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

UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education ADDITIONAL MATHEMATICS 0606/01 Paper 1 October/November 2005 2 hours Additional Materials: Answer Booklet/Paper Electronic calculator Graph paper (2 sheets) 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 6 printed pages and 2 blank pages. SP (NF/KN) S96217/1 © UCLES 2005 [Turn over www.XtremePapers.com

2 0606/01/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 0606/01/O/N/05 © UCLES 2005 [Turn over 1 Find the set of values of x for which (x – 6)2 > x. [3] 2 (a) (i) (ii) For each of the Venn diagrams above, express the shaded region in set notation. [2] (b) (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 A' ∩(B∪C). [1] 3 Find the values of the constant c for which the line 2y = x + c is a tangent to the curve y = 2x + . [4] 4 A cuboid has a square base of side (2 – )m and a volume of (2 – 3)m3. Find the height of the cuboid in the form (a + b )m, where a and b are integers. [4] 5 The diagram, which is not drawn to scale, shows a horizontal rectangular surface. One corner of the surface is taken as the origin O and i and j are unit vectors along the edges of the surface. A fly, F, starts at the point with position vector (i + 12j)cm and crawls across the surface with a velocity of (3i + 2j)cm s–1. At the instant that the fly starts crawling, a spider, S, at the point with position vector (85i + 5j)cm, sets off across the surface with a velocity of (–5i + kj)cm s–1, where k is a constant. Given that the spider catches the fly, calculate the value of k. [6] j i O 3 3 3 6–x A B C
A B
A B

4 0606/01/O/N/05 © UCLES 2005 6 A particle starts from rest at a fixed point O and moves in a straight line towards a point A. The velocity, vms–1, of the particle, t seconds after leaving O, is given by v = 6 – 6e–3t. Given that the particle reaches A when t = ln2, find (i) the acceleration of the particle at A, [3] (ii) the distance OA. [4] 7 (a) Solve log7(17y + 15) = 2 + log7(2y – 3). [4] (b) Evaluate logp8 × log16p. [3] 8 A curve has the equation y = (x + 2) . (i) Show that = , where k is a constant, and state the value of k. [4] (ii) Hence evaluate
2 5 dx. [4] 9 (a) Find all the angles between 0° and 360° which satisfy the equation 3cosx = 8tanx. [5] (b) Given that 4
y
6, find the value of y for which 2cos  + = 0. [3] 3 2y ––3 x −1 x x −1 kx dy ––– dx x −1

5 0606/01/O/N/05 © UCLES 2005 [Turn over 10 Solutions to this question by accurate drawing will not be accepted. The diagram, which is not drawn to scale, shows a quadrilateral ABCD in which A is (0, 10), B is (2, 16) and C is (8, 14). (i) Show that triangle ABC is isosceles. [2] The point D lies on the x-axis and is such that AD = CD. Find (ii) the coordinates of D, [4] (iii) the ratio of the area of triangle ABC to the area of triangle ACD. [3] 11 A function f is defined by f : x  2x – 3 – 4, for –2
x
3. (i) Sketch the graph of y = f(x). [2] (ii) State the range of f. [2] (iii) Solve the equation f(x) = –2. [3] A function g is defined by g: x  2x – 3 – 4, for –2
x
k. (iv) State the largest value of k for which g has an inverse. [1] (v) Given that g has an inverse, express g in the form g: x ax + b, where a and b are constants. [2] y x B (2, 16) C (8, 14) D A (0, 10) O

6 0606/01/O/N/05 © UCLES 2005 12 Answer only one of the following two alternatives. EITHER Variables x and y are related by the equation yxn = a, where a and n are constants. The table below shows measured values of x and y. (i) On graph paper plot lg y against lgx, using a scale of 2cm to represent 0.1 on the lg x axis and 1cm to represent 0.1 on the lg y axis. Draw a straight line graph to represent the equation yxn = a. [3] (ii) Use your graph to estimate the value of a and of n. [4] (iii) On the same diagram, draw the line representing the equation y = x2 and hence find the value of x for which xn + 2 = a. [3] OR The diagram shows a semicircle, centre O, of radius 8cm. The radius OC makes an angle of 1.2 radians with the radius OB. The arc CD of a circle has centre A and the point D lies on OB. Find the area of (i) sector COB, [2] (ii) sector CAD, [5] (iii) the shaded region. [3] A B C O D 8 cm 8 cm 1.2 rad x 1.5 2 2.5 3 3.5 y 7.3 3.5 2.0 1.3 0.9

8 0606/01/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 ADDITIONAL MATHEMATICS 0606/01 Paper 1 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 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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