# Pearson Edexcel Centre Umber Candidate Umber International Advanced .

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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*

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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*

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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*

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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*

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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)

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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*

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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*

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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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