Cambridge IGCSE Mathematics - Additional 0606 — 2009 May/June Paper 2 · Variant 1

This document consists of 6 printed pages and 2 blank pages. SP (SM) V07190/1 © UCLES 2009 [Turn over UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education READ THESE INSTRUCTIONS FIRST If you have been given an Answer Booklet, follow the instructions on the front cover of the Booklet. Write your Centre number, candidate number and name on all the work you hand in. Write in dark blue or black pen. You may use a soft pencil for any diagrams or graphs. Do not use staples, paper clips, highlighters, glue or correction fluid. Answer all the questions. Write your answers on the separate Answer Paper provided. Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question. The use of an electronic calculator is expected, where appropriate. You are reminded of the need for clear presentation in your answers. At the end of the examination, fasten all your work securely together. 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 80. * 3 0 5 8 0 2 7 7 3 2 * ADDITIONAL MATHEMATICS 0606/02 Paper 2 May/June 2009 2 hours Additional Materials: Answer Paper Graph paper (1 sheet) Electronic calculator Mathematical tables www.theallpapers.com

2 0606/02/M/J/09 © UCLES 2009 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation ax2 + bx + c = 0, x b b ac a = − − 2 4 2 . Binomial Theorem (a + b)n = an + ( n 1) an–1 b + ( n 2) an–2 b2 + … + ( n r) an–r br + … + bn, where n is a positive integer and ( n r ) = n! (n – r)!r! . 2. TRIGONOMETRY Identities sin2 A + cos2 A = 1. sec2 A = 1 + tan2 A. cosec2 A = 1 + cot2 A. Formulae for ∆ABC a sin A = b sin B = c sin C . a2 = b2 + c2 – 2bc cos A. ∆ = 1 2 bc sin A. www.theallpapers.com

3 0606/02/M/J/09 © UCLES 2009 [Turn over 1 (a) X
Y Express, in set notation, the set represented by the shaded region. [1] (b) In a class of 30 students, 17 are studying politics, 14 are studying economics and 10 are studying both of these subjects. (i) Illustrate this information using a Venn diagram. [1] Find the number of students studying (ii) neither of these subjects, [1] (iii) exactly one of these subjects. [1] 2 Given that A =
7 6 3 4, find A 1 − and hence solve the simultaneous equations 7x + 6y = 17, 3x + 4y = 3. [4] 3 Sketch the graph of y =  x2 – 8x + 12 . [4] 4 Find the coefficient of x4 in the expansion of (i) (1 + 2x)6, [2] (ii)
1 – x 4(1 + 2x)6. [3] www.theallpapers.com

4 0606/02/M/J/09 © UCLES 2009 5 Two variables, x and y, are related by the equation y = 6x2 + 32 x3 . (i) Obtain an expression for dy dx . [2] (ii) Use your expression to find the approximate change in the value of y when x increases from 2 to 2.04. [3] 6 The function f is defined by f(x) = 2 + x – 3 for x
3. Find (i) the range of f, [1] (ii) an expression for f –1(x). [2] The function g is defined by g(x) = 12 x + 2 for x  0. Find (iii) gf(12). [2] 7 Given that logp X = 9 and logpY = 6, find (i) logp X , [1] (ii) logp ( 1 X) , [1] (iii) logp (XY ), [2] (iv) logY X. [2] www.theallpapers.com

5 0606/02/M/J/09 © UCLES 2009 [Turn over 8 y x t units y = 27 – x2 P Q O S R The diagram shows part of the curve y = 27 – x2. The points P and S lie on this curve. The points Q and R lie on the x-axis and PQRS is a rectangle. The length of OQ is t units. (i) Find the length of PQ in terms of t and hence show that the area, A square units, of PQRS is given by A = 54t – 2t 3. [2] (ii) Given that t can vary, find the value of t for which A has a stationary value. [3] (iii) Find this stationary value of A and determine its nature. [3] 9 A musician has to play 4 pieces from a list of 9. Of these 9 pieces 4 were written by Beethoven, 3 by Handel and 2 by Sibelius. Calculate the number of ways the 4 pieces can be chosen if (i) there are no restrictions, [2] (ii) there must be 2 pieces by Beethoven, 1 by Handel and 1 by Sibelius, [3] (iii) there must be at least one piece by each composer. [4] 10 The line 2 x + y = 12 intersects the curve x2 + 3xy + y2 = 176 at the points A and B. Find the equation of the perpendicular bisector of AB. [9] 11 (a) Find all the angles between 0º and 360º which satisfy (i) 2sin x – 3cos x = 0, [3] (ii) 2sin2 y – 3cos y = 0. [5] (b) Given that 0  z  3 radians, find, correct to 2 decimal places, all the values of z for which sin(2z + 1) = 0.9. [3] www.theallpapers.com

6 0606/02/M/J/09 © UCLES 2009 12 Answer only one of the following two alternatives. EITHER The point P(0, 5) lies on the curve for which dy dx = e 1 2 x. The point Q, with x-coordinate 2, also lies on the curve. (i) Find, in terms of e, the y-coordinate of Q. [5] The tangents to the curve at the points P and Q intersect at the point R. (ii) Find, in terms of e, the x-coordinate of R. [5] OR B O A C D y y = e + 5 x x ¹² The diagram shows part of the curve y = e 1 2 x+ 5 crossing the y-axis at A. The normal to the curve at A meets the x-axis at B. (i) Find the coordinates of B. [4] The line through B, parallel to the y-axis, meets the curve at C. The line through C, parallel to the x-axis, meets the y-axis at D. (ii) Find the area of the shaded region. [6] www.theallpapers.com

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8 0606/02/M/J/09 BLANK PAGE 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. www.theallpapers.com

UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education MARK SCHEME for the May/June 2009 question paper for the guidance of teachers 0606 ADDITIONAL MATHEMATICS 0606/02 Paper 2, maximum raw mark 80 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. • CIE will not enter into discussions or correspondence in connection with these mark schemes. CIE is publishing the mark schemes for the May/June 2009 question papers for most IGCSE, GCE Advanced Level and Advanced Subsidiary Level syllabuses and some Ordinary Level syllabuses. www.theallpapers.com

Page 2 Mark Scheme: Teachers’ version Syllabus Paper IGCSE – May/June 2009 0606 02 © UCLES 2009 Mark Scheme Notes Marks are of the following three types: M Method mark, awarded for a valid method applied to the problem. Method marks are not lost for numerical errors, algebraic slips or errors in units. However, it is not usually sufficient for a candidate just to indicate an intention of using some method or just to quote a formula; the formula or idea must be applied to the specific problem in hand, e.g. by substituting the relevant quantities into the formula. Correct application of a formula without the formula being quoted obviously earns the M mark and in some cases an M mark can be implied from a correct answer. A Accuracy mark, awarded for a correct answer or intermediate step correctly obtained. Accuracy marks cannot be given unless the associated method mark is earned (or implied). B Accuracy mark for a correct result or statement independent of method marks. • When a part of a question has two or more "method" steps, the M marks are generally independent unless the scheme specifically says otherwise; and similarly when there are several B marks allocated. The notation DM or DB (or dep*) is used to indicate that a particular M or B mark is dependent on an earlier M or B (asterisked) mark in the scheme. When two or more steps are run together by the candidate, the earlier marks are implied and full credit is given. • The symbol √ implies that the A or B mark indicated is allowed for work correctly following on from previously incorrect results. Otherwise, A or B marks are given for correct work only. A and B marks are not given for fortuitously "correct" answers or results obtained from incorrect working. • Note: B2 or A2 means that the candidate can earn 2 or 0. B2, 1, 0 means that the candidate can earn anything from 0 to 2. www.theallpapers.com

Page 3 Mark Scheme: Teachers’ version Syllabus Paper IGCSE – May/June 2009 0606 02 © UCLES 2009 The following abbreviations may be used in a mark scheme or used on the scripts: AG Answer Given on the question paper (so extra checking is needed to ensure that the detailed working leading to the result is valid) BOD Benefit of Doubt (allowed when the validity of a solution may not be absolutely clear) CAO Correct Answer Only (emphasising that no "follow through" from a previous error is allowed) ISW Ignore Subsequent Working MR Misread PA Premature Approximation (resulting in basically correct work that is insufficiently accurate) SOS See Other Solution (the candidate makes a better attempt at the same question) Penalties MR -1 A penalty of MR -1 is deducted from A or B marks when the data of a question or part question are genuinely misread and the object and difficulty of the question remain unaltered. In this case all A and B marks then become "follow through √" marks. MR is not applied when the candidate misreads his own figures – this is regarded as an error in accuracy. OW -1,2 This is deducted from A or B marks when essential working is omitted. PA -1 This is deducted from A or B marks in the case of premature approximation. S -1 Occasionally used for persistent slackness – usually discussed at a meeting. EX -1 Applied to A or B marks when extra solutions are offered to a particular equation. Again, this is usually discussed at the meeting. www.theallpapers.com

Page 4 Mark Scheme: Teachers’ version Syllabus Paper IGCSE – May/June 2009 0606 02 © UCLES 2009 1 (a) ' ' or ) ( Y X Y X ∩ ∪ ' B1 (b) (i) B1 (ii) 9 B1 (iii) 11 B1 4 2       − − = − 7 3 6 4 10 1 1 A B2,1,0 premultiply       3 17 M1 x = 5 , y = –3 A1 4 3 Understands modulus M1 Curve from x I 2 to x K 6 A1 Cusp A1 Position correct A1 4 4 (i) 15 or 24 B1 240 B1 (ii) 160 B1 ( ) ( ) 160 4 1 240 ×    − + M1 200 A1 5 10 4 7 P E 9 www.theallpapers.com

Page 5 Mark Scheme: Teachers’ version Syllabus Paper IGCSE – May/June 2009 0606 02 © UCLES 2009 5 (i) 4 96 12 x x − oe B1+B1 (ii) uses x x y y ∂ × = ∂ d d with x = 2 M1 Substitute ∂x = 0.04 DM1 0.72 A1 5 6 (i) f(x) A 2 B1 (ii) Method for inverse M1 (x – 2)2 + 3 or x2 – 4x + 7 A1 (iii) g(5) or gf(x) ( ) 2 3 2 12 + − + = x B1 4.4 only B1 5 7 (i) 4.5 B1 (ii) –9 B1 (iii) log X + log Y M1 15 A1 (iv) Y X log log M1 1.5 A1 6 8 (i) 27 – t2 B1 2t × (27 – t2) B1 (ii) 2 6 54 d d t t A − = B1 Solve 0 d d = t A M1 t = 3 A1 (iii) Substitute for t in expression for A M1 A = 108 only A1 completely correct method and maximum B1 8 www.theallpapers.com

Page 6 Mark Scheme: Teachers’ version Syllabus Paper IGCSE – May/June 2009 0606 02 © UCLES 2009 9 (i) )1 ( 2 3 4 6 7 8 9 × × × × × × M1 126 A1 (ii) )1 ( 2 3 4 × × B1 2 3 )1 ( 2 3 4 × ×       × × M1 36 A1 (iii) adds number of arrangements of 2,1,1 and 1,2,1 and 1,1,2 only M1 multiplies for each selection M1 (36) + 4 × 3 × 2 + 4 × 3 (× 1) A1√ 72 A1 9 10 Eliminate y (or x) M1 x2 + 12x + 32 = 0 (or y2 – 48y + 560 = 0) A1 solve 3-term quadratic M1 x = –4 and x = –8 (or y = 20 and y = 28) A1 (–4, 20) and (–8, 28) A1√ M (–6, 24) M1 Grad AB = –2 B1 Uses grad perpendicular = AB m 1 − and coordinates of a point M1 27 2 1 + = x y or ) 6 ( 2 1 24 + = − x y or 0 54 2 = + − y x A1 9 11 (a) (i) tan x = 1.5 B1 56.3 B1 236.3 B1√ (ii) uses sin2 y = 1 – cos2 y M1 2cos2 y + 3cos y – 2 = 0 A1 solve 3-term quadratic M1 60 A1 300 A1√ (b) 1.12 B1 0.06 or 0.0599 B1 0.51 or 0.511 B1 11 www.theallpapers.com

Page 7 Mark Scheme: Teachers’ version Syllabus Paper IGCSE – May/June 2009 0606 02 © UCLES 2009 12 EITHER (i) integrates to find y M1 ) ( e 2 2 1 c x + A1 substitute (0, 5) into c k y x + = 2 1 e M1 c = 3 A1 (y-coord of Q is ) 2e + 3 A1√ (ii) uses y = mx + c M1 y = x + 5 A1 y = ex + 3 A1 solve linear equations M1 1 e 2 − = x A1 10 OR (i) A (0, 6) B1 x x y 2 1 e 2 1 d d = B1 uses m1m2 = –1 M1 B (3, 0) B1√ (ii) Integrates for area below curve M1 x x 5 2e 2 1 + A1 uses limits of 0 and xB M1 13 + 2e1.5 or 21.96 or 22 A1 Area rectangle = 3(e1.5 + 5) or 28.4(4..) M1 Area = e1.5 + 2 or 6.45 to 6.5 A1 10 www.theallpapers.com