NSWDepartmentofEducationMathematicsStage5DiagnosticTaskAnswers1
MathematicsStage5DiagnosticTasksAnswerswithCommonErrorsAnswers5.1 CommonErrors5.1FinancialMathematics QUESTION1 a) $540b) $13.05(2decimalplaces)c) $847.60d)
i) $2942.31(2decimalplaces)ii) $6375
e) i) $908.65(2decimalplaces)
(using52weeksinayear)ii) $6100.96(2decimalplaces)
d) Incorrectidentifyingthefactortodivideby(i.e.,12or26or52)
(ii)Studentsmaymultiplyanswer(i)by2thinkingthatthereis2fortnightsinamonthratherthangoingbacktotheoriginalsalary.
e) Notdistinguishingthedifferencebetweenholidayloadingandholidaypay.
QUESTION2 a) $1250b) $61
Findingthevaluetocalculatecommissionon(i.e.Determiningcommissioncalculatedon$30000).
QUESTION3 a) $25466.33b)
i) $6672.50ii) Refund.$1077.50iii) Taxableincome=$39867.50
Medicarelevy=$698.10(2decimalplaces)
a) Addingthedeductionsinsteadofsubtractingb) ii)Notspecifyingtherefundamount.iii)mayworkoutthelevyusingthetaxamountinquestionspart(ii)
QUESTION4 a) $1560b) $3805.50c) $65.63(2decimalplaces)d)
i) $3900ii) $998.40iii) $4898.40iv) $6398.40
b)Workingouttheinterestandforgettingtoaddontheprincipalc)Incorrectlyconvertingrateortimeperiod.d)i) Notexcludingthedeposit.ii) UsingthewrongPrincipaliii) Notaddingtheprincipalamountiv) Notaddingtheinitialdeposittothetotal
pricepaid
NSWDepartmentofEducationMathematicsStage5DiagnosticTaskAnswers2
5.1FinancialMathematics CommonErrorsQUESTION5 a)
i) $3675.13(2decimalplaces)ii) $675.13
b) $6486.25(2decimalplaces)
c)
i) $5728.86ii) $5373.45InvestmentA I=5000×0.03×6 =$900 A=$5900InvestmentBA=5000×1.1941 =$5970.50Therefore,InvestmentBisthebetterinvestmentby$70.50
a) andb)incorrectlydistinguishingbetweeninterestearnedandfinalvalue.
c)Notrecognisingthetableisonlyfor$1andyouneedtomultiplybytheprincipalamount.
NSWDepartmentofEducationMathematicsStage5DiagnosticTaskAnswers3
5.1Indices CommonErrorsQUESTION1 a)
i) a4ii) 25
b) i) 25ii) m9iii) 6x6iv) 5p4v) 32x3y5
a) Addingratherthanmultiplyingespeciallywith
(i)andgetting4aasananswerb) Multiplyingtheindicesinsteadofadding
iii) Multiplyingthebasenumberstoget4"thisisonlyacommonerrormadewhenthebasesandnumerical.
QUESTION2 a) y3b) 2xc) -6kd) 4xy2
Dividetheindicesratherthansubtract
QUESTION3 a) 56b) 23q9=8q9
c) #$%
&= ()$*
%*
a) Squaringthe5toget25-thisisonlyacommonerrormadewhenthebasesandnumerical.
b) andc)notraisingeverythinginsidethebrackettothepoweri.e.getting2𝑞/and#$
*
%*or#$
*
%
QUESTION4 a) 1b) 2c) 1
b)Thinkingthattheentireexpressionistothepowerofzeroandgivingtheanswer1
QUESTION5 a)
i) 0&*
ii) 0
"1
b) i) 2-4ii) -4-3
c) i)
0)
ii) − 0"1= − 0
0&"
iii) − 00-+ 0
(= /
0-
NSWDepartmentofEducationMathematicsStage5DiagnosticTaskAnswers4
5.1LinearRelationships CommonErrorsQUESTION1 a) positiveb) (1.5,1)c) (
"
d) 41 ≈ 6.4(1decimalplace)
QUESTION2 a)
x 2 2 2 2 2 2y -2 -1 0 1 2 3
b)
x -2 -1 0 1 2 3y -2 -2 -2 -2 -2 -2
Gettingtheorderofthecoordinatesmixedup,doingthey-ordinateacrossandthex-ordinateup/downNotextendingthelinebeyondtheplottedpoints.Notusingarrowsonbothendsoftheline.
NSWDepartmentofEducationMathematicsStage5DiagnosticTaskAnswers5
5.1LinearRelationships CommonErrorsQUESTION3 a) x-intercept:
𝑦 = 0𝑥 + 3 = 0𝑥 = −3
y-intercept:𝑥 = 0𝑦 = 0 + 3𝑦 = 3
b)
a)Incorrectlysolvingtheequationswhenx=0andy=0
b)Notextendingthelinesufficientlytoshowthex-andy-intercepts.
Notusingarrowsonbothendsoftheline.
QUESTION4 3 = 3 2 − 23 = 6 − 23 ≠ 4No,(2,3)doesnotlieontheliney=3x–2
Substitutingthexandyvaluesintothewrongvariable.
QUESTION5 y=3xandy=3x+1
Includingy=-3xinthesolutionasitalsohasa3
NSWDepartmentofEducationMathematicsStage5DiagnosticTaskAnswers6
5.1Non-LinearRelationships CommonErrorsQUESTION1 i)
x -2 -1 0 1 2 3
y 4 1 0 1 4 9
ii)
x -2 -1 0 1 2 3
y14
12
1 2 4 8
i) Caremustbetakenbystudentswhenputting(-2)2and(-1)2intothecalculatorasiftheydon’tusethebracketsitwillcomeoutwithanegativenumber
i)andii)pointsmustbejoinedusingasmoothcurvenotastraightline
NSWDepartmentofEducationMathematicsStage5DiagnosticTaskAnswers7
5.1Non-LinearRelationships CommonErrorsQUESTION2 i) Aii) Ciii) B
QUESTION3 i) y=x2+1
x -2 -1 0 1 2 3
y 5 2 1 2 5 10
y=x2−1
x -2 -1 0 1 2 3
y 3 0 -1 0 3 8
y=x2+1crossestheyaxisaty=1y=x2–1crossestheyaxisaty=-1
ii) Thevalueoftheconstantisequaltowhere
thegraphcrossestheyaxis.
NSWDepartmentofEducationMathematicsStage5DiagnosticTaskAnswers8
5.1AreaandSurfaceArea CommonErrorsQUESTION1
a) 𝐴 = 𝑠& + 0&𝜋𝑟&
= 8×8 +12×𝜋×4
&= 89.13𝑐𝑚&(correctto2decimalplaces)
b) 𝐴 = G&𝑎 + 𝑏
=6.32 10 + 18.4
= 89.46𝑐𝑚&c) 𝐴 = 𝑏ℎ= 12×8= 96𝑐𝑚&
d) 𝐴 = 0/K/-K
𝜋𝑟&
=130360×𝜋×10
&= 133.45𝑐𝑚&(correctto2decimalplaces)
e) 𝐴 = 𝜋𝑅& − 𝜋𝑟&= 𝜋×9& − 𝜋×4&= 204.20𝑐𝑚&(correctto2decimalplaces)
f) 𝐴 = 0&𝑏ℎ − 𝑙𝑤
=12×25×14 − 4×3
= 163𝑐𝑚&
Acrossallquestionsinvolvingareaandsurfacearea
Ø MixingupareaandperimeterØ Mixingupsurfaceareaandvolume
a) Confusingtheuseofthecircumference
formulaandtheareaofacircleformulaorevenacombinationofboth(2𝜋𝑟&isacommonerrorfortheareaformula)orforgettingtohalvethecircle.
e) using18cmastheradiusofthelargercircle
insteadof9cm.
QUESTION2 a) 𝑇𝑜𝑝 = 22×6.5= 143𝑐𝑚&𝐹𝑟𝑜𝑛𝑡 = 22×10= 220𝑐𝑚&𝑆𝑖𝑑𝑒 = 6.5×10= 65𝑐𝑚&𝑆𝑢𝑟𝑓𝑎𝑐𝑒𝐴𝑟𝑒𝑎 = 2× 143 + 220 + 65 = 856𝑐𝑚&
b) Findingtheperpendicularheightofthetriangle:ℎ& = 13& − 5&ℎ = 12𝑐𝑚
𝐹𝑟𝑜𝑛𝑡 =12×10×12
= 60𝑐𝑚&𝑆𝑙𝑜𝑝𝑖𝑛𝑔𝑠𝑖𝑑𝑒 = 13×18= 234𝑐𝑚&𝐵𝑎𝑠𝑒 = 10×18= 180𝑐𝑚&𝑆𝑢𝑟𝑓𝑎𝑐𝑒𝐴𝑟𝑒𝑎 = 2×60 + 2×234 + 180= 768𝑐𝑚&
b)using13cmastheheightofthetriangle
NSWDepartmentofEducationMathematicsStage5DiagnosticTaskAnswers9
5.1AreaandSurfaceArea CommonErrorsQUESTION3 a) 5b) 𝐹𝑟𝑜𝑛𝑡 = 36×42= 1512𝑐𝑚&𝑆𝑖𝑑𝑒 = 25×42= 1050𝑐𝑚&𝐵𝑎𝑠𝑒 = 36×25= 900𝑐𝑚&𝑆𝑢𝑟𝑓𝑎𝑐𝑒𝐴𝑟𝑒𝑎 = 2×1512 + 2×1050 + 900= 6024𝑐𝑚&
Notrecognisingthattheboxisopenandhasnottopandincludingitinthecalculation.
NSWDepartmentofEducationMathematicsStage5DiagnosticTaskAnswers10
5.1NumbersofAnyMagnitude CommonErrorsQUESTION1 a) 2430b) 0.238c) 1.29
a) Getting243,becausetheydon’tputthezerointoholdplacevalue.
b) Justcuttingthedecimaloffat3decimalplacesandnotroundingupi.e.getting0.237.
c) Roundingto3decimalplacesinsteadof3sigfigsi.e.1.288/1.287
QUESTION2 a)
i) 0.78cmii) 2.34miii) 1.86×10-6miv) 60μm
b) i) 7850Lii) 78.26mL
c) i) 7MB=7×1024KB=7168KBii) 5200GB=5200÷1024=5.078TB(correctto3decimalplaces)iii) 2.1TB=2.1×1024×1024×1024=2254858000KB(tothenearestthousand)
d) i) 7.8×60×1000=468000msii) 170000μs=170000÷1000000s
=0.17÷60min=0.0028min(correctto2significantfigures)
iii) 1fortnight=1×14days=14×24hours=336×60minutes=20160×60seconds=1209600s
iv) 897÷10=89.7decades
a),b)andd)mixingupthemultiplicationanddivisionbypowersof10i.e.multiplyingwhenitshouldbedivisionandviceversa.Orusingthewrongpowerof10c)Note:210=1024usingmultiplesof10forconvertingratherthanthepowersof2requiredfordigitalconversions.
Prefix Decimal/Analogvalue
Binary/Digitalvalue
k(kilo) 103
(1000)210
(1024)
M(mega) 106
(1000000)220
(1048576)G(Giga) 10
9
(1000000000)
230
(1073741824)
T(Tera) 1012
(1000000000000)
240
(1099511627776)
QUESTION3 a)
i) 2800=2.8×103ii) 1240000=1.24×106iii) 0.03456=3.456×10-2iv) 0.0000541=5.41×10-5
b) i) 2.6×102=260ii) 4.56×10-4=0.000456
NSWDepartmentofEducationMathematicsStage5DiagnosticTaskAnswers11
5.1NumbersofAnyMagnitude CommonErrorsQUESTION4 a) 6.3x108,1.2x109,3.5x109b) 4.8x10-3,4.1x10-2,3.8x10-1
Comparingonlythedecimalpartofthenumberandnottakingthepowerof10intoconsideration
QUESTION5
a) 𝑡 = %^= 0."&×0K_
/×0K`= 507𝑠(correcttothe
nearestwhole)b) 28000×365=10220000words
=1.022×107words
QUESTION6 a) ±0.5kgb) 15.5kgto16.5kg
QUESTION7 a) 25+15=40b) 140+20=160(Answersmayvaryslightly)
NSWDepartmentofEducationMathematicsStage5DiagnosticTaskAnswers12
5.1Trigonometry–RightAngledTriangles CommonErrorsQUESTION1 a)
i) ABii) BCiii) AC
b)
i) ("
ii) /"
iii) (/
c) i) 0.29ii) 0.21iii) 1.17
d) i) 82°ii) 57°iii) 37°
AcrossallquestionsinvolvingtrigonometryØ UsingthewrongratiosØ Gettingconfusedwhentheunknownisinthe
denominatorforexampleif:tan 30° = 0&e
rearrangingincorrectlytoget𝑦 = 12𝑡𝑎𝑛30°
Ø Notusingtheinversefunctionwhencalculatinganunknownangle(i.e.onthecalculatorstudentsneedtopressSHIFTbeforethetrigratio)
Ø Mixinguptheoppositeandadjacentsidesiscommon
c)incorrectrounding
QUESTION2 a) i) tan 52 = e
f
𝑦 = 8× tan 52𝑦 = 10.2cm(roundedto1decimalplace)
ii) cos 46 = #."j
𝑥 = 7.5 ÷ cos 46𝑦 = 10.8cm(roundedto1decimalplace)
b) i) tan 𝜃 = (
/
𝜃 = 53°ii) cos 𝜃 = -
0K
𝜃 = 53°c) tan 54 = G
"&K
ℎ = 520× tan 54ℎ = 716mtheheightofthecliffis716m.
NSWDepartmentofEducationMathematicsStage5DiagnosticTaskAnswers13
5.1PropertiesofGeometricFigures CommonErrorsQUESTION1 Twoshapesaresimilariftheirshapeisthesamebuttheirsizeisdifferent.
QUESTION2 AandCMatchinganglesareequal
QUESTION3 a) AC-PR,CD-RS,DB-SQ,BA-QPb) Ð A-Ð P,Ð C-Ð R,ÐD-Ð S,Ð B-ÐQ
QUESTION4 a)
i) scalefactor = imageoriginal
= 0K(= 2.5
ii) scalefactor = imageoriginal
= 00((= 0.25
b)
i) scalefactor = imageoriginal
= /-= 0.5
ii) 5× 0&= 2.5cm
Notrecognisingthattheobjectisonthelefthandsideandtheimageisontherighthandsidei.e.gettingthescalefactorbackwards.Oralwaysthinkingthatascalefactorhastobelargerthan1andhavingSF=4for(ii)
QUESTION5 i) scale= /(
f= 4.25
ii) 6.7×scale=6.7×4.25=28.5cm(correctto1decimalplace)
QUESTION6a)
NSWDepartmentofEducationMathematicsStage5DiagnosticTaskAnswers14
5.1PropertiesofGeometricFigures CommonErrorsb)
NSWDepartmentofEducationMathematicsStage5DiagnosticTaskAnswers15
5.1SingleVariableDataAnalysis CommonErrorsQUESTION1 Positivelyskewed,negativelyskewed,bi-modal,symmetric
QUESTION2 i)
GIRLS LEAF BOYS
9 0 56 1 2
542 2 56763 3 345
0 4 ii) boys
iii) Girls:mean=28.25,median=29,range=31
Boys:mean=22,median=25.5,range=30
Thegirlsachievedthebetterresultswithahighermedianandmeanalthoughtheirrangeislargeritisonlybyonemarkandisnegligiblewhenothermeasuresaretakenintoconsideration.
iv) Negativelyskewed
Somestudentsmaynotorderthestem-and-leafplotwhichwillaffecttheanswerstothequestionsthatfollow.
QUESTION3 a)
i) Datesourceandsizeofsamplenotmentioned.
ii) Fromacensus,thisdataismeaningful.
NSWDepartmentofEducationMathematicsStage5DiagnosticTaskAnswers16
5.1Probability CommonErrorsQUESTION1 i) (
"K
ii) &K"K+ "
"K= &"
"K= 0
&
Mixinguprelativefrequencywithfrequencyi.e.answering4for(i)
QUESTION2 54150×500 = 180
QUESTION3 a)
i) 89
ii) 5
iii) 0ff)
iv) --f)
b)
i) 54
ii) )Kf(K
iii) (/Kf(K
a) Notseeingthe5outsideoftheellipses
NSWDepartmentofEducationMathematicsStage5DiagnosticTaskAnswers17
5.2Answers CommonErrors5.2FinancialMathematics
QUESTION1a)
iii) = $5624.32
iv) A=624.32
b)TypeA:𝐴 = 10000 1 + 0.05 "= 12762.815625≈ $12762.82
TypeB:
𝐴 = 10000 1 +0.0512
-K
= 12833.586785…≈ $12833.59
TypeBisthebetterinvestment.c)≈ 18.6%𝑑)𝐴 = 7500 1 + 0.07 qTryn=5,𝐴 = 7500 1.07 " = $10519.14Tryn=4,𝐴 = 7500 1.07 ( = $9830.97Therefore,itwilltake5years.
a) v) Mixingupsimpleandcompoundinterestand
usingtheincorrectformulavi) Notsubtractingtheprincipal
b) Incorrectlyconvertingrateortimeperiodc) Notconvertingtheratebacktoapercentaged) Tryingtosolvewithoutusingtrialanderror
QUESTION2a) $1412.49b)
i) $11339.90ii) 23000 − 11339.90= $11660.10Thecardepreciatedinvalueby$11660.10
NSWDepartmentofEducationMathematicsStage5DiagnosticTaskAnswers18
5.2RatesandRatio CommonErrorsQUESTION1
a) $"."K0h
= $"."K-Kmin
= $0.09/min
b) &"KL
0h= &"KKKKmL
0h
= 250000mL/hc) -"km
0h= -"KKKm
-Kmin
=65000m3600s
= 18.1m/s
Usingincorrectconversionfactor,especiallywhenitcomestotime
QUESTION2
a) speed= &"Kkm"h
= 50km/h
b) rateofleak= &."L
-h
= 0.417L/h= 417mL/h
c) fuelconsumption= ("L
/KKkm
=15L
100km
= 15L/100km
c)calculatingperkmratherthanper100km
QUESTION3a) directb) inversec) direct
QUESTION4 a) 70°Fb) 35°Cc) positived) Yes,asonevariableincreasessodoesthe
other.
a) andb)readingfromthewrongaxis
QUESTION5 a) 𝐹 = 𝑘×𝑑b) 0
f𝑜𝑟(0.125)
c) 320km
Studentswillneedtoeitherusethetwopairsofinformationfrompartb)andc)orrecognisethat𝐹 = 0
f𝑑astheequationoftheline.
NSWDepartmentofEducationMathematicsStage5DiagnosticTaskAnswers19
5.2RatesandRatio CommonErrorsd)Sue’sfuelconsumption
NSWDepartmentofEducationMathematicsStage5DiagnosticTaskAnswers20
5.2AlgebraicTechniques CommonErrorsQUESTION1a)
i) "j-
ii) &e
0"
iii) u
*
(
iv) 3𝑥
v) 00
&q
b) i) 2𝑦& + 12ii) 40𝑎& − 32𝑎 + 4iii) 𝑥& + 6𝑥 + 8iv) 2𝑘& − 𝑘 − 15
i) andii)NotcreatingacommondenominatorAdd/subtractingdenominatorsiii)andiv)notsimplifyingfullyiv)dividingthenumeratorsanddenominators
QUESTION2
i) 4𝑑/ − 4𝑑 = 4𝑑 𝑑& − 1 ii) 21𝑥𝑦 − 3𝑥 + 9𝑥& = 3𝑥 7𝑦 − 1 + 3𝑥 iii) ℎ& + 8ℎ + 12 = ℎ + 2 ℎ + 6
NSWDepartmentofEducationMathematicsStage5DiagnosticTaskAnswers21
5.2Indices CommonErrorsQUESTION1a)
i) 0j*= 𝑥v&
ii) &w1 = 2𝑚v/
iii) 0(u*
= 0(𝑎v&
=𝑎v&
4 b)
i) 𝑥v/ = 0j1
ii) 4𝑚v" = (w`
c) 1v/ = 0
01= 1
−3×1 = −3𝑥v/isnotequivalentto−3𝑥
a) iii)raisingthe4tothenegativepowertoget4𝑎v&𝑜𝑟(4𝑎)v&
b) ii)puttingthe4inthedenominator,not
recognisingthatonlythemistoanegativepower,i.e.getting 0
(w`
QUESTION2a) i) 6𝑚v0Kii) 𝑥v0Kiii) 3𝑦/iv) 3𝑘0Kv) 64𝑚v-
b) i) 4𝑥K = 4,∴false
ii) 𝑎- ÷ 𝑎f = 𝑎v&,∴false
iii) 6𝑚 K = 1,∴true
iv) 3𝑥- ÷ 3𝑥- = 1,∴false
v) 4𝑐v/ = ($1,∴false
vi) 12𝑚0K×2𝑚v& = 24𝑚f,∴true
a) i)andiii)multiplyingtheindicesinsteadof
addingiv)dividingtheindicesinsteadofsubtractingv)notraisingeverythinginsidethebracketstothepowerof3i.e.getting4𝑚v-b) similarissuestoparta)
QUESTION3a) −3𝑚& 5𝑚& + 2𝑚(𝑛 = −15𝑚( − 6𝑚-𝑛
b) 0-j*
(j= 4𝑥
c) &Ku
1y*
0"uy1= (u*
/y
d) /je
&× &j
-= j*e
&
b)notsimplifyingfullye.g.getting0-j
(or(j
*
j
c)–f)notsimplifyingfully
NSWDepartmentofEducationMathematicsStage5DiagnosticTaskAnswers22
5.2Indices CommonErrorse)
𝑏&𝑐/
4 ×12𝑏𝑐& = 3𝑏𝑐
f)(q1
/w* ÷0&q/w`
=4𝑛/
3𝑚& ×3𝑚"
12𝑛
=𝑚/𝑛&
3
f)dividingthenumeratorsandthedenominatorstogete.g./q
*
w1 𝑜𝑟/q*
wz1
NSWDepartmentofEducationMathematicsStage5DiagnosticTaskAnswers23
5.2Equations CommonErrorsQUESTION1a) 𝑦 = f
"
b) 𝑦 = −12c) 𝑘 = 0"
0(
AcrossallquestionsinvolvingequationØ ErrorsinalgebraØ Notperformingtheinverseoperationto
bothsidesØ Tryingtosolveequationsbeforecollection
liketermsafterexpansionsØ Whendividingbothsidesbythesame
number,subtractingratherthandividinge.g.5𝑦 = 8"e
"= f
"
𝑦 = 3Ø Whenfacedwithadivisionquestione.g.
j"= 20,studentsmaysolveas
𝑥5 = 20𝑥 = 20 ÷ 5𝑥 = 4
QUESTION2a) 𝑦 = 00
/
b) 𝑦 = 10c) 𝑥 = 0K
/
d) 𝑚 = 3e) 𝑦 = f
/
c)multiplyingby5firstwillstillgivethesameanswer,butmakestheequationmoredifficulttosolve.
QUESTION3a)i) 𝑦 = ±2ii) 𝑚 = ±4iii) 𝑥 = ± 8
𝑥 = ±2.6b)i)𝑘 = ± 20
ii)𝑘 = ± &/
a)–b)notincludingthe±intheanswerb)squarerootingbeforedividingby6
NSWDepartmentofEducationMathematicsStage5DiagnosticTaskAnswers24
5.2Equations CommonErrorsQUESTION4i) 𝑥 = −1ii) 𝑦 = 3𝑜𝑟𝑦 = 2iii) 𝑧 = 10𝑜𝑟𝑧 = −1
QUESTION5a) i) B = 7.5ii) L = 16
b) 𝐶 = 27℃
QUESTION6 a) 13,14and15b) 9cm
a)studentsmayonlysolvetheequationandnotgivetheanswertothequestion
QUESTION7a) 𝑥 > 5
b)𝑏 > 15
c)𝑏 ≥ 8
d)𝑦 < −1e)𝑚 ≥ − (
"
b)Leavingansweras15 < 𝑏orincorrectlyinterpretingthisas𝑏 < 15.e)notreversingtheinequalitysignwiththedivisionof-5
NSWDepartmentofEducationMathematicsStage5DiagnosticTaskAnswers25
5.2Equations CommonErrorsQUESTION8a)
(212 , 4)
b) x=3andy=7c) x=4andy=1
a)notaccuratelygraphingandthereforthepointofintersectioncannotbeaccuratelyreadoffthegraphb)andc)errorsinsolvingequations
NSWDepartmentofEducationMathematicsStage5DiagnosticTaskAnswers26
5.2LinearRelationships CommonErrorsQUESTION1a) Gradient Y-intercept
i) 3 -4
ii) -4 -5
iii) 12
1
iv) 1 32
b) 𝑦 = −2𝑥 + 6c) m=-4andy-int=2d)
i) 2ii) -1iii) 𝑦 = −𝑥 + 2
d)i)2ii)-1iii)𝑦 = −𝑥 + 2
ii)Maymixupanswersasthetermsintheequationarewritteninadifferentorder.iii)andiv)studentsmaystruggleastheequationsarenotwritteniny=mx+bform.Errorsmaybemadeintherearrangingoftheequation.d)ii)notrecognisingthegradientasnegative
QUESTION2
a) 𝐲 = −𝟐𝐱 + 𝟐𝐚𝐧𝐝𝐲 = 𝟏𝟐𝐱 + 𝟒
b)i)𝑦 = −4𝑥 − 2ii)𝑦 = − /
&𝑥 − 2
c)𝑦 = − /&𝑥 + 3
𝑦 = &/𝑥+2
𝑚0×𝑚& = −32×
23
= −1∴line1andline2areperpendicular
NSWDepartmentofEducationMathematicsStage5DiagnosticTaskAnswers27
5.2Non-LinearRelationships CommonErrorsQUESTION1a)i)ii)
iii)Theco-efficientofthex2affectstheconcavity
andthewidthoftheparabola.Theconstanttermshiftstheparabolaupanddowntheyaxis(i.e.changingthey-intercept)
b) 𝑦 = 𝑎𝑥& − 1(anyvalueof‘a’isacceptablee.g.𝑦 = 2𝑥& − 1)
c) 𝑥 = ± 6(𝑥 = ±2.4)d) 𝑦 = −𝑥& + 1
Acrossallquestionsinvolvinggraphingnon-linearrelationships
Ø NosmoothcurveØ Coefficientinfrontofthebasicfunction
widensthecurveasitsvalueincreasesØ Mixingupthedirectionthattheconstant
shiftsthecurve
c)incorrectsolvingofequationsandforgettingthe±
NSWDepartmentofEducationMathematicsStage5DiagnosticTaskAnswers28
5.2Non-LinearRelationships CommonErrorsQUESTION2a)
𝑦 = −2jisareflectionof𝑦 = 2jinthex-axis.𝑦 =2vjIsareflectionof𝑦 = 2jinthey-axisb)
Theconstantshiftstheexponentialfunctionup/downtheyaxiswhichalsochangeswheretheasymptoteis.
Acrossallquestionsinvolvinggraphingnon-linearrelationships
Ø NosmoothcurveØ Coefficientinfrontofthebasicfunction
widensthecurveasitsvalueincreasesØ Mixingupthedirectionthattheconstant
shiftsthecurvea)incorrectreflection,mixingup𝑦 = −2jand𝑦 =2vjb)noshiftingtheasymptotewiththechangeinthey-intercept
QUESTION3a) Centre(0,0)
Radius3unitsb) 𝑥& + 𝑦& = 4
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5.2Non-LinearRelationships CommonErrorsQUESTION4a) i)Bii)Ciii)Eiv)Av)Dvi)F
NSWDepartmentofEducationMathematicsStage5DiagnosticTaskAnswers30
5.2AreaandSurfaceArea CommonErrorsQUESTION1a) 915.4cm2b) 932.6cm2c) 143.1cm2
Acrossallquestionsinvolvingareaandsurfacearea
Ø MixingupareaandperimeterØ Mixingupsurfaceareaandvolume
a)usingtheincorrectformulafortheareaofacircle(𝐴 = 2𝜋𝑟&isacommonerrorstudentsusefortheareaofacircle)b)nothalvingtheareasforthecylinderorforgettingtoaddtherectangleonthebottom
QUESTION2a)
i) 691.2cm2ii) 789.2cm2
b) S/A=1078.8m2Cost=$48546c)
i) 75.4m2ii) 96m
a)ii)notrecognisingthatitisatintoputstationaryinandthereforhasnotop(i.e.onlyonecircularface)b)difficultywiththetrapeziumface,orincludingthetopinthecalculation(i.e.tilingthetopofthepool)c)i)calculatingthesurfaceareaoftheentiregreenhouseratherthanjusttheroof
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5.2Volume CommonErrorsQUESTION1a) 1680m3b) 900cm3c) 424.1cm3d) 220cm3
AcrossallquestionsinvolvingvolumeØ Mixingupvolumeandsurfacearea
c)notconvertingallmeasurementstothesameunits
QUESTION2384KL
QUESTION3a) 249505.29mm3b) 317680mm3c) 21.5%
a)notconvertingallmeasurementstothesameunitsc)calculatingthepercentageofspacethatthecrackerstakesupinsteadoftheemptyspace(i.e.78.5%)
NSWDepartmentofEducationMathematicsStage5DiagnosticTaskAnswers32
5.2RIGHT-ANGLEDTRIANGLES(TRIGONOMETRY)
CommonErrors
QUESTION1a) 13.67cmb) 32.04cm
AcrossallquestionsinvolvingtrigonometryØ UsingthewrongratiosØ Gettingconfusedwhentheunknownisin
thedenominatorforexampleif:tan 30° = 0&
e
rearrangingincorrectlytoget𝑦 =12𝑡𝑎𝑛30°
Ø Notusingtheinversefunctionwhencalculatinganunknownangle(i.e.onthecalculatorstudentsneedtopressSHIFTbeforethetrigratio)
QUESTION2a) 4701’b) 33024’
QUESTION3a) 13.2mb) 200mc) 2340d) i)
ii)419miii) 326041’
a)incorrectlyplacingthe15mastheheightitreachesupthewallratherthanthelengthoftheladderd)iii)stoppingatcalculatingtheangleandnotrecognisingthatthisisnotthebearing.
NSWDepartmentofEducationMathematicsStage5DiagnosticTaskAnswers33
5.2PropertiesofGeometricFigures CommonErrorsQUESTION1a) SSS,SAS,AAS,RHSb) i)Yes(SSS)ii)YES(RHS)c) i)RHS
ii)correspondinganglesincongruenttriangles
a)mixingupthecongruenttriangletestwiththetestforsimilartriangles,especiallytheequiangulartestforsimilarityc)notgivingreasonsorstatingreasonsthatarenotonthediagram
QUESTION2i) AASii) needtoproveoppositeanglesareequal
Noreasonsstatedornotstatingtheparallellinesthatformthealternateangles
QUESTION31. Equiangular2. Threesidesofonetriangleareproportionalto
thethreesidesoftheothertriangle3. Anangleofonetriangleiscongruenttoan
angleofasecondtriangleandthelengthsofthesidesincludingtheseanglesareproportional
4. Thelengthofthehypotenuseandasecondsideofarightangledtriangleareproportionaltohypotenuseandasecondsideoftheothertriangle
mixingupthetestforsimilartriangleswithcongruenttriangletestwith
QUESTION4a) Equiangularb) Threesidesofonetriangleareproportionalto
thethreesidesoftheothertriangle
QUESTION5y=4x=15
ErrorsinsolvingequationsIncorrectlysettingupequalitystatemente.g. e
0K=
"&
QUESTION6i) 32400ii) 1620iii) 180
i)using20×180°toget3600°insteadof18×180°
NSWDepartmentofEducationMathematicsStage5DiagnosticTaskAnswers34
5.2SingleVariableDataAnalysis CommonErrorsQUESTION1a) Lower=2,Upper=12,Median=5.5
Q1=4,Q3=7b) 3c) 50%d) e) 25%
a)statingthemedianas5ad6ratherthanfindingthemeanof5and6d)notincludingascale
QUESTION2a) ii)b) i)
QUESTION3a) Year9=7Year12=8b) Year9=2Year12=3c) Year9=3Year12=7d) Year12spentthelongest,theyhaveahigher
medianand75%ofstudentsspentmorethan6hourscomparedwithYear9whohadonly25%ofstudentsspendmorethan4hours.
e)50%
Incorrectlyreadingthescale
NSWDepartmentofEducationMathematicsStage5DiagnosticTaskAnswers35
5.2BivariateDataAnalysis CommonErrorsQUESTION1a) Dependent:Rainfall
Independent:monthb) Dependent:handspan
Independent:height
QUESTION2Linegraph,asyoucantracksmallchangesovertimemoreaccurately
QUESTION3a)
b) Septemberc) Januaryd) TherainfallisthehighestinJanuaryandthen
slowlyfallsthroughouttheyeartillSeptemberandthenslowlyincreasesagainthroughtoDecember
QUESTION4Strongpositive,nocorrelation,weakpositive,strongnegative
QUESTION5a) Linearandapositivecorrelationb) Approx.52c) Astheresultsinthetopictestsincrease,so
doestheresultinthefinalexam.
b)tryingtoestimatetheresultwithoutconstructingalineofbestfit.
NSWDepartmentofEducationMathematicsStage5DiagnosticTaskAnswers36
5.2BivariateDataAnalysis CommonErrorsQUESTION6a)
b) Therelationshipislinearandindirect,the
furtheryoutravelthelessfuelyouhaveinthetank
NSWDepartmentofEducationMathematicsStage5DiagnosticTaskAnswers37
5.2Probability CommonErrorsQUESTION1a) {H1,H2,H3,H4,H5,H6,T1,T2,T3,T4,T5,T6}b) 0
0&
b)calculatingtheprobabilityofaheadora6andgetting #
0&
QUESTION2
a) (#
b) 0-()
c) 0&()
QUESTION3a)
b) 0
f
c) #f
QUESTION4
a) &)
b) /f
c)
(#
b)andc)notdecreasingthesamplespaceforeachsuccessiveevent.(whenyoueatthejellybeanitisnolongerinthebag)
QUESTION5a) Independentb) Independentc) Dependent
NSWDepartmentofEducationMathematicsStage5DiagnosticTaskAnswers38
5.2Probability CommonErrorsQUESTION6a) {3,4,5,6}b) 0
(
c) 0
a)listingthesamplespaceofthedieas{1,2,3,4,5,6}b)andc)usingtheincorrectsamplespace
QUESTION7a) Thesecondtossisindependentofthefirsttoss
thereforeverytimeyourolladiethereisa0-
chanceofobtainingaoneb) Eachtooisanindependentevent,everytime
youtossacointhereisanevenchanceofobtainingahead.