Upload
ngokien
View
216
Download
2
Embed Size (px)
Citation preview
MathematicsSt.BonaventureCollegeandHighSchool
Form4,Quiz3(ProblemLearningtoQuadraticEquation,NatureofRootsofaQuadraticEquation&RelationbetweenRootsandCoefficients)
1
Name: Class: ()Date: Mark: /36
Time:54min,FullMarks:361. HKCEE2007Q5
2. HKCEE2000Q10
MathematicsSt.BonaventureCollegeandHighSchool
Form4,Quiz3(ProblemLearningtoQuadraticEquation,NatureofRootsofaQuadraticEquation&RelationbetweenRootsandCoefficients)
2
3. Findaquadraticequationinxwhoserootsareone-thirdtherootsof 3𝑥 − 5 𝑥 + 6 = 0.Express
youransweringeneralform.(3marks)
MathematicsSt.BonaventureCollegeandHighSchool
Form4,Quiz3(ProblemLearningtoQuadraticEquation,NatureofRootsofaQuadraticEquation&RelationbetweenRootsandCoefficients)
3
4. Considerthequadraticequation𝑘𝑥! − 2𝑘 + 5 𝑥 + 9 = 0.Ifthesumofrootsisequaltotheproductofroots,(a)findthevalueofk,(b)hence,solvetheequation.(6marks)
MathematicsSt.BonaventureCollegeandHighSchool
Form4,Quiz3(ProblemLearningtoQuadraticEquation,NatureofRootsofaQuadraticEquation&RelationbetweenRootsandCoefficients)
4
5. If𝛼and𝛽aretherootsofthequadraticequation4𝑥! + 10𝑥 + 5 = 0,findthevaluesofthefollowingexpressions.(a)𝛼! + 𝛽!,(b)(𝛼 − 𝛽)!(5marks)
MathematicsSt.BonaventureCollegeandHighSchool
Form4,Quiz3(ProblemLearningtoQuadraticEquation,NatureofRootsofaQuadraticEquation&RelationbetweenRootsandCoefficients)
5
6. Inthefigure,thegraphof 5)5( 2 −−−−= kxxky cutsthex-axisattwopointsA(α,0)andB(β,0).If
OA+OB=6units,findthevalueofk.(5marks)
MathematicsSt.BonaventureCollegeandHighSchool
Form4,Quiz3(ProblemLearningtoQuadraticEquation,NatureofRootsofaQuadraticEquation&RelationbetweenRootsandCoefficients)
6
7. Itisgiventhatthequadraticequation
232 kxx =+ hastwodistinctrealroots.
(a) Findtherangeofpossiblevaluesofk.
(b) Hence,findtherootsoftheequationforthesmallestintegralvalueofkin(a).
MathematicsSt.BonaventureCollegeandHighSchool
Form4,Quiz3(ProblemLearningtoQuadraticEquation,NatureofRootsofaQuadraticEquation&RelationbetweenRootsandCoefficients)
7
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
MathematicsSt.BonaventureCollegeandHighSchool
Form4,Quiz3(ProblemLearningtoQuadraticEquation,NatureofRootsofaQuadraticEquation&RelationbetweenRootsandCoefficients)
8
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