F.4 Quiz 3- Problem Learning to Quadratic Equation, … St. Bonaventure College and High School Form...

MathematicsSt.BonaventureCollegeandHighSchool

Form4,Quiz3(ProblemLearningtoQuadraticEquation,NatureofRootsofaQuadraticEquation&RelationbetweenRootsandCoefficients)

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Name: Class: ()Date: Mark: /36

Time:54min,FullMarks:361. HKCEE2007Q5

2. HKCEE2000Q10



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3. Findaquadraticequationinxwhoserootsareone-thirdtherootsof 3𝑥 − 5 𝑥 + 6 = 0.Express

youransweringeneralform.(3marks)



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4. Considerthequadraticequation𝑘𝑥! − 2𝑘 + 5 𝑥 + 9 = 0.Ifthesumofrootsisequaltotheproductofroots,(a)findthevalueofk,(b)hence,solvetheequation.(6marks)



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5. If𝛼and𝛽aretherootsofthequadraticequation4𝑥! + 10𝑥 + 5 = 0,findthevaluesofthefollowingexpressions.(a)𝛼! + 𝛽!,(b)(𝛼 − 𝛽)!(5marks)



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6. Inthefigure,thegraphof 5)5( 2 −−−−= kxxky cutsthex-axisattwopointsA(α,0)andB(β,0).If

OA+OB=6units,findthevalueofk.(5marks)



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7. Itisgiventhatthequadraticequation

232 kxx =+ hastwodistinctrealroots.

(a) Findtherangeofpossiblevaluesofk.

(b) Hence,findtherootsoftheequationforthesmallestintegralvalueofkin(a).



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Solutions:1. HKCEE2007Q5

2. HKCEE2000Q10

3. Therootsof 3𝑥 − 5 𝑥 + 6 = 0are5/3and-6.

Therootsofanotherequationare(5/3)(1/3)=5/9and(-6)(1/3)=-2.…1ATheequationis(x-5/9)(x-(-2))=0…1Mi.e.(x-5/9)(x+2)=0𝑥! + !"

!𝑥 − !"

!= 0

9𝑥! + 13𝑥 − 10 = 0…1A4. (a)Fortheequation,𝑘𝑥! − 2𝑘 + 5 𝑥 + 9 = 0

Thesumofroots= !!!!!

Theproductofroots= !!…1A

Thesumofroots=Theproductofroots!!!!!

= !!…1M

𝑘 2𝑘 + 5 = 9𝑘2𝑘! + 5𝑘 = 9𝑘2𝑘! − 4𝑘 = 0𝑘 2𝑘 − 4 = 0𝑘 = 0 𝑜𝑟 2𝑘 − 4 = 0𝑘 = 0(𝑟𝑒𝑗. ) 𝑜𝑟 𝑘 = !

!= 2…1A

(b)Fork=0,theequationisnotaquadraticequation….1A



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1A

1A

1A

Fork=2,theequationis2𝑥! − 9𝑥 + 9 = 0,…1Mthesolutionare3and3/2.…1A

5. Fortheequationof4𝑥! + 10𝑥 + 5 = 0,thesumofroots=𝛼+𝛽=-10/4=-5/2,theproductofroots=𝛼𝛽=5/4.….1A(a)𝛼! + 𝛽! = 𝛼! + 2𝛼𝛽 + 𝛽! − 2𝛼𝛽…1M = 𝛼 + 𝛽 ! − 2𝛼𝛽

= − !!

!− 2 !

!= !"

!…1A

(b) 𝛼 − 𝛽 ! = 𝛼! − 2𝛼𝛽 + 𝛽! = 𝛼! + 𝛽! − 2𝛼𝛽 = 𝛼 + 𝛽 ! − 2𝛼𝛽 − 2𝛼𝛽….1M = 𝛼 + 𝛽 ! − 4𝛼𝛽

= − !!

!− 4 !

!= !

!…1A

6.

1M+1A



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1A

1A

1A

1A

1A

7.

1M+1A

F.4 Quiz 3- Problem Learning to Quadratic Equation, … St. Bonaventure College and High School Form...

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