Paper 3 Exploration Questions SIX QUESTIONS

Paper3ExplorationQuestions

Mathematics IB Higher Level

Analysis and Approaches

(For first examination in 2021). Author: Andrew Chambers All rights reserved. No part of this publication may be reproduced, distributed, or transmitted in any form or by any means, including recording, or other electronic or mechanical methods, without the prior written permission of the publisher, except in the case of brief quotations embodied in critical reviews and certain other noncommercial uses permitted by copyright law. These questions are licensed for non-commercial use only. For the full worked solutions please contact me through ibmathsresources.com.

CopyrightAndrewChambers2020.Licensedfornon-commercialuseonly.Visitibmathsresources.comtodownloadthefullworkedmark-schemeandfor300explorationideas.

2

Author’snoteI’vemadetheseinvestigationsspecificallyfortheIBHLAnalysisPaper3(firstexam2021). Myaimineach investigationwastohavesomeelementofdiscovery–sonewmathematicsandnewideasmaybeintroduced(astheywillontherealexam).I’vealsocreatedafulltypedmarkscheme–whichyoucandownloadfrommysiteifyougetstuck. DanielHwanghasalsocreatedasetofexcellent(andchallenging)Paper3questions–sobesuretoalsotrythese.Goodluck!AndrewChambers

TableofContents Page6:Rotatingcurves.[AlsosuitableforApplications]Themathematicsusedhereistrigonometry(identitiesandtriangles),functionsandtransformations.StudentsexploretheuseofparametricandCartesianequationstorotateacurvearoundtheorigin.Page8:WhokilledMr.Potato?[MostpartssuitableforApplications]Themathematicsusedhere is logs laws, linear regression and solvingdifferentialequations.StudentsexploreNewton’sLawofCoolingtopredictwhenapotatowasremovedfromanoven.Note: Some of the data on a cooling potatowas adapted from the real experimentcarriedoutatbyTomWoodsonetal.Page 10: Graphically understanding complex roots [Some parts suitable forApplications]The mathematics used here is complex numbers (finding roots), the sum andproductof roots, factorand remainder theorems, equationsof tangents. Studentsexploregraphicalmethodsforfindingcomplexroots.Page12:Avoidingamagicalbarrier[AlsosuitableforApplications]The mathematics used here is creating equations, optimization and probability.Students explore a scenario that requires them to solve increasingly difficultoptimizationproblems.Note:TheoriginalideaforthispuzzlewasfromaMindYourDecisionsvideo:AvoidingaTroll


3Page14:Circlepackingdensity[AlsosuitableforApplications]The mathematics used here is trigonometry and using equations of tangents.Students explore differentmethods of filling a spacewith circles to find differentcirclepackingdensities.Note:FellowIBmathsteacherDanielHwanghasalsomadealargenumberofpracticepaper3s.Heexploresasimilartopicwithfocuson3Dpacking.Page16:Aslidingladderinvestigation[MostpartssuitableforApplications]Themathematicsusedhere is trigonometryanddifferentiation. Students find thegeneralequationofthemidpointofaslippingladderandcalculatethelengthoftheastroidformed.Note:ThiswasalsofirstinspiredbythevideofromaMindYourDecisionsvideo:CanyousolveanOxfordinterviewquestion?Page19:ExploringtheSi(x)function[NotsuitableforApplications]Themathematics used here is Maclaurin series, integration, summation notation,sketching graphs. Students explore methods for approximating non-integrablefunctionsandconcludebyapproximating!

!

!".

Page 21: Volume optimization of a cuboid [Most parts suitable forApplications]The mathematics used here is optimization, graph sketching, extended binomialseries,limitstoinfinity.Studentsstartwithasimplevolumeoptimizationproblembutextendthistoageneralcase.Note:ThistaskwasoriginallydiscussedinanNrichproblemnumber6399.


4Rotatingcurves[40marks]

1.[Maximummarks:15]Thisquestionasksyoutoinvestigatetherotationofacoordinatepoint.(a) The line𝑦 = 0is rotated𝜃radians anti-clockwise about the origin. Find the

equationofthenewlineintermsof𝑦, 𝑥 𝑎𝑛𝑑 𝜃. [2](b) Ifwehaveacoordinatepoint(𝑎, 𝑏)rotated𝜃radiansanti-clockwiseaboutthe

originwecanfindthe𝑥,𝑦coordinatesofthenewpointbyusingthefollowingparametricequations:

𝑥 = 𝑎𝑐𝑜𝑠𝜃 − 𝑏𝑠𝑖𝑛𝜃 (1)

𝑦 = 𝑎𝑠𝑖𝑛𝜃 + 𝑏𝑐𝑜𝑠𝜃 (2)

Rotate the point(1,1)anti-clockwise around the origin by!!, !!𝑎𝑛𝑑 !

!radians.

Giveyouranswersascoordinates. [4](c) Draw a sketch of your points. What is the locus of points for this

transformation? [4](d) Bysquaringbothequations(1)and(2)obtainanequationintermsof𝑥and𝑦

only.Whatisthegeometricalsignificanceofthisequation? [5]2.[Maximummarks:25]This question expands this method to rotating a curve. We can derive the generalequationasfollows:We start with a function𝑓(𝑡), and draw a rectangle centred at the origin throughpoint P:(𝑡, 𝑓 𝑡 ). We then rotate the curve and the rectangle by𝜃radians anti-clockwisefromthehorizontal.AnglesABCandCDParerightangles.(a) ExplainwhyangleCABandPCDareequal. [2]


5(b) ShowthatthexandycoordinatesofPcanbewrittenas:

𝑥 = 𝑡𝑐𝑜𝑠𝜃 − 𝑓 𝑡 𝑠𝑖𝑛𝜃 (3)

𝑦 = 𝑡𝑠𝑖𝑛𝜃 + 𝑓 𝑡 𝑐𝑜𝑠𝜃 (4)[4]

(c) By multiplying equation (3) by𝑐𝑜𝑠 𝜃 and equation (4) by𝑠𝑖𝑛 𝜃 show that

thiscanbewrittenas:

𝑦𝑐𝑜𝑠𝜃 − 𝑥𝑠𝑖𝑛𝜃 = 𝑓(𝑦𝑠𝑖𝑛𝜃 + 𝑥𝑐𝑜𝑠𝜃)[6]

(d) By taking𝑓 𝑡 = 2𝑡 + 3, showthat theequationwhen𝑦 = 2𝑥 + 3is rotated!

!

radiansanticlockwisearound(0,0)isgivenby𝑦 = −3𝑥 − 3 2. [5]

For 2 straight lines𝑎!𝑥 + 𝑏!𝑦 + 𝑐! = 0 and 𝑎!𝑥 + 𝑏!𝑦 + 𝑐! = 0 , the acute anglebetweenthemisgivenby:

𝑡𝑎𝑛𝜃 =𝑎!𝑏! − 𝑏!𝑎!𝑎!𝑎! + 𝑏!𝑏!

(e) Showthattheanglebetween𝑦 = 2𝑥 + 3and𝑦 = −3𝑥 − 3 2isindeed!

!.[3]

(f) By taking𝑓 𝑡 = 𝑡!find the equation of y =𝑥! when it is rotated!

!radians

anticlockwisearound(0,0).

Leaveyouranswerintheform:

𝑎𝑥! + 𝑏𝑥 + 𝑐𝑦! + 𝑑𝑦 + 𝑔𝑥𝑦 = 0[5]


6

WhokilledMr.Potato?[36marks]1.[Maximummarks:12]Coolingratesareanessentialtoolinforensicsindeterminingtimeofdeath. Wecanuse differential equations to model the cooling of a body using Newton’s Law ofCooling. The rate of change in the body temperature with respect to time isproportionaltothedifferenceintemperaturebetweenthebodyandtheambientroomtemperature.

Inthefollowinginvestigationwewilldeterminewhenapotatowasremovedfromanoven. If the ambient room temperature is 75 degrees Fahrenheit and𝑇 is thetemperatureofthepotatoattime𝑡,thenwehave:

𝑑𝑇𝑑𝑡 = −𝑘(𝑇 − 75)

a) Showthatthesolutiontothisdifferentialequationis:

𝑇 − 75 = 𝐴𝑒!!"[4]

Wearriveataroomatmidday12:00todiscoverapotatoonthekitchencounter.Theambient room temperature is 75 degrees Fahrenheit and we know the initialtemperature of the potato before being taken out of the oven was 194 degreesFahrenheit.Wetakethefollowingmeasurements:Timeafter12:00(tmins) Temperature of potato (T

degreesFahrenheit)Difference from ambient roomtemperature𝑇! (T–75)

0 133 5830 116 4160 104 2990 98 23120 91 16(b) By using the measurements when𝑡 = 0 and𝑡 = 120, find𝐴and𝑘. Use your

modeltopredictwhenthepotatowasfirstremovedfromtheoven.[8]


72.[Maximummarks:12]Inthisquestionweuseanalternativemethodtoestimate𝑘.Ifwetake𝑇! = 𝑇 − 75,thenwehave:

𝑇! = 𝐵𝑒!!"(a) Byfindingtheregressionlineof𝑙𝑛 𝑇! on𝑡,findanestimatefor𝑘,𝐵.Statethe

correlationcoefficientandcommentonthisvalue.[8]

(b) Use your model to predict to the nearest minute when the potato was first

removed from the oven. Compare your answerwith (1c). Whichmethod offinding𝑘islikelytobemoreaccurate?

[4]3.[Maximummarks:14]In this questionweuseamodified versionofNewton’s lawof cooling introducedbyMarshallandHoare.Themodified equation includes an extra exponential term to account for an initial,morerapidcooling.Wecanwritethisas:

𝑑𝑇!𝑑𝑡 + 𝑘𝑇! = 119𝑘𝑒!!"

(a) Showthatthesolutiontothisdifferentialequationis:

𝑇! =119𝑘𝑘 − 𝑝 𝑒

!!" + 𝐴𝑒!!"

[7](b) Thismodel requires thatwe set𝑡 = 0 as the timewhen the potatowas first

removed from the oven (at this time𝑇! = 119). We are given that𝑘 ≈0.012,𝑝 ≈ 9.5.Find𝐴anduseyourmodeltopredicthowmanyminutesittookfor 𝑇! toreach58.Compareyour3results.

[7]


8

Graphicallyunderstandingcomplexroots:aninvestigation[35marks]

1.[Maximummarks:8]Inthisquestionwewillexplorethecomplexrootsof𝑓 𝑥 = 𝑥! + 4𝑥 + 5(a) Write𝑓 𝑥 intheform(𝑥 − 𝑎)! + 𝑏.

[1](b) Bysolving 𝑓 𝑥 =0,findthecomplexrootsof𝑓 𝑥 .

[1](c) Reflect𝑓 𝑥 intheline𝑦 = 𝑏.Callthisnewgraph𝑔 𝑥 .

[1](d) Findtheroots𝑥!, 𝑥!of𝑔 𝑥 .

[1](e) Rotate𝑥!, 𝑥!90degreesanticlockwisearoundthepoint(𝑎, 0).Writedownthecoordinatesofthepoints.Howisthisrelatedtopart(a)?

[2](f) For a graph𝑔 𝑥 with real roots𝑥! = 𝑎 + 𝑏 , 𝑥! = 𝑎 − 𝑏, what are the

coordinatesthatrepresentthecomplexrootsof𝑓 𝑥 inthecomplexplane?[2]

2.[Maximummarks:11]Inthisquestionwewillexplorethecomplexrootsof𝑓 𝑥 = 𝑥! − 9𝑥! + 𝛽𝑥 − 17(a) 𝑓(𝑥)has only 1 real root,𝑥! = 1. Write down equations for the sum and

productoftherootsof𝑓(𝑥),[2]

(b) Hencefindthe2complexroots,𝑥!, 𝑥!.

[3](c) Byusingthefactortheoremorotherwise,showthat𝛽 = 25.

[2]


9(d) By method of polynomial long division, factorise𝑓 𝑥 into a linear and

quadraticpolynomial.[2]

(e) Henceshowanalternativemethodforfindingthe2complexroots,𝑥!, 𝑥!.

[2]3.[Maximummarks:16]In this questionwewill investigate a graphicalmethod of finding complex roots ofcubics.

Ifwehaveacubic𝑓 𝑥 withonerealrootwecangraphicallyfindthecomplexrootsbythefollowingmethod:Drawthelinethroughtherealrootwhichisalsotangenttothecurve𝑓(𝑥)atanotherpoint. If the x-coordinate of the point of intersection between the tangent and thecurveisa,andthegradientofthetangentism,thenthecomplexrootsare𝑎 ± 𝑚𝑖(a) Thegraphaboveshowsacubic,witha tangent to thecurvedrawnso that it

also passes through its real root. Write down the 3 roots of this cubic andhencefindtheequationofthiscubic.

[6](b) Forthecurve𝑓 𝑥 = 𝑥! − 4𝑥! + 6𝑥 − 4,verifythat(𝑥 − 2)isafactor

[1]

(ii)Giventhatthelinethroughtherealrootisatangenttothecurveat𝑥 = 𝑎,findtheequationofthetangentandshowthat:

0 = −2𝑎! + 10𝑎! − 16𝑎 + 8

[6]

(iii)Hencefindthecomplexrootsof𝑓(𝑥)[3]


10Avoidingamagicalbarrier.[30marks]

1.[Maximummarks:30]In this question we explore a scenario where we walk from Town A to Town B, adistanceof2km.50%of the time there isnoobstructionalong this route. However,50%of the timethereisamagicalbarrierperpendiculartotherouteexactlyhalfwaybetweenAandB,extendingfor1kminbothdirections.Thisbarrierisinvisibleandcanonlybesensedwhenyoumeetit.

Yourtaskistoinvestigatetheoptimumstrategyforminimizingyouraveragejourney.(a) You setoutdirectlyon the routeAC. You thenwalk in the lineCB. Find the

distancetravelled.[1]

(b) YousetoutdirectlyontherouteAB.Ifthereisnobarrieryoucontinueonthisroute toB. Ifyoumeet thebarrieryouwalkup toCandthen in the lineCB.Findtheaveragedistancetravelled.

[3]

(c) This time you set off from A in a straight line to a point 0.5 km along theperpendicular bisector of AB. If you meet the barrier you continue up to CbeforetravellinginthelineCB.Ifyoudon’tmeetthebarrieryouheadstraightforpointB.

Findtheaveragedistancetravelled.[3]


11(d) This time you set off from A in a straight line to a point x km along the

perpendicular bisector of AB. If you meet the barrier you continue up to CbeforetravellinginthelineCB.Ifyoudon’tmeetthebarrieryouheadstraightforpointB.

(i)Findanequationfortheaveragedistancetravelledintermsofx.[3]

(ii) Use calculus to find the exact value of x which minimizes the averagedistancetravelled.

[5]

(iii) Sketch a graph to verify your result graphically. What is theminimumaveragedistance?

[2]

(e) The barrier nowappears n%of the time. Use calculus to find the optimumdistancex,intermsofn.

[7](f) For a given integer value of n, the optimum strategy is simply to head in a

straightlinefromAtoC.Findthisvalueofn.[6]


12

Circlepackingdensity[31marks]

1.[Maximummarks:31]Inthisquestionweinvestigatethedensityofcirclesinagivenspace.Below we have 3 circles tangent to each other, each with radius 1. Point A hascoordinates (0,0). Point D is at the intersection of the three angle bisectors of thetriangle.

(a) FindthecoordinatesofPointB,PointCandPointD

[5](b) FindthepercentageofthetriangleABCthatisnotfilledbythecircles.

[4](c) Thegeneralequationofacirclewithradiusrandcenteredat(a,b)isgivenby:

(𝑥 − 𝑎)! + (𝑦 − 𝑏)! = 𝑟!

Writedowntheequationofthethreecircles.[2]

(d) ThetangentstothethreecirclesmakeanequilateraltriangleOPQ.

ShowthatthecoordinatepointFhascoordinates(! !

!, !!).

[7]


13(e) ByfirstfindingtheequationofthetangentPQ,findthecoordinatesofPandO,

[7]

(f) HencefindtheareaofthetriangleOPQ.[3]

(g) Find the percentage of the triangle OPQ that is not filled by the circles.

Commentonwhichtrianglehasahighercirclepackingdensity.[3]


14

Aslidingladderinvestigation[40marks]1.[Maximummarks:16]Inthisquestionyouinvestigatethemotionofthemidpointofaslidingladder.Youhavea5metrelongladderwhichendsrestagainstaperpendicularwallat(0,4)and(3,0).

(a) Explainwhythemidpointoftheladderisgivenbythecoordinate(1.5,2).

[1](b) Theladderbeginstoslipdownthewallsuchthatthenewbasecoordinateis

(3+t,0).Findthenewheightintermsoft.[1]

(c) Find thenewmidpoint (x,y) in termsof t. Writedownan equation for the x

coordinateintermsoftandanequationfortheycoordinateintermsoft.[2]

(d) Eliminatetandshowthatanequationforthemidpointcanbewrittenas

𝑦! + 𝑥! = !"!

𝑥 ≥ 0, 𝑦 ≥ 0. [3](e) Sketchthisequationandcommentonitsgeometricalsignificance.

[3](f) Findthegeneralequationofthemidpointofaslippingladderwhentheladder

restsagainstaperpendicularwallat(0,a)and(b,0).

[6]


152.[Maximummarks:14]Thistimewetakea ladderwith length5anddrawthe family(envelope)of lineswegetasitslipsdownthewall.Theboundaryoftheselinescreatesthefirstquadrantofanastroid.

Theequationofthisparticularastroidisgivenby:

𝑥!! + 𝑦

!! = 5

!!

(a) Findanequationfor!"

!"intermsofxonly.

[5](b) Sketch!"

!" 𝑥 > 0.

[3](c) Findthegradientoftheastroidwhen𝑦 = 𝑥,𝑥 > 0,𝑦 > 0.

[4](d) What is the significanceof the liney=x in relation to theastroid in the first

quadrant?[2]

Thisquestioncontinuesonthenextpage.


163.[Maximummarks:10]Westartwiththeastroidformedbyaladderoflengthm

𝑥!! + 𝑦

!! = 𝑚

!!

Wecandefinethecurveparametricallyas:

𝑥 = 𝑚𝑐𝑜𝑠!𝜃

𝑦 = 𝑚𝑠𝑖𝑛!𝜃withthelengthLoftheastroidgivenby:

𝐿4 = (

𝑑𝑥𝑑𝜃)

! + (𝑑𝑦𝑑𝜃)

!

!!

! 𝑑𝜃

(a) Find!"

!"and!"

!"

[2]

(b) Henceshowthat:

𝐿4 = 3𝑚 𝑠𝑖𝑛𝜃𝑐𝑜𝑠𝜃

!!

! 𝑑𝜃

[4](c) Find an equation for L in terms of𝑚. Find the first quadrant length of the

astroidmadebyaladderoflength5.[4]


17ExploringtheSi(x)function[36marks]

(GraphdrawnfromWolframAlpha)

InthisquestionweinvestigatekeyfeaturesabouttheSi(x)function.1.[Maximummarks:9]The𝑆𝑖 𝑥 ,whichisdrawnabovecanbedefinedas:

𝑠𝑖𝑛 (𝑥)𝑥 𝑑𝑥 = 𝑆𝑖 𝑥 + 𝑐

(a) Writedown !

!"(𝑆𝑖 𝑥 )

[1](b) Findthe𝑥coordinatesofthelocalmaximumsof𝑆𝑖 𝑥 for𝑥 > 0

[3](c) Byfinding !

!

!!!(𝑆𝑖 𝑥 )findthe𝑥coordinateofthefirstpointofinflectionof

𝑆𝑖 𝑥 ,for𝑥 > 0. [5]

2.[Maximummarks:9](a) By considering the Maclaurin expansion of 𝑠𝑖𝑛 𝑥 give a polynomial

approximationfor!"# (!)!

uptothe𝑥!term. Explainwhythisapproximationisnotvalidforall𝑥.

[4](b) Drawasketchof𝑦 = !"# (!)

!andyourpolynomialapproximationonthegraph

paperbelowfor 0 <𝑥 ≤ 4. [4]


18(c) Forwhatdomaindoesyourpolynomialcloselyapproximate!"# (!)

!?

[1]3.[Maximummarks:9](a) Findapolynomialapproximationfor𝑆𝑖 𝑥 uptothe𝑥!term.

[3](b) Ifwehaveourintegralstartfrom𝑥 = 0 thenwecanwrite:

𝑠𝑖𝑛 (𝑥)𝑥

!

!𝑑𝑥 = 𝑆𝑖 𝑎

Usingyourpolynomialfindanapproximationfor𝑆𝑖 !

!to6significantfigures.

[3](c) Wecanuseacomputertocalculate𝑆𝑖 !

!≈ 1.37076.Whatisthepercentage

errorinusingyourapproximation?[3]

4.[Maximummarks:9](a) Showthat

𝑙𝑛 (1+ 𝑥)𝑥

!

!𝑑𝑥 = (−1)!!!

1𝑛!

!

!!!

[5]

(b) Byconsideringthefirst5termsofthisseriesfindanapproximationto !" (!!!)

!!! 𝑑𝑥to6significantfigures.

[2](c) Giventhat:

(−1)!!!1𝑛!

!

!!!

=𝜋!

12

Findthepercentageerrorwhenusingyourapproximationtofind!!

!".

[2]


19Volumeoptimizationofacuboid[37marks]

1.[Maximummarks:11]Apieceofpaperhas4squareedgesofsize𝑥cutout.Itisthenfoldeduptomakeacuboidwithanopentop.

(a) Find an expression for the volumeof the cuboid formedwhen the original

paperwasasquareoflength20cm.[1]

(b) Usecalculustofindthevalueof𝑥thatgivesthemaximumvolume.[3]

(c) By sketching a graph, find the value of𝑥that gives the maximum volumewhentheoriginalpaperisasquarewithsides:

(i)10cm.

(ii)30cm.

[5](d) Hence find the value of𝑥which gives a maximum volume of an𝑚 𝑏𝑦 𝑚

square.[2]

2.[Maximummarks:26]Nextwewillexplorea10 𝑏𝑦 𝑛rectangle.(a) Showthatthevalueof𝑥whichgivesthemaximumvolumeisgivenby:

𝑥 =40+ 4𝑛 − 4 (𝑛 − 5)! + 75

24 [8]

(b) Usethisequationtofindthevalueof𝑥whichgivesthemaximumvolumefora10 𝑏𝑦 30rectangle.

[2]


20(c) We will now investigate what happens to this value of𝑥as𝑛tends to

infinity,ie:

lim!→!

40+ 4𝑛 − 4 (𝑛 − 5)! + 7524

Bymakingthesubstitution𝑛 = !"!+ 5showthatthiscanbewrittenas:

lim!→!

15+ 75𝑢 − 75 1𝑢 1+ 𝑢!

6 [4]

(d) Findthebinomialexpansionof 1+ 𝑢!untilthe𝑢!term

[3]

(e) Hence find the value that𝑥approaches as𝑛tends to infinity for an10 𝑏𝑦 𝑛rectangle

[5]

(f) By a similarprocesswe can arrive at the following equation for anm 𝑏𝑦 𝑛rectangle

lim!→!

4𝑚 + 4(3𝑚!

4𝑢 +𝑚2 ) − 4 3𝑚!

41𝑢 1+ 𝑢!

24

Find the value that𝑥 approaches as𝑛 tends to infinity for an m 𝑏𝑦 𝑛rectangle

[4]

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Paper 3 Exploration Questions SIX QUESTIONS