IBSLMathYear1
DecemberExamination
PRACTICEEXAM#1
1.Thefollowingdiagramshowspartofthegraphof𝑓(𝑥) = 5 − 𝑥 ! withvertex𝑉(0, 5).
ℎ
Itsimage𝑦 = 𝑔(𝑥)afteratranslationwithvector
hasvertex𝑇(3, 6).
𝑘
(a)Writedownthevalueofℎand𝑘.
(b)Writedownanexpressionfor𝑔(𝑥).
2.(5marks)Thequadraticequation2𝑘𝑥 ! + 4𝑘𝑥 + 1 = 0,𝑘 > 0hasexactlytworealsolutions
for𝑥.Findthevalueof𝑘.
3.Let𝑓(𝑥) = 3𝑥 + 5and𝑔(𝑥) = 2(1 − 𝑥).
(a)(3marks)Find𝑓 !! (2).
(b)(3marks)Solvetheequation(𝑔 ∘ 𝑓)(−4)
!
4.(5marks)Solvetheequation𝑙𝑜𝑔! 81 + 𝑙𝑜𝑔! ! +𝑙𝑜𝑔! 3 = 𝑙𝑜𝑔! 𝑥for𝑥.
5.Apopulationofbacteriaisgrowingattherateof2.3%perminute.Howlongwillittakefor
thesizeofthepopulationtodouble?Giveyouranswertothenearestminute.
6.(5marks)Considerthebinomial(3𝑥 ! − 1)! (a)(2marks)Howmanytermsinthisexpansion?
(b)(4marks)Findtheconstantterm.
7.(a) Onthefollowingdiagram,sketchthegraphsof
𝑦 = 𝑒 ! and𝑦 = 𝑐𝑜𝑠 𝑥 for– 2 ≤ 𝑥 ≤ 1.
(b)Theequation𝑒 ! = 𝑐𝑜𝑠(𝑥)hasasolution
between–2and–1.Findthissolution.
8.Partofthegraphofafunction𝑓isshowninthediagrambelow.
(a)(3marks)Onthesamediagramsketchthe
graphof𝑦 = − 𝑓(𝑥).
!
(b)(3marks)Let𝑔(𝑥) = ! 𝑓(𝑥 + 3).Describe
fullythetransformationthatmapsthegraph
of𝑓tothegraphof𝑔.
9.Awomanplacesasinglegrainofriceononesquareofachessboard.Shethenplaces2grains
ofriceonthenextsquare.Shethenplaces4grainsonthenextsquare.Thisprocessisrepeated
untilall64squaresonthechessboardhavebeenused.
(a)(2marks)Howmanygrainsofriceareonthe10thsquareofthechessboards.
(b)(3marks)Howmanygrainsofriceintotalhavetofitonthechessboard?.
10.Solvetheequationfor𝑥onthegivendomain:
(a)(3marks)𝑡𝑎𝑛(𝑥) = 3,for0 ≤ 𝑥 ≤ 2𝜋
!
!!
(b)(3marks)𝑥 = 𝑐𝑜𝑠 ! ( ! ) − 𝑠𝑖𝑛( ! ),for0 ≤ 𝑥 ≤ 2𝜋
11.(marks)Thediagramshowstwoconcentriccircleswith
centerO.
Theradiusofthesmallercircleis8cmandtheradiusofthe
largercircleis10cm.
PointsA,BandCareonthecircumferenceofthelarger
circlesuchthatAOˆBisπ3radians.
(a)FindthelengthofthearcACB.
(b)Findtheareaoftheshadedregion.
!"!
12.(4marks)Algebraicallyconvertandsimplifyfully ! toanexactnumberofdegrees.
13.Inthediagram𝑃isthemidpointof𝐴𝐵and𝑄isthemidpointof𝐴𝐶.Calculatetheshaded
area.
14.Considerthediagram
(a)(3marks)Findangle𝐵𝐴𝐶.
(b)(3marks)Find𝐴𝐶
(c)(3marks)Henceorotherwise,findthelengthofarc𝐴𝐵𝐶.
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