Sample Assessment Materials September 2007 GCE Mathematics Edexcel Advanced Subsidiary GCE in Mathematics (8371) Edexcel Advanced Subsidiary GCE in Further Mathematics (8372) Edexcel Advanced Subsidiary GCE in Pure Mathematics (8373) Edexcel Advanced Subsidiary GCE in Further Mathematics (Additional) (8374) First examination 2009 Edexcel Advanced GCE in Mathematics (9371) Edexcel Advanced GCE in Further Mathematics (9372) Edexcel Advanced GCE in Pure Mathematics (9373) Edexcel Advanced GCE in Further Mathematics (Additional) (9374) First examination 2009 This document contains the new specimen assessment materials for the amended units FP1, FP2, FP3, D1 and D2 The specimen assessment materials for Core, Statistics and Mechanics units are contained in the previous issue of the specimen papers UA014392 (2004) Edexcel GCE e-Spec Your free e-Spec Everything you need in one CD Easy-to-use Contents A Introduction B Sample question papers 6667/01: Further Pure Mathematics FP1 6668/01: Further Pure Mathematics FP2 11 6669/01: Further Pure Mathematics FP3 15 6689/01: Decision Mathematics D1 19 6689/01: Decision Mathematics D1 Answer Booklet 31 6690/01: Decision Mathematics D2 43 6690/01: Decision Mathematics D2 Answer Booklet 51 C Sample mark schemes 63 Notes on marking principles 65 6667/01: Further Pure Mathematics FP1 67 6668/01: Further Pure Mathematics FP2 71 6669/01: Further Pure Mathematics FP3 77 6689/01: Decision Mathematics D1 83 6690/01: Decision Mathematics D2 89 Edexcel GCE in Mathematics © Edexcel Limited 2007 Sample Assessment Materials Sample Assessment Materials © Edexcel Limited 2007 Edexcel GCE in Mathematics A Introduction These sample assessment materials have been prepared to support the specification Their aim is to provide the candidates and centres with a general impression and flavour of the actual question papers and mark schemes in advance of the first operational examinations = Edexcel GCE in Mathematics © Edexcel Limited 2007 Sample Assessment Materials = Sample Assessment Materials © Edexcel Limited 2007 Edexcel GCE in Mathematics B Sample question papers 6667/01: Further Pure Mathematics FP1 6668/01: Further Pure Mathematics FP2 11 6669/01: Further Pure Mathematics FP3 15 6689/01: Decision Mathematics D1 19 6689/01: Decision Mathematics D1 Answer Booklet 31 6690/01: Decision Mathematics D2 43 6690/01: Decision Mathematics D2 Answer Booklet 51 = Edexcel GCE in Mathematics © Edexcel Limited 2007 Sample Assessment Materials Sample Assessment Materials © Edexcel Limited 2007 Edexcel GCE in Mathematics Paper Reference(s) 6667/01 Edexcel GCE Further Pure Mathematics FP1 Advanced Subsidiary/Advanced Sample Assessment Material Time: hour 30 minutes Materials required for examination Mathematical Formulae Items included with question papers Nil 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 formulas stored in them Instructions to Candidates In the boxes on the answer book, write your centre number, candidate number, your surname, initial(s) and signature Check that you have the correct question paper When a calculator is used, the answer should be given to an appropriate degree of accuracy Information for Candidates A booklet ‘Mathematical Formulae and Statistical Tables’ is provided Full marks may be obtained for answers to ALL questions The marks for individual questions and the parts of questions are shown in round brackets: e.g (2) There are questions in this question paper The total mark for this paper is 75 There are pages in this question paper Any blank pages are indicated Advice to Candidates You must ensure that your answers to parts of questions are clearly labelled You should show sufficient working to make your methods clear to the Examiner Answers without working may not gain full credit Printer’s Log No N31066A *N31066A* Turn over W850/????/57570 2/2/2/2/ This publication may be reproduced only in accordance with Edexcel Limited copyright policy ©2007 Edexcel Limited Edexcel GCE in Mathematics © Edexcel Limited 2007 Sample Assessment Materials f(x) = x3 – 3x2 + 5x – (a) Use differentiation to find f´(x) (2) The equation f(x) = has a root α in the interval 1.4 < x < 1.5 (b) Taking 1.4 as a first approximation to α, use the Newton-Raphson procedure once to obtain a second approximation to α Give your answer to decimal places (4) (Total marks) The rectangle R has vertices at the points (0, 0), (1, 0), (1, 2) and (0, 2) (a) Find the coordinates of the vertices of the image of R under the transformation given by the matrix A = a , where a is a constant 1 (3) (b) Find det A, giving your answer in terms of a (1) Given that the area of the image of R is 18, (c) find the value of a (3) (Total marks) The matrix R is given by R = 2 2 (a) Find R2 (2) (b) Describe the geometrical transformation represented by R2 (2) (c) Describe the geometrical transformation represented by R (1) (Total marks) Sample Assessment Materials © Edexcel Limited 2007 Edexcel GCE in Mathematics 6669/01: Further Pure Mathematics FP3 Question number (a) Scheme Marks AB = (− 1, 3, − 1) ; AC = (− 1, 3, 1) i j M1 A1 k AB × AC = − − −1 = i (3 + 3) + j (1 + 1) + k (− + 3) M1 A1 = 6i + j A1 Area of Δ ABC = = (b) AB × AC 36 + = 10 square units Volume of tetrahedron = M1 A1ft ( AD AB × AC = cubic units ⎯ ⎯→ (c) M1 A1 (2) ) − 12 + = (7) ⎯ ⎯→ Unit vector in direction AB × AC i.e perpendicular to plane containing A, B, and C is n= 40 (6i + j) = p = n ⋅ AD = = 10 (3i + j) M1 (3i + j) ⋅ (− 2i + j) 10 −6+4 = units 10 10 M1 A1 (3) = Edexcel GCE in Mathematics © Edexcel Limited 2007 Sample Assessment Materials 81 6669/01: Further Pure Mathematics FP3 Question number (a) Scheme x2 − a2 2x a − y2 b2 Marks =1 y dy =0 b dx M1 A1 dy x b b a secθ b = = = dx a y a b tan θ a sin θ M1 A1 a Gradient of normal is then − sin θ b a Equation of normal: ( y − b tan θ ) = − sin θ ( x − a secθ ) b ( ) ax sin θ + by = a + b tan θ (b) M1 A1 M: A normal cuts x = at y = (a ) + b2 tan θ b (6) M1 A1 a + b2 B normal cuts y = at x = tan θ a sin θ (a = + b2 a cos θ (a Hence M is ) ( ) A1 ) M1 + b2 a2 + b2 secθ , tan θ 2a 2b Eliminating θ M1 sec2 θ = + tan2 θ 2aX a2 + b2 2 2bY = 1+ A1 a2 + b2 [ 4a X − 4b 2Y = a + b ] A1 (7) = 82 Sample Assessment Materials © Edexcel Limited 2007 Edexcel GCE in Mathematics 6689/01: Decision Mathematics D1 Question Number Scheme Marks M1 + 10 reject top of list + 10 = Nicky = Trevor reject bottom of list 7+8 = Steve A1 A1 reject bottom of list [7] = Preety reject A1 Nigel not in list (a) (4) G–3=J–4=L–5 Change status: M1 G= 3–J=4–L=5 Improved matching: A1 E=2 G=3 J=4 L=5 B1 (3) (1) (b) e.g George and Yi Wen may both only be assigned to B1 (c) Y–3=G–2=E–4=J–1 M1 Change status: A1 Y = – G = – E = –J = Complete Matching E=4 G=2 J=1 L=5 Y=3 A1 (3) (7 marks) (a) (i) FH, AD, DE, CE, (not DC), BC , (not AC), CF, HI, (not EG M1 A1 A1 (3) FI), IJ (ii) AD, DE, EC, (b) BC , CF, FH, HI, IJ stop EG M1 A1 A1 (3) Start off the tree with AB and FI, then apply Kruskal M1 A1 (2) (8 marks) Edexcel GCE in Mathematics © Edexcel Limited 2007 Sample Assessment Materials 83 6689/01: Decision Mathematics D1 Question Number (a) Scheme Marks E.g: 650 650 650 710 710 431 643 710 650 650 245 710 643 643 643 643 455 455 455 455 455 431 431 431 452 710 245 245 452 431 234 234 452 245 245 162 162 234 234 234 452 452 162 162 162 M1 A1 134 134 134 134 134 A1 ft A1 ft A1 (5) (b) Bin 643 + 162 + 134 Bin 650 + 234 (c) Bin 710 + 245 Bin 431 Bin 455 + 452 M1 A1 A1 A1(ft) (4) 4116 = 4.116 bins needed optimal 1000 M1 A1(ft) (2) (11 marks) (a) e.g Each edge contributes to the sum of degree, hence this sum must be even B2, 1, Therefore there must be an even (or zero) number of vertices of odd degree (2) Hence there cannot be an odd number of vertices of odd degree (b) CD + FH = 200 + 220 = 420 M1 A1 CF + DH = 180 + 380 = 560 A1 CH + DF = 400 + 160 = 560 Repeat CA, AD and FH (c) A1 (4) Length = 4180 + 420 = 4600 m B1 (ft) (1) (9 marks) 84 Sample Assessment Materials © Edexcel Limited 2007 Edexcel GCE in Mathematics Edexcel GCE in Mathematics © Edexcel Limited 2007 Sample Assessment Materials 85 15 18 15 18 17 10 15 18 F 35 33 E 27 28 32 32 13 12 17 H 45 44 H 44 38 G 28 (44) 13 15 44 11 38 29 28 32 28 Eg A D F E G J or A C E G J; length 54 km 29 (c) 53 km Length: 14 10 53 53 (54) (55) J General explanation - trace back from J - Include arc XY if Y is already on path and if difference in trial labels equals length of arc Specific explanation 53 – 15 = 38 GJ 38 – = 32 EG 32 – = 28 FE 28 – 10 = 18 CF 18 – 18 = AC ACFEGJ 15 D 18 C Route: A 17 17 B Scheme (b) Question Number (a) 6689/01: Decision Mathematics D1 (5) = (9 marks) B1; B1 ft ( 2) B 2ft 1ft (2) A1 A1 ft A1 ft A1 M1 Marks 6689/01: Decision Mathematics D1 Question Number Scheme Marks (b) (c) B1 There must be fewer than 10 children B1 B2, 1, 2x + 3y ≥ 24 B1 x ≤ 2y (a) To show a strict inequality There must be between and 10 adults inclusive B1 (1) (3) (2) B1 ft (2x + 3y = 24) (d) B1 ft (x = 2y) B1 ft (shading) Children Adults - Passengers B1 M1 A1 (4) Minimum Maximum (e) Children 10 Adults - 19 Passengers B1 B1 (4) (14 marks) = 86 Sample Assessment Materials © Edexcel Limited 2007 Edexcel GCE in Mathematics Edexcel GCE in Mathematics © Edexcel Limited 2007 Sample Assessment Materials 87 (d) (e) (c) (b) (a) Question Number D(5) 21 C(11) 14 42 39 N(12) A1 M1 A1 (1) (2) C G, H, E, F Gantt Chart Day 15: Day 25: (3) (4) = (15 marks) B2, 1, B4 B1 M1 A1 ft (3) L(6) K(21) 54 54 (2) Float on F = 42 – 20 – 14 = F(20) 33 33 M(9) M1 A1 B1 ft 16 E(15) H(12) 45 45 Marks G–I–M A – H–K C Float on D = 21 – – 14 = B(14) I(1) 21 I(16) J(10) 29 10 G(8) 29 Scheme 10 A(16) 6689/01: Decision Mathematics D1 88 Sample Assessment Materials © Edexcel Limited 2007 Edexcel GCE in Mathematics 6690/01: Decision Mathematics D2 Question number (a) Scheme (By conservation of flow at B, C and D) x = 11 y=5 B3, 2ft 1ft z = 12 (√x – 6) (b) Marks (3) (√y + 7) Flow is 31 B1 (max flow = cut), cut through AB, AC and SD B1 (2) (5 marks) (a) b.v x y z r s Value Raw ops z 1 20 R1 ÷ s 120 R2 P 8 400 R3 + 20R1 M1 A1 2R1 M1 A1 ft A1 ft (5) (b) P + 8x 8y + 5r = 400 B1 ft (1) (c) Not optimal since there is a negative number in the profit row B1 ft (1) (7 marks) (a) D A E 20 F 26 B A1 14 C (b) M1 SA = SB = DD = 21 Sc = DE = 24 I13 = IAF = 16 DF = 18 M1 18 = A1 I21 = IBD = 18 + 21 = M1 I31 = ICD = 15 21 = A1 ft I32 = ICE = 19 (c) (2) 24 = 12 Eg CD(+) AD( ) D A E A1 ft AE(+) BE( ) BF(+) 18 12 B C 13 * CF( ) (5) = 14 M1 A1 ft F 20 Cost £1384 A1 ft A1 (4) 14 (11 marks) = Edexcel GCE in Mathematics © Edexcel Limited 2007 Sample Assessment Materials 89 6690/01: Decision Mathematics D2 Question number (a) Scheme Marks Deleting F leaves r.s.t M1 B r.s.t length = 86 A So lower bound = 86 + 16 + 19 = 14 18 15 E C 11 121 A1 28 G D M1 A1 Best LB is 129 by deleting C (c) B1 ft (1) Add 33 to BF and FB B1 Add 31 to DE and ED (b) (4) B1 Tour, visits each vertex, order correct using table of least M1 A1 distances e.g F C D A B E G F (2) (actual route F C D C A B E G F) A1 Upper bound of 138 km A1 (4) (11 marks) Let xij be number of units transported from i to j Where i {W, X, Y} and j {J, K, L] Warehouse Supermarket Objective minimise “c” = 3xWJ + 6xWK + 3xWL + 5xXJ + 8xXK + 4xXL + 2xYJ + 5xYK + 7xYL Subject to B1 xWJ + xWK + xWL = 34 B1 M1 A1 xXJ + xXK + xXL = 57 xYJ + xYK + xYL = 25 A1 xWJ + xXJ + xYJ = 20 xWK + xXK + xYK = 56 xWL + xXL + xYL = 40 xij ≥ ∀ i {W, X, Y} and j {J, K, L} B1 (6 marks) 90 Sample Assessment Materials © Edexcel Limited 2007 Edexcel GCE in Mathematics 6690/01: Decision Mathematics D2 Question number Scheme Stage Marks HK 18 * I IK 19 * J JK 21 * FH min(16, 18) = 16 FI (23, 19) = 19 * FJ min(17, 21) = 17 GH min(20, 18) = 18 GI min(15, 19) = 15 GJ min(28, 21) = 21 * B BG min(18, 21) = 18 * C CF min(25, 19) = 19 * CG min(16, 21) = 16 DF min(22, 19) = 19 * DG min(19, 21) = 19 * E EF min(14, 19) = 14 * A AB min(24, 18) = 18 AC min(25, 19) = 19 * AD min(27, 19) = 19 * AE Action H State min(23, 14) = 14 F G D Value Routes A C F I K, or A D F I K or A D G J K M1 A1 M1 A1 A1 A1 M1 A1 ft A1 ft A1 ft (9 marks) = Edexcel GCE in Mathematics © Edexcel Limited 2007 Sample Assessment Materials 91 6690/01: Decision Mathematics D2 Question number (a) Scheme Marks To maximise, subtract all entries from n ≥ 30 0 e.g 5 M1 Minimise uncovered element is 0 So 4 0 M1 A2ft 1ft M1 el = 0 0 0 el = 2 0 A B C D B C D 0 A2 ft ft A M1 A1 ft (2) (b) £1160 000 B2, 1, (2) (c) Gives other solution M1 A1 ft (2) (15 marks) = 92 Sample Assessment Materials © Edexcel Limited 2007 Edexcel GCE in Mathematics 6690/01: Decision Mathematics D2 Question number (a) Scheme I II I Max Since (b) Marks M1 A1 II 5 III 4 not stable Min Min max A1 (3) M1 A1 (2) Let A play I with probability p Let A play II with probability (1 If B plays I A’s gain are 5p + 3(1 If B plays II A’s gain are 2p + 5(1 If B plays III A’s gain are 3p + 4(1 p) p) = 2p + p) = p) = 3p p A 2, 1, (2) 6 2p + I 4 p III 3p II II III I p A should play I of time and II of time; value (to A) = Intersection of 2p + and p p= M1 A1ft A1 ft A1 ft (2) (15 marks) Edexcel GCE in Mathematics © Edexcel Limited 2007 Sample Assessment Materials 93 Further copies of this publication are available from Edexcel Publications, Adamsway, Mansfield, Notts, NG18 4FN Telephone: 01623 467467 Fax: 01623 450481 Email: publications@linneydirect.com Publications code UA019583 September 2007 For more information on Edexcel and BTEC qualifications please contact Customer Services on 0870 240 9800 or enquiries.edexcel.org.uk or visit our website: www.edexcel.org.uk Edexcel Limited Registered in England and Wales No 4496750 Registered Office: One90 High Holborn, London WC1V 7BH VAT Reg No 780 0898 07 ... Mathematics © Edexcel Limited 2007 Sample Assessment Materials Sample Assessment Materials © Edexcel Limited 2007 Edexcel GCE in Mathematics A Introduction These sample assessment materials have been prepared... Edexcel GCE in Mathematics © Edexcel Limited 2007 Sample Assessment Materials = Sample Assessment Materials © Edexcel Limited 2007 Edexcel GCE in Mathematics B Sample question papers 6667/01: Further... 28 Sample Assessment Materials © Edexcel Limited 2007 Edexcel GCE in Mathematics BLANK PAGE Edexcel GCE in Mathematics © Edexcel Limited 2007 Sample Assessment Materials 29 BLANK PAGE 30 Sample