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Name: Higher Level Mathematical Problem-Solving Exam | Date: [Date] | Time: 1 hour 30 minutes | Total Marks Available: 8

This Math exam contains challenging questions involving simultaneous equations, button packet quantities, foot length data analysis, and angle calculations. Test your problem-solving skills! |

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Name: Higher Level Mathematical Problem-Solving Exam | Date: [Date] | Time: 1 hour 30 minutes | Total Marks Available: 8

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  1. Name: Paper 3 Higher Predicted Calculator Date: Time:1 hour 30 minutes Total marksavailable:80 Total marksachieved:

  2. Q1. Lizbuyspackets ofcolouredbuttons. There are 8redbuttonsin each packet ofredbuttons. There are 6silverbuttonsin eachpacketofsilverbuttons. There are 5goldbuttons in each packet of gold buttons. Lizbuysequal numbersofredbuttons, silverbuttonsandgoldbuttons. Howmanypacketsofeach colour ofbuttonsdidLizbuy? ........................................................... packetsof redbuttons ........................................................... packetsofsilverbuttons ........................................................... packetsof gold buttons (Totalforquestion= 3marks) Q2. Solve thesimultaneousequations 2x–4y= 19 3x+ 5y=1 x= ........................................................... y=........................................................... (Totalforquestion= 4marks)

  3. Q1. Lizbuyspackets ofcolouredbuttons. There are 8redbuttonsin each packet ofredbuttons. There are 6silverbuttonsin eachpacketofsilverbuttons. There are 5goldbuttons in each packet of gold buttons. Lizbuysequal numbersofredbuttons, silverbuttonsandgoldbuttons. Howmanypacketsofeach colour ofbuttonsdidLizbuy? 8 16 24 32 40 48 56 64 72 80 88 96 104 112 120 6 12 18 24 30 36 42 48 54 60 66 72 78 84 90 96 102 108 114 120 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 15 20 120 ÷ 8 = 15 120 ÷ 6 = 20 120 ÷ 5 = 24 ........................................................... packetsof redbuttons ........................................................... packetsofsilverbuttons 24 ........................................................... packetsof gold buttons (Totalforquestion= 3marks) Q2. Solve thesimultaneousequations 2x–4y= 19 3x+ 5y=1 6x-12y = 57 6x+10y= 2 -22y=55 y= -2.5 3x + - 12.5=1 3x = 13.5 x = 4.5 4.5 x= ........................................................... y=........................................................... -2.5 (Totalforquestion= 4marks)

  4. Q3. Thetable showssome information about thefootlengthsof40adults. (a) Writedown themodal class interval. ........................................................... (1) (b) Calculateanestimatefor themeanfoot length. ........................................................... cm (3) (Totalforquestion= 4marks) Q4. Thediagramshowsaregularoctagonandaregularhexagon.

  5. Q3. Thetable showssome information about thefootlengthsof40adults. (a) Writedown themodal class interval. 22≤f<24 ........................................................... (1) (b) Calculateanestimatefor themeanfoot length. (17 x 3)+(19 x 6)+(21 x 10)+(23 x 12)+(25x9) = 876 876÷40 = 21.9 21.9 ........................................................... cm (3) (Totalforquestion= 4marks) Q4. Thediagramshowsaregularoctagonandaregularhexagon.

  6. Find thesize oftheangle markedx You mustshowall your working. x=...........................................................º (Totalforquestion= 3marks) Q5. Steve is askedtosolve theequation5(x+ 2)= 47. Hereishisworking. 5(x+2) =47 5x+ 2=47 5x= 45 x= 9 Steve'sansweriswrong. (a)What mistake did hemake? ............................................................................................................................................. ............................................................................................................................................. (1) Liz isaskedtosolve theequation3x2+8= 83 Hereisherworking. 3x2+8= 83 x2=25 x= 5 3x2=75 (b) Explain whatiswrong with Liz'sanswer. ............................................................................................................................................. ............................................................................................................................................. (1) (Totalforquestion= 2marks)

  7. Find thesize oftheangle markedx You mustshowall your working. (8-2) x 180 = 1080 ÷8= 135 (6-2) x 180 = 720 ÷ 6 = 120 360 -135 – 120 = 105 105 x=...........................................................º (Totalforquestion= 3marks) Q5. Steve is askedtosolve theequation5(x+ 2)= 47. Hereishisworking. 5(x+2) =47 5x+ 2=47 5x= 45 x= 9 Steve'sansweriswrong. (a)What mistake did hemake? ............................................................................................................................................. ............................................................................................................................................. Expanded the brackets incorrectly should be 5x + 10 (1) Liz isaskedtosolve theequation3x2+8= 83 Hereisherworking. 3x2+8= 83 x2=25 x= 5 3x2=75 (b) Explain whatiswrong with Liz'sanswer. ............................................................................................................................................. ............................................................................................................................................. A square root has both positive and negative roots eg +5 and -5 (1) (Totalforquestion= 2marks)

  8. Q6. Thegroupedfrequencytable gives informationabout theheightsof30students. (a) Writedown themodal class interval. ........................................................... (1) Thisincorrect frequencypolygonhasbeendrawnfor theinformationinthetable. (b) Writedown two thingswrong with thisincorrect frequencypolygon. 1 ............................................................................................................................................. 2 ............................................................................................................................................. (2) (Totalforquestionis3marks)

  9. Q6. Thegroupedfrequencytable gives informationabout theheightsof30students. 160 < h ≤ 170 (a) Writedown themodal class interval. ........................................................... (1) Thisincorrect frequencypolygonhasbeendrawnfor theinformationinthetable. (b) Writedown two thingswrong with thisincorrect frequencypolygon. 1 ............................................................................................................................................. 2 ............................................................................................................................................. Points should be plotted at mid points Polygon should not be closed (2) (Totalforquestionis3marks)

  10. Q7. Thegraphshows thedepth,dcm, ofwaterina tankaftert seconds. (a) Findthegradient ofthisgraph. ........................................................... (2) (b) Explain whatthis gradientrepresents. ............................................................................................................................................. ............................................................................................................................................. (1) (Totalforquestionis3marks)

  11. Q7. Thegraphshows thedepth,dcm, ofwaterina tankaftert seconds. (a) Findthegradient ofthisgraph. 210 ÷ 140 = -1.5 ........................................................... (2) (b) Explain whatthis gradientrepresents. ............................................................................................................................................. ............................................................................................................................................. Rate of change of depth of water in the tank (1) (Totalforquestionis3marks)

  12. Q8. Herearetheequationsof four straight lines. LineA Line B Line C LineD y=2x+4 2y= x+4 2x+2y=4 2x− y=4 Two oftheselinesareparallel. Writedown thetwo parallel lines? Line ................................ and line................................ (Totalforquestionis1mark) Q9. ABCDisa rhombus. M is themidpoint ofBD. E is thepoint onBD such thatDE=CE. Calculatethesize ofangleMCE. (Totalforquestion= 3marks)

  13. Q8. A has gradient 2 B has gradient C has gradient 1 D has gradient 2 Herearetheequationsof four straight lines. LineA Line B Line C LineD y=2x+4 2y= x+4 2x+2y=4 2x− y=4 Two oftheselinesareparallel. Writedown thetwo parallel lines? a d Line ................................ and line................................ (Totalforquestionis1mark) Q9. ABCDisa rhombus. M is themidpoint ofBD. E is thepoint onBD such thatDE=CE. Calculatethesize ofangleMCE. EDC = (180-116) ÷ 2 = 32 MCE = 58 – 32 = 26 (Totalforquestion= 3marks)

  14. Q10. Thegraphofy= f(x) istransformedtogive thegraph ofy = −f(x+ 3) ThepointAonthegraphofy= f(x)ismappedto thepointPonthegraphofy= −f(x + 3) ThecoordinatesofpointA are(9, 1) Find thecoordinatesofpointP. (............................ , ............................) (Totalforquestionis2marks) Q11. Simplifyfully(√a+ √4b)(√a–2√b) ........................................................... (Totalforquestion= 3marks)

  15. Q10. Thegraphofy= f(x) istransformedtogive thegraph ofy = −f(x+ 3) ThepointAonthegraphofy= f(x)ismappedto thepointPonthegraphofy= −f(x + 3) ThecoordinatesofpointA are(9, 1) Find thecoordinatesofpointP. 9-3 =6 because of f(x+3) 1 becomes -1 because of –f(x) 6 -1 (............................ , ............................) (Totalforquestionis2marks) Q11. Simplifyfully(√a+ √4b)(√a–2√b) a – 2 √a√b+ √4b√a - √4b2√b a - 2 √a √b + 2 √a √b -2 √b2√b a – 4b a – 4b ........................................................... (Totalforquestion= 3marks)

  16. Q12. Sami asked50people which drinksthey likedfrom tea, coffeeandmilk. All 50people like at least one of thedrinks 19people likeall threedrinks. 16people liketeaandcoffeebut donotlikemilk. 21people likecoffeeandmilk. 24people liketeaandmilk. 40people likecoffee. 1 personlikesonlymilk. Sami selects at randomone ofthe50people. (a) Work out theprobabilitythatthispersonlikestea. ........................................................... (4) (b) Given that theperson selectedat random from the50people likestea, findtheprobabilitythat this personalsolikesexactly one otherdrink. ........................................................... (2) (Totalforquestion= 6marks)

  17. Q12. Sami asked50people which drinksthey likedfrom tea, coffeeandmilk. All 50people like at least one of thedrinks 19people likeall threedrinks. 16people liketeaandcoffeebut donotlikemilk. 21people likecoffeeandmilk. 24people liketeaandmilk. 40people likecoffee. 1 personlikesonlymilk. Sami selects at randomone ofthe50people. (a) Work out theprobabilitythatthispersonlikestea. t c 4 16 3 19 2 5 1 21-19 = 2 24 – 19 = 5 5+19+2+16+3+1 = 46 50 – 46 = 4 4+5+19+16 = 44 m ........................................................... (4) (b) Given that theperson selectedat random from the50people likestea, findtheprobabilitythat this personalsolikesexactly one otherdrink. 5 + 16 = 21 ........................................................... (2) (Totalforquestion= 6marks)

  18. Q13. Onthegrid, enlargethetriangle byscalefactor–1½,centre(0, 2) (Totalforquestion= 2marks)

  19. Q13. Onthegrid, enlargethetriangle byscalefactor–1½,centre(0, 2) (Totalforquestion= 2marks)

  20. Q14. Solvex2−5x+ 3=0 Give yoursolutions correct to 3significantfigures. ........................................................... (Totalforquestion= 3marks) Q15. (a) Showthat theequation3x2− x3+ 3=0canbe rearrangedtogive (2) (b) Using findthe valuesofx1,x2andx3 ........................................................... (3) (c) Explain whatthevaluesofx1,x2andx3represent. ............................................................................................................................................. ............................................................................................................................................. (1) (Totalforquestionis6marks)

  21. Q14. Solvex2−5x+ 3=0 Give yoursolutions correct to 3significantfigures. 4.30 or 0.697 ........................................................... (Totalforquestion= 3marks) Q15. (a) Showthat theequation3x2− x3+ 3=0canbe rearrangedtogive x³-3x²=3 x² (x-3)=3 x-3 = (2) (b) Using findthe valuesofx1,x2andx3 x₁ = 3.29296875 x₂ = 3.276659786 x₃ = 3.279420685 3.28 ........................................................... (3) (c) Explain whatthevaluesofx1,x2andx3represent. ............................................................................................................................................. ............................................................................................................................................. iteration is an estimate of the solution like trial and error (1) (Totalforquestionis6marks)

  22. Q16. Afrustumismadebyremovingasmall conefromalargecone asshown in thediagram. Thefrustumismadefrom glass. Theglass hasa densityof2.5g /cm3 Workout themassof thefrustum. Give youranswerto anappropriatedegreeofaccuracy. ...........................................................g (Totalforquestion= 5marks)

  23. Q16. Afrustumismadebyremovingasmall conefromalargecone asshown in thediagram. Thefrustumismadefrom glass. Theglass hasa densityof2.5g /cm3 Workout themassof thefrustum. Give youranswerto anappropriatedegreeofaccuracy. 15 ÷ 5 = 3 12 ÷ 3 = 4 radius of small cone is 2cm radius of large cone is 6cm x π x 2² x 5 = π 180π- π = 544.5427266 544.5427266 x 2.5 1361 ...........................................................g (Totalforquestion= 5marks)

  24. Q17. Thegraphshowsinformation about the velocity,vm/s,ofaparachutistt secondsafter leavingaplane. (a) Workout anestimatefor theaccelerationof theparachutist att = 6 ........................................................... m/s2 (2) (b) Workout anestimatefor thedistancefallenbytheparachutist inthefirst 12secondsafter leaving theplane. Use 3stripsofequal width. ........................................................... m (3) (Totalforquestionis5marks)

  25. Q17. Thegraphshowsinformation about the velocity,vm/s,ofaparachutistt secondsafter leavingaplane. (a) Workout anestimatefor theaccelerationof theparachutist att = 6 =3.5 ........................................................... m/s2 (2) (b) Workout anestimatefor thedistancefallenbytheparachutist inthefirst 12secondsafter leaving theplane. Use 3stripsofequal width. (4x35)÷2 +4x(35+51)÷2 + 4x(51+54)÷2= 452 ........................................................... m (3) (Totalforquestionis5marks)

  26. Q18. VABCDisa solid pyramid. ABCDisa squareofside 20cm. Theangle between anyslopingedgeandtheplaneABCDis55° Calculatethesurfacearea ofthepyramid. Give youranswercorrect to2 significantfigures. ...........................................................cm2 (Totalforquestion= 5marks)

  27. Q18. VABCDisa solid pyramid. ABCDisa squareofside 20cm. Theangle between anyslopingedgeandtheplaneABCDis55° Calculatethesurfacearea ofthepyramid. Give youranswercorrect to2 significantfigures. AC²=20²+20²=800 AX²=10²+10²=200 VX= √200 x tan 55=20.19… VM=√(20.19²+10²)=22.54 4xx22.54x20+20² = 1300 1300 ...........................................................cm2 (Totalforquestion= 5marks)

  28. Q19. (a) Expandand simplifyx(x+ 1)(x− 1) (2) Inalistofthreeconsecutivepositiveintegersatleastoneofthenumbersisevenandoneofthenumbersis a multiple of3 nisa positive integer greater than1 (b) Provethatn3− nisa multiple of6forall possible valuesofn. (2) 261−1 isa prime number. (c) Explain why261+1 is amultiple of3 (2) (Totalforquestion= 6marks)

  29. Q19. (a) Expandand simplifyx(x+ 1)(x− 1) x (x²-1) x³-x (2) Inalistofthreeconsecutivepositiveintegersatleastoneofthenumbersisevenandoneofthenumbersis a multiple of3 nisa positive integer greater than1 (b) Provethatn3− nisa multiple of6forall possible valuesofn. (n-1)n(n+1)=(n²-n)(n+1)=n³-n²+n²-n=n³-n n-1 and n and n+1 are consecutive numbers If middle one is even then the next one or previous one would be a multiple of three therefore a multiple of 6 (2) 261−1 isa prime number. (c) Explain why261+1 is amultiple of3 261−1 , 261 and 261+1 These are 3 consecutive numbers the first is odd as all prime numbers are odd except 2 so the next is even so the next is a multiple of 3 (2) (Totalforquestion= 6marks)

  30. Q20. IntriangleRPQ, RP =8.7 cm PQ =5.2 cm AnglePRQ = 32° (a) Assuming that anglePQRisanacuteangle, calculatetheareaoftriangleRPQ. Give youranswercorrect to3 significantfigures. ...........................................................cm2 (4) (b) IfyoudidnotknowthatanglePQRisanacuteangle,whateffectwouldthishaveonyourcalculationof theareaof triangleRPQ? ............................................................................................................................................. ............................................................................................................................................. (1) (Totalforquestion= 5marks) Q21. Herearethefirst five termsofanarithmeticsequence. 7 13 19 25 31 Prove that thedifferencebetween the squaresof anytwo termsofthesequence is alwaysa multiple of24 (Totalforquestionis6marks)

  31. Q20. IntriangleRPQ, RP =8.7 cm PQ =5.2 cm AnglePRQ = 32° (a) Assuming that anglePQRisanacuteangle, calculatetheareaoftriangleRPQ. Give youranswercorrect to3 significantfigures. Sin Q = x8.7 Q =62.4285… Angle RPQ = 180 -62-32=86 x8.7x5.2xsin 86 =22.56…. ..........................................................cm2 (4) (b) IfyoudidnotknowthatanglePQRisanacuteangle,whateffectwouldthishaveonyourcalculationof theareaof triangleRPQ? ............................................................................................................................................. ............................................................................................................................................. (1) (Totalforquestion= 5marks) If it were obtuse you would need to find the area of 2 triangles Q21. Herearethefirst five termsofanarithmeticsequence. 7 13 19 25 31 Prove that thedifferencebetween the squaresof anytwo termsofthesequence is alwaysa multiple of24 x²- y 6m+1 and 6n+1 are two terms in the sequence (6m+1)²-(6n+1)²=36m²+12m+1-36n²-12n-1 36(m²-n²)+12(m-n) 3(m²-n²)+(m-n) (n-m)(3(n+m)+1 (Totalforquestionis6marks)

  32. MarkScheme Q1. Q2. Q3. Q4.

  33. Q5. Q6. Q7.

  34. Q8. Q9. Q10.

  35. Q11. Q12. Q13.

  36. Q14. Q15. Q16.

  37. Q17. Q18.

  38. Q19.

  39. Q20. Q21.

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