CBSENCERTSolutionsforClass10mathematicsChapter2

Exercise2.1

Q.1. Thegraphofy=p(x)isgiveninthefigurebelow,forsomepolynomialp(x).Findthenumberofzeroesofp(x).

Solution: Weknowthat,

Ifapolynomialpxisofdegreen,thenthegraphofy=pxintersectsthex-axisatmostnpoints.

Therefore,apolynomialpxofdegreenhasatmostnzeroes.

Here,thegraphofp(x)passesthroughonlyy-axisanddoesnotcutthex-axisatall.Therefore,thenumberofzeroesis0.

Q.2. Thegraphofy=p(x)isgiveninthefigurebelow,forsomepolynomialsp(x).Findthenumberofzeroesofp(x),ineachcase.

Solution: Giventhegraphofy=p(x)

WeknowthatthepointofintersectionsofagraphwiththeX-axisarecalledrootsorzeroesofthegivenequationorpolynomial.Here,asthegraphofp(x)intersectsthex-axisatonly1point.Therefore,thenumberofzeroesis1.

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Q.3. Thegraphofy=p(x)isgivenbelowforapolynomialp(x).Findthenumberofzeroesofp(x).

Solution: Givengraphis:

Wecanseethat,thegraphofy=pxisintersectingthex-axisatthreepoints.Thenumberofpointsofintersectionofthegraphwiththex-axisdenotesthenumberofzeroesofthepolynomial.Hence,thegivenpolynomialhasthreezeroes.

Q.4. Thegraphsofy=p(x)aregiveninthefigurebelow,forsomepolynomialsp(x).Findthenumberofzeroesofp(x),ineachcase.

Solution: Considerthefigurebelow,

Asthegraphofpxintersectsthex-axisatpointaandb.Hence,thenumberofzeroesistwo.

Q.5. Thegraphofy=p(x)isgiveninthefigurebelow.Findthenumberofzeroesofp(x).

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Solution: Weknowthat,thepointofintersectionsofagraphwithX-axisarecalledrootsorzeroesofthegivenequationorpolynomial.

Thegraphofy=p(x)wherep(x)isapolynomial.

Here,thegivengraphofp(x)intersectstheX-axisat4points.Therefore,thenumberofzeroesare4.

Q.6. Thegraphsofy=p(x)isgivenbelowforsomepolynomialsp(x).Findthenumberofzeroesofp(x).

Solution: Weknowthat,thepointofintersectionsofagraphwithX-axisarecalledrootsorzeroesofthegivenequationorpolynomial.

Thegraphp(x)intersectstheX-axisat3points.Acubicequationcanhave3zeroes,asthehighestpowerofxis3orasthedegreeis3Therefore,thenumberofzeroesare3.

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Exercise2.2

Q.1. Findthezeroesofthefollowingquadraticpolynomialandverifytherelationshipbetweenthezeroesandthecoefficients.x2-2x-8

Solution: Given:

x2-2x-8

=x2-4x+2x-8[Factorisationbysplittingthemiddleterm]=xx-4+2(x-4)=x-4x+2Weknowthatthezeroesofthequadraticpolynomialax2+bx+carethesameastherootsofthequadraticequationax2+bx+c=0.

Therefore,byequatingthegivenpolynomialtozero.Weget,

x2-2x-8=0

⇒x-4x+2=0⇒x-4=0orx+2=0⇒x=4orx=-2Therefore,thezeroesofx²-2x-8are4and-2.Sumofzeroes=4-2=2=--21=-CoefficientofxCoefficientofx2Productofzeroes=4×-2=-8=-81=ConstanttermCoefficientofx2Hence,therelationshipbetweenthezeroesandthecoefficientsisverified.

Q.2. Findthezeroesofthefollowingquadraticpolynomialandverifytherelationshipbetweenthezeroesandthecoefficients.

4s2-4s+1

Solution: Given:4s2-4s+1

Weknowthatthezeroesofthequadraticpolynomialax2+bx+carethesameastherootsofthequadraticequationax2+bx+c=0.Therefore,byequatingthegivenpolynomialtozero,weget ,4s2-4s+1=0⇒(2s-1)2=0⇒2s-1=0⇒s=12

Therefore,thezeroesof4s2-4s+1are12and12

Sumofzeroes=12+12=1=(-(-4))4=-(Coefficient of s)(Coefficient of s2)Productofzeroes=12×12=14=Constant termCoefficient of s2Hence,therelationshipbetweenthezeroesandthecoefficientsisverified.

Q.3. Findthezeroesofthefollowingquadraticpolynomialandverifytherelationshipbetweenthezeroesandthecoefficients.

6x2-3-7x

Solution: Given:6x2-3-7x

=6x2-7x-3

=6x2-9x+2x-3=3x(2x-3)+(2x-3)=(3x+1)(2x-3)Weknowthatthezeroesofthequadraticpolynomialax2+bx+carethesameastherootsofthequadraticequationax2+bx+c=0.Therefore,byequatingthegivenpolynomialtozero.Weget,6x2-3-7x=0⇒3x+1=0or2x-3=0⇒x=-13orx=32Therefore,thezeroesof6x2-3-7xare-13and32.

Sumofzeroes=-13+32=76=-(-7)6=(-Coefficient of x)(Coefficient of x2)Productofzeroes=-13×32=-12=-36=Constant termCoefficient of x2

Hence,therelationshipbetweenthezeroesandthecoefficientsisverified.

Q.4. Findthezeroesofthefollowingquadraticpolynomialandverifytherelationshipbetweenthezeroesandthecoefficients.4u2+8u

Solution: Given:4u2+8u=4u2+8u+0=4u(u+2)Weknowthatthezeroesofthequadraticpolynomialax2+bx+carethesameastherootsofthequadraticequationax2+bx+c=0.

Therefore,byequatingthegivenpolynomialtozero.Weget4u2+8u=0⇒4u=0oru+2=0⇒u=0oru=-2

So,thezeroesof4u2+8uare0and-2Sumofzeroes=0+(-2)=-2=(-8)4=-Coefficient of uCoefficient of u2

Productofzeroes=0×(-2)=0=04=ConstanttermCoefficientofu2Hence,therelationshipbetweenthezeroesandthecoefficientsisverified.

Q.5. Findthezeroesofthefollowingquadraticpolynomialandverifytherelationshipbetweenthezeroesandthecoefficients.

t2-15

Solution: Given:t2-15=t2-(15)2=(t-15)(t+15)Weknowthat,thezeroesofthequadraticpolynomialax2+bx+carethesameastherootsofthequadraticequationax2+bx+c=0.Therefore,byequatingthegivenpolynomialtozero.Weget:t2-15=0⇒t-15=0⇒t=15ort=-15

So,zeroesoft2-15are15and-15.Sumofzeroes=15+(-15)=0=(-0)1=-CoefficientoftCoefficientoft2Productofzeroes=(15)(-15)=-15=(-15)1=ConstanttermCoefficientoft2Hence,therelationshipbetweenthezeroesandthecoefficientsisverified.

Q.6. Findthezeroesofthefollowingquadraticpolynomialandverifytherelationshipbetweenthezeroesandthecoefficients.

3x2-x-4

Solution: Considerthegivenpolynomial,3x2-x-4=3x2-4x+3x-4=x(3x-4)+1(3x-4)=(3x-4)(x+1)

Weknowthatthezeroesofthequadraticpolynomialax2+bx+carethesameastherootsofthequadraticequationax2+bx+c=0.Therefore,byequatingthegivenpolynomialtozero.Weget,3x2-x-4=0⇒3x-4=0orx+1=0⇒x=43orx=-1

Verification:Thezeroesof3x2-x-4are43and-1.

Sumofzeroes=43+(-1)=13=-(-1)3=-Coefficient of xCoefficient of x2Productofzeroes=43(-1)=(-4)3=ConstanttermCoefficientofx2Hence,therelationshipbetweenthezeroesandthecoefficientsisverified.

Q.7. Findaquadraticpolynomialeachwiththegivennumbersasthesumandproductofitszeroesrespectively.14,-1

Solution: Weknowthatifαandβarethezeroesofaquadraticpolynomialpx,then,thepolynomialp(x)canbewrittenaspx=ax2-α+βx+αβor,p(x)=ax2-(Sumofthezeroes)x+Productofthezeroes,whereaisanon-zerorealnumber.Therefore,sumoftheroots=α+β=14andproductoftheroots=αβ=-1Hence,thequadraticpolynomialpxcanbewrittenas:p(x)=a{x2-14x-1}=a4x2-x-44Bytakinga=4,wegetoneofthequadraticpolynomialswhichsatisfythegivenconditions.Hence,thequadraticpolynomialis4x2-x-4.

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Q.8. Findaquadraticpolynomialwith2,13asthesumandproductofitszeroesrespectively.

Solution: Weknowthatifαandβarethezeroesofaquadraticpolynomialpxthen,thepolynomialp(x)canbewrittenaspx=ax2-α+βx+αβor,

p(x)=ax2-(Sumofthezeroes)x+Productofthezeroes,whereaisanon-zerorealnumber.

Here,sumoftheroots=α+β=2andproductoftheroots=αβ=13Hence,thequadraticpolynomialpxcanbewrittenas:px=a{x2-2x+13}=a3x2-32x+13Bytakinga=3,wegetoneofthequadraticpolynomialswhichsatisfythegivenconditions.Therefore,thequadraticpolynomialis3x2-32x+1.

Q.9. Findaquadraticpolynomialeachwiththegivennumbersasthesumandproductofitszeroesrespectively.

0,5

Solution: Weknowthatifαandβarethezeroesofaquadraticpolynomialpxthen,thepolynomialp(x)canbewrittenaspx=ax2-α+βx+αβ

or,p(x)=ax2-(Sumofthezeroes)x+Productofthezeroes,whereaisanon-zerorealnumber.

Therefore,sumoftheroots=α+β=0andtheproductoftheroots=αβ=5.Hence,thequadraticpolynomialpxcanbewrittenas:px=a{x2-0×x+5}=a{x2+5}Bytakinga=1,wegetoneofthequadraticpolynomialswhichsatisfythegivenconditions.Hence,thequadraticpolynomialisx2+5.

Q.10. Findaquadraticpolynomialwiththegivennumbersasthesumandproductofitszeroesrespectively.1,1

Solution: Weknowthatifαandβarethezeroesofaquadraticpolynomialpx,then,thepolynomialp(x)canbewrittenas

px=ax2-α+βx+αβor,

p(x)=ax2-(Sumofthezeroes)x+Productofthezeroes,whereaisanon-zerorealnumber.Therefore,sumoftheroots=α+β=1andproductoftheroots=αβ=1Hence,thequadraticpolynomialpxcanbewrittenas:px=a{x2-1.x+1}=a{x2-x+1}Bytakinga=1,wegetoneofthequadraticpolynomialswhichsatisfythegivenconditions. Hence,thequadraticpolynomialisx2-x+1.

Q.11. Findaquadraticpolynomialwiththegivennumbersasthesumandproductofitszeroesrespectively.

-14,14

Solution: Weknowthatifαandβarethezeroesofaquadraticpolynomialpx,then,thepolynomialp(x)canbewrittenas

px=ax2-α+βx+αβor,

p(x)=ax2-(Sumofthezeroes)x+Productofthezeroes,whereaisanon-zerorealnumber.Therefore,sumoftheroots=α+β=-14andproductoftheroots=αβ=14Hence,thequadraticpolynomialpxcanbewrittenas:p(x)=a{x2+14x+14}=a4x2+x+14Bytakinga=4,wegetoneofthequadraticpolynomialswhichsatisfythegivenconditions. Hence,thequadraticpolynomialis4x2+x+1.

Q.12. Findaquadraticpolynomialwiththegivennumbersasthesumandproductofitszeroesrespectively.

4,1

Solution: Weknowthatifαandβarethezeroesofaquadraticpolynomialpx,then,thepolynomialp(x)canbewrittenas

px=ax2-α+βx+αβor,

p(x)=ax2-(Sumofthezeroes)x+Productofthezeroes,whereaisanon-zerorealnumber.Therefore,sumoftheroots=α+β=4andproductoftheroots=αβ=1Hence,thequadraticpolynomialpxcanbewrittenas:p(x)=a{x2-4x+1}Bytakinga=1,wegetoneofthequadraticpolynomialswhichsatisfythegivenconditions. Hence,thequadraticpolynomialisx2-4x+1.

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Exercise2.3

Q.1. Dividethepolynomialpxbythepolynomialgxandfindthequotientandremainder.px=x3-3x2+5x-3,gx=x2-2

Solution: Given,px=x3-3x2+5x-3,gx=x2-2

Bylongdivisionmethod,thepolynomialpxcanbedividedbythepolynomialgxasfollows:

Here,Quotient=x-3andRemainder=7x-9

Q.2. Dividethepolynomialpxbythepolynomialgxandfindthequotientandremainderineachofthefollowing:

px=x4-3x2+4x+5,gx=x2+1-x

Solution: Given:px=x4-3x2+4x+5=x4+0.x3-3x2+4x+5.

Here,thepolynomialp(x)isalreadyarrangedinthedescendingpowersofvariable.

gx=x2+1-xHere,thepolynomialg(x)isnotarrangedinthedescendingpowersofvariable.Now,gx=x2-x+1Thepolynomialpxcanbedividedbythepolynomialgxasfollows:x2+x-3x2-x+1x4+0·x3-3x2+4x+5x4-x3+x2-+-x3-4x2+4x+5+x3-x2+x-+--3x2+3x+5-3x2+3x-3+-+8Quotient=x2+x-3Remainder=8.

Q.3. Dividethepolynomialpxbythepolynomialgxandfindthequotientandremainder.px=x4-5x+6,gx=2-x2

Solution: Given,

px=x4-5x+6=x4+0.x2-5x+6

gx=2-x2Here,thepolynomialg(x)isnotarrangedinthedescendingpowersofvariable.Now,gx=-x2+2Thepolynomialpxcanbedividedbythepolynomialgxasfollows:

Quotient=-x2-2Remainder=-5x+10

Q.4. Checkwhetherthefirstpolynomialisafactorofthesecondpolynomialbydividingthesecondpolynomialbythefirstpolynomial:t2-3,2t4+3t3-2t2-9t-12

Solution: Thepolynomial2t4+3t3-2t2-9t-12canbedividedbythepolynomialt2-3=t2+0.t-3asfollows:

Sincetheremainderis0,hencet2-3isafactorof2t4+3t3-2t2-9t-12.

Q.5. Checkwhetherthefirstpolynomialisafactorofthesecondpolynomialbydividingthesecondpolynomialbythefirstpolynomial:x2+3x+1,3x4+5x3-7x2+2x+2

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Solution: Given:Thepolynomial3x4+5x3-7x2+2x+2canbedividedbythepolynomialx2+3x+1asfollows:

Sincetheremainderis0,hencex2+3x+1isafactorof3x4+5x3-7x2+2x+2.

Q.6. Checkwhetherthefirstpolynomialisafactorofthesecondpolynomialbydividingthesecondpolynomialbythefirstpolynomial.x3-3x+1,x5-4x3+x2+3x+1

Solution: Thepolynomialx5-4x3+x2+3x+1canbedividedbythepolynomialx3-3x+1asfollows:

Since,theremainderisnotequalto0,hencex3-3x+1isnotafactorofx5-4x3+x2+3x+1.

Q.7. Obtainallotherzeroesof3x4+6x3-2x2-10x-5,iftwoofitszeroesare53and-53.

Solution: Letpx=3x4+6x3-2x2-10x-5

Itisgiventhatthetwozeroesofpxare53and-53

∴x-53x+53=x2-53isafactorofpxTherefore,ondividingthegivenpolynomialbyx2-53,weobtainremainderas0.3x2+6x+3x2+0x+533x4+6x3-2x2-10x-53x4+0x3-5x2--+6x3+3x2-10x-56x3+0x2-10x--+3x2+0x-53x2+0x-5--+0

Hence,3x4+6x3-2x2-10x-5=x2-533x2+6x+3

=3x2-53x2+2x+1

Now,x2+2x+1=0⇒x+1x+1=0⇒x=-1,-1Thus,thetwozeroesofx2+2x+1are-1and-1Therefore,thezeroesofthegivenpolynomialare53,-53,-1and-1.

Q.8. Ondividingx3-3x2+x+2byapolynomialgx,thequotientandremainderwerex-2and-2x+4,respectively.Findgx.

Solution: Itisgiventhatpx=x3-3x2+x+2,Quotient=x-2andremainder=-2x+4

g(x)isthedivisor

Accordingtothedivisionalgorithm,Dividend=Divisor×Quotient+Remainderx3-3x2+x+2=gx×x-2+-2x+4⇒x3-3x2+x+2+2x-4=gxx-2⇒x3-3x2+3x-2=gxx-2

Now,gxisthequotientwhenx3-3x2+3x-2isdividedbyx-2.(Since,Remainder=0)

∴gx=x2-x+1.

Q.9. Giveexamplesofpolynomialspx,g(x),qxandrx,whichsatisfythedivisionalgorithmanddegpx=degq(x)

Solution: Accordingtothedivisionalgorithm,ifpxandgxaretwopolynomialswithgx≠0,thenwecanfindpolynomialsqxandrxsuchthatpx=gx×qx+rxDegreeofquotientwillbeequaltodegreeofdividendwhendivisorisconstant.Letusconsiderthedivisionof2x2+2x-16by2Here,px=2x2+2x-16andqx=x2+x-8andrx=0

Clearly,thedegreeofpxandqxisthesamewhichis2.Verification:px=gx×qx+rxp(x)=2(x2+x-8)+0=2x2+2x-16Thus,thedivisionalgorithmissatisfied.

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Q.10. Giveexampleofpolynomialspx,gx,qxandrx,whichsatisfythedivisionalgorithmanddegqx=degrx.

Solution: Accordingtothedivisionalgorithm,ifpxandgxaretwopolynomialswithgx≠0,thenwecanfindpolynomialsqxandrxsuchthat

px=gx×qx+rx,whererx=0ordegreeofr(x)<degreeofgx.

Letusconsiderthedivisionof4x+3by(x+2)Here,px=4x+3andgx=x+2,qx=4andr(x)=-5Here,degreeofqxandrxisthesamewhichis0.Verification:px=gx×qx+rx⇒p(x)=(x+2)×4+(-5)⇒p(x)=4x+3Thus,thedivisionalgorithmissatisfied.

Q.11. Giveexamplesofpolynomialspx,gx,qxandrx,whichsatisfythedivisionalgorithmanddegr(x)=0.

Solution: Accordingtothedivisionalgorithm,ifpxandgxaretwopolynomialswithgx≠0,thenwecanfindpolynomialsqxandrxsuchthatpx=gx×qx+rx,whererx=0ordegreeofr(x)<degreeofgx.

Degreeofremainderwillbe0whenremainderobtainedondivisionisaconstant.

Letusconsiderthedivisionof4x+3.Here,px=4x+3,gx=x+2,qx=4andr(x)=-5.

Here,wegetremainderasaconstant.Therefore,thedegreeofrxis0.

Verification:

px=gx×qx+rx∴p(x)=(x+2)×4+(-5)⇒p(x)=4x+3Thus,thedivisionalgorithmissatisfied.

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Exercise2.4

Q.1. Verifythatthenumbersgivenalongsideofthecubicpolynomialsbelowaretheirzeroes.Alsoverifytherelationshipbetweenthezeroesandthecoefficientsineachcase:

2x3+x2-5x+2;12,1,-2

Solution: Givenpolynomialis:

p(x)=2x3+x2-5x+2

Now,p(12)=2×(12)3+122-5(12)+2=2×18+14-52+2=0p(1)=2+1-5+2=0And,p(-2)=2(-2)3+(2)2+10+2=-16+4+12=0Therefore,12,1,-2arethezeroesof2x3-x2-5x+2

Wecancomparethegivenpolynomial2x3-x2-5x+2bystandardpolynomialax3+bx2+c+d,weget:

a=2,b=1,c=-5,d=2

Takeα=12,β=1,γ=-2α+β+γ=12+1-2=-12=-baαβ+βγ+γα=12×(1)+1×(-2)+(-2)×12=-52=caαβγ=12×1×(-2)=-22=-1=-daHence,therelationshipbetweenthezeroesandthecoefficientsisverified.

Q.2. Verifythatthenumbersgivenalongsideofthecubicpolynomialbelowisitszeroes.Also,verifytherelationshipbetweenthezeroesandthecoefficientsinthegivencase:

x3-4x2+5x-2;2,1,1

Solution: Givenpolynomialis:p(x)=x3-4x2+5x-2

Now,p(2)=(2)3-422+5(2)-2=8-16+10-2=0.

And,p(1)=1-4+5-2=6-6=0Therefore,2,1,1arethezeroesofx3-4x2+5x-2.

Uponcomparingthegivenpolynomialx3-4x2+5x-2withthestandardpolynomialax3+bx2+cx+d,weget:

a=1,b=-4,c=5,d=-2

Takeα=2,β=1,γ=1α+β+γ=2+1+1=4=-(-4)1=-baαβ+βγ+γα=2+1+2=5=51=caαβγ=2=-(-2)1=-daHence,therelationshipbetweenthezeroesandthecoefficientsisverified.

Q.3. Findacubicpolynomialwherethesum,sumoftheproductofitszeroestakentwoatatimeandtheproductofitszeroesare 2,-7,-14respectively.

Solution: Given:

α+β+γ=2

αβ+βγ+γα=-7αβγ=-14Weknowthat,x3-(α+β+γ)x2+(αβ+βγ+γα)x-(αβγ)=x3-2x2-7x+14istherequiredcubicpolynomial.

Q.4. Ifthezeroesofthepolynomialx3-3x2+x+1area-b,a,a+b,findaandb.

Solution: Givenpolynomialis:

x3-3x2+x+1

Zeroesarea-b,a,a+bComparingthegivenpolynomialwitha1x3+b1x2+c1x+d1,weget:a1=1,b1=-3,c1=1,d1=1Wehave,α+β+γ=-b1a1⇒a-b+a+a+b=-b1a1⇒a=1

αβ+βγ+γα=c1a1

⇒(a-b)a+a(a+b)+(a+b)(a-b)=1

Puttinga=1,weget:b=±2Hence,a=1,b=±2

Q.5. Iftwozeroesofthepolynomialx4-6x3-26x2+138x-35are2±3,findotherzeroes.

Solution: Let,p(x)=x4-6x3-26x2+138x-35.

2+3,2-3arethetwoofthezeroesofp(x).

Therefore,(x-2-3)and(x-2+3)arethefactorsofp(x).Thus,(x-2)2-(3)2isafactorofp(x)=x2-4x+4-3=x2-4x+1Therefore,x2-4x+1isafactorofp(x).

Dividingx4-6x3-26x2+138x-35byx2-4x+1,weget:

Fromthedivisionalgorithmwehave: Dividend=Divisor×Quotient+Remainder.

x4-6x3-26x2+138x-35=(x2-4x+1)×(x2-2x-35)+0

Thus,itisobservedthatx2-2x-35isalsoafactorofthegivenpolynomial.

Factorisingx2-2x-35bythemiddletermsplittingmethod,weget:x2-2x-35=x2-(7-5)x-35=x2-7x+5x-35=x(x-7)+5(x-7)=(x-7)(x+5)Therefore,x+5=0⇒x=-5And,x-7=0⇒x=7Hence,7and-5aretheotherzeroesofthegivenpolynomial.

Q.6. Ifthepolynomialx4-6x3+16x2-25x+10isdividedbyanotherpolynomialx2-2x+k,theremaindercomesouttobex+a,findkanda.

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Solution: Givenpolynomialisx4-6x3+16x2-25x+10.

Usingthedivisionalgorithm,weget

Dividend=Divisor×Quotient+Remainder∴Dividend−Remainder=Divisor×Quotientx4-6x3+16x2-25x+10-x-a=x4-6x3+16x2-26x+10-awillbeperfectlydivisiblebyx2-2x+k.Now,divide

x2-4x+(8-k)x2-2x+kx4-6x3+16x2-26x+10-a-x4-2x3+kx2-4x3+(16-k)x2-26x--4x3+8x2-4kx

Wecanseethat,-10+2kx+10-a-8k+k2willbe0.

Thus,-10+2k=0and10-a-8k+k2=0.

Now,for-10+2k=0⇒2k=10⇒k=5Andfor10-a-8k+k2=010-a-8×5+25=0⇒a=-5Therefore,k=5anda=-5.

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