Cambridge IGCSE Mathematics - International 0607 — 2011 May/June Paper 2 · Variant 2

This document consists of 8 printed pages. IB11 06_0607_22/3RP © UCLES 2011 [Turn over *9433607138* For Examiner's Use UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/22 Paper 2 (Extended) May/June 2011 45 minutes Candidates answer on the Question Paper Additional Materials: Geometrical Instruments READ THESE INSTRUCTIONS FIRST Write your Centre number, candidate number and name on all the work you hand in. Write in dark blue or black pen. Do not use staples, paper clips, highlighters, glue or correction fluid. You may use a pencil for any diagrams or graphs. DO NOT WRITE IN ANY BARCODES. Answer all the questions. CALCULATORS MUST NOT BE USED IN THIS PAPER. All answers should be given in their simplest form. You must show all the relevant working to gain full marks and you will be given marks for correct methods even if your answer is incorrect. The number of marks is given in brackets [ ] at the end of each question or part question. The total number of marks for this paper is 40. www.XtremePapers.com

2 © UCLES 2011 0607/22/M/J/11 Formula List For the equation ax2 + bx + c = 0 x = 2 _ ± 4 2 _ b b ac a Curved surface area, A, of cylinder of radius r, height h. A = 2πrh Curved surface area, A, of cone of radius r, sloping edge l. A = πrl Curved surface area, A, of sphere of radius r. A = 4πr2 Volume, V, of pyramid, base area A, height h. V= 1 3 Ah Volume, V, of cylinder of radius r, height h. V = πr2h Volume, V, of cone of radius r, height h. V = 1 3 πr2h Volume, V, of sphere of radius r. V = 4 3 πr3 = = sin sin sin a b c A B C a2 = b2 + c2 – 2bc cos A Area = 1 2 bc sin A A a B C b c

3 © UCLES 2011 0607/22/M/J/11 [Turn over For Examiner's Use Answer all the questions. 1 (a) Work out 2 3 27 . Answer(a) [1] (b) Simplify 1 3 2 2 (9 ) c c × . Answer(b) [2] 2 The first four terms of a sequence are 1, 3, 9, 27. (a) Write down the next term of this sequence. Answer(a) [1] (b) Find an expression for the nth term of this sequence. Answer(b) [2] 3 The size of one interior angle of a regular polygon is 156°. Find the number of sides of the polygon. Answer [2]

4 © UCLES 2011 0607/22/M/J/11 For Examiner's Use 4 U = { x | 1 Y x Y 16, x ∈ k } A = { factors of 12 } B = { factors of 16 } Complete the following. (a) A = { } [1] (b) n (A ∩ B' ) = [1] 5 (a) Find the value of log2 8. Answer(a) [1] (b) Write the following as a single logarithm. 3log 2 – log 4 + 2log 5 Answer(b) [3]

5 © UCLES 2011 0607/22/M/J/11 [Turn over For Examiner's Use 6 Simplify fully 2 3 9 3 a a a a ÷ − − . Answer [3] 7 p = 2 3 −       q = 5 7 −       (a) Find p + q . Answer(a)           [2] (b) Work out │p + q│. Answer(b) [2]

6 © UCLES 2011 0607/22/M/J/11 For Examiner's Use 8 (a) Simplify 8 2 + 2 8 . Answer(a) [2] (b) Simplify by rationalising the denominator. 3 2 3 2 − Answer(b) [2] 9 The equation of a line passing through the point (2, 3) is ax + by = d, where a, b, d ∈ k. This line is perpendicular to the line y = 2x + 5. Find the values of a, b and d. Answer a = b = d = [3]

7 © UCLES 2011 0607/22/M/J/11 [Turn over For Examiner's Use 10 The cost of a mango is $m. The cost of a pineapple is $p. (a) Write an expression, in terms of m and p, for the cost of 2 mangoes and 3 pineapples. Answer(a) $ [1] (b) The cost of 2 mangoes and 3 pineapples is $13. The cost of 6 mangoes and 2 pineapples is $18. Write down two equations and solve them to find the cost of one mango and the cost of one pineapple. Answer(b) mango = $ pineapple = $ [4] 11 x is an obtuse angle and sin x = 1 2 . Find the exact value of cos x. Answer [2]

8 Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. University of Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. © UCLES 2011 0607/22/M/J/11 For Examiner's Use 12 The graph of y = f(x) where f(x) = ax2 + bx + 3 crosses the x-axis at (−3, 0) and (1, 0). The y coordinate of the vertex is 4. (a) On the axes, sketch the graph of y = f(x), for –4 Y x Y 4. 5 –5 –4 4 y x 0 [2] (b) Find the values of a and b. Answer(b) a = [1] b = [1] (c) Write down the range of f(x) when the domain is o. Answer(c) [1]

UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education MARK SCHEME for the May/June 2011 question paper for the guidance of teachers 0607 CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/22 Paper 2 (Extended), maximum raw mark 40 This mark scheme is published as an aid to teachers and candidates, to indicate the requirements of the examination. It shows the basis on which Examiners were instructed to award marks. It does not indicate the details of the discussions that took place at an Examiners’ meeting before marking began, which would have considered the acceptability of alternative answers. Mark schemes must be read in conjunction with the question papers and the report on the examination. • Cambridge will not enter into discussions or correspondence in connection with these mark schemes. Cambridge is publishing the mark schemes for the May/June 2011 question papers for most IGCSE, GCE Advanced Level and Advanced Subsidiary Level syllabuses and some Ordinary Level syllabuses. www.XtremePapers.com

Page 2 Mark Scheme: Teachers’ version Syllabus Paper IGCSE – May/June 2011 0607 22 © University of Cambridge International Examinations 2011 1 (a) (b) 9 3c2 B1 B2 B1 for 3ck or kc2 [3] 2 (a) (b) 81 3n – 1 oe B1 B2 Ignore extra terms If B0 award SC1 for any power of 3 in terms of n E.g. 3n or 3n + 1 [3] 3 15 B2 If B0 award M1 for 360 ÷ 24 soi or n n 180 ) 2 ( × − = 156 oe seen [2] 4 (a) (b) 1, 2, 3, 4, 6, 12 3 B1 B1ft Strict ft from (a). [2] 5 (a) (b) 3 For correct use of nloga = logan For correct use of loga + logb = log ab or B1 M1 E.g. log23, log8, log52, log25 loga – logb = log b a M1 Using their figures log50 www3 A1 [4] 6 a a a a a ) 3 ( ) 3 )( 3 ( 3 − × + − M1M1 M1 for correct factorizing of a2 – 9, M1 for inverting second fraction (with x) 3 3 + a final answer www3 A1 [3] 7 (a)       −4 3 B2 Award B1 for each correct number (b) 5 B2ft Not ±5 If B0 award M1 for their 32 + their (±4)2 Ignore absence of brackets. Final answer of ±5 or 25 implies M1. [4] 8 (a) 2 12 B2 If B0 award B1 for 2 4 seen (b) 7 6 2 9 + or ( ) 7 2 2 3 3 + B2 If B0 award M1 for intention of multiplying numerator and denominator by 2 3 + [4]

Page 3 Mark Scheme: Teachers’ version Syllabus Paper IGCSE – May/June 2011 0607 22 © University of Cambridge International Examinations 2011 9 a = 1, b = 2, d = 8 or positive multiples of the above. E.g. a = 2, b = 4, d = 16 B3 After B0, award wwwSC2 for a, b, d such that a:b:d = 1:2:8. E.g. a = 2 1 , b = 1, d = 4 or a = –1, b = –2, d = –8. After SC0, award M1 for a : b = 1 : 2, OR M1 for gradient = 2 1 − , (implied by y = x b d 2 1 − oe or y = c x + −2 1 oe (condone c = d) or a = 2 1 ± ) OR M1 for substitution of (2, 3) in their equation form E.g. 2a + 3b = d, (oe) or 3 = c x + − 2 2 1 , (oe) [3] 10 (a) (b) 2m + 3p m = 2, p = 3 B1 B4 B1 for 6m + 2p = 18 seen M1ft for correct method to eliminate one letter. Allow one numerical slip. A1, A1 www for answers [5] 11 2 3 − B2 If B0 scored, award B1 for (x =) 150, or SC1 for 2 3 or 0.866 [2] 12 (a) Sketch y x 3 0 1 -3 4 P2 For P marks condone absence of numbers only if axes are sectioned. P1 for x and y intercepts given P1 for smooth curve with maximum at y = 4. (Curve must go below y = 0 and be the correct shape) If P0, P0 scored then award SC1 for correct shape correctly positioned.

Page 4 Mark Scheme: Teachers’ version Syllabus Paper IGCSE – May/June 2011 0607 22 © University of Cambridge International Examinations 2011 (b) (c) a = −1 b = −2 f(x) ≤ 4 B1 B1 If B0, B0, award M1 for two of 9a – 3b + 3 = 0 oe, a + b + 3 = 0 oe or 4 = a – b + 3 and correct method to eliminate a or b. (Allow one numerical slip.) OR M1 for complete correct alternative method to evaluate a or b. (Allow one numerical slip.) E.g. y = a(x – p)(x – q) y = a(x – –3)(x – 1) 3 = ax3x – 1 a = –1 Accept y ≤ 4 Not x ≤ 4 or y < 4 [5]