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2016 YOUMUST BRI NGTHI S BOOKLETWI TH YOUTO ENROLMENT Mathematics Bridging Material 2016 Youmustcompletetheexercisesinthisbookletandbringyourwork withyoutoenrolment. IfyouhaveappliedtostudyMathematicsorUseofMathematicsthen pleasecompleteexercisesAtoM. IfyouhaveappliedtostudyFurtherMathematicsthenplease completetheentirebooklet. Name:_________________ 1 Contents 3. AguidetoA‐LevelMathematicsCoursesatOSFC 5. Indices 6. Indices‐ExerciseA 7. Fractionalindices–ExerciseB 8. FractionalIndices–ExerciseC 9. NegativeIndices–ExerciseD 10. Twospecialpowers–ExerciseE 11. Surds 12. Surds‐ExerciseF 13. Quadratics–ExerciseG 14. TheDifferenceofTwoSquares‐ExerciseH 14. FactorisingOtherQuadraticExpressions–ExerciseI 15. SolvingQuadraticEquations–ExerciseJ 16. SolvingQuadraticEquations‐ExerciseK 17.SimultaneousEquations‐ExL 18. CommonMisconceptions–ExerciseM 19. NoteforFurtherMathematicians 19. SolvingComplexEquations–ExerciseN 20. GraphsofQuadraticFunctions 21. GraphsofQuadraticFunctions‐ExerciseO 22. ChangingtheSubjectofaFormula‐ExerciseP 23. Answers 2 AGuidetoA‐LevelMathematicsCoursesat OldhamSixthFormCollege TheMathsDepartmentoffers3differentA‐Levelsdesignedtomatchtheabilities,interests andambitionsofstudents.Experienceshowsthatenjoymentofthesubjectandeventual examsuccessdependstoalargeextentonyourbeingenrolledontothecorrectoneof thesethreeA‐Levels. AllthreeofourA‐Levelsrequireyoutopurchaseagraphicalcalculator Casio fx‐9860GII . Thesecanbepurchasedthroughthecollegeatthestartoftheyear.Studentswhoqualifyfor thebursaryreceivea50%discountonthecostofthecalculator. A‐LevelMathematics You will continue to develop your existing knowledge of topics from GCSE including algebra, geometry, trigonometry and vectors. Additionally you will study an applied module of either Statistics or Mechanics (if you choose to study A-Level Physics). ASMathematicsisachallenginganddemandingsubject.Algebraicfluencyisextremely important,sincetheCoreunitsrelyheavilyonthisskill.AhighgradeatGCSEisnotalways anautomaticindicatorofalgebraicreadiness,andtobesuccessfulevenstudentsachieving anAorA*atGCSEwillneedtoworkespeciallyhardwiththeiralgebratosucceed. A‐LevelMathscombineswellwithmostothersubjects. A‐LevelUseofMaths YouwillstudyanAlgebramodule,aDecisionmoduleandeitheraStatisticsmoduleora Dynamicsmodule if you choose to do Physics.AlloftheappliedmodulesareFSMQs Free StandingMathsQualifications ,which,ifpassed,countasqualificationsintheirownright, gainingUCASpoints. Ineachmodulethereisastrongemphasisonrelatingthetheorycoveredtoreallife situations.Thismakesthecourseespeciallysuitableforthosewholiketoseethepractical relevanceofwhattheylearnorwhomightbestudyingthesubjecttosupporttheirotherA‐ Levelchoices. A‐LevelUseofMathscombinesparticularlywellwithSciences,Humanities,SocialSciences andArts. 3 A‐LevelFurtherMathematics InadditiontothetopicscoveredinA‐LevelMathematicsyouwillstudyadditionalmodules inalgebra,mechanicsandstatistics. FurtherMathematicsmustbestudiedalongsideA‐LevelMathematics.Itissuitableonlyfor thosestudentswithapassionandnaturalflairforMathematics.Typically,studentsonthe coursewillhaveachievedveryhighgradesinalloftheirmathematicalandscientificsubjects atGCSE. ItisimportanttorememberthatchoosingMathematicsandFurtherMathematicscountsas twosubjectsonyourtimetable.FurtherMathematicsisanessentialchoiceforthosewho want to study Mathematics at university and a good choice for those who want to study relatedcoursessuchasEngineeringorPhysics AnyqueriespleasecontactClaudiaBroad,HeadofMathematics,email:[email protected] 4 Indices Anindexisanothernameforpower. Theindexisthenumberoftimesa base numberismultipliedbyitself. indexorpower aⁿ base E.g53 5 5 5 125 Thefollowingrulesonlyapplywhenmultiplyingordividingpowersofthe samenumberorvariable samebase . Ruleformultiplyingnumbersinindexform addthepowers : 1. 34 35 3 4 5 39 2. 104 10‐2 10 4 ‐2 102 3. am an a m n Rulefordividingnumbersinindexform subtractthepowers : 1. 45 42 4 5‐2 43 2. 10‐2 10‐4 10 ‐2‐‐4 102 3. am an a m‐n Ruleforraisingapowertermtoafurtherpower multiplythepowers : 1. 62 4 62x4 68 2. 7‐2 4 7‐2x4 7‐8 3. am n amn 5 Indices ExerciseA Non‐calculator Writeasasinglepower: 1. 56 5‐3 2. 4 42 3. 64 6‐2 4. 6‐3 64 5. 42 3 6. 43 ‐2 7. 4‐2 ‐3 8. 47 0 9. 42 25 10. 22 3 25 Simplifythefollowingexpressions: 1. 6 2 2. 2 4 3. 4. 5.Canyousimplifythis? 6 FractionalIndices misthepowerandnistheroot Whenm 1 i.eIndicesoftheform 1 ‐thismeans‘thenthrootof’. n Examples √64 1. 64 √4 4 4 2. 9 √9 3. √ 4 3 ExerciseB Non‐calculator Evaluate: 1. 27 2. 81 3. 125 4. 100000 5. 6. 7 FractionalIndices Indicesoftheform Examples: :thenthrootof raisedtothepowerofm 1. 16 √16 4 64 2. 32 √32 2 16 3. √ √ ExerciseC Non‐calculator Evaluate 1. 1252/3 4. 27p3 2/3 2. 163/2 5. x4y10 3/2 3. 2434/5 Rewriteinindexform: 6. √ 7. P O W E R n2 n3 n4 n5 n6 n7 N 2 4 8 16 32 64 128 U M B E R n 3 4 5 6 7 8 9 10 11 9 16 25 36 49 64 81 100 121 27 64 125 216 343 512 729 1000 1331 81 256 625 1296 243 1024 729 8 NegativeIndices Considertheindexrulefordividing:52 52–3 5‐1 53 Numerically,52 53 25 125 ⅕ Therefore,5‐1 ⅕ i.eThenegativesignmeans‘oneover’ reciprocal 1. 4 2. 9 3. 4. 4 4 4 ExerciseD. Non‐calculator Writeeachoftheseinfractionform. 1 10‐5 2 8‐2 3 t‐1 4 4q‐4 5 9 Indices–TwoSpecialPowers Anumberraisedtothepower1staysthesamenumber. e.g51 5 Anumberraisedtothepower0isalwaysequalto1. e.g40 1 Because: 43 43 43‐3 40and 64 64 1 ExerciseE. Non‐calculator Simplifythefollowing 1. 81 2. 27 3. 4 4. 5. 6. 10 Surds Surdsarerootsofnumbers:i.e.√2, √5,√10 Rulesformultiplyingnumbersinsurdform. 1. √4 √9 √4 9 √36 6 2. 2√3 5√7 2 5√3 7 10√21 3. √ √ √ 4. √ √ √ Rulesfordividingnumbersinsurdform. 1. √4 √9 √ √ 2. 2√18 √ 5√2 √9 √ 3 1 3. √ √ 4. √ √ InMaths,wewouldprefertowritethesquarerootof300insurdformrather thanasadecimalbecauseitisEXACT. √300 √ 100 3 √100√3 10√3 Similarly√50 √ 25 2 √25√2 5√2 and√80 √ 16 5 √16√5 4√5 11 Surds ExerciseF Non‐calculator Simplifyeachexpression.Leaveyouranswerinsurdformwherenecessary donotusedecimals . 1 √2 √3 2 √6 √6 3 √54 √6 4 √12 √3 5 √5 √8 √8 6 4√2 5√3 7 4 2√2 3√18 8 √75 9 √700 10 √128 Canyousimplifythese? 11 2√5 3√6 √30 12 √ √ NBitisacceptabletowritethesquarerootof10 as√10 or√ 10 butnot √10 ;thelineaboveactsasabracket! 12 QuadraticExpressions QuadraticExpansion Expressionssuchas5p 2p‐3 and 3y 2 4y–5 canbeexpanded multipliedtogether togivequadraticexpressions. 5p 2p–3 10p2–15pand 3y 2 4y‐5 12y2‐7y‐10 Therearemanymethodsusedforexpandingsuchexpressionsas t 5 3t–4 buttheruleistomultiplyeverythinginonebracketby everythingintheotherbracket. Example:Expand x 3 x 4 x² 4x 3x 12 Nowsimplify: x² 7x 12 Youmustbecarefulwithnegativesigns. Rememberpositivexpositive positive positivexnegative negative negativexpositive negative negativexnegative positive Example:Expand x‐3 x 5 Nowsimplify: Example:Expand x‐7 x‐4 Nowsimplify: ExerciseG. Non‐calculator Expandthefollowingexpressions: 1. 2. 3. 4. 5. 6. 7. x² 5x‐3x‐15 x² 2x‐15 x²‐4x‐7x 28 x²‐11x 28 2x x‐1 ‐3x 5–2x x 3 x 2 x–3 x 4 3x–2 2x 5 2‐x 1‐3x 7x–1 7x 1 13 QuadraticExpressions‐Factorising Aquadraticexpressionwithonlytwoterms,bothofwhichareperfectsquares separatedbyaminussign,iscalledthedifferenceoftwosquares. Eg.x²‐9,x²‐25,x²‐100,etc. Recognisethepatternasx²minusasquarenumbern² Itsfactorsare x n x–n Factorisex²‐36.Recognise36as6²,thisthenbecomes x 6 x–6 Similarly,recognise9x²‐100as3xsquaredminustensquared whichwillfactoriseto 3x 10 3x–10 ExerciseH Non‐calculator Factorisethefollowingexpressions. 1.x²‐9 2k²–100 3.16x²‐9 4.16y²‐25x² Nowconsiderthisexample: Factorise3x² 8x 4 Bothsignsarepositive,sobothbracketsignsmustbepositive 3hasonly3 1asfactors,thebracketsmuststart 3x x Factorsof4are4 1and2 2 Wecouldtry 3x 4 x 1 or 3x 1 x 4 or 3x 2 x 2 Therearemanywaystotestwhichoftheseworksbutthesafestistoexpand themandsee!Thecorrectfactorisationis 3x 2 x 2 Note:Youshouldalwayscheckyoursolutionbyexpandingthebrackets. **Becarefulwithnegativesigns! ExerciseI Factorisethefollowingexpressions. 1 x2 6x 8 5 3x2 3x‐36 9 4x2‐25 2 z2 10z 16 6 5x2‐9x‐2 10 5x2 10x 3 y2 8y 7 7 3y2‐14y 8 11 ¼x2–9 4 2x² 5x 2 8 7x² 8x 1 12 3t²‐16t–12 14 SolvingQuadraticEquations Therearethreemainmethodsusedtosolvequadraticequations algebraically: 1. Byfactorising. 2. Usingthequadraticformula atASthisneedstobememorised 3. Bycompletingthesquare totaughtinAS 1.ByFactorising. Example: Solve2x2 5x 3firstlyrearrangetheequationtomakeitequalzero 2x2 5x‐3 0nowfactorisethelefthandside 2x‐1 x 3 0 2x‐1 0orx 3 0 eitherx ½orx ‐3 ExerciseJ Solvethefollowingquadraticequations: 1 x2‐6x 5 0 6 13x2 11–2x 2 x2‐4x 5 7 4x2‐4x 35 8 7x2 5x 4 2x2‐x 3 3 4x2 4x‐35 0 9 4x2 2x‐25 2x 4 4x2‐25 0 2 5 3x ‐8x‐3 0 10 2x2‐x 6 2.Usingthequadraticformula. Manyquadraticequationscannotbesolvedbyfactorisation.Onewayof solvingthistypeofequationistousethisformula: Thesolutionoftheequationax² bx c 0isgivenby b b 2 4ac x 2a 15 Example:Solvethefollowingequationusingthequadraticformula. Giveyour answerinsurdform. 2x² x–8 0 note:a 2,b 1,c ‐8 Substitutingintotheformulagives x 1 1 (4 2 8) 1 4 4 65 thisisinsurdform ExerciseK Solvethefollowingequations,givingyouranswersinsurdform. 1. x²‐x–10 0 2. 4x² 9x 3 0 3. 6x² 12x 5 0 4. 4x²‐9x 4 0 16 SimultaneousEquations Thereareseveralmethodsforsolvingsimultaneousequations.Wheretwo linearequationsareinvolved,eliminationisusuallythebestmethod. Example:Solvethefollowing 3x 4y 26 1 7x–y 9 2 Firstmakeeitherthecoefficientofthex’sorthey’sthesame.Inthiscaseitis easiesttomaketheycoefficientsthesame.Somultiplyequation 2 by4.So weget 28x–4y 36 3 Becausethesignsinfrontofthey’saredifferent,weaddtheequations ifthey werethesamewewouldsubtract . 3 1 31x 62 Nowsolveforxbydividingby31. x 2 Substituteintooneoftheoriginalequationstofindy. 7x2–y 9,soy 5 Checkbysubstitutingintotheotheroriginalequation. 3x2 4x5 26 correct ExerciseL:Solvethefollowing 1 5x 3y 1 2 3x y 19 3x–y 9 5x 2y 32 3 2x 3y ‐1 4 2x 3y ‐8 7x 4y 16 3x–4y 5 17 CommonMisconceptions Amisconceptionisafalseviewofhowthingsare.Inmathematics,somethings thatlooksensibleare,infact,completelywrong.Itisimportantthatyouare awareofcommonmistakessomestudentsmakeanddon’tmakethem yourself! Agoodwaytocheckistosubstitutevalues. Forexample,toshowthat:.1 .1isNOTequalto. 1. xy x y substitutex 1andy 1. Then 1 1 2 and 1 1 NOTthesame. 1 1 11 2 ExerciseM Arethefollowingtrueorfalse?Iffalsewritedownthecorrectanswer. 1. 3 3 3 2. 3. 5 5 5 4. 1 5. 1 1 6. 7. 8. 6 6 9. 2 18 Note: ThefollowingpagescoverworkthatyourASMathsteacherwillbe introducinginthefirsttwoweeksofthecourse.Doasmuchasyoucanfrom theseexercises–theremaybesomequestionsyouareunabletoanswer simplybecauseyouhavenotcoveredtheworkatschool. If you have applied to study Further Maths at OSFC then we strongly recommend you work through these exercises. As a Further Maths student you must be quicker to take on board new concepts, particularly in manipulating algebraic expressions and rearranging formulae. The ability to sketch graphs and recognise the graphs of the equations you have worked with at GCSE is also important. This may be daunting to many students but if you are likely to achieve a grade A* or a high grade A at GCSE Maths and you enjoy the challenge of a more demanding Maths then Further Maths might be for you ! HarderEquations ExerciseN: Usingallyouralgebraicskillstosolvethefollowingequations: 1. 4 4 5 2. 3. 4 64 4. 3 2 3 5 4 3 5. √ 9 5 6. 4 7. √4 50 8. 9. 10. 3 2 8 4 19 GraphsofQuadraticFunctions Aquadraticfunctionisusuallywrittenintheformax2 bx cwherea,band careconstants.Allquadraticsareparabolasandlooklike: a<0 Maximum point a>0 Minimum point NOTICE–thegraphsaresymmetricalandhaveoneturningpoint amaximumoraminimum Wecanfindthepointofintersectionwiththey‐axisbysubstitutingx 0 intotheequation. Fortheequationy x2 2x–8atx 0weobtainy ‐8. Wecanfindanypointsofintersectionwiththex‐axisbysubstituting y 0intotheequation. Fortheequationy x2 2x–8 ify 0weobtain whichcanbefactorised 0 x2 2x–8 0 x 4 x–2 andsolved togive x ‐4andx 2 thetwopointsonthex‐axis. Thepicturebelowisy x2 2x–8showingthexaxisinterceptsof‐4and2 aswellasbeingabletoseethattheyaxisinterceptis‐8. 20 GraphsofQuadraticFunctions ExerciseO Sketchthefollowingcurves,showingwheretheyintersectwitheachaxis. 1. y x2 5x 6 2. y x2 x–2 y y x x 3. y 25–x24.y ‐x2 5x‐6 y y x x 21 ChangingtheSubjectofaFormula Examples Make thesubjectofthefollowingformulae: 1. subtracting frombothsides dividingby squarerooting 2. 3. expandingthebrackets subtracting frombothsides dividingby 3 4 5 5 5 4 4 3Ensureall termsareonthesameside 3Factorise Divideby 5 ExerciseP Make thesubjectofthefollowingformulae: 1. 2. 3. 6. 7. 5. 9. 4 5 7 10. 22 4. 8. Answers 1. ExerciseA: 53 1. 2 43 2. 3. 66 3. ExerciseD: 1 64 1 4. 6‐7 4. 4 5. 46 5. 6. 7. 8. 9. 10. 4‐6 46 40 29 211 3 4 1. 2. 3. Simplify: 4. 5. 1. 3a 6. 2. 3. 4. 5. 1. 2. 3. 4. 5. 8a5b4 2a3 3a3b2 3a2c4 ExerciseB: 3 3 5 10 5 6 3 5 ExerciseC: 1. 2. 3. 4. 5. 6. 25 64 81 9p2 x6y15 Rewriteas: t2/37.m3/4 6. 1. 2. 3. 4. 5. 6. 7. 8. 9. ExerciseE: 3 1 1 8 y16 5 2 27 64 ExerciseF: √6 6 3 2 8√5 20√6 144 5√3 10√7 10. 8√2 11. 6 12. 23 1. 2. 3. 4. 5. 6. ExerciseG: 2x2–2x ‐15x 6x2 x2 5x 6 x2 x–12 6x2 11x–10 2‐7x 3x2 1. 2. 3. 4. 5. 6. 7. 49x2–1 7. 8. x 1. 2. 3. ExerciseH: x 3 x‐3 k 10 k‐10 4x 3 4x‐3 9. 10. x 2.5or x ‐2.5 x 2or x ‐1.5 4. ExerciseJ: x 5or x 1 x 5or x ‐1 x 2.5or x ‐3.5 x 2.5or x ‐2.5 x 3or x ‐1/3 x or x ‐1 x 3.5or x ‐2.5 or x ‐1 ExerciseK: 1. 1 4y 5x 4y–5x 2. 3. ExerciseI: 4. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 1. x 4 x 2 z 8 z 2 y 7 y 1 2x 1 x 2 3x–9 x 4 5x 1 x–2 3y–2 y–4 7x 1 x 1 2x 5 2x–5 5x x 2 ½x–3 ½x 3 3t 2 t–6 ExerciseM: True6.False 2. 3. False7.True False8.False 2. 3. 4. True9.False 4. 24 √41 2 9 √33 8 12 √24 12 9 √17 8 1. 2. 3. 4. ExerciseL: x 2,y ‐3 x 6,y 1 x 4,y ‐3 x ‐1,y ‐2 1. ExerciseN: x ‐2andx ‐6 6.x x 4.5 7.x 5 x 3 8.x 5 2√10 x ‐8 9.x 6 5. False 5. x 34 10.x Exercise O : 1. 2. 3. 4. Exercise P: 1. 2. 5. 6. 9. 10. 3. 7. 25 4. 8. √