Cambridge A Level Mathematics 9709 — 2007 Oct/Nov Paper 1 · Variant 1

UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS General Certificate of Education Advanced Subsidiary Level and Advanced Level MATHEMATICS 9709/01 Paper 1 Pure Mathematics 1 (P1) October/November 2007 1 hour 45 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. 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. 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 75. Questions carrying smaller numbers of marks are printed earlier in the paper, and questions carrying larger numbers of marks later in the paper. This document consists of 4 printed pages. ©UCLES 2007 [Turn over *1843791734*

2 1 Determine the set of values of the constant k for which the line y = 4x + k does not intersect the curve y = x2. [3] 2 Find the area of the region enclosed by the curve y = 2 √x, the x-axis and the lines x = 1 and x = 4. [4] 3 (i) Find the first three terms in the expansion of (2 + u)5 in ascending powers of u. [3] (ii) Use the substitution u = x + x2 in your answer to part (i) to find the coefficient of x2 in the expansion of
2 + x + x2 5. [2] 4 The 1st term of an arithmetic progression is a and the common difference is d, where d ≠0. (i) Write down expressions, in terms of a and d, for the 5th term and the 15th term. [1] The 1st term, the 5th term and the 15th term of the arithmetic progression are the first three terms of a geometric progression. (ii) Show that 3a = 8d. [3] (iii) Find the common ratio of the geometric progression. [2] 5 (i) Show that the equation 3 sin x tan x = 8 can be written as 3 cos2x + 8 cos x −3 = 0. [3] (ii) Hence solve the equation 3 sinx tan x = 8 for 0◦≤x ≤360◦. [3] 6 The three points A (3, 8), B (6, 2) and C (10, 2) are shown in the diagram. The point D is such that the line DA is perpendicular to AB and DC is parallel to AB. Calculate the coordinates of D. [7] © UCLES 2007 9709/01/O/N/07

3 7 In the diagram, AB is an arc of a circle, centre O and radius r cm, and angle AOB = θ radians. The point X lies on OB and AX is perpendicular to OB. (i) Show that the area, A cm2, of the shaded region AXB is given by A = 1 2r2(θ −sin θ cos θ). [3] (ii) In the case where r = 12 and θ = 1 6π, find the perimeter of the shaded region AXB, leaving your answer in terms of √3 and π. [4] 8 The equation of a curve is y = (2x −3)3 −6x. (i) Express dy dx and d2y dx2 in terms of x. [3] (ii) Find the x-coordinates of the two stationary points and determine the nature of each stationary point. [5] 9 A curve is such that dy dx = 4 −x and the point P (2, 9) lies on the curve. The normal to the curve at P meets the curve again at Q. Find (i) the equation of the curve, [3] (ii) the equation of the normal to the curve at P, [3] (iii) the coordinates of Q. [3] © UCLES 2007 9709/01/O/N/07 [Turn over

4 10 The diagram shows a cube OABCDEFG in which the length of each side is 4 units. The unit vectors i, j and k are parallel to −−→ OA, −−→ OC and −−→ OD respectively. The mid-points of OA and DG are P and Q respectively and R is the centre of the square face ABFE. (i) Express each of the vectors −−→ PR and −−→ PQ in terms of i, j and k. [3] (ii) Use a scalar product to find angle QPR. [4] (iii) Find the perimeter of triangle PQR, giving your answer correct to 1 decimal place. [3] 11 The function f is defined by f : x →2x2 −8x + 11 for x ∈. (i) Express f(x) in the form a(x + b)2 + c, where a, b and c are constants. [3] (ii) State the range of f. [1] (iii) Explain why f does not have an inverse. [1] The function g is defined by g : x →2x2 −8x + 11 for x ≤A, where A is a constant. (iv) State the largest value of A for which g has an inverse. [1] (v) When A has this value, obtain an expression, in terms of x, for g−1(x) and state the range of g−1. [4] 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 Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. © UCLES 2007 9709/01/O/N/07

UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS GCE Advanced Subsidiary Level and GCE Advanced Level MARK SCHEME for the October/November 2007 question paper 9709 MATHEMATICS 9709/01 Paper 1, maximum raw mark 75 This mark scheme is published as an aid to teachers and candidates, 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. • CIE will not enter into discussions or correspondence in connection with these mark schemes. CIE is publishing the mark schemes for the October/November 2007 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.

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