Applications of Advanced Mathematics (C4)

1 This question paper consists of 5 printed pages and 3 blank CAMBRIDGE AND RSA EXAMINATIONSA dvanced Subsidiary General Certificate of EducationAdvanced General Certificate of EducationMEI STRUCTURED MATHEMATICS4754(A) Applications of Advanced Mathematics (C4) paper AMonday12 JUNE 2006 Afternoon1 hour 30 minutesAdditional materials:8 page answer bookletGraph paperMEI Examination Formulae and Tables (MF2)TIME1 hour 30 minutesINSTRUCTIONS TO CANDIDATES Write your name, centre number and candidate number in the spaces provided on the answerbooklet. Answer all the questions. You are permitted to use a graphical calculator in this paper . Final answers should be given to a degree of accuracy appropriate to the FOR CANDIDATES The number of marks is given in brackets [ ] at the end of each question or part question .

2 You are advised that an answer may receive no marksunless you show sufficient detail of theworking to indicate that a correct method is being used. The total number of marks for this paper is This paper will be followed by paper B: OCR 2006 [T/102/2653]Registered Charity 1066969[Turn over2 Section A(36 marks)1 Fig. 1 shows part of the graph of Fig. 1 Express in the form where and Hence write down the exact coordinates of the turning point P.[6]2(i)Given that where A, Band Care constants, find Band C, and show that[4](ii)Given that xis sufficiently small, find the first three terms of the binomial expansions ofand Hence find the first three terms of the expansion of[4]3 Given that show that Hence solve the equationfor[7]0 q 360 .sin (q 40 ) 2sin q,tan q sin a2 cos (q a) 2sin q,3 2x2(1 x)2(1 4x).(1 4x) 1.(1 x) 2A 2x2(1 x)2(1 4x) A1 x B(1 x)2 C1 4x,0 a 0R sin (x a),sincosxx-3yxPyx x=-sincos.]

3 34754(A) June 20064754(A) June 2006[Turn over34(a)The number of bacteria in a colony is increasing at a rate that is proportional to the square rootof the number of bacteria present. Form a differential equation relating x, the number ofbacteria, to the timet.[2](b)In another colony, the number of bacteria, y, after time tminutes is modelled by the differentialequationFind yin terms of t, given that when Hence find the number of bacteria after10 minutes.[6]5(i)Show that [3]Avase is made in the shape of the volume of revolution of the curve about the x-axisbetween and (see Fig. 5). Fig. 5(ii)Show that this volume of revolution is[4]14415p- 2x 0y=x12e-x xe 2x dx 14e 2x(1 2x) 900ddyty= B(36 marks)6 Fig. 6 shows the arch ABCD of a 6 The section from B to C is part of the curve OBCE with parametric equationsfor where ais a constant.]

4 (i)Find, in terms of a, (A)the length of the straight line OE, (B)the maximum height of the arch.[4](ii)Find in terms of[3]The straight line sections AB and CD are inclined at 30 to the horizontal, and are tangents to thecurve at B and C respectively. BC is parallel to the x-axis. BF is parallel to the y-axis.(iii)Show that at the point B the parameter satisfies the equation Verify that is a solution of this equation. Hence show thatand find OF in terms of a, giving your answer exactly.[6](iv)Find BC and AF in terms of a. Given that the straight line distance AD is 20 metres, calculate the value of a.[5]BF 32a,q 23psin( cos ).qq= q 2p,x a(q sin q), y a(1 cos q)30 30 yxOAFEDBC4754(A) June 20067 Fig. 7 Fig. 7 illustrates a house. All units are in metres. The coordinates of A, B, C and E are as is horizontal and parallel to AE.

5 (i)Find the length AE.[2](ii)Find a vector equation of the line BD. Given that the length of BD is 15 metres, find thecoordinates of D.[4](iii)Verify that the equation of the plane ABC isWrite down a vector normal to this plane.[4](iv)Show that the vector is normal to the plane ABDE. Hence find the equation of the plane ABDE.[4](v)Find the angle between the planes ABC and ABDE.[4]435 3x 4y 5z ( 1, 7, 11)C( 8, 6, 6)OyxzA(0, 0, 6)E(15, 20, 6)54754(A) June 2006 OXFORD CAMBRIDGE AND RSA EXAMINATIONSA dvanced Subsidiary General Certificate of EducationAdvanced General Certificate of EducationMEI STRUCTURED MATHEMATICS4754(B) Applications of Advanced Mathematics (C4) paper B: ComprehensionMonday12 JUNE 2006 AfternoonUp to 1 hourAdditional materials:Rough paperMEI Examination Formulae and Tables (MF2)HN/2 OCR 2006 [T/102/2653]Registered Charity 1066969[Turn overThis question paper consists of 4 printed pages and an NameCentre Number NumberTIMEUp to 1 hourINSTRUCTIONS TO CANDIDATES Write your name, centre number and candidate number in the spaces at the top of this page.]

6 Answer all the questions. Write your answers in the spaces provided on the question paper . You are permitted to use a graphical calculator in this FOR CANDIDATES The number of marks is given in brackets [ ] at the end of each question or part question . The insert contains the text for use with the questions. You may find it helpful to make notes and do some calculations as you read the passage. You are notrequired to hand in these notes with your question paper . You are advised that an answer may receive no marksunless you show sufficient detail of theworking to indicate that a correct method is being used. The total number of marks for this paper is Examiner s UseQu. Mark123456 Total1 The marathon is 26 miles and 385 yards long (1 mile is 1760 yards). There are now severalmen who can run 2 miles in 8 minutes. Imagine that an athlete maintains this average speedfor a whole marathon.

7 How long does the athlete take?[2]..2 According to the linear model, in which calendar year would the record for the men s milefirst become negative?[3]..3 Explain the statement in line 93 According to this model the 2-hour marathon will never berun. [1]..24754(B) June 2006 ForExaminer sUse4 Explain how the equation in line 49, is consistent with Fig. 2 (i)initially,[3] (ii)for large values of t.[2](i)..(ii)..[Questions 5 and 6 are printed overleaf.]R L (U L)e kt,3 ForExaminer sUse4754(B) June 20065 Amodel for an athletics record has the formwhere and (i)Sketch the graph of Ragainst t, showing Aand Bon your graph.[3](ii)Name one event for which this might be an appropriate model.[1](i)(ii)..6 Anumber of cases of the general exponential model for the marathon are given in Table of these is .(i)What is the value of tfor the year 2012?

8 [1](ii)What record time does this model predict for the year 2012?[2](i)..(ii)..R 115 (175 115)e B 0R A (A B)e kt44754(B) June 2006 ForExaminer sUseINSTRUCTIONS TO CANDIDATES This insert contains the text for use with the insert consists of 11 printed pages and 1 blank OCR 2006 [T/102/2653]Registered Charity 1066969[Turn over OXFORD CAMBRIDGE AND RSA EXAMINATIONSA dvanced Subsidiary General Certificate of EducationAdvanced General Certificate of EducationMEI STRUCTURED MATHEMATICS4754(B) Applications of Advanced Mathematics (C4) paper B: ComprehensionINSERTM onday12 JUNE 2006 AfternoonUp to 1 hourModelling athletics recordsIntroductionIn the 1900 Olympic Games, shortly before world records were first kept, the record time forthe marathon was almost exactly 3 hours. One hundred years later, in 2000, the world recordstood at 2 hours 5 minutes and 42 seconds; it had been set during the previous year by KhalidKannouchi of Morocco.]

9 At the time of writing this article, the world marathon record for menis 2 hours 4 minutes and 55 seconds, set by Paul Tergat of Kenya. When will the marathon record fall below 2 hours?It is clearly not possible to predict exactly when any world record will be broken, or when aparticular time, distance or height will be achieved. It depends, among other things, on whichathletes are on form at any time. However, it is possible to look at overall trends and so to makejudgements about when new records are likely to be set. Prediction inevitably involves extrapolating beyond existing data, and so into the unknown. Ifthis is to be more than guesswork, it must be based on a suitable mathematical is reasonable to hope that a general model can be found, one that can be adapted to manyathletics events. Such a model will take the form of a formula involving several parameters;these will take different values for different events.

10 The parameter values will take account ofthe obvious distinction that, whereas records for track events (like the marathon and the mile)decrease with time, those for field events (like the long jump and the javelin) article looks at possible formulae for such a linear modelThe simplest type of model is linear and this is well illustrated by the men s mile. The graphin Fig. 1 shows the world record for the mile plotted against the year from 1915 to of these records are given in Appendix 11915 1925 1935 1945 1955 1965 1975 1985 1995 2005220225230235240245250255 Year, TR (seconds)24754(B) Insert June 200651015204754(B) Insert June 2006[Turn overAline of best fit has been drawn on Fig. 1. Its equation iswhere Ris the record time in seconds Tis the calendar year. (This equation was calculated using a standard statistical technique.)]