Pearson Edexcel Centre umber Candidate umber Level 3 GCE ...

1 Centre NumberCandidate NumberWrite your name hereSurnameOther namesTotal MarksPaper Reference*P58349A0144*P58349A 2018 Pearson Education over Candidates may use any calculator allowed by the regulations of the Joint Council for Qualifications. Calculators must not have the facility for symbolic algebra manipulation, differentiation and integration, or have retrievable mathematical formulae stored in Use black ink or ball-point pen. If pencil is used for diagrams/sketches/graphs it must be dark (HB or B). Fill in the boxes at the top of this page with your name, Centre number and Candidate number. Answer all questions and ensure that your answers to parts of questions are clearly labelled. Answer the questions in the spaces provided there may be more space than you need.

2 You should show sufficient working to make your methods clear. Answers without working may not gain full credit. Answers should be given to three significant figures unless otherwise A booklet Mathematical Formulae and Statistical Tables is provided. There are 14 questions in this question paper. The total mark for this paper is 100. The marks for each question are shown in brackets use this as a guide as to how much time to spend on each Read each question carefully before you start to answer it. Try to answer every question. Check your answers if you have time at the endMathematicsAdvanced Paper 2: pure mathematics 2 Wednesday 13 June 2018 MorningTime: 2 hours9MA0/02 You must have:Mathematical Formulae and Statistical Tables, calculatorPearson Edexcel Level 3 GCE2*P58349A0244* DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREAA nswer ALL questions.

3 Write your answers in the spaces provided. (x) = 253xx+ x 5 (a) Find gg(5).(2) (b) State the range of g.(1) (c) Find g 1(x), stating its domain.(3)_____3*P58349A0344*Turn over DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREAQ uestion 1 continued_____(Total for Question 1 is 6 marks)4*P58349A0444* DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA2. Relative to a fixed origin O, the point A has position vector (2i + 3j 4k), 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 D is the point such that AB = BD.

4 (a) Find the position vector of D. (2) Given |AC | = 4 (b) find the value of a.(3)_____5*P58349A0544*Turn over DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREAQ uestion 2 continued_____(Total for Question 2 is 5 marks)6*P58349A0644* DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA3. (a) If m and n are irrational numbers, where m n, then mn is also irrational. Disprove this statement by means of a counter example.(2) (b) (i) Sketch the graph of y = |x| + 3 (ii) Explain why |x| + 3 |x + 3| for all real values of x.

5 (3)_____7*P58349A0744*Turn over DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREAQ uestion 3 continued_____(Total for Question 3 is 5 marks)8*P58349A0844* DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA4. (i) Show that 352131798116++()== rrr(4) (ii) A sequence u1, u2, u3, .. is defined by uuunn+==11123, Find the exact value of urr= 1100(3)_____9*P58349A0944*Turn over DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREAQ uestion 4 continued_____(Total for Question 4 is 7 marks)10*P58349A01044* DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA5.

6 The equation 2x3 + x2 1 = 0 has exactly one real root. (a) Show that, for this equation, the Newton-Raphson formula can be written xxxxxnnnnn+=+++13224162(3) Using the formula given in part (a) with x1 = 1 (b) find the values of x2 and x3(2) (c) Explain why, for this question, the Newton-Raphson method cannot be used with x1 = 0(1)_____11*P58349A01144*Turn over DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREAQ uestion 5 continued_____(Total for Question 5 is 6 marks)12*P58349A01244* DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS (x) = 3x3 + 8x2 9x + 10, x (a)

7 (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 = 0 (2) (c) deduce the number of real solutions, for 7 < 10 , to the equation3 tan3 8 tan2 + 9 tan 10 = 0(1)_____13*P58349A01344*Turn over DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREAQ uestion 6 continued_____14*P58349A01444* DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREAQ uestion 6 continued_____15*P58349A01544*Turn over DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREAQ uestion 6 continued_____(Total for Question 6 is 6 marks)

8 16*P58349A01644* DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA7. (i) Solve, for 0 x < 2 , the equation 4 sin x = sec x(4) (ii) Solve, for 0 < 360 , the equation5 sin 5 cos = 2 giving your answers to one decimal place. (Solutions based entirely on graphical or numerical methods are not acceptable.)(5)_____17*P58349A01744*Turn over DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREAQ uestion 7 continued_____18*P58349A01844* DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREAQ uestion 7 continued_____19*P58349A01944*Turn over DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREAQ uestion 7 continued_____(Total for Question 7 is 9 marks)

9 20*P58349A02044* DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS 1 Figure 1 is a graph showing the trajectory of a rugby ball. The height of the ball above the ground, H metres, has been plotted against the horizontal distance, x metres, measured from the point where the ball was kicked. The ball travels in a vertical plane. The ball reaches a maximum height of 12 metres and hits the ground at a point 40 metres from where it was kicked. (a) Find a quadratic equation linking H with x that models this situation.(3) The ball passes over the horizontal bar of a set of rugby posts that is perpendicular to the path of the ball. The bar is 3 metres above the ground.

10 (b) Use your equation to find the greatest horizontal distance of the bar from O.(3) (c) Give one limitation of the model.(1)_____21*P58349A02144*Turn over DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREAQ uestion 8 continued_____22*P58349A02244* DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREAQ uestion 8 continued_____23*P58349A02344*Turn over DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREAQ uestion 8 continued_____(Total for Question 8 is 7 marks)