Cambridge IGCSE Mathematics - Additional 0606 — 2010 May/June Paper 1 · Variant 1

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 Booklet/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 9 1 4 3 7 7 1 9 9 * ADDITIONAL MATHEMATICS 0606/11 Paper 1 May/June 2010 2 hours Additional Materials: Answer Booklet/Paper Electronic calculator This document consists of 5 printed pages and 3 blank pages. DC (LEO/KN) 25698/1 © UCLES 2010 [Turn over www.XtremePapers.com

2 0606/11/M/J/10 © UCLES 2010 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.

3 0606/11/M/J/10 © UCLES 2010 [Turn over 1 Differentiate with respect to x (i) 1 + x3 , [2] (ii) x2 cos 2x. [3] 2 (i) Find the first 3 terms of the expansion, in ascending powers of x, of (1 + 3x)6. [2] (ii) Hence find the coefficient of x2 in the expansion of (1 + 3x)6 (1 – 3x – 5x2). [3] 3 Find the set of values of k for which the equation x2 + (k – 2)x + (2k – 4) = 0 has real roots. [5] 4 (a) A B C  (i) Copy the Venn diagram above and shade the region that represents (A ∩ B) ∪ C. [1] (ii) Copy the Venn diagram above and shade the region that represents Aʹ ∩ Bʹ. [1] (iii) Copy the Venn diagram above and shade the region that represents (A ∪ B) ∩ C. [1] (b) It is given that the universal set  = {x : 2  x  20, x is an integer}, X = {x : 4 < x < 15, x is an integer}, Y = {x : x  9, x is an integer}, Z = {x : x is a multiple of 5}. (i) List the elements of X ∩ Y. [1] (ii) List the elements of X ∪ Y. [1] (iii) Find (X ∪ Y)ʹ ∩ Z. [1] 5 Solve the equation 3x(x2 + 6) = 8 – 17x2. [6]

4 0606/11/M/J/10 © UCLES 2010 6 Given that log8 p = x and log8 q = y, express in terms of x and/or y (i) log8 p + log8 q2, [2] (ii) log8  q–8, [2] (iii) log2 (64p). [3] 7 The function f is defined by f(x) = (2x + 1)2 – 3 for x  – 1 2 . Find (i) the range of f, [1] (ii) an expression for f–1 (x). [3] The function g is defined by g(x) = 3 1 + x for x > –1. (iii) Find the value of x for which fg(x) = 13. [4] 8 (a) Solve the equation (23 – 4x) (4x + 4) = 2. [3] (b) (i) Simplify 108 – 12 –– 3 , giving your answer in the form k 3 , where k is an integer. [2] (ii) Simplify 5 + 3 ––––– 5 – 2 , giving your answer in the form a 5 + b, where a and b are integers. [3] 9 (a) Variables x and y are related by the equation y = 5x + 2 – 4e–x. (i) Find dy –– dx . [2] (ii) Hence find the approximate change in y when x increases from 0 to p, where p is small. [2] (b) A square of area A cm2 has a side of length x cm. Given that the area is increasing at a constant rate of 0.5 cm2 s–1, find the rate of increase of x when A = 9. [4]

5 0606/11/M/J/10 © UCLES 2010 10 Solve (i) 4 sin x = cos x for 0° < x < 360°, [3] (ii) 3 + sin y = 3 cos2 y for 0° < y < 360°, [5] (iii) sec  z–3 = 4 for 0 < z < 5 radians. [3] 11 Answer only one of the following two alternatives. EITHER A curve has equation y = ln x ––– x2 , where x > 0. (i) Find the exact coordinates of the stationary point of the curve. [6] (ii) Show that d2y ––– dx2 can be written in the form a ln x + b –––––––– x4 , where a and b are integers. [3] (iii) Hence, or otherwise, determine the nature of the stationary point of the curve. [2] OR A curve is such that dy –– dx = 6 cos 2x + π–2 for – π–4  x  5π ––4 . The curve passes through the point  π–4, 5. Find (i) the equation of the curve, [4] (ii) the x-coordinates of the stationary points of the curve, [3] (iii) the equation of the normal to the curve at the point on the curve where x = 3π ––4 . [4]

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8 0606/11/M/J/10 © UCLES 2010 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.

UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education MARK SCHEME for the May/June 2010 question paper for the guidance of teachers 0606 ADDITIONAL MATHEMATICS 0606/11 Paper 11, 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 2010 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 2010 0606 11 © UCLES 2010 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 2010 0606 11 © UCLES 2010 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 2010 0606 11 © UCLES 2010 1 (i) 2 1 3 2 ) 1( 3 2 1 − + x x (ii) 2x cos2x – 2x2 sin2x B1,B1 [2] M1 A2,1,0 [3] B1 for 2 1 3) 1( 2 1 − + x B1 for × 3x2 M1 for attempt to differentiate a product –1 each error 2 (i) 1 + 18x + 135x2… (ii) (1 × –5) + (18 × –3) + (135 × 1) = 76 B1,B1 [2] M1,A1ft A1 [3] B2, 1, 0 –1 for each error M1 for a correct method using their (i) A1ft on their 3 terms unsimplified 3 (k – 2)2 – 4(2k – 4) k2 – 12k + 20 = 0 critical values 2 and 10 k Y 2 and k [ 10 M1 A1 M1 A1 A1 [5] M1 for use of discriminant for 3 term quadratic in k M1 for attempt to solve quadratic A1 for critical values A1 for range 4 (i), (ii) and (iii) (b) (i) {9,10,11,12,13,14} (ii) {5,6,7,8,9,10,11,12,13,14,15,16,17,18,19, 20} (iii) x or { } B1 B1 B1 [3] B1 B1 B1 [3] B1 for each correct Venn diagram Or equivalent Or equivalent 5 3x3 + 17x2 + 18x – 8 = 0 f(–2) = 0 (or other roots) (x + 2)(3x2 + 11x – 4)(= 0) (x + 2)(3x – 1)(x + 4)(= 0) x = –2, –4, 3 1 M1 M1 M1 DM1 B1, A1 [6] M1 for simplification = 0 M1 for attempt to find a root M1 for attempt to obtain quadratic factor DM1 for obtaining linear factors or use of quadratic formula B1 for first solution A1 for the other pair www.theallpapers.com

Page 5 Mark Scheme: Teachers’ version Syllabus Paper IGCSE– May/June 2010 0606 11 © UCLES 2010 6 (i) 2 1 x + 2y (ii) y – 1 (iii) 2 log log 2 log 64 log 8 8 8 8 p + = 6 + 3x B1 B1 [2] M1 A1 [2] M1 B1 A1 [3] B1 for each term M1 for difference of 2 logarithms M1 for attempt at a valid method B1 for 6 A1 for + 3x 7 (i) f [ –3 (ii) f–1 = 2 1 3 − + x (iii) 13 3 1 1 3 2 2 = −       +       + x 16 1 7 2 =       + + x x x = 1 B1 [1] M1 M1 A1 [3] M1 A1 M1 B1 [4] M1 for correct order of operations M1 for ‘interchange’ of x and y M1 for correct order A1 for correct simplification M1 for solution B1 for one solution only 8 (a) 23 – 4x 22x + 8 = 2 3 – 4x + 2x + 8 = 1 x = 5 (b) (i) 3 2 (ii) ( ) ( ) 2 5 2 5 2 5 5 3 + − + + leading to 1 11 5 5 + M1 DM1 A1 [3] M1 A1 [2] M1 A1 A1 [3] M1 for to obtain powers of 2, 4 or 8 DM1 for attempt to equate powers of 2, 4 or 8, using addition M1 for attempt to obtain each term in terms of 3 M1 for attempt to rationalise A1 for numerator A1 for denominator (can be implied) www.theallpapers.com

Page 6 Mark Scheme: Teachers’ version Syllabus Paper IGCSE– May/June 2010 0606 11 © UCLES 2010 9 (a) (i) x y d d = 5 + 4e–x (ii) when x = 0, x y d d = 9 use of dy ≈ x y d d dx leading to dy ≈ 9 p M1 A1 [2] M1 A1 [2] M1 for attempt to differentiate M1 for attempt to use small changes (b) td dA = 0.5 A = x2, x A d d = 2x, x = 3 t x d d = 0.5 × x 2 1 = 12 1 B1 M1 DM1 A1 [4] M1 for attempt to get x A d d DM1 for correct use of rates of change 10 (i) tan x = 0.25 x = 14.0°, 194.0° (ii) 3 + sin y = 3(1 – sin2 y) 3sin2 y + sin y = 0 sin y(3sin y + 1) = 0 sin y = 0, sin y = – 3 1 y = 180°, y = 199.5°, 340.5° (iii) cos 4 1 3 = z 3 z = 1.3181 leading to z = 3.95 Allow 3.96, 1.25π, 1.26π M1 A1,√A1 [3] M1 DM1 B1 A1 √A1 [5] B1 M1 A1 [3] M1 for use of tan M1 for use of correct identity and attempt to simplify DM1 for attempt to solve quadratic B1 for 180° A1 for 189.5° Ft on their 189.5° B1 for cos 4 1 3 = z or equivalent in terms of cos M1 for a correct order of operations (allow π) www.theallpapers.com

Page 7 Mark Scheme: Teachers’ version Syllabus Paper IGCSE– May/June 2010 0606 11 © UCLES 2010 11 EITHER (i) x y d d = 4 1 2 ln 2 ) ( x x x x x − 3 ln 2 1 x x − = when x y d d = 0, ln x = 2 1 , x = e 2 1 , y = e 2 1 , y = e 2 1 (ii) ( ) ( ) 6 2 2 3 2 2 3 ln 2 1 d d x x x x x y x − − − = = 4 ln 6 5 x x + − (iii) when x = e 2 1 , 2 2 d d x y is –ve (= 2 e 2 − ), max B3,2,1,0 M1 A1 A1 [6] M1 A1, A1 [3] M1 A1 [2] –1 each error M1 for attempt to solve x y d d = 0 M1 for attempt at 2nd derivative A1 for a, A1for b M1 for a correct method A1 must be from correct working only 11 OR (i) y = 3sin ( )( ) c x + + 2 π 2 5 = 3sin π + c, c = 5 y = 3sin ( ) 2 π 2 + x + 5 (ii) cos ( ) 2 π 2 + x = 0 x = 0, 2 π , π (iii) when x = 4 3π , y = 5 x y d d = 6 normal y – 5 = – 6 1 ( ) 4 3π − x       + − = 39 .5 6 1 x y M1, A1 M1, A1 [4] M1 A2,1,0 [3] M1 M1 DM1 A1 [4] M1 for sin ( ) 2 π 2 + x M1 for attempt to find c M1 for attempt to solve x y d d = 0 M1 for attempt to obtain y M1 for attempt to obtain x y d d and perp gradient DM1 for attempt at straight line (Must have (i) correct) www.theallpapers.com