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www.igexams.comPlease check the examination details below before entering your candidate informationCandidate surnamePearson EdexcelOther namesCentre NumberCandidate NumberInternationalAdvanced LevelWednesday 22 January 2020Morning (Time: 1 hour 30 minutes)Paper Reference WMA13/01MathematicsInternational Advanced LevelPure Mathematics P3You must have:Mathematical Formulae and Statistical Tables (Lilac), calculatorTotal MarksCandidates may use any calculator permitted by Pearson regulations.Calculators must not have the facility for symbolic algebra manipulation,differentiation and integration, or have retrievable mathematicalformulae 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, centre number and Fillcandidate number.Answer all questions and ensure that your answers to parts of questions are clearlylabelled.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.Inexactanswersshouldbe given to three significant figures unless otherwise stated.Informationbooklet ‘Mathematical Formulae and Statistical Tables’ is provided. AThereare 9 questions in this question paper. The total mark for this paper is 75.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.your answers if you have time at the end. CheckIf you change your mind about an answer, cross it out and put your new answerand any working underneath.Turn overP60568A 2020 Pearson Education Ltd.1/1/1/1/*P60568A0132*

www.igexams.com1.LeaveblankA population of a rare species of toad is being studied.The number of toads, N, in the population, t years after the start of the study, is modelledby the equation900e0.12tN 0.12t2e 1t 0, t According to this model,(a) calculate the number of toads in the population at the start of the study,(1)(b) find the value of t when there are 420 toads in the population, giving your answer to2 decimal places.(4)(c) Explain why, according to this model, the number of toads in the population can neverreach 500(1)2*P60568A0232*

www.igexams.comLeaveblankQuestion 1 continuedQ1(Total 6 marks)*P60568A0332*3Turn over

www.igexams.com2.LeaveblankThe function f and the function g are defined byf(x) 12x 1x 0, x g(x) 5ln x2x 0, x (a) Find, in simplest form, the value of fg(e2)(b) Find f –1(2)(3)(c) Hence, or otherwise, find all real solutions of the equationf –1 (x) f(x)(3)4*P60568A0432*

www.igexams.comLeaveblankQuestion 2 continued*P60568A0532*5Turn over

www.igexams.comLeaveblankQuestion 2 continued6*P60568A0632*

www.igexams.comLeaveblankQuestion 2 continuedQ2(Total 8 marks)*P60568A0732*7Turn over

www.igexams.comLeaveblank3.log10 y(0, 4)(6, 0)Olog10 xFigure 1Figure 1 shows a linear relationship between log10 y and log10 xThe line passes through the points (0, 4) and (6, 0) as shown.(a) Find an equation linking log10 y with log10 x(2)(b) Hence, or otherwise, express y in the form px q, where p and q are constants to befound.(3)8*P60568A0832*

www.igexams.comLeaveblankQuestion 3 continuedQ3(Total 5 marks)*P60568A0932*9Turn over

www.igexams.comLeaveblank4.(i)f (x) (a) Find f ′(x) in the formexpressions.(2 x 5)x 32x 3P( x)where P(x) and Q(x) are fully factorised quadraticQ( x)(b) Hence find the range of values of x for which f(x) is increasing.(ii)g(x) x sin 4x0 x (6)π4The curve with equation y g(x) has a maximum at the point M.Show that the x coordinate of M satisfies the equationtan 4x kx 0where k is a constant to be found.(5)10*P60568A01032*

www.igexams.comLeaveblankQuestion 4 continued*P60568A01132*11Turn over

www.igexams.comLeaveblankQuestion 4 continued12*P60568A01232*

www.igexams.comLeaveblankQuestion 4 continuedQ4(Total 11 marks)*P60568A01332*13Turn over

www.igexams.com5.Leaveblank(a) Use the substitution t tan x to show that the equation12 tan 2x 5 cot x sec2 x 0can be written in the form5t 4 – 24t 2 – 5 0(4)(b) Hence solve, for 0 x 360 , the equation12 tan 2x 5 cot x sec2 x 0Show each stage of your working and give your answers to one decimal place.(4)14*P60568A01432*

www.igexams.comLeaveblankQuestion 5 continued*P60568A01532*15Turn over

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www.igexams.comLeaveblankQuestion 5 continuedQ5(Total 8 marks)*P60568A01732*17Turn over

www.igexams.com6.Leaveblankyy f(x)POxFigure 2Figure 2 shows part of the graph with equation y f(x), wheref(x) 2 ½ 2x – 5 ½ 3x 0The vertex of the graph is at point P as shown.(a) State the coordinates of P.(2)(b) Solve the equation f(x) 3x – 2(4)Given that the equationf (x) kx 2where k is a constant, has exactly two roots,(c) find the range of values of k.(3)18*P60568A01832*

www.igexams.comLeaveblankQuestion 6 continued*P60568A01932*19Turn over

www.igexams.comLeaveblankQuestion 6 continued20*P60568A02032*

www.igexams.comLeaveblankQuestion 6 continuedQ6(Total 9 marks)*P60568A02132*21Turn over

www.igexams.comLeaveblank7.yOPxQRFigure 3Figure 3 shows a sketch of part of the curve with equationy 2 cos 3x – 3x 4x 0where x is measured in radians.The curve crosses the x‑axis at the point P, as shown in Figure 3.Given that the x coordinate of P is α,(a) show that α lies between 0.8 and 0.9(2)The iteration formulaxn 1 1arccos (1.5xn – 2)3can be used to find an approximate value for α.(b) Using this iteration formula with x1 0.8 find, to 4 decimal places, the value of(i) x2(ii) x5(3)The point Q and the point R are local minimum points on the curve, as shown in Figure 3.Given that the x coordinates of Q and R are β and λ respectively, and that they are the twosmallest values of x at which local minima occur,(c) find, using calculus, the exact value of β and the exact value of λ.22*P60568A02232*(6)

www.igexams.comLeaveblankQuestion 7 continued*P60568A02332*23Turn over

www.igexams.comLeaveblankQuestion 7 continued24*P60568A02432*

www.igexams.comLeaveblankQuestion 7 continuedQ7(Total 11 marks)*P60568A02532*25Turn over

www.igexams.com8.Leaveblank(i) Find, using algebraic integration, the exact value of 4232dx3x 1giving your answer in simplest form.(ii)h(x) Given h(x) Ax B C( x 1)2(4)2 x3 7 x 2 8 x 1( x 1)2x 1where A, B and C are constants to be found, find h ( x) d x(6)26*P60568A02632*

www.igexams.comLeaveblankQuestion 8 continued*P60568A02732*27Turn over

www.igexams.comLeaveblankQuestion 8 continued28*P60568A02832*

www.igexams.comLeaveblankQuestion 8 continuedQ8(Total 10 marks)*P60568A02932*29Turn over

www.igexams.comLeaveblank9.f(θ) 5 cos θ – 4 sin θθ (a) Express f(θ) in the form R cos (θ α), where R and α are constants, R 0 andπ0 α . Give the exact value of R and give the value of α, in radians, to 3 decimal2places.(3)The curve with equation y cos θ is transformed onto the curve with equation y f(θ) bya sequence of two transformations.Given that the first transformation is a stretch and the second a translation,(b) (i) describe fully the transformation that is a stretch,(ii) describe fully the transformation that is a translation.(2)Giveng(θ) (c) find the range of g.904 (f(θ ))2θ (2)30*P60568A03032*

www.igexams.comLeaveblankQuestion 9 continued*P60568A03132*31Turn over

www.igexams.comLeaveblankQuestion 9 continued(Total 7 marks)END32TOTAL FOR PAPER IS 75 MARKS*P60568A03232*Q9

International Advanced Level Pure Mathematics P3 Morning (Time: 1 hour 30 minutes) Paper Reference WMA13/01 Wednesday 22 January 2020 Pearson Edexcel International Advanced Level P60568A *P60568A0132* 2020 Pearson Education Ltd. 1/1/1/1/ Candidates may use any calculator permitted by Pearson regulations.

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