Cambridge A Level Mathematics 9709 — 2006 Oct/Nov Paper 2 · Variant 1

UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS General Certificate of Education Advanced Subsidiary Level MATHEMATICS 9709/02 Paper 2 Pure Mathematics 2 (P2) October/November 2006 1 hour 15 minutes Additional Materials: Answer Booklet/Paper Graph paper List of Formulae (MF9) 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. 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 50. Questions carrying smaller numbers of marks are printed earlier in the paper, and questions carrying larger numbers of marks later in the paper. At the end of the examination, fasten all your work securely together. This document consists of 3 printed pages and 1 blank page. © UCLES 2006 [Turn over

2 1 Solve the inequality |2x −1| > |x|. [4] 2 (i) Express 4x in terms of y, where y = 2x. [1] (ii) Hence find the values of x that satisfy the equation 3
4x −10
2x + 3 = 0, giving your answers correct to 2 decimal places. [5] 3 The polynomial 4x3 −7x + a, where a is a constant, is denoted by p(x). It is given that (2x −3) is a factor of p(x). (i) Show that a = −3. [2] (ii) Hence, or otherwise, solve the equation p(x) = 0. [4] 4 (i) Prove the identity tan(x + 45◦) −tan(45◦−x) ≡2 tan 2x. [4] (ii) Hence solve the equation tan(x + 45◦) −tan(45◦−x) = 2, for 0◦≤x ≤180◦. [3] 5 The diagram shows a chord joining two points, A and B, on the circumference of a circle with centre O and radius r. The angle AOB is α radians, where 0 < α < π. The area of the shaded segment is one sixth of the area of the circle. (i) Show that α satisfies the equation x = 1 3π + sin x. [3] (ii) Verify by calculation that α lies between 1 2π and 2 3π. [2] (iii) Use the iterative formula xn+1 = 1 3π + sin xn, with initial value x1 = 2, to determine α correct to 2 decimal places. Give the result of each iteration to 4 decimal places. [3] © UCLES 2006 9709/02/O/N/06

3 6 The diagram shows the part of the curve y = e2x x for x > 0, and its minimum point M. (i) Find the coordinates of M. [5] (ii) Use the trapezium rule with 2 intervals to estimate the value of  2 1 e2x x dx, giving your answer correct to 1 decimal place. [3] (iii) State, with a reason, whether the trapezium rule gives an under-estimate or an over-estimate of the true value of the integral in part (ii). [1] 7 (i) Given that y = tan 2x, find dy dx. [2] (ii) Hence, or otherwise, show that  1 6π 0 sec2 2x dx = 1 2 √3, and, by using an appropriate trigonometrical identity, find the exact value of  1 6π 0 tan2 2x dx. [6] (iii) Use the identity cos 4x ≡2 cos2 2x −1 to find the exact value of  1 6π 0 1 1 + cos 4x dx. [2] © UCLES 2006 9709/02/O/N/06

4 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. 9709/02/O/N/06

UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS GCE Advanced Subsidiary Level and GCE Advanced Level MARK SCHEME for the October/November 2006 question paper 9709 MATHEMATICS 9709/02 Paper 2, maximum raw mark 50 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.

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 marks indicated in the scheme may not be subdivided. If there is genuine doubt whether a candidate has earned a mark, allow the candidate the benefit of the doubt. Unless otherwise indicated, marks once gained cannot subsequently be lost, e.g. wrong working following a correct form of answer is ignored. • Wrong or missing units in an answer should not lead to the loss of a mark unless the scheme specifically indicates otherwise. • For a numerical answer, allow the A or B mark if a value is obtained which is correct to 3 s.f., or which would be correct to 3 s.f. if rounded (1 d.p. in the case of an angle). As stated above, an A or B mark is not given if a correct numerical answer arises fortuitously from incorrect working. For Mechanics questions, allow A or B marks for correct answers which arise from taking g equal to 9.8 or 9.81 instead of 10.

• The following abbreviations may be used in a mark scheme or used on the scripts: AEF Any Equivalent Form (of answer is equally acceptable) 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) SR Special Ruling (detailing the mark to be given for a specific wrong solution, or a case where some standard marking practice is to be varied in the light of a particular circumstance) 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. An MR-2 penalty may be applied in particular cases if agreed at the coordination meeting. • PA -1 This is deducted from A or B marks in the case of premature approximation. The PA -1 penalty is usually discussed at the meeting.

Page 3 Mark Scheme Syllabus Paper GCE A/AS LEVEL - OCT/NOV 2006 9709 02 © UCLES 2006 M1 (=22x) (A1 √ if ± 1.59)

Page 4 Mark Scheme Syllabus Paper GCE A/AS LEVEL - OCT/NOV 2006 9709 02 © UCLES 2006 B1 * (* allow √ y = 2e if x = 2 1 ‘illicitly’ obtained)