Cambridge IGCSE Mathematics - Additional 0606 — 2010 Oct/Nov Paper 2 · Variant 3

This document consists of 7 printed pages and 1 blank page. DC (CW/DJ) 34220 © UCLES 2010 [Turn over UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education * 8 5 7 9 3 0 0 3 9 0 * ADDITIONAL MATHEMATICS 0606/23 Paper 2 October/November 2010 2 hours Additional Materials: Answer Booklet/Paper Graph paper (1 sheet) Electronic calculator 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. www.theallpapers.com

2 0606/23/O/N/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 www.theallpapers.com

3 0606/23/O/N/10 © UCLES 2010 [Turn over 1 The two variables x and y are such that y = 10 (x + 4)3 . (i) Find an expression for dy d x . [2] (ii) Hence find the approximate change in y as x increases from 6 to 6 + p, where p is small. [2] 2 Find the equation of the curve which passes through the point (4, 22) and for which dy d x = 3x(x – 2). [4] 3 (a) 0 0º 60º 120º 180º 240º 300º 360º x 1 2 3 4 5 6 7 8 9 10 y The diagram shows the curve y = A cos Bx + C for 0°  x  360°. Find the value of (i) A, (ii) B, (iii) C. [3] (b) Given that f(x) = 6 sin 2x + 7, state (i) the period of f, [1] (ii) the amplitude of f. [1] www.theallpapers.com

4 0606/23/O/N/10 © UCLES 2010 4 (i) Find, in ascending powers of x, the first 4 terms of the expansion of (1 + x)6. [2] (ii) Hence find the coefficient of p3 in the expansion of (1 + p – p2)6. [3] 5 (a) Given that A = (2 –4 1) and B =  3 –1 0 5 –2 7 , find the matrix product AB. [2] (b) Given that C =  3 5 –2 –4 and D =  6 –4 2 8 , find (i) the inverse matrix C–1, [2] (ii) the matrix X such that CX = D. [2] 6 (a) A  B Copy the diagram above and shade the region which represents the set A  B. [1] (b) The sets P, Q and R are such that P  Q =  and P  Q  R. Draw a Venn diagram showing the sets P, Q and R. [2] (c) In a group of 50 students F denotes the set of students who speak French and S denotes the set of students who speak Spanish. It is given that n(F) = 24, n(S) = 18, n(F  S) = x and n(F  S) = 3x. Write down an equation in x and hence find the number of students in the group who speak neither French nor Spanish. [3] 7 The line y = 2x – 6 meets the curve 4x2 + 2xy – y2 = 124 at the points A and B. Find the length of the line AB. [7] www.theallpapers.com

5 0606/23/O/N/10 © UCLES 2010 [Turn over 8 (i) Show that (5 + 3√⎯2)2 = 43 + 30√⎯2. [1] Hence find, without using a calculator, the positive square root of (ii) 86 + 60√⎯2, giving your answer in the form a + b√⎯2, where a and b are integers, [2] (iii) 43 – 30√⎯2, giving your answer in the form c + d√⎯2, where c and d are integers, [1] (iv) 1 43 + 30√⎯2, giving your answer in the form f + g√⎯2 h , where f, g and h are integers. [3] 9 X A B C O D 30 cm 8 cm The diagram shows a rectangle ABCD and an arc AXB of a circle with centre at O, the mid-point of DC. The lengths of DC and BC are 30 cm and 8 cm respectively. Find (i) the length of OA, [2] (ii) the angle AOB, in radians, [2] (iii) the perimeter of figure ADOCBXA, [2] (iv) the area of figure ADOCBXA. [2] 10 The equation of a curve is y = x2ex. The tangent to the curve at the point P(1, e) meets the y-axis at the point A. The normal to the curve at P meets the x-axis at the point B. Find the area of the triangle OAB, where O is the origin. [9] www.theallpapers.com

6 0606/23/O/N/10 © UCLES 2010 11 B X Q P A O b a In the diagram ⎯→ OA = a, ⎯→ OB = b, ⎯→ OP = 2a and ⎯→ OQ = 3b. (i) Given that ⎯→ AX = l ⎯→ AQ, express ⎯→ OX in terms of l, a and b. [3] (ii) Given that ⎯→ BX = k ⎯→ BP, express ⎯→ OX in terms of k, a and b. [3] (iii) Hence find the value of l and of k. [3] www.theallpapers.com

7 0606/23/O/N/10 © UCLES 2010 12 Answer only one of the following two alternatives. EITHER The table shows values of the variables v and p which are related by the equation p = a v2 + b v , where a and b are constants. v 2 4 6 8 p 6.22 2.84 1.83 1.35 (i) Using graph paper, plot v2 p on the y-axis against v on the x-axis and draw a straight line graph. [2] (ii) Use your graph to estimate the value of a and of b. [4] In another method of finding a and b from a straight line graph, 1 v is plotted along the x-axis. In this case, and without drawing a second graph, (iii) state the variable that should be plotted on the y-axis, [2] (iv) explain how the values of a and b could be obtained. [2] OR The table shows experimental values of two variables r and t. t 2 8 24 54 r 22 134 560 1608 (i) Using the y-axis for ln r and the x-axis for ln t, plot ln r against ln t to obtain a straight line graph. [2] (ii) Find the gradient and the intercept on the y-axis of this graph and express r in terms of t. [6] Another method of finding the relationship between r and t from a straight line graph is to plot lg r on the y-axis and lg t on the x-axis. Without drawing this second graph, find the value of the gradient and of the intercept on the y-axis for this graph. [2] www.theallpapers.com

8 0606/23/O/N/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. www.theallpapers.com

UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education MARK SCHEME for the October/November 2010 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. • CIE will not enter into discussions or correspondence in connection with these mark schemes. CIE is publishing the mark schemes for the October/November 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 – October/November 2010 0606 23 © 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 – October/November 2010 0606 23 © 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 – October/November 2010 0606 23 © UCLES 2010 1 ( ) 4 4 d d − + = x k x y k = –30 Use x x y y ∂ × = ∂ d d with x = 6 and p x = ∂ –0.003p or 1000 3 − p M1 A1 M1 A1√ [4] 2 Integrate 3x2 – 6x (y =)x3 – 3x2 (+ c) Substitute x = 4, y = 22 y = x3 – 3x2 + 6 M1 A1 M1 A1 [4] 3 (a) (i) 4 B1 (ii) 3 B1 (iii) 5 B1 (b) (i) π or 180 B1 (ii) 6 B1 [5] 4 (i) 1 + 6x + 15x2 + 20x3 B2,1,0 (ii) Substitute x = p – p2 Multiply out brackets to obtain terms in p3. (–30 p3 + 20p3) –10 M1 M1 A1 [5] 5 (a) ( ) 15 4 − B1+B1 (b) (i)     − − = − 3 2 5 4 2 1 1 C or     − − − = − 3 2 5 4 2 1 1 C B1+B1 (ii) X = C–1D evaluated       − − 8 9 12 17 M1 A1 [6] www.theallpapers.com

Page 5 Mark Scheme: Teachers’ version Syllabus Paper IGCSE – October/November 2010 0606 23 © UCLES 2010 6 (a) B1 (b) B1+B1 (c) 24 – x + x + 18 – x + 3x = 50 or 24 + 18 – x + 3x = 50 Solve for x (4) 12 B1 M1 A1 [6] 7 Eliminate y (or x) 4x2 + 12x – 160 = 0 (or y2 + 18y – 88 = 0) Factorise 3 term quadratic x = 5 and –8 (or y = –22 and 4) y = –22 and 4 (or x = 5 and –8) Use Pythagoras 29.1 or 845 or 5 13 M1 A1 M1 A1 A1√ M1 A1 [7] R P B A Q www.theallpapers.com

Page 6 Mark Scheme: Teachers’ version Syllabus Paper IGCSE – October/November 2010 0606 23 © UCLES 2010 8 (i) ( ) 18 2 30 25 2 3 5 2 + + = + or 18 2 15 2 15 25 + + + B1 AG (ii) ( ) ( ) 2 3 5 2 2 30 43 2 + = + 2 5 6 + M1 A1 (iii) ( ) 2 3 5 − B1 (iv) 2 3 5 1 + 2 3 5 2 3 5 2 3 5 1 − − × + 7 2 3 5 − alternative for last 3 marks 2 30 43 2 30 43 2 30 43 1 − − × + 49 2 30 43− 7 2 3 5 − B1 M1 A1 (M1) (A1) (A1) [7] 9 (i) 2 2 15 8 + AO = 17 M1 A1 (ii) ( ) 15 8 arctan 2 − = π AOB or       × × − + 17 17 2 30 17 17 arccos 2 2 2 AOB = 2.16 M1 A1 (iii) Complete, correct plan with s = rθ 82.7 M1 A1 (iv) Complete, correct plan with θ 2 2 1 r A = 432 M1 A1 [8] www.theallpapers.com

Page 7 Mark Scheme: Teachers’ version Syllabus Paper IGCSE – October/November 2010 0606 23 © UCLES 2010 10 Use product rule 2xex + x2ex Evaluate gradient of tangent at P e + 2e = 3e Equation tangent y – e = 3e(x – 1) or y = 3ex – 2e At A, e 2 0 − = ⇒ = y x or –5.44 Use m1m2 = –1, equation normal is ( )1 e 3 1 e − − = − x y At B, 1 e 3 0 2 + = ⇒ = x y or 23.2 Area OAB = e e 3 3 + or 63(.0) or 63.1 M1 A1 M1 A1 M1 A1 M1 A1 A1 [9] 11 (i) a b AQ − = 3 AQ OA OX µ + = ( ) a b a − + 3 µ M1 M1 A1 (ii) b a BP − = 2 BP OB OX λ + = ( ) b a b − + 2 λ M1 M1 A1 (iii) Equate vectors and solve       − = = − λ µ λ µ 1 3 2 1 2.0 = µ 4.0 = λ M1 A1 A1 [9] 12E (i) Plot with (attempt at) linear scales v 2 4 6 8 v2p 24.9 45.4 65.9 86.4 M1 A1 (ii) v2p = a + bv Valid attempt at gradient b = 10.2 to 10.3 a = 4 to 4.5 B1 M1 A1 B1 (iii) Attempt to rearrange to ... + = v a Y pv on y-axis M1 A1 (iv) Gradient is a y intercept is b A1 A1 [10] www.theallpapers.com

Page 8 Mark Scheme: Teachers’ version Syllabus Paper IGCSE – October/November 2010 0606 23 © UCLES 2010 12O (i) Plot with (attempt at) linear scales ln t 0.69 2.08 3.18 3.99 ln r 3.09 4.90 6.33 7.38 M1 A1 (ii) Gradient is 1.3 Intercept is 2.2 ln r = 1.3ln t + 2.2 ln r = 1.3ln t + ln 9 ln r = ln t1.3 + ln 9 r = 9t1.3 B1 B1 M1 M1 M1 A1√ (iii) Gradient is 1.3 Intercept 0.95 or lg9 B1√ B1√ [10] www.theallpapers.com