnotinjective.

(b)Everynumberofco-domainhaspre-imageinitsdomaine.g.,1hastwopre-images1and

2.

So, isonto,therefore, isnotbijective.

10. Let A = R – {3} and B = R – {1}. Consider the function : A B defined by

Is one-oneandonto?Justifyyouranswer.

Ans.A=R–{3}andB=R–{1}and



Let A,then and

Now,for

isone-onefunction.

Now

=

Therefore, isanontofunction.



11.Let bedefinedas Choosethecorrectanswer:

(A) isone-oneonto

(B) ismany-oneonto

(C) isone-onebutnotonto

(D) isneitherone-onenoronto

Ans. and

Let ,then and

Therefore, isnotone-onefunction.

Now,

and

Therefore, isnotontofunction.

Therefore,option(D)iscorrect.

12.Let bedefinedas Choosethecorrectanswer:

(A) isone-oneonto

(B) ismany-oneonto



(C) isone-onebutnotonto

(D) isneitherone-onenoronto

Ans.Let Rsuchthat

Therefore, isone-onefunction.

Now,consider R(co-domainof )certainly R(domainof )

Thusforall R(co-domainof )thereexists R(domainof )suchthat

Therefore, isontofunction.

Therefore,option(A)iscorrect.



CBSEClass–12MathematicsNCERTsolutionChapter-1


1.Let :{1,3,4} {1,2,5}and :{1,2,5} {1,3}begivenby ={(1,2),(3,5),(4,1)}and ={(1,3),(2,3),(5,1)}.Writedown

Ans. ={(1,2),(3,5),(4,1)}and ={(1,3),(2,3),(5,1)}

Now, and

and

Hence, {(1,3),(3,1),(4,3)}

2.Let and befunctionsfromR R.Showthat:

Ans.(a)Toprove:

L.H.S.= = = =R.H.S.

(b)Toprove:

L.H.S.= = = =R.H.S.

3.Find and ,if:



(i) and

(ii) and

Ans.Tofind: and

(i) and

and = =

(ii) and

and =

4. If show that for all What is the

inverseof

Ans.Given:

L.H.S.= = = =

= =R.H.S.



Now,

Henceinverseof

5.Statewithreasonwhetherfollowingfunctionshaveinverse:

(i) :{1,2,3,4} {10}with ={(1,10),(2,10),(3,10),(4,10)}

(ii) :{5,6,7,8} {1,2,3,4}with ={(5,4),(6,3),(7,4),(8,2)}

(iii) :{2,3,4,5} {7,9,11,13}with ={(2,7),(3,9),(4,11),(5,13)}

Ans.(i) ={(1,10),(2,10),(3,10),(4,10)}

Itismany-onefunction,therefore hasnoinverse.

(ii) ={(5,4),(6,3),(7,4),(8,2)}

Itismany-onefunction,therefore hasnoinverse.

(iii) ={(2,7),(3,9),(4,11),(5,13)}

isone-oneontofunction,therefore, hasaninverse.



6.Showthat Rgivenby isone-one.Findtheinverse

ofthefunction Range

Ans.PartI: Rgivenby

Let ,then and

When then

isone-one.

PartII:Let Rangeof

forsome in

As

isonto.

Therefore,



7.Consider :R Rgivenby Showthat isinvertible.Findtheinverseof [Hint: ]

Ans.Consider :R Rgivenby

Let R,then and

Now,for ,then isone-one.

Let Rangeof

isonto.

Therefore, isinvertibleandhence, .

8.Consider givenby Showthat is invertiblewiththeinverse of givenby where isthesetofallnon-negativerealnumbers.

Ans.Consider and

Let R ,then and

isone-one.



Now

as

isonto.

Therefore, isinvertibleand .

9. Consider given by Show that is

invertiblewith

Ans.Consider and

Let R ,then and

Now, then

isone-one.

Now,again



= = =

=

= isonto.

Therefore, isinvertibleand .

10.Let beaninvertiblefunction.Showthat hasuniqueinverse.

(Hint: Suppose and are two inverses of Then for allUseone-onenessof ).

Ans.Given: beaninvertiblefunction.

Thus is1–1andontoandtherefore exists.

Let and betwoinversesof Thenforall Y,

Theinverseisuniqueandhence hasauniqueinverse.



11.Consider :{1,2,3} givenby and

Find andshowthat

Ans. ,thenitisclearthat is1–1andontoandtherefore

exists.

Also, and

Hence,

12.Let beaninvertiblefunction.Showthattheinverseof is

,i.e.,

Ans.Let beaninvertiblefunction.

Then isone-oneandonto

Xwhere isalsoone-oneandontosuchthat

and

Now, and

13.If :R Rgivenby then is:



(A)

(B)

(C)

(D)

Ans. :R Rand

=

= = =


14.Let :R– Rbeafunctiondefinedas Theinverseof

isthemap :Rangeof givenby:

(A)

(B)

(C)

(D)



Ans.Given: :R– Rand

Now,Rangeof

Let

f-1(y)=g(y)=





NCERTsolution

Chapter-1


1. Determine whether or not each of the definition of * given below gives a binary

operation.Intheeventthat*isnotabinaryoperation,givenjustificationforthis.

(i)On define*by

(ii)On define*by

(iii)OnR,define*by

(iv)On define*by

(v)On define*by

Ans.(i)On ={1,2,3,…..},

Let

Therefore,operation*isnotabinaryoperationon .

(ii)On ={1,2,3,…..},

Let

Therefore,operation*isabinaryoperationon .

(iii)onR(setofrealnumbers)

Let R

Therefore,operation*isabinaryoperationonR.

(iv)On ={1,2,3,…..},



Let


(v)On ={1,2,3,…..},

Let


2.Foreachbinaryoperation*definedbelow,determinewhether*iscommutativeor

associative:

(i)On define

(ii)OnQ,define

(iii)OnQ,define

(iv)On define

(v)On define

(vi)OnR–{–1},define

Ans.(i)Forcommutativity: and =

Forassociativity: =

Also, =

Therefore,theoperation*isneithercommutativenorassociative.

(ii)Forcommutativity: and



Forassociativity: =

Also, =

Therefore,theoperation*iscommutativebutnotassociative.

(iii)Forcommutativity: and =

Forassociativity: =

Also, =

Therefore,theoperation*iscommutativeandassociative.

(iv)Forcommutativity: and

Forassociativity: =

Also, =

Therefore,theoperation*iscommutativebutnotassociative.

(v)Forcommutativity: and

Forassociativity: =



Also, =


(vi)Forcommutativity: and

Forassociativity: =

Also, =


3. Consider the binary operation on the set {1, 2, 3, 4, 5} defined by

Writetheoperationtableoftheoperation

Ans.LetA={1,2,3,4,5}definedby i.e.,minimumof and

1 2 3 4 5

1 1 1 1 1 1

2 1 2 2 2 2

3 1 2 3 3 3

4 1 2 3 4 4

5 1 2 3 4 5

4. Consider a binary operation * on the set {1, 2, 3, 4, 5} given by the following

multiplicationtable(table1.2).

(i)Compute(2*3)*4and2*(3*4)



(ii)Is*commutative?

(iii)Compute(2*3)*(4*5)

(Hint:Usethefollowingtable)

Table1.2

* 1 2 3 4 5

1 1 1 1 1 1

2 1 2 1 2 1

3 1 1 3 1 1

4 1 2 1 4 1

5 1 1 1 1 5

Ans.(i)2*3=1and3*4=1

Now(2*3)*4=1*4=1and2*(3*4)=2*1=1

(ii)2*3=1and3*4=1

2*3=3*2andotherelementofthegivenset.

Hencetheoperationiscommutative.

(iii)(2*3)*(4*5)=1*1=1

5.Let*’bethebinaryoperationontheset{1,2,3,4,5}definedby H.C.F.of

and Istheoperation*’sameastheoperation*definedinExercise4above?Justify

youranswer.

Ans.LetA={1,2,3,4,5}and H.C.F.of and

*’ 1 2 3 4 5

1 1 1 1 1 1

2 1 2 1 2 1

3 1 1 3 1 1

4 1 2 1 4 1



5 1 1 1 1 5

Weobservethattheoperation*'isthesameastheoperation*inExp.4.

6.Let*bethebinaryoperationonNgivenby L.C.M.of and Find:

(i)5*7,20*16

(ii)Is*commutative?

(iii)Is*associative?

(iv)Findtheidentityof*inN.

(v)WhichelementsofNareinvertiblefortheoperation*?

Ans. L.C.M.of and

(i)5*7=L.C.M.of5and7=35

20*16=L.C.M.of20and16=80

(ii) L.C.M.of and =L.C.M.of and =

Therefore,operation*iscommutative.

(iii) =

=

Similarly,

Thus,

Therefore,theoperationisassociative.

(iv)Identityof*inN=1because =L.C.M.of and1=

(v)Onlytheelement1inNisinvertiblefortheoperation*because



7.Is*definedontheset{1,2,3,4,5}by L.C.M.of and abinaryoperation?

Justifyyouranswer.

Ans.LetA={1,2,3,4,5}and L.C.M.of and

* 1 2 3 4 5

1 1 2 3 4 5

2 2 2 6 4 10

3 3 x 3 12 15

4 4 4 12 4 20

5 5 x 15 20 5

Here,2*3=6 A

Therefore,theoperation*isnotabinaryoperation.

8. Let * be the binary operation on N defined by H.C.F. of and Is *

commutative?Is*associative?DoesthereexistidentityforthisbinaryoperationonN?

Ans. H.C.F.of and

(i) H.C.F.of and =H.C.F.of and =

Therefore,operation*iscommutative.

(ii) =

=

Therefore,theoperationisassociative.

Therefore,theredoesnotexistanyidentityelement.

9.Let*beabinaryoperationonthesetQofrationalnumbersasfollows:



(i) (ii)

(iii) (iv)

(v) (vi)

Findwhichofthebinaryoperationsarecommutativeandwhichareassociative.

Ans.(i) operation*isnotcommutative.

And

Here, operation*isnotassociative.

(ii) operation*iscommutative.

And


(iii) and

Therefore,operation*isnotcommutative.

And


(iv) operation*iscommutative.



And


(v) operation*iscommutative.

And

Here, operation*isassociative.

(vi) and operation*isnotcommutative.

And


10.Showthatnoneoftheoperationsgivenabovetheidentity.

Ans.LettheidentitybeI.

(i)

(ii)

(iii)

(iv)



(v)

(vi)

Therefore,noneoftheoperationsgivenabovehasidentity.

11. Let A = N x N and * be the binary operation on A defined by

Showthat*iscommutativeandassociative.Findtheidentityelementfor*onA,ifany.

Ans.A=NxNand*isabinaryoperationdefinedonA.

The operation is

commutative

Again,

And

Here, Theoperationisassociative.

Letidentityfunctionbe ,then

Foridentityfunction

Andfor

As0 N,therefore,identity-elementdoesnotexist.

12.Statewhetherthefollowingstatementsaretrueorfalse.Justify:

(i)Foranarbitrarybinaryoperation*onasetN,

(ii)If*isacommutativebinaryoperationonN,then



Ans.(i)*beingabinaryoperationonN,isdefinedas

Henceoperation*isnotdefined,therefore,thegivenstatementisfalse.

(ii)*beingabinaryoperationonN.

Thus, ,thereforethegivenstatementistrue.

13.Considerabinaryoperation *onNdefinedas .Choose the correct

answer:

(A)Is*bothassociativeandcommutative?

(B)Is*commutativebutnotassociative?

(C)Is*associativebutnotcommutative?

(D)Is*neithercommutativenorassociative?

Ans. Theoperation*iscommutative.

Again,

And

Theoperation*isnotassociative.





NCERTsolution

Chapter-1

Relations&Functions-MiscellaneousExercise

1. Let be defined as Find the function

suchthat

Ans.Given:

Now and

2.Let bedefinedas if isoddand if is

even.Showthat isinvertible.Findtheinverseof Here,Wisthesetofallwhole

numbers.

Ans.Given: definedas

Injectivity:Let beanytwooddrealnumbers,then

Again,let beanytwoevenwholenumbers,then



Is isevenand isodd,then

Also,if oddand iseven,then

Hence,

isaninjectivemapping.

Surjectivity:Let beanarbitrarywholenumber.

If isanoddnumber,thenthereexistsanevenwholenumber suchthat

If isanevennumber,thenthereexistsanoddwholenumber suchthat

Therefore,every Whasitspre-imageinW.

So, isasurjective.Thus isinvertibleand exists.

For :

and

Hence,

3.If isdefinedby find



Ans.Given:

=

4. Show that the function defined by

Risone-oneandontofunction.

Ans.Itisgiventhatf: isdefinedasf(x)=

Forone-one

Supposef(x)=f(y),wherex,y R

Itcanbeobservedthatifxispositiveandyisnegative.

Then,wehave

Since,xispositiveandyisnegative,

x>y ,but,2xyisnegative

Then,2xy x-y.

Thus,thecaseofxbeingpositiveandybeingnegativecanberuledout.

Underasimilarargument,xbeingnegativeandybeingpositivecanalsoberuledout.

xandyhavetobeeitherpositiveornegative.

Whenxandyarebothpositive,wehave

f(x)=f(y)

Whenxandyarebothnegative,wehave

f(x)=f(y)

fisone-one

ForOnto



Now,lety Rsuchthat-1<y<1.

Ifyisnegativethen,thereexistsx= suchthat

f(x)=

Ifyispositive,then,thereexistsx= suchthat

f(x)=

Therefore,fisonto.Hence,fisone-oneandonto.

5.Showthatthefunction givenby isinjective.

Ans.Let Rbesuchthat

Therefore, isone-onefunction,hence isinjective.

6. Give examples of two functions and such that is

injectivebut isnotinjective.

(Hint:Consider and )

Ans.Given:twofunctions and

Let and

Therefore, isinjectivebut isnotinjective.

7. Give examples of two functions and such that is

ontobut isnotonto.



(Hint:Consider and )

Ans.Let

Thesearetwoexamplesinwhich isontobut isnotonto.

8.GivenanonemptysetX,considerP(X)whichisthesetofallsubsetsofX.

DefinetherelationRinP(X)asfollows:

ForsubsetsA,BinP(X),ARBifandonlyifA B.IsRanequivalencerelationonP(X)?

Justifyyouranswer.

Ans.(i)A A Risreflexive.

(ii)A B B A Risnotcommutative.

(iii)IfA B,B C,thenA C Ristransitive.

Therefore,Risnotequivalentrelation.

9.Givenanon-emptysetX,considerthebinaryoperation*:P(X)xP(X) P(X)given

byA*B=A B A,BinP(X),whereP(X)isthepowersetofX.ShowthatXisthe

identityelement for thisoperationandX is theonly invertibleelement inP (X)with

respecttotheoperation*.

Ans.LetSbeanon-emptysetandP(S)beitspowerset.LetanytwosubsetsAandBofS.

A B S

A B P(S)

Therefore, isanbinaryoperationonP(S).



Similarly,ifA,B P(S)andA–B P(S),thentheintersectionofsets anddifferenceof

setsarealsobinaryoperationonP(S)andA S=A=S AforeverysubsetAofsets

A S=A=S AforallA P(S)

Sistheidentityelementforintersection onP(S).

10.Findthenumberofallontofunctionsfromtheset{1,2,3,……., }toitself.

Ans.The number of onto functions that can be defined from a finite set A containing

elementsontoafinitesetBcontaining elements=

11.LetS= andT={1,2,3}.Find ofthefollowingfunctionsFfromStoT,if

itexists.

(i)F=

(ii)F=

Ans.S= andT={1,2,3}

(i)F=

(ii)

Fisnotone-onefunction,sinceelement and havethesameimage1.

Therefore,Fisnotone-onefunction.



12. Consider the binary operation * : R x R R and o = R x R R defined as

and R.Showthat*iscommutativebutnotassociative,

o is associative but not commutative. Further, show that R,

[Ifitisso,wesaythattheoperation*distributesoverthe

operationo].Doesodistributeover*?Justifyyouranswer.

Ans.PartI: also operation*iscommutative.

Now,

And


PartII: R

And,

operation isnotcommutative.

Now and

Here operation isassociative.

PartIII:L.H.S. =

R.H.S. = =L.H.S.Proved.

Now,anotherdistributionlaw:

L.H.S.

R.H.S.

AsL.H.S. R.H.S.



Therefore,theoperation doesnotdistributeover.

13.Givenanon-emptysetX,let*:P(X)xP(X) P(X)bedefinedasA*B=(A–B)

(B–A), A,B P(X).Showthattheemptyset istheidentityfortheoperation*

andalltheelementsAofP(X)areinvertiblewithA-1=A.(Hint:

and )

Ans.ForeveryA P(X),wehave

=

And =

istheidentityelementfortheoperation*onP(X).

AlsoA*A=(A–A) (A–A)=

EveryelementAofP(X)isinvertiblewith =A.

14. Define binary operation * on the set {0, 1, 2, 3, 4, 5} as

Show that zero is the identity for this operation and each element of the set is

invertiblewith beingtheinverseof

Ans.Abinaryoperation(orcomposition)*ona(non-empty)setisafunction*:AxA A.

Wedenote by foreveryorderedpair AxA.

Abinaryoperationonano-emptysetAisarulethatassociateswitheveryorderedpair

ofelements (distinctorequal)ofAsomeuniqueelement ofA.

* 0 1 2 3 4 5

0 0 1 2 3 4 5

1 1 2 3 4 5 0



2 2 3 4 5 0 1

3 3 4 5 0 1 2

4 4 5 0 1 2 3

5 5 0 1 2 3 4

Forall A,wehave (mod6)=0

And and

0istheidentityelementfortheoperation.

Alsoon0=0–0=0*

2*1=3=1*2

15.LetA={–1,0,1,2},B={–4,–2,0,2}and bethefunctionsdefinedby

Aand A.Are and equal?Justifyyour

answer.

(Hint: One may note that two functions and such that

A,arecalledequalfunctions).

Ans.When then and

At and

At and



At and

Thusforeach A,

Therefore, and areequalfunction.

16. Let A = {1, 2, 3}. Then number of relations containing (1, 2) and (1, 3) which are

reflexiveandsymmetricbutnottransitiveis:

(A)1

(B)2

(C)3

(D)4

Ans.Itisclearthat1isreflexiveandsymmetricbutnottransitive.

Therefore,option(A)iscorrect.

17.LetA={1,2,3}.Thennumberofequivalencerelationscontaining(1,2)is:

(A)1

(B)2

(C)3

(D)4

Ans.2


18.Let betheSignumFunctiondefinedas and

betheGreatestFunctiongivenby where isgreatestintegerlessthan



orequalto Then,does and coincidein(0,1)?

Ans.Itisclearthat and

Consider whichlieon(0,#1)

Now,

And

in(0,1]

No,fogandgofdon'tcoincidein(0,1].

19.Numberofbinaryoperationontheset are:

(A)10

(b)16

(C)20

(D)8

Ans.A=

AxA=

=4

Numberofsubsets= =16

Hencenumberofbinaryoperationis16.





NCERTsolution

Chapter-2

InverseTrigonometricFunctions-Exercise2.1

Findtheprincipalvaluesofthefollowing:

1.

Ans.Let

Since,theprincipalvaluebranchof is

Therefore,Principalvalueof is

2.

Ans.Let





3.

Ans.Let



4.

Ans.Let





5.

Ans.Let



6.

Ans.Let





7.

Ans.Let



8.

Ans.Let





9.

Ans.Let



10.

Ans.Let





Findthevalueofthefollowing:

11.

Ans.

=

=

= =

12.

Ans.

=

=

=

13.If then:

A)



(B)

(C)

(D)

Ans.Bydefinitionofprincipalvaluefor


14. isequalto:

(A)

(B)

(C)

(D)

Ans.

=

=

=





NCERTsolution

Chapter-2

InverseTrigonometricFunctions-Exercise2.2

Provethefollowing:

1.

Ans.Weknowthat:

Putting

Putting ,

Proved.

2.

Ans.Weknowthat:

Putting



Putting ,

Proved.

3.

Ans.L.H.S.=

=

=

=

=

=R.H.S.

Proved.

4.

Ans.L.H.S.=



=

=

=

=

= =R.H.S.

Proved.

Writethefollowingfunctionsinthesimplestform:

5.

Ans.Putting sothat



=

=

=

=

=

=

=

6.

Ans.Putting sothat

=



=

=

=

=

=

7.

Ans.

=

=

=

8.



Ans.

Dividingthenumeratoranddenominatorby

=

=

=

=

9.

Ans.Putting sothat

=

=



=

=

=

10.

Ans.

= [Dividingnumeratoranddenominatorby ]

Putting sothat

=

=

= =

Findthevaluesofeachofthefollowing:

11.



Ans.

=

=

=

=

=

=

12.

Ans.

=

13.

Ans.Putting and



=

=

=

=

=

=

14.If thenfindthevalueof

Ans.Given:



15.If thenfindthevalueof

Ans.Given:

FindthevaluesofeachoftheexpressionsinExercises16to18.

16.



Ans.

=

=

=

=

17.

Ans.

=

=

=

=

=



18.

Ans.Putting and sothat and

Now, =

And

and

=

= =

= =

19. isequalto:

(A)

(B)



(C)

(D)

Ans.

=

=


20. isequalto:

(A)

(B)

(C)

(D)1

Ans.

=



=

=

=


21. isequalto:

(A)

(B)

(C)

(D)

Ans.

=

=

=

=

=

=





NCERTsolution

Chapter-2

InverseTrigonometricFunctions-MiscellaneousExercise

Findthevalueofthefollowing:

1.

Ans.

=

=

=

= =

2.

Ans.

=



=

=

= =

3.Provethat:

Ans.Let sothat

Since

= = =



4.Provethat:

Ans.Let sothat

Again,Let sothat

Since

=

=



5.Provethat:

Ans.Let sothat

Again,Let sothat

Since =

=

6.Provethat:

Ans.Let sothat

Again,Let sothat



Since =

=

7.Provethat:

Ans.Let sothat

Again,Let sothat



Since =

=

8.Provethat:

Ans.L.H.S.=

=

=

=



=

=

=

R.H.S.

9.Provethat:

Ans.Let sothat

=

=

=

=

10.Provethat:



Ans.Weknowthat =

Again,

=

L.H.S.=

=

=

=

11.Provethat:

Ans.Putting sothat

L.H.S.=



=

=

=

Dividingeverytermby

=

=

= =

= =R.H.S.

12.Provethat:

Ans.L.H.S.=

=



= …(i)

Now,let sothat

Fromeq.(i), =R.H.S.

13.Solvetheequation:

Ans.

14.Solvetheequation:

Ans.Putting



15. isequalto:

(A)



(B)

(C)

(D)

Ans.Let where sothat

Putting


16. then isequalto:

(A)

(B)

(C)0

(D)



Ans.Putting sothat

or

or

But doesnotsatisfythegivenequation.


17. isequalto:



(A)

(B)

(C)

(D)

Ans.

=

=

=

=

=

=



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