Please write clearly in block capitals. Centre number Candidate number Surname _________________________________________________________________________ Forename(s) _________________________________________________________________________ Candidate signature _________________________________________________________________________ I declare this is my own work. A-level FURTHER MATHEMATICS Paper 2 Thursday 4 June 2020 Afternoon Time allowed: 2 hours Materials l l You must have the AQA formulae and statistical tables booklet for A‑level Mathematics and A‑level Further Mathematics. You should have a scientific calculator that meets the requirements of the specification. (You may use a graphical calculator.) Instructions l l l l l l l Use black ink or black ball‑point pen. Pencil should only be used for drawing. Fill in the boxes at the top of this page. Answer all questions. You must answer each question in the space provided for that question. If you require extra space for your answer(s), use the lined pages at the end of this book. Write the question number against your answer(s). Do not write outside the box around each page. Show all necessary working; otherwise marks for method may be lost. Do all rough work in this book. Cross through any work you do not want to be marked. Information l l l Mark 1 2 3 4 5 6 7 8 9 10 12 13 Unless stated otherwise, you may quote formulae, without proof, from the booklet. You do not necessarily need to use all the space provided. (JUN207367201) Question 11 The marks for questions are shown in brackets. The maximum mark for this paper is 100. Advice l For Examiner’s Use PB/Jun20/E7 14 15 TOTAL 7367/2 2 Do not write outside the box Answer all questions in the spaces provided. 1 Three of the four expressions below are equivalent to each other. Which of the four expressions is not equivalent to any of the others? Circle your answer. [1 mark] a (a þ b) 2 (a þ b) b (a b) b a (a b) Given that arg (a þ bi) ¼ j , where a and b are positive real numbers and 0 < j < p 2 , three of the following four statements are correct. Which statement is not correct? Tick (3) one box. [1 mark] arg ( a bi) ¼ p j arg (a bi) ¼ j arg (b þ ai) ¼ p 2 j arg (b ai) ¼ j (02) p 2 Jun20/7367/2 3 3 Do not write outside the box Find the gradient of the tangent to the curve y ¼ sin1 x at the point where x ¼ 1 5 Circle your answer. [1 mark] pffiffiffi 5 6 12 pffiffiffi 2 6 5 pffiffiffi 4 3 25 25 24 Turn over for the next question s (03) Turn over Jun20/7367/2 4 4 Do not write outside the box The matrices A and B are defined as follows: " A¼ 2 # x þ 2 3 " B¼ xþ1 x4 x2 0 # 2 Show that there is a value of x for which AB ¼ k I , where I is the 2 2 identity matrix and k is an integer to be found. [3 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (04) Jun20/7367/2 5 5 Do not write outside the box Solve the inequality 2x þ 3 xþ5 x1 [5 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ s (05) Turn over Jun20/7367/2 6 6 Find the sum of all the integers from 1 to 999 inclusive that are not square or cube numbers. [5 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (06) Jun20/7367/2 Do not write outside the box 7 7 The diagram shows part of the graph of y ¼ Do not write outside the box cos1 x y 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 – 0.2 O – 0.2 0.2 0.4 0.6 0.8 1.0 1.2 x The finite region enclosed by the graph of y ¼ cos1 x , the y-axis, the x-axis and the line x ¼ 0:8 is rotated by 2p radians about the x-axis. Use Simpson’s rule with five ordinates to estimate the volume of the solid formed. Give your answer to four decimal places. [5 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ s (07) Turn over Jun20/7367/2 8 8 (a) Do not write outside the box 2a þ b þ x x þ b x 2 þ b 2 2 as fully as possible. Factorise 0 a a 2 aþb b b [6 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (08) Jun20/7367/2 9 Do not write outside the box _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 8 (b) The matrix M is defined by 2 6 M¼4 13 þ x x þ 3 x 2 þ 9 0 5 8 3 3 7 25 5 9 Under the transformation represented by M, a solid of volume 0.625 m3 becomes a solid of volume 300 m3 Use your answer to part (a) to find the possible values of x. [3 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ s (09) Turn over Jun20/7367/2 10 9 a b The matrix C ¼ , where a and b are positive real numbers, b a 2p 3 3 1 6 2 27 6 7 2 and C ¼ 6 7 4 1 p3 5 2 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi pffiffiffiffi p mþn Use C to show that cos , where m and n are can be written in the form 2 12 integers. [7 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (10) Jun20/7367/2 Do not write outside the box 11 Do not write outside the box 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_____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Turn over for the next question s (11) Turn over Jun20/7367/2 12 10 Do not write outside the box The sequence u1 , u2 , u3 , . .. is defined by u1 ¼ 0 unþ1 ¼ 5 6 un Prove by induction that, for all integers n 1 , un ¼ 5n 5 5n 1 [6 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (12) Jun20/7367/2 13 Do not write outside the box _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ Turn over for the next question s (13) Turn over Jun20/7367/2 14 11 (a) Starting from the series given in the formulae booklet, show that the general term of the Maclaurin series for sin x x cos x is (1) rþ1 2r x 2r (2 r þ 1)! [4 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (14) Jun20/7367/2 Do not write outside the box 15 11 (b) Do not write outside the box Show that 2 sin x cos x 3 7 2 lim 6 x 5¼ 3 1 cos x x!0 4 [4 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ s (15) Turn over Jun20/7367/2 16 Do not write outside the box ðb 12 (a) Given that I ¼ e 2t sin t dt , show that a I ¼ [ qe 2t sin t þ re 2t cos t ]ba where q and r are rational numbers to be found. [6 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (16) Jun20/7367/2 17 12 (b) Do not write outside the box A small object is initially at rest. The subsequent motion of the object is modelled by the differential equation dv þ v ¼ 5 e t sin t dt where v is the velocity at time t. Find the speed of the object when t ¼ 2p , giving your answer in exact form. [6 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ s (17) Turn over Jun20/7367/2 18 13 Do not write outside the box Charlotte is trying to solve this mathematical problem: Find the general solution of the differential equation d 2 y dy þ 2 y ¼ 10 e 2x d x2 d x Charlotte’s solution starts as follows: Particular integral: y ¼ le 2x so dy ¼ 2 le 2x dx and d 2y ¼ 4 le 2x 2 dx 13 (a) Show that Charlotte’s method will fail to find a particular integral for the differential equation. [2 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (18) Jun20/7367/2 19 13 (b) Explain how Charlotte should have started her solution differently and find the general solution of the differential equation. [8 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ s (19) Turn over Jun20/7367/2 Do not write outside the box 20 14 Do not write outside the box The diagram shows the polar curve C1 with equation r ¼ 2 sin y The diagram also shows part of the polar curve C2 with equation r ¼ 1 þ cos 2 y O 14 (a) 2 Initial line On the diagram above, complete the sketch of C2 [2 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 14 (b) Show that the area of the region shaded in the diagram is equal to kp þ ma sin 2a þ q sin 4a where a ¼ sin 1 ! pffiffiffi 51 , and k, m and q are rational numbers. 2 [9 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (20) Jun20/7367/2 21 Do not write outside the box _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ s (21) Turn over Jun20/7367/2 22 15 The points A(7, 2, 8), B(7, 4, 0) and C(3, 3.2, 9.6) all lie in the plane P. 15 (a) Find a Cartesian equation of the plane P. Do not write outside the box [3 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (22) Jun20/7367/2 23 2 3 2 Do not write outside the box 3 15 (b) 15 5 6 7 6 7 The line L1 has equation r ¼ 4 0:4 5 þ m 4 3 5 4 4:8 15 (b) (i) Show that L1 lies in the plane P. [2 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 15 (b) (ii) Show that every point on L1 is equidistant from B and C. [4 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ s (23) Turn over Jun20/7367/2 24 15 (c) The line L2 lies in the plane P, and every point on L2 is equidistant from A and B. Find an equation of the line L2 [4 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (24) Jun20/7367/2 Do not write outside the box 25 15 (d) Do not write outside the box The points A, B and C all lie on a circle G. The point D is the centre of circle G. Find the coordinates of D. [3 marks] _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ END OF QUESTIONS (25) Jun20/7367/2 26 Do not write outside the box There are no questions printed on this page DO NOT WRITE ON THIS PAGE ANSWER IN THE SPACES PROVIDED (26) Jun20/7367/2 27 Do not write outside the box (27) Jun20/7367/2 28 Do not write outside the box Copyright information For confidentiality purposes, all acknowledgements of third-party copyright material are published in a separate booklet. This booklet is published after each live examination series and is available for free download from www.aqa.org.uk. Permission to reproduce all copyright material has been applied for. In some cases, efforts to contact copyright-holders may have been unsuccessful and AQA will be happy to rectify any omissions of acknowledgements. If you have any queries please contact the Copyright Team. Copyright ª 2020 AQA and its licensors. All rights reserved. (28) Jun20/7367/2 (206A7367/2)
