Cambridge IGCSE Mathematics - Additional 0606 — 2005 Oct/Nov Paper 2 · Variant 1

UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education ADDITIONAL MATHEMATICS 0606/02 Paper 2 October/November 2005 2 hours Additional Materials: Answer Booklet/Paper Electronic calculator Graph paper Mathematical tables 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 on both sides of the paper. 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. 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. The use of an electronic calculator is expected, where appropriate. You are reminded of the need for clear presentation in your answers. This document consists of 5 printed pages and 3 blank pages. SP (NF/KN) S93693 © UCLES 2005 [Turn over www.XtremePapers.com

2 0606/02/O/N/05 © UCLES 2005 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation ax2 + bx + c = 0, . Binomial Theorem (a + b)n = an + an–1 b + an–2 b2 + … + an–r br + … + bn, where n is a positive integer and = . 2. TRIGONOMETRY Identities sin2 A + cos2 A = 1. sec2 A = 1 + tan2 A. cosec2 A = 1 + cot2 A. Formulae for ∆ABC = = . a2 = b2 + c2 – 2bc cos A. ∆= bc sin A. 1 2 c sin C b sin B a sin A n! (n – r)!r! ) n r( ) n r( ) n 2( ) n 1( x b b ac a = −± − 2 4 2

3 0606/02/O/N/05 © UCLES 2005 [Turn over 1 Variables V and t are related by the equation V = 1000e–kt, where k is a constant. Given that V = 500 when t = 21, find (i) the value of k, [2] (ii) the value of V when t = 30. [2] 2 The line x + y = 10 meets the curve y2 = 2x + 4 at the points A and B. Find the coordinates of the mid-point of AB. [5] 3 (i) Given that y = 1 + ln (2x – 3), obtain an expression for . [2] (ii) Hence find, in terms of p, the approximate value of y when x = 2 + p, where p is small. [3] 4 The function f is given by f : x 2 + 5 sin 3x for 0°
x
180°. (i) State the amplitude and period of f. [2] (ii) Sketch the graph of y = f(x). [3] 5 The binomial expansion of (1 + px)n, where n > 0, in ascending powers of x is 1 – 12x + 28p2x2 + qx3 + ... . Find the value of n, of p and of q. [6] 6 It is given that A =
 and that A + A–1 = kI, where p and k are constants and I is the identity matrix. Evaluate p and k. [6] 3 1 5 p dy ––– dx

4 0606/02/O/N/05 © UCLES 2005 7 In the diagram OP → = p, OQ → = q, PM → = PQ → and ON → = OQ →. (i) Given that OX → = mOM →, express OX → in terms of m, p and q. [2] (ii) Given that PX → = nPN →, express OX → in terms of n, p and q. [3] (iii) Hence evaluate m and n. [2] 8 (a) Find the value of each of the integers p and q for which
 – = 2p × 5q. [2] (b) (i) Express the equation 4x – 2x + 1 = 3 as a quadratic equation in 2x. [2] (ii) Hence find the value of x, correct to 2 decimal places. [3] 9 The function f(x) = x3 – 6x2 + ax + b, where a and b are constants, is exactly divisible by x – 3 and leaves a remainder of –55 when divided by x + 2. (i) Find the value of a and of b. [4] (ii) Solve the equation f(x) = 0. [4] 10 A curve is such that = 6x – 2. The gradient of the curve at the point (2, –9) is 3. (i) Express y in terms of x. [5] (ii) Show that the gradient of the curve is never less than – . [3] 11 (a) Each day a newsagent sells copies of 10 different newspapers, one of which is The Times. A customer buys 3 different newspapers. Calculate the number of ways the customer can select his newspapers (i) if there is no restriction, [1] (ii) if 1 of the 3 newspapers is The Times. [1] (b) Calculate the number of different 5-digit numbers which can be formed using the digits 0,1,2,3,4 without repetition and assuming that a number cannot begin with 0. [2] How many of these 5-digit numbers are even? [4] 16 –– 3 d2y –––– dx2 3–2 25 –– 16 2–5 1–3 P M Q O N X p q

5 0606/02/O/N/05 © UCLES 2005 12 Answer only one of the following two alternatives. EITHER The diagram, which is not drawn to scale, shows part of the curve y = x2 – 10x + 24 cutting the x-axis at Q(4, 0). The tangent to the curve at the point P on the curve meets the coordinate axes at S(0, 15) and at T (3.75, 0). (i) Find the coordinates of P. [4] The normal to the curve at P meets the x-axis at R. (ii) Find the coordinates of R. [2] (iii) Calculate the area of the shaded region bounded by the x-axis, the line PR and the curve PQ. [5] OR A curve has the equation y = 2cosx – cos2x, where 0 < x
. (i) Obtain expressions for and . [4] (ii) Given that sin 2x may be expressed as 2sin x cos x, find the x-coordinate of the stationary point of the curve and determine the nature of this stationary point. [4] (iii) Evaluate ∫ ydx. [3] d2y –––– dx2 dy ––– dx π–2 S (0, 15) Q (4, 0) P R O T (3.75, 0) x y y = x2 – 10x + 24 π/2 π/3

8 0606/02/O/N/05 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 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 November 2005 question paper 0606/02 ADDITIONAL MATHEMATICS 0606/02 Paper 2 maximum raw mark 80 This mark scheme is published as an aid to teachers and students, to indicate the requirements of the examination. It shows the basis on which Examiners were initially instructed to award marks. It does not indicate the details of the discussions that took place at an Examiners’ meeting before marking began. Any substantial changes to the mark scheme that arose from these discussions will be recorded in the published Report on the Examination. All Examiners are instructed that alternative correct answers and unexpected approaches in candidates’ scripts must be given marks that fairly reflect the relevant knowledge and skills demonstrated. Mark schemes must be read in conjunction with the question papers and the Report on the Examination. The minimum marks in these components needed for various grades were previously published with these mark schemes, but are now instead included in the Report on the Examination for this session. • CIE will not enter into discussion or correspondence in connection with these mark schemes. CIE is publishing the mark schemes for the November 2005 question papers for most IGCSE and GCE Advanced Level and Advanced Subsidiary Level syllabuses and some Ordinary Level syllabuses www.XtremePapers.com

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 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. 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) CWO Correct Working Only – often written by a ‘fortuitous' answer ISW Ignore Subsequent Working MR Misread A 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.

Page 1 Mark Scheme Syllabus Paper IGCSE – November 2005 0606 2 © University of Cambridge International Examinations 2005

Page 2 Mark Scheme Syllabus Paper IGCSE – November 2005 0606 2 © University of Cambridge International Examinations 2005

Page 3 Mark Scheme Syllabus Paper IGCSE – November 2005 0606 2 © University of Cambridge International Examinations 2005