# Pearson Edexcel Centre Umber Candidate Umber Level 3

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Answer ALL questions. Write your answers in the spaces provided.DO NOT WRITE IN THIS AREA1.yRO1x3Figure 1x, x 01 xDO NOT WRITE IN THIS AREAFigure 1 shows a sketch of the curve with equation y The finite region R, shown shaded in Figure 1, is bounded by the curve, the line withequation x 1, the x‑axis and the line with equation x 3The table below shows corresponding values of x and y for y x1 xx11.522.53y0.50.67420.82840.96861.0981(a) Use the trapezium rule, with all the values of y in the table, to find an estimate for thearea of R, giving your answer to 3 decimal places.(1)(c) Giving your answer to 3 decimal places in each case, use your answer to part (a) todeduce an estimate for(i)(ii)2 3 35xdx1 1 x1x 6 dx1 x *S60736A0232*(2)DO NOT WRITE IN THIS AREA(b) Explain how the trapezium rule can be used to give a better approximation for thearea of R.(3)

DO NOT WRITE IN THIS AREAQuestion 1 continuedDO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREA(Total for Question 1 is 6 marks)*S60736A0332*3Turn over

2. (a) Show that the binomial expansion of1in ascending powers of x, up to and including the term in x 2 is2 5x kx 24giving the value of the constant k as a simplified fraction.(b) (i) Use the expansion from part (a), with x Give your answer in the formpq1, to find an approximate value for10(4)2DO NOT WRITE IN THIS AREA(4 5x) 2where p and q are integers.(4)DO NOT WRITE IN THIS AREA1(ii) Explain why substituting x into this binomial expansion leads to a valid10approximation.4*S60736A0432*DO NOT WRITE IN THIS AREA

DO NOT WRITE IN THIS AREAQuestion 2 continuedDO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREA(Total for Question 2 is 8 marks)*S60736A0532*5Turn over

3. A sequence of numbers a1, a2, a3,. is defined byan 1 100(a) Findan 3,an 2n arr 1(3)100(b) Hence find99 a arr 1rr 1(1)DO NOT WRITE IN THIS AREAa1 3DO NOT WRITE IN THIS AREA6*S60736A0632*DO NOT WRITE IN THIS AREA

DO NOT WRITE IN THIS AREAQuestion 3 continuedDO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREA(Total for Question 3 is 4 marks)*S60736A0732*7Turn over

4. Relative to a fixed origin O,Given that ABCD is a parallelogram,(a) find the position vector of point D.(2) The vector AX has the same direction as AB. Given that ½ AX ½ 10 2 ,(b) find the position vector of X.DO NOT WRITE IN THIS AREAthe point A has position vector i 7j – 2k,the point B has position vector 4i 3j 3k,and the point C has position vector 2i 10j 9k.(3)DO NOT WRITE IN THIS AREA8*S60736A0832*DO NOT WRITE IN THIS AREA

DO NOT WRITE IN THIS AREAQuestion 4 continuedDO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREA(Total for Question 4 is 5 marks)*S60736A0932*9Turn over

5.f(x) x 3 ax 2 – ax 48, where a is a constant(a) (i) show that a 4(ii) express f(x) as a product of two algebraic factors.(4)Given that 2log2(x 2) log2 x – log2(x – 6) 3(b) show that x 3 4x 2 – 4x 48 0(4)DO NOT WRITE IN THIS AREAGiven that f (–6) 0(c) hence explain why2log2(x 2) log2 x – log2(x – 6) 3has no real roots.DO NOT WRITE IN THIS AREA(2)10*S60736A01032*DO NOT WRITE IN THIS AREA

DO NOT WRITE IN THIS AREAQuestion 5 continuedDO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREA(Total for Question 5 is 10 marks)*S60736A01132*11Turn over

6.C5mA6mFigure 2OBxFigure 3DO NOT WRITE IN THIS AREAyFigure 2 shows the entrance to a road tunnel. The maximum height of the tunnel ismeasured as 5 metres and the width of the base of the tunnel is measured as 6 metres.Figure 3 shows a quadratic curve BCA used to model this entrance.(a) Find an equation for curve BCA.(3)A coach has height 4.1 m and width 2.4 m.(b) Determine whether or not it is possible for the coach to enter the tunnel.(c) Suggest a reason why this model may not be suitable to determine whether or not thecoach can pass through the tunnel.(2)DO NOT WRITE IN THIS AREAThe points A, O, B and C are assumed to lie in the same vertical plane and the groundAOB is assumed to be horizontal.(1)12*S60736A01232*DO NOT WRITE IN THIS AREA

DO NOT WRITE IN THIS AREAQuestion 6 continuedDO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREA(Total for Question 6 is 6 marks)*S60736A01332*13Turn over

7.yDO NOT WRITE IN THIS AREARxOFigure 4Figure 4 shows a sketch of part of the curve with equationy 2e2x – xe2x,x Use calculus to show that the exact area of R can be written in the form pe4 q, wherep and q are rational constants to be found.(Solutions based entirely on graphical or numerical methods are not acceptable.)(5)DO NOT WRITE IN THIS AREAThe finite region R, shown shaded in Figure 4, is bounded by the curve, the x‑axis andthe y‑axis.14*S60736A01432*DO NOT WRITE IN THIS AREA

DO NOT WRITE IN THIS AREAQuestion 7 continuedDO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREA(Total for Question 7 is 5 marks)*S60736A01532*15Turn over

8. There were 2100 tonnes of wheat harvested on a farm during 2017.(a) Find the total mass of wheat expected to be harvested from 2017 to 2030 inclusive,giving your answer to 3 significant figures.(2)Each year it costs 5.15 per tonne to harvest the first 2000 tonnes of wheat 6.45 per tonne to harvest wheat in excess of 2000 tonnes(b) Use this information to find the expected cost of harvesting the wheat from 2017 to2030 inclusive. Give your answer to the nearest 1000DO NOT WRITE IN THIS AREAThe mass of wheat harvested during each subsequent year is expected to increase by1.2% per year.(3)DO NOT WRITE IN THIS AREA16*S60736A01632*DO NOT WRITE IN THIS AREA

DO NOT WRITE IN THIS AREAQuestion 8 continuedDO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREA(Total for Question 8 is 5 marks)*S60736A01732*17Turn over

9. The curve C has equationThe curve C passes through the point P(1, 7).Use differentiation from first principles to find the value of the gradient of the tangentto C at P.(5)DO NOT WRITE IN THIS AREAy 2x 3 5DO NOT WRITE IN THIS AREA18*S60736A01832*DO NOT WRITE IN THIS AREA

DO NOT WRITE IN THIS AREAQuestion 9 continuedDO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREA(Total for Question 9 is 5 marks)*S60736A01932*19Turn over

10. The function f is defined by3x 5,x 1x , x –1(a) Find f –1(x).(3)(b) Show thatff(x) x a,x 1x , x 1where a is an integer to be found.(4)DO NOT WRITE IN THIS AREAf:x The function g is defined byg : x x2 – 3x,x , 0 x 5(d) Find the range of g.(2)(3)(e) Explain why the function g does not have an inverse.(1)DO NOT WRITE IN THIS AREA(c) Find the value of fg(2).20*S60736A02032*DO NOT WRITE IN THIS AREA

DO NOT WRITE IN THIS AREAQuestion 10 continuedDO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREA(Total for Question 10 is 13 marks)*S60736A02132*21Turn over

11.yOBDO NOT WRITE IN THIS AREAAxFigure 5Given thatf′(x) k – 4x – 3x 2where k is constant,(a) show that C has a point of inflection at x –23(3)DO NOT WRITE IN THIS AREAFigure 5 shows a sketch of the curve C with equation y f(x).The curve C crosses the x‑axis at the origin, O, and at the points A and B as shown inFigure 5.Given also that the distance AB 4 2(b) find, showing your working, the integer value of k.(7)22*S60736A02232*DO NOT WRITE IN THIS AREA

DO NOT WRITE IN THIS AREAQuestion 11 continuedDO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREA(Total for Question 11 is 10 marks)*S60736A02332*23Turn over

12. Show that0sin 2θdθ 2 – 2 ln 21 cos θ(7)DO NOT WRITE IN THIS AREA π2DO NOT WRITE IN THIS AREA24*S60736A02432*DO NOT WRITE IN THIS AREA

DO NOT WRITE IN THIS AREAQuestion 12 continuedDO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREA(Total for Question 12 is 7 marks)*S60736A02532*25Turn over

13. (a) Express 2 sinθ – 1.5 cosθ in the form R sin(θ – α), where R 0 and 0 α State the value of R and give the value of α to 4 decimal places.(3)Tom models the depth of water, D metres, at Southview harbour on 18th October 2017by the formula 4πt 4πt – 1.5 cos ,D 6 2 sin 25 25 0 t 24where t is the time, in hours, after 00:00 hours on 18th October 2017.DO NOT WRITE IN THIS AREAπ2Use Tom’s model to(b) find the depth of water at 00:00 hours on 18th October 2017,(c) find the maximum depth of water,(1)(d) find the time, in the afternoon, when the maximum depth of water occurs.Give your answer to the nearest minute.(3)Tom’s model is supported by measurements of D taken at regular intervals on18th October 2017. Jolene attempts to use a similar model in order to model thedepth of water at Southview harbour on 19th October 2017.Jolene models the depth of water, H metres, at Southview harbour on 19th October 2017by the formula 4πx 4πx – 1.5 cos ,H 6 2 sin 25 25 DO NOT WRITE IN THIS AREA(1)0 x 24where x is the time, in hours, after 00:00 hours on 19th October 2017.(e) (i) explain why Jolene’s model is not correct,(ii) hence find a suitable model for H in terms of x.(3)26*S60736A02632*DO NOT WRITE IN THIS AREABy considering the depth of water at 00:00 hours on 19th October 2017 for both models,

DO NOT WRITE IN THIS AREAQuestion 13 continuedDO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREA*S60736A02732*27Turn over

Question 13 continuedDO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREA28*S60736A02832*DO NOT WRITE IN THIS AREA

DO NOT WRITE IN THIS AREAQuestion 13 continuedDO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREA(Total for Question 13 is 11 marks)*S60736A02932*29Turn over

14.yODO NOT WRITE IN THIS AREACxFigure 6Figure 6 shows a sketch of the curve C with parametric equations0 t 2πShow that a Cartesian equation of C can be written in the form(x y) 2 ay 2 bwhere a and b are integers to be found.(5)DO NOT WRITE IN THIS AREAπ x 4 cos t , y 2 sin t, 6 30*S60736A03032*DO NOT WRITE IN THIS AREA

DO NOT WRITE IN THIS AREAQuestion 14 continuedDO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREA*S60736A03132*31Turn over

Question 14 continuedDO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREA(Total for Question 14 is 5 marks)TOTAL FOR PAPER IS 100 MARKS32*S60736A03232*DO NOT WRITE IN THIS AREA

Pearson Edexcel Level 3 GCE. 2 *S60736A0232* T TE T AEA T TE T AEA T TE T AEA Answer ALL questions. Write your answers in the spaces provided. 1. y O 1 3 R x Figure 1 Figure 1 sh

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