Cambridge IGCSE Mathematics - Additional 0606 — 2012 May/June Paper 2 · Variant 3

This document consists of 16 printed pages. DC (SLM) 57480 © UCLES 2012 [Turn over UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education * 1 1 9 4 7 1 7 3 6 9 * ADDITIONAL MATHEMATICS 0606/23 Paper 2 May/June 2012 2 hours Candidates answer on the Question Paper. Additional Materials: Electronic calculator 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. You may use a pencil for any diagrams or graphs. Do not use staples, paper clips, highlighters, glue or correction fluid. DO NOT WRITE IN ANY BARCODES. Answer all the questions. 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. For Examiner’s Use 1 2 3 4 5 6 7 8 9 10 11 12 Total www.theallpapers.com

2 0606/23/M/J/12 © UCLES 2012 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 0606/23/M/J/12 © UCLES 2012 [Turn over For Examiner’s Use 1 (i) Given that A =  4 –3 2 5, find the inverse matrix A–1. [2] (ii) Use your answer to part (i) to solve the simultaneous equations 4x – 3y = –10, 2x + 5y = 21. [2] www.theallpapers.com

4 0606/23/M/J/12 © UCLES 2012 For Examiner’s Use 2 A cuboid has a square base of side (2 + 3 ) cm and a volume of (16 + 9 3 ) cm3. Without using a calculator, find the height of the cuboid in the form (a + b 3 ) cm, where a and b are integers. [4] www.theallpapers.com

5 0606/23/M/J/12 © UCLES 2012 [Turn over For Examiner’s Use 3 (a) 12 10 8 6 4 2 O 180° y x The diagram shows a sketch of the curve y = asin(bx) + c for 0°  x  180°. Find the values of a, b and c. [3] (b) Given that f(x) = 5cos3x + 1, for all x, state (i) the period of f, [1] (ii) the amplitude of f. [1] www.theallpapers.com

6 0606/23/M/J/12 © UCLES 2012 For Examiner’s Use 4 (i) Find d dx(x2 lnx). [2] (ii) Hence, or otherwise, find xlnx dx. [3] www.theallpapers.com

7 0606/23/M/J/12 © UCLES 2012 [Turn over For Examiner’s Use 5 (a) Solve the equation 32x = 1000, giving your answer to 2 decimal places. [2] (b) Solve the equation 362y–5 63y = 62y–1 216 y+6. [4] www.theallpapers.com

8 0606/23/M/J/12 © UCLES 2012 For Examiner’s Use 6 By shading the Venn diagrams below, investigate whether each of the following statements is true or false. State your conclusions clearly. (i) A B = (A B) [2] B A  B A  A B (A B) (ii) X Y = X Y [2] Y X  Y X  X Y X Y (iii) (P Q) (Q R) = Q (P R) [2] Q R P  Q R P  (P Q) (Q R) Q (P R) www.theallpapers.com

9 0606/23/M/J/12 © UCLES 2012 [Turn over For Examiner’s Use 7 Given that f(x) = x2 – 648 x , find the value of x for which f (x) = 0. [6] www.theallpapers.com

10 0606/23/M/J/12 © UCLES 2012 For Examiner’s Use 8 Relative to an origin O, the position vectors of the points A and B are 2i – 3j and 11i + 42j respectively. (i) Write down an expression for → AB . [2] The point C lies on AB such that → AC = 1 3 → AB . (ii) Find the length of → OC. [4] The point D lies on → OA such that → DC is parallel to → OB. (iii) Find the position vector of D. [2] www.theallpapers.com

11 0606/23/M/J/12 © UCLES 2012 [Turn over For Examiner’s Use 9 A particle moves in a straight line so that, t s after passing through a fixed point O, its velocity, v ms–1, is given by v = 2t – 11 + 6 t + 1 . Find the acceleration of the particle when it is at instantaneous rest. [7] www.theallpapers.com

12 0606/23/M/J/12 © UCLES 2012 For Examiner’s Use 10 Solutions to this question by accurate drawing will not be accepted. O D C (–3, 2) B (7, 7) A (11, 4) y x The diagram shows a trapezium ABCD with vertices A(11, 4), B(7, 7), C(–3, 2) and D. The side AD is parallel to BC and the side CD is perpendicular to BC. Find the area of the trapezium ABCD. [9] www.theallpapers.com

13 0606/23/M/J/12 © UCLES 2012 [Turn over For Examiner’s Use 11 B A C D 1 rad 12 cm The diagram shows a right-angled triangle ABC and a sector CBDC of a circle with centre C and radius 12 cm. Angle ACB = 1 radian and ACD is a straight line. (i) Show that the length of AB is approximately 10.1 cm. [1] (ii) Find the perimeter of the shaded region. [5] (iii) Find the area of the shaded region. [4] www.theallpapers.com

14 0606/23/M/J/12 © UCLES 2012 For Examiner’s Use 12 Answer only one of the following two alternatives. EITHER The equation of a curve is y = 2x2 – 20x + 37. (i) Express y in the form a(x + b)2 + c, where a, b and c are integers. [3] (ii) Write down the coordinates of the stationary point on the curve. [1] A function f is defined by f : x  2x2 – 20x + 37 for x  k. Given that the function f–1(x) exists, (iii) write down the least possible value of k, [1] (iv) sketch the graphs of y = f(x) and y = f–1(x) on the axes provided, [2] (v) obtain an expression for f–1. [3] OR A function g is defined by g : x  5x2 + px + 72, where p is a constant. The function can also be written as g : x  5(x – 4)2 + q. (i) Find the value of p and of q. [3] (ii) Find the range of the function g. [1] (iii) Sketch the graph of the function on the axes provided. [2] (iv) Given that the function h is defined by h : x  ln x, where x  0, solve the equation gh(x) = 12. [4] Start your answer to Question 12 here. Indicate which question you are answering. EITHER OR … … … … … … … … www.theallpapers.com

15 0606/23/M/J/12 © UCLES 2012 [Turn over Continue your answer here. … … … … … … … … … … … … … … … … … O y x www.theallpapers.com

16 0606/23/M/J/12 © UCLES 2012 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. Continue your answer here if necessary. … … … … … … … … … … … … … … … … … … … … … … … … … www.theallpapers.com

UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education MARK SCHEME for the May/June 2012 question paper for the guidance of teachers 0606 ADDITIONAL MATHEMATICS 0606/23 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. • Cambridge will not enter into discussions or correspondence in connection with these mark schemes. Cambridge is publishing the mark schemes for the May/June 2012 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 2012 0606 23 © University of Cambridge International Examinations 2012 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 2012 0606 23 © University of Cambridge International Examinations 2012 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 2012 0606 23 © University of Cambridge International Examinations 2012 1 (i) 7 ∈ P B1 (ii) 8 ∉ S B1 (iii) ( ) n 6 N S ∩ = B1 2 (i) 4 1 k x + oe 6 k = allow unsimplified M1 A1 (ii) Use ( 6) x dy y p dx = ∂= × 30p M1 A1√ 3 Eliminate y ( ) 2 3 9 0 x m x + − + = Use 3 m = − or 9b2 – 4ac ≈ 0 Solve for 2 values of m Solve for 2 values of m OR Eliminate y and m ( ) ( ) 2 2 3 5 3 4 x x x x + − = + + Solve quadratic for x Solve for 2 values of m M1 A1 M1 M1 A1 4 (i) 5 4 1 7 3 2 5 1 1         or transpose 8 2 5 2 4 4 3 6 2    +     or transpose B1 B1+B1 (ii) 30 40 or 26 56             or 30 40 and y y             or and 26 56 x x             from correct part (i) Claire 70 and Denise 82 B1 B1 5 (i) ( ) (2) 8 4 16 8 0 f k = + − − = 4 k = M1 A1 (ii) Find quadratic factor 2 6 4 x x + + Use quadratic formula or completing square ( ) 2 6 6 4 1 4 2 −± − × × 3 5 −± M1 A1 M1 A1 B1√ www.theallpapers.com

Page 5 Mark Scheme: Teachers’ version Syllabus Paper IGCSE – May/June 2012 0606 23 © University of Cambridge International Examinations 2012 6 (a) (i) ( ) 3 2 or 35 − 280 − (ii) ( ) 2 2 2 21 x × ( ) ( ) 3 280 4 84 × − + × –504 B1 B1 B1 M1 A1 (b) Identify 2 4 2 3 x x   ×    15 × only with 0x 135( ) 0x B1 B1 B1 7 (i) ln ln ln y a b x = + OR lg lg lg y a b x = + may be implied Plot ln / lg y y against ln / lg x x with attempt at linear scale ln x 1.61 3.40 5.01 5.99 ln y 2.19 3.09 3.89 4.39 lg x 0.70 1.48 2.18 2.60 lg y 0.95 1.34 1.69 1.91 (Marks for points and line on graph NOT for table) B1 M1 A2, 1, 0 (ii) Calculates gradient of straight line log graph 0.5 0.03 b = ± Intercept is ln a or lg a on straight line log graph 4 0.3 a = ± M1 A1 M1 A1 (iii) Uses suitable graph or formula 32 to 49 M1 A1 8 2 256 x h = 2 4 A x xh = + 2 1024 A x x = + 2 d 1024 2 d A x x x = − oe Equate to 0 and solve 8 4 x h = = OR 256 64 A h h = + 2 d 256 32 d A h h h − = + 4 8 h x = = B1 M1 A1 A1√ M1 A1 A1 www.theallpapers.com

Page 6 Mark Scheme: Teachers’ version Syllabus Paper IGCSE – May/June 2012 0606 23 © University of Cambridge International Examinations 2012 9 (a) (i) 5 tan 3 x = 59(.0) x = 239(.0) x = and no others B1 B1 B1√ (ii) Use sin2 y = 1 – cos2 y 2 5cos 9cos 2 0 y y − − = Solve 3 term quadratic (in cosy) 101.5 258.5 and no others B1 B1 M1 A1 B1√ (b) ( ) 3 z − = 0.927 or 0.93 2.07 ( ) 3 z − = 0.927 π −  0.786 or 0.785 or 0.79 and no others B1 B1 M1 A1 10 (a) (i) 792 B1 (ii) 4W, 3M and 5W, 2M 5 35 × or (1) 21 × 196 M1 B1 A1 (b) (i) 4 5 4 3 × × × or 2 6 5 4 3 3 × × × × 240 M1 A1 (ii) 4 4 3 1 × × × or ( ) 1 240 5 × 48 M1 A1 www.theallpapers.com

Page 7 Mark Scheme: Teachers’ version Syllabus Paper IGCSE – May/June 2012 0606 23 © University of Cambridge International Examinations 2012 11 E (i) d 1 cos d 2 y k x x = 1 1 cos 2 2 x       Gradient tangent 1 2 4 − or –0.35… 2 1 3 2 2 4 2 y x π   − = − −     3 0, 2 or 6.71 2 y x π = = + M1 A1 M1 A1 (ii) METHOD A 1 1 sin d 2cos 2 2 x x x = − ∫ Identify 2π Use limits of 1.5π and ( ) 2π on 1 cos 2 k x ( ) 2 2 or 0.586 − Attempt at area of triangle 2 0.707 2   = =       Idea of area of triangle subtract area under curve. Plan completely correct 3 2 2 2 − or 0.121 B1 B1 M1 M1 M1 M1 A1 METHOD B Using integral of (equation of line −equation of curve) 1 1 sin d 2cos 2 2 x x x = − ∫ Identify 2π Use limits of 1.5π and ( ) 2π on 1 cos 2 k x Use limits of 1.5π and ( ) Q x on integral of equation of line Plan completely correct 3 2 2 2 − or 0.121 M1 B1 B1 M1 M1 M1 A1 www.theallpapers.com

Page 8 Mark Scheme: Teachers’ version Syllabus Paper IGCSE – May/June 2012 0606 23 © University of Cambridge International Examinations 2012 11 O (i) Uses product rule ( ) 1 e x x − − ( ) 1 e d e x x x x x − − − = ∫ e d e e d x x x x x x x − − − = − + ∫ ∫ = e e x x x − − = − − M1 A1 M1 A1ag (ii) gradient tangent 2 1 e = − or 0.135 = − ) 2 ( e 1 e 2 2 2 − − = − x y Uses line cuts y-axis at 2 4 e or 0.541 Area trapezium 2 6 or 0.812 e   =     Uses limits of 2 and 0 on e e x x x − − − − (= 2 3 1 e − or 0.594) Evaluate area of trapezium subtract area under curve 2 9 1 e − or 0.218 B1 M1 A1 M1 M1 M1 A1 www.theallpapers.com