Download Maths (MATH,FURTHER,USE) - Oldham Sixth Form College

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
11
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.
√