Cambridge IGCSE Mathematics - Additional 0606 — 2006 May/June Paper 1 · Variant 1

This document consists of 5 printed pages and 3 blank pages. SPA (NF/KN) T04535 © UCLES 2006 [Turn over UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education ADDITIONAL MATHEMATICS 0606/01 Paper 1 May/June 2006 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. The use of an electronic calculator is expected, where appropriate. You are reminded of the need for clear presentation in your answers. 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. At the end of the examination, fasten all your work securely together. www.XtremePapers.com

2 0606/01/M/J/06 © UCLES 2006 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 0606/01/M/J/06 © UCLES 2006 [Turn over 1 A curve has the equation y = (x – 1)(2x – 3)8. Find the gradient of the curve at the point where x = 2. [4] 2 The line y + 4x = 23 intersects the curve xy + x = 20 at two points, A and B. Find the equation of the perpendicular bisector of the line AB. [6] 3 A plane flies due north from A to B, a distance of 1000km, in a time of 2 hours. During this time a steady wind, with a speed of 150kmh–1, is blowing from the south-east. Find (i) the speed of the plane in still air, [4] (ii) the direction in which the plane must be headed. [2] 4 The diagram shows part of the curve y = f(x), where f(x) = p – ex and p is a constant. The curve crosses the y-axis at (0, 2). (i) Find the value of p. [2] (ii) Find the coordinates of the point where the curve crosses the x-axis. [2] (iii) Copy the diagram above and on it sketch the graph of y = f–1(x). [2] 5 The matrices A and B are given by A =
, B =
 . Find matrices P and Q such that (i) P = B2 – 2A, [3] (ii) Q = B(A–1). [4] 6 The cubic polynomial f(x) is such that the coefficient of x3 is 1 and the roots of f(x) = 0 are –2, 1 + and 1 – . (i) Express f(x) as a cubic polynomial in x with integer coefficients. [3] (ii) Find the remainder when f(x) is divided by x – 3. [2] (iii) Solve the equation f(–x) = 0. [2] 3 3 0 –1 4 3 –2 –1 6 2 1 2 1 2 y x O

4 0606/01/M/J/06 © UCLES 2006 7 A particle moves in a straight line, so that, ts after leaving a fixed point O, its velocity, v ms–1, is given by v = pt2 + qt + 4, where p and q are constants. When t = 1 the acceleration of the particle is 8ms–2. When t = 2 the displacement of the particle from O is 22m. Find the value of p and of q. [7] 8 (i) Given that y = , show that = . [5] (ii) The diagram shows part of the curve y = . Using the result given in part (i), find the area of the shaded region bounded by the curve, the x-axis and the lines x = and x = . [3] 9 (a) Given that u = log4 x, find, in simplest form in terms of u, (i) x, (ii) log4
, (iii) logx 8. [5] (b) Solve the equation (log3 y)2 + log3(y2) = 8. [4] 10 The function f is defined, for 0°
x
180°, by f(x) = 3cos4x – 1. (i) Solve the equation f(x) = 0. [3] (ii) State the amplitude of f. [1] (iii) State the period of f. [1] (iv) State the maximum and minimum values of f. [2] (v) Sketch the graph of y = f(x). [3] 16 ––x 5π ––– 4 3π ––– 4 2 ––––––– 1 – sin x y x O y = 2 –––––– 1 – sinx 3π ––– 4 5π ––– 4 1 ––––––– 1 – sin x dy –– dx 1 + sin x ––––––– cos x

5 0606/01/M/J/06 © UCLES 2006 11 Answer only one of the following two alternatives. EITHER The table below shows values of the variables x and y which are related by the equation y = , where a and b are constants. (i) Using graph paper, plot y against xy and draw a straight line graph. [3] (ii) Use your graph to estimate the value of a and of b. [4] An alternative method for obtaining a straight line graph for the equation y = is to plot x on the vertical axis and on the horizontal axis. (iii) Without drawing a second graph, use your values of a and b to estimate the gradient and the intercept on the vertical axis of the graph of x plotted against . [3] OR The diagram, which is not drawn to scale, shows a quadrilateral ABCD in which A is (6, –3), B is (0, 6) and angle BAD is 90°. The equation of the line BC is 5y = 3x + 30 and C lies on the line y = x. The line CD is parallel to the y-axis. (i) Find the coordinates of C and of D. [6] (ii) Show that triangle BAD is isosceles and find its area. [4] y x O B C D A (6, –3) (0, 6) y = x 5y = 3x + 30 1–y 1––y a ––––– x + b a ––––– x + b x 0.1 0.4 1.0 2.0 3.0 y 8.0 6.0 4.0 2.6 1.9

8 0606/01/M/J/06 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 May/June 2006 question paper 0606 ADDITIONAL MATHEMATICS 0606/01 Paper 1, maximum raw mark 80 These mark schemes are published as an aid to teachers and students, to indicate the requirements of the examination. They show the basis on which Examiners were initially instructed to award marks. They do 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 May/June 2006 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 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.

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