Cambridge IGCSE Mathematics - Additional 0606 — 2006 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 2006 2 hours Additional Materials: Answer Paper Electronic calculator 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 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. This document consists of 5 printed pages and 3 blank pages. SP (SLM) T26710/1 © UCLES 2006 [Turn over www.XtremePapers.com

2 0606/02/O/N/06 © UCLES 2006 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 0606/02/O/N/06 © UCLES 2006 [Turn over 1 The functions f and g are defined for x

by f : x
x3, g : x
x + 2. Express each of the following as a composite function, using only f, g, f –1 and/or g–1 : (i) x
x3 + 2, [1] (ii) x
x3 – 2, [1] (iii) x
(x + 2) 1 3. [1] 2 Prove the identity cos x cot x + sin x
cosec x . [4] 3 Evaluate ∫0 π–6sin(2x + π–6 )dx. [4] 4 90 m 2 ms–1 A B The diagram shows a river 90 m wide, flowing at 2 ms–1 between parallel banks. A ferry travels in a straight line from a point A to a point B directly opposite A. Given that the ferry takes exactly one minute to cross the river, find (i) the speed of the ferry in still water, [3] (ii) the angle to the bank at which the ferry must be steered. [2] 5 The straight line 2x + y = 14 intersects the curve 2x2 – y2 = 2xy – 6 at the points A and B. Show that the length of AB is 24 5 units. [7]

4 0606/02/O/N/06 © UCLES 2006 6 A curve has equation y = x3 + ax + b, where a and b are constants. The gradient of the curve at the point (2, 7) is 3. Find (i) the value of a and of b, [5] (ii) the coordinates of the other point on the curve where the gradient is 3. [2] 7 (a) Find the value of m for which the line y = mx – 3 is a tangent to the curve y = x + 1 x and find the x-coordinate of the point at which this tangent touches the curve. [5] (b) Find the value of c and of d for which {x : – 5 < x < 3} is the solution set of x2 + cx < d. [2] 8 Given that A = 4 1 3 2 − − ⎛ ⎝⎜ ⎞ ⎠⎟ , use the inverse matrix of A to (i) solve the simultaneous equations y – 4x + 8 = 0, 2y – 3x + 1 = 0, (ii) find the matrix B such that BA = − − ⎛ ⎝⎜ ⎞ ⎠⎟ 2 3 9 1 . [8] 9 (a) Express 2 5 8 3 5 2 − ( ) − − in the form p + q 5, where p and q are integers. [4] (b) Given that a b b a ab x x y y 3 1 2 6 − + × ( ) = , find the value of x and of y. [4] 10 (a) How many different four-digit numbers can be formed from the digits 1, 2, 3, 4, 5, 6, 7, 8, 9 if no digit may be repeated? [2] (b) In a group of 13 entertainers, 8 are singers and 5 are comedians. A concert is to be given by 5 of these entertainers. In the concert there must be at least 1 comedian and there must be more singers than comedians. Find the number of different ways that the 5 entertainers can be selected. [6] 11 The equation of a curve is y x x = − e 2. (i) Show that d d e y x x x = − ( ) − 1 2 2 2. [3] (ii) Find an expression for d2y dx2 . [2] The curve has a stationary point at M. (iii) Find the coordinates of M. [2] (iv) Determine the nature of the stationary point at M. [2]

5 0606/02/O/N/06 © UCLES 2006 12 Answer only one of the following two alternatives. EITHER r cm
rad L O M N The diagram shows a sector of a circle, centre O and radius r cm. Angle LOM is θ radians. The tangent to the circle at L meets the line through O and M at N. The shaded region shown has perimeter P cm and area A cm2. Obtain an expression, in terms of r and θ, for (i) P, [4] (ii) A. [3] Given that θ = 1.2 and that P = 83, find the value of (iii) r, [2] (iv) A. [1] OR Solutions to this question by accurate drawing will not be accepted. y x A (3, 3) B (6, 3) C O E (10, k) D The diagram shows an isosceles triangle ABC in which A is the point (3, 3), B is the point (6, 3) and C lies below the x-axis. Given that the area of triangle ABC is 6 square units, (i) find the coordinates of C. [3] The line CB is extended to the point D so that B is the mid-point of CD. (ii) Find the coordinates of D. [2] A line is drawn from D, parallel to AC, to the point E (10, k) and C is joined to E. (iii) Find the value of k. [3] (iv) Prove that angle CED is not a right angle. [2]

8 0606/02/O/N/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 October/November 2006 question paper 0606 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 instructed to award marks. It does not indicate the details of the discussions that took place at an Examiners’ meeting before marking began. 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 grade thresholds for various grades are published in the report on the examination for most IGCSE, GCE Advanced Level and Advanced Subsidiary Level syllabuses. • CIE will not enter into discussions or correspondence in connection with these mark schemes. CIE is publishing the mark schemes for the October/November 2006 question papers for most IGCSE, 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.

Page 4 Mark Scheme Syllabus Paper IGCSE - OCT/NOV 2006 0606 02 © UCLES 2006 M1 A1 cos 2 1 − x 7 = 8 + 2a + b B1 Solving for a and b       = − = ⇒ 17 9 b a

Page 5 Mark Scheme Syllabus Paper IGCSE - OCT/NOV 2006 0606 02 © UCLES 2006 and M1 A1 M1 B1 A1 of relevant two terms only = 910 [≈0.736] or MN = rsecθ

Page 6 Mark Scheme Syllabus Paper IGCSE - OCT/NOV 2006 0606 02 © UCLES 2006 or 2 1 4 = M x or 2 1 4 = C x