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Write your name hereSurnameOther namesPearson EdexcelLevel 3 GCECentre NumberCandidate NumberMathematicsAdvancedPaper 2: Pure Mathematics 2Wednesday 13 June 2018 – MorningTime: 2 hoursYou must have:Mathematical Formulae and Statistical Tables, calculatorPaper Reference9MA0/02Total MarksCandidates may use any calculator allowed by the regulations of theJoint Council for Qualifications. Calculators must not have the facilityfor symbolic algebra manipulation, differentiation and integration, orhave retrievable mathematical formulae stored in them.Instructionsblack ink or ball-point pen. UseIf pencil is used for diagrams/sketches/graphs it must be dark (HB or B).in the boxes at the top of this page with your name, Fillcentre number and candidate number.Answer all questions and ensure that your answers to parts of questions areclearly labelled.Answerthe questions in the spaces provided – there maybe more space than you need.Youshouldshowsufficient working to make your methods clear. Answers without working maynot gain full credit.Answersshouldbegivento three significant figures unless otherwise stated. Informationbooklet ‘Mathematical Formulae and Statistical Tables’ is provided. AThereare 14 questions in this question paper. The total mark for this paper is 100.Themarkseach question are shown in brackets – use this asfora guideas to how much time to spend on each question.Adviceeach question carefully before you start to answer it. ReadTry to answer every question. Check your answers if you have time at the endP58349A 2018 Pearson Education Ltd.1/1/1/1/1/*P58349A0144*Turn over

g(x) 2x 5x 5x 3(a) Find gg(5).(2)(b) State the range of g.(1)(c) Find g 1(x), stating its domain.DO NOT WRITE IN THIS AREA1.DO NOT WRITE IN THIS AREAAnswer ALL questions. Write your answers in the spaces provided.(3)DO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREA2*P58349A0244*

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the point B has position vector (4i 2j 3k),and the point C has position vector (ai 5j 2k), where a is a constant and a 0 BD .D is the point such that AB(a) Find the position vector of D.(2) 4Given AC(b) find the value of a.DO NOT WRITE IN THIS AREAthe point A has position vector (2i 3j 4k),DO NOT WRITE IN THIS AREA2. Relative to a fixed origin O,(3)DO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREA4*P58349A0444*

DO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREAQuestion 2 continuedDO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREA(Total for Question 2 is 5 marks)*P58349A0544*5Turn over

(2)(b) (i) Sketch the graph of y x 3(ii) Explain why x 3 x 3 for all real values of x.(3)DO NOT WRITE IN THIS AREADisprove this statement by means of a counter example.DO NOT WRITE IN THIS AREA3. (a) “If m and n are irrational numbers, where m n, then mn is also irrational.”6*P58349A0644*DO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREA

DO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREAQuestion 3 continuedDO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREA(Total for Question 3 is 5 marks)*P58349A0744*7Turn over

16 (3 5r 2 ) 131 798r(ii) A sequence u1, u2, u3, is defined byu n 1 1,unu1 23100Find the exact value of ur 1r(3)DO NOT WRITE IN THIS AREA(4)r 1DO NOT WRITE IN THIS AREA4. (i) Show thatDO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREA8*P58349A0844*

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xn 1 4 xn3 xn2 16 xn2 2 xn(3)Using the formula given in part (a) with x1 1(b) find the values of x2 and x3(c) Explain why, for this question, the Newton-Raphson method cannot beused with x1 0(2)DO NOT WRITE IN THIS AREA(a) Show that, for this equation, the Newton-Raphson formula can be writtenDO NOT WRITE IN THIS AREA5. The equation 2x3 x2 1 0 has exactly one real root.(1)DO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREA10*P58349A01044*

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(a) (i) Calculate f (2)(ii) Write f (x) as a product of two algebraic factors.(3)Using the answer to (a)(ii),(b) prove that there are exactly two real solutions to the equation 3y 6 8y 4 9y 2 10 0DO NOT WRITE IN THIS AREAf (x) 3x3 8x2 9x 10,   x DO NOT WRITE IN THIS AREA6.(2)DO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREA(1)DO NOT WRITE IN THIS AREA3 tan3θ 8 tan2θ 9 tanθ 10 0DO NOT WRITE IN THIS AREA(c) deduce the number of real solutions, for 7π θ 10π, to the equation12*P58349A01244*

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4 sin x sec x(4)(ii) Solve, for 0 θ 360 , the equation5 sin θ 5 cos θ 2giving your answers to one decimal place.(Solutions based entirely on graphical or numerical methods are not acceptable.)(5)DO NOT WRITE IN THIS AREAπ, the equation2DO NOT WRITE IN THIS AREA7. (i) Solve, for 0 x DO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREA16*P58349A01644*

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OxFigure 1DO NOT WRITE IN THIS AREAHDO NOT WRITE IN THIS AREA8.Figure 1 is a graph showing the trajectory of a rugby ball.(3)The ball passes over the horizontal bar of a set of rugby posts that is perpendicular to thepath of the ball. The bar is 3 metres above the ground.(b) Use your equation to find the greatest horizontal distance of the bar from O.(c) Give one limitation of the model.(3)(1)20*P58349A02044*DO NOT WRITE IN THIS AREA(a) Find a quadratic equation linking H with x that models this situation.DO NOT WRITE IN THIS AREAThe ball reaches a maximum height of 12 metres and hits the ground at a point 40 metresfrom where it was kicked.DO NOT WRITE IN THIS AREAThe ball travels in a vertical plane.DO NOT WRITE IN THIS AREAThe height of the ball above the ground, H metres, has been plotted against the horizontaldistance, x metres, measured from the point where the ball was kicked.

DO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREAQuestion 8 continuedDO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREA*P58349A02144*21Turn over

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You may assume the formula for cos (A B) and that as h 0,sin hcos h 1 1 and 0hh(5)DO NOT WRITE IN THIS AREAd(cos θ ) sin θdθDO NOT WRITE IN THIS AREA9. Given that θ is measured in radians, prove, from first principles, thatDO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREA24*P58349A02444*

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Using this model and all the information given,(a) find an equation linking the radius of the mint and the time.(You should define the variables that you use.)(b) Hence find the total time taken for the mint to completely dissolve. Give youranswer in minutes and seconds to the nearest second.(c) Suggest a limitation of the model.(5)DO NOT WRITE IN THIS AREAIn a simple model, the rate of decrease of the radius of the mint is inversely proportionalto the square of the radius.DO NOT WRITE IN THIS AREA10. A spherical mint of radius 5 mm is placed in the mouth and sucked.Four minutes later, the radius of the mint is 3 mm.(2)(1)DO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREA26*P58349A02644*

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(a) Find the values of the constants A, B and C.f (x) (4)1 11x 6 x 2x 3( x 3)(1 2 x)(b) Prove that f (x) is a decreasing function.(3)DO NOT WRITE IN THIS AREA1 11x 6 x 2BC A ( x 3)(1 2 x)( x 3) (1 2 x)DO NOT WRITE IN THIS AREA11.DO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREA30*P58349A03044*

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(b) Hence solve, for (2n 1)π,  n 2(3)ππ x , the equation22(sec2 x 5)(1 cos 2x) 3 tan2 x sin 2xGive any non-exact answer to 3 decimal places where appropriate.(6)DO NOT WRITE IN THIS AREA1 cos 2θ tan θ sin 2θ,  θ DO NOT WRITE IN THIS AREA12. (a) Prove thatDO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREA34*P58349A03444*

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13.DO NOT WRITE IN THIS AREAClP(e, e)RDO NOT WRITE IN THIS AREAyxOFigure 2DO NOT WRITE IN THIS AREAShow that the exact area of R is Ae2 B where A and B are rational numbers to be found.(10)DO NOT WRITE IN THIS AREAThe region R, shown shaded in Figure 2, is bounded by the curve C, the line l and the x-axis.DO NOT WRITE IN THIS AREAThe line l is the normal to C at the point P(e, e)DO NOT WRITE IN THIS AREAFigure 2 shows a sketch of part of the curve C with equation y x ln x,  x 038*P58349A03844*

DO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREAQuestion 13 continuedDO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREA*P58349A03944*39Turn over

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N 900,  t ,  t 03 7e 0.25t(a) Find the number of mice in the population at the start of the study.(b) Show that the rate of growthdNdNN (300 N )is given by 1200dtdt(1)(4)DO NOT WRITE IN THIS AREAThe number of mice, N, in the population, t months after the start of the study, ismodelled by the equationDO NOT WRITE IN THIS AREA14. A scientist is studying a population of mice on an island.The rate of growth is a maximum after T months.(1)DO NOT WRITE IN THIS AREA(d) State the value of P.DO NOT WRITE IN THIS AREAAccording to the model, the maximum number of mice on the island is P.DO NOT WRITE IN THIS AREA(4)DO NOT WRITE IN THIS AREA(c) Find, according to the model, the value of T.42*P58349A04244*

DO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREAQuestion 14 continuedDO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREA*P58349A04344*43Turn over

Question 14 continuedDO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREA(Total for Question 14 is 10 marks)TOTAL FOR PAPER IS 100 MARKS44*P58349A04444*DO NOT WRITE IN THIS AREA

20 *P58349A02044* T TE T AEA T TE T AEA T TE T AEA T TE T AEA T TE T AEA T TE T AEA 8. H O x Figure 1 Figure 1 is a graph showing the trajectory of a rugby ball. The height of the ball above the ground, metres, has been plotted against the horizontal H distance, x metres, measured from the point where the ball was kicked. The ball travels in a vertical plane.

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