Cambridge IGCSE Mathematics - Additional 0606 — 2008 May/June Paper 1 · Variant 1

This document consists of 6 printed pages and 2 blank pages. SPA (NF/DT) T56698 © UCLES 2008 [Turn over UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education 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. 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. 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. * 8 5 7 5 6 6 7 4 3 1 * ADDITIONAL MATHEMATICS 0606/01 Paper 1 May/June 2008 2 hours Additional Materials: Answer Booklet/Paper Electronic calculator Mathematical tables www.XtremePapers.com

2 0606/01/M/J/08 © UCLES 2008 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 0606/01/M/J/08 © UCLES 2008 [Turn over 1 Express 8 3 2 4 3 2 − + in the form a b + 2 , where a and b are integers. [3] 2 A committee of 5 people is to be selected from 6 men and 4 women. Find (i) the number of different ways in which the committee can be selected, [1] (ii) the number of these selections with more women than men. [4] 3 The line y = 3x + k is a tangent to the curve x2 + xy + 16 = 0. (i) Find the possible values of k. [3] (ii) For each of these values of k, find the coordinates of the point of contact of the tangent with the curve. [2] 4 Variables x and y are such that, when e y is plotted against x2, a straight line graph passing through the points (0.2, 1) and (0.5, 1.6) is obtained. (i) Find the value of e y when x = 0. [2] (ii) Express y in terms of x. [3] 5 Variables x and y are connected by the equation y = x tan x . Given that x is increasing at the rate of 2 units per second, find the rate of increase of y when x = π 4. [5] 6 Solve the equation x2 (2x + 3) = 17x – 12. [6]

4 0606/01/M/J/08 © UCLES 2008 7 O B C A 4 cm The diagram shows a sector OAB of a circle, centre O, radius 4 cm. The tangent to the circle at A meets the line OB extended at C. Given that the area of the sector OAB is 10 cm2, calculate (i) the angle AOB in radians, [2] (ii) the perimeter of the shaded region. [4] 8 (i) Given that log9 x = a log3 x, find a. [1] (ii) Given that log27 y = b log3 y, find b. [1] (iii) Hence solve, for x and y, the simultaneous equations 6log9 x + 3 log27 y = 8, log3 x + 2 log9 y = 2. [4] 9 A curve is such that dy dx = 2 cos
2x – π 2. The curve passes through the point
π 2, 3. (i) Find the equation of the curve. [4] (ii) Find the equation of the normal to the curve at the point where x = 3π 4 . [4]

5 0606/01/M/J/08 © UCLES 2008 [Turn over 10 In this question, i is a unit vector due east and j is a unit vector due north. At 0900 hours a ship sails from the point P with position vector (2i + 3j) km relative to an origin O. The ship sails north-east with a speed of 15 2 km h–1. (i) Find, in terms of i and j, the velocity of the ship. [2] (ii) Show that the ship will be at the point with position vector (24.5i + 25.5j) km at 1030 hours. [1] (iii) Find, in terms of i, j and t, the position of the ship t hours after leaving P. [2] At the same time as the ship leaves P, a submarine leaves the point Q with position vector (47i – 27j) km. The submarine proceeds with a speed of 25 km h–1 due north to meet the ship. (iv) Find, in terms of i and j, the velocity of the ship relative to the submarine. [2] (v) Find the position vector of the point where the submarine meets the ship. [2] 11 Solve the equation (i) 3 sin x + 5 cos x = 0 for 0° < x < 360°, [3] (ii) 3 tan2 y – sec y – 1 = 0 for 0° < y < 360°, [5] (iii) sin(2z – 0.6) = 0.8 for 0 < z < 3 radians. [4] [Question 12 is printed on the next page.]

6 0606/01/M/J/08 © UCLES 2008 12 Answer only one of the following two alternatives. EITHER A curve has equation y = (x2 – 3)e–x. (i) Find the coordinates of the points of intersection of the curve with the x-axis. [2] (ii) Find the coordinates of the stationary points of the curve. [5] (iii) Determine the nature of these stationary points. [3] OR A particle moves in a straight line such that its displacement, s m, from a fixed point O at a time t s, is given by s = ln(t + 1) for 0
t
3, s = 1 2ln (t – 2) – ln(t + 1) + ln 16 for t > 3. Find (i) the initial velocity of the particle, [2] (ii) the velocity of the particle when t = 4, [2] (iii) the acceleration of the particle when t = 4, [2] (iv) the value of t when the particle is instantaneously at rest, [2] (v) the distance travelled by the particle in the 4th second. [2]

8 0606/01/M/J/08 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 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.

UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education MARK SCHEME for the May/June 2008 question paper 0606 ADDITIONAL MATHEMATICS 0606/01 Paper 1, maximum raw mark 80 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 May/June 2008 question papers for most IGCSE, GCE Advanced Level and Advanced Subsidiary Level syllabuses and some Ordinary Level syllabuses. www.XtremePapers.com

Page 2 Mark Scheme Syllabus Paper IGCSE – May/June 2008 0606 01 © UCLES 2008 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 Accuracy 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.

Page 3 Mark Scheme Syllabus Paper IGCSE – May/June 2008 0606 01 © UCLES 2008 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) 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 – May/June 2008 0606 01 © UCLES 2008 1 ( ) ( ) 4 3 2 8 3 2 4 3 2 4 3 2 32 12 2 24 2 18 16 18 50 36 2 2 − − + − − − + − − − a = –25, b = 18 M1 DM1 A1 [3] M1 for attempt to rationalise DM1 for attempt to expand out and simplify Allow A1 at this stage 2 (i) 10 5 252 C = (ii) 4 women, 1 man: 6 3 women, 2 men: 4 6 3 2 C C × = 60 Total = 66 B1 [1] M1 B1 B1 A1 [4] M1 for a plan B1 for 6 B1 for 60 A1 for total Allow marks for other valid methods 3 (i) ( ) ( ) 2 2 2 4 16 0 4 5 144 0 x kx y ky k + + = − + + = M1 M1 for attempt to get a quadratic in terms of one variable 16 256, , 4 2 2 ± = = = k k ac b DM1, A1 [3] DM1 for use of b2 – 4ac A1 for both (ii) using 2 b x a = − , or equivalent When k = –16, (2, –10) When k = 16, (–2, 10) B1 B1 [2] B1 for each pair Allow B1 for x values only 4 (i) gradient = 2, equation of line of form Y = mX + c, where c = 0.6 e 0.6 y ∴ = (ii) 2 2 e 2 0.6 ln(2 0.6) y x y x = + ∴ = + M1 A1 [2] A1 M1 A1 [3] M1 for attempt to get equation of straight line A1 for correct form (allow if seen in (i)) M1 for attempt to take ln

Page 5 Mark Scheme Syllabus Paper IGCSE – May/June 2008 0606 01 © UCLES 2008 5 2 2 d tan sec d tan y x x x x x − = When π d π , 1 4 d 2 y x x = = − Using d d d d , 2 π d d d d y y x y t x t t = × = − (-1.14) M1 A1 M1 M1 A1 [5] M1 for correct attempt to differentiate a quotient A1 all correct M1 for attempt to sub π 4 x = in to their d d y x M1 for attempt to use rates of change 6 3 2 2 3 17 12 0 x x x + − + = f(1) = 0, (x – 1) is a factor ( )( ) 0 12 5 2 1 2 = − + − x x x ( )( )( ) 0 34 3 2 1 = + − − x x x 4 2 3 1, − = , x M1 M1 M1 DM1 B1,A1 [6] M1 for simplification M1 for attempt to find a root M1 for attempt to get quadratic factor DM1 for factorising on all previous M marks B1 for solution from first root A1 for the other pair 7 (i) 2 1 4 10 2 θ = , leading to θ = 1.25 rads (ii) 5 4tan1.25, 12.038 4 4, 8.685 cos1.25 AB AC AC BC BC = = = = − = Perimeter = 25.7, allow 25.8 M1 A1 [2] B1 M1 M1 A1 [4] M1 for use of 2 1 2 r θ M1 for attempt to get AC M1 for attempt to get BC 8 (i) 1 2 a = (ii) 1 3 b = (allow 0.33 or better) (iii) 8 log log 3 3 3 = + y x 2 log log 3 3 = + y x 27 3, log3 = = x x 3 1 1, log3 = − = y y Allow solutions using index notation B1 [1] B1 [1] M1 DM1 A1 A1 [4] M1 for reducing equations to terms of base 3 logs DM1 for dealing with simultaneous equations and logs to get final answers A1 for each

Page 6 Mark Scheme Syllabus Paper IGCSE – May/June 2008 0606 01 © UCLES 2008 9 (i) π sin 2 2 y x c   = − +     c = 2 (ii) at 3π d , 2 4 d y x x = = − Grad of normal = 1 2 When 3π 4 x = , y = 2 normal 1 3π 2 2 4 y x   − = −     M1 A1 M1, A1 [4] M1 M1 M1, A1 [4] M1 for π sin 2 2 x   −     A1 correct M1 for attempt to get c Allow A1 for c = 2 M1 for attempt to get d d y x and for ⊥gradient M1 for attempt to obtain y using 3π 4 x = in answer to (i) M1 for attempt to obtain normal, must be using ⊥gradient – allow unsimplified 10 (i) v = ( ) 2 2 15 j i + v = 15i + 15j (ii) (2i + 3j) + (15i + 15j)1.5 24.5i + 25.5j (iii) (2 + 15t)i + (3 + 15t)j Allow (2i +3j) + (15i + 15j)t (iv) relative velocity (15i + 15j) – 25j = 15i – 10j (v) relative displacement (47i – 27j) – (2i + 3j) = 45i – 30j Time taken = 3 hours Position vector at interception 47i + 48j or ( ) ( ) 2 3 15 15 47 27 25 t t + + + = − + i j i j i j or equivalent Allow solutions to (v) by drawing M1 A1 [2] B1 [1] M1, √A1 [2] M1, A1 [2] M1 A1 [2] M1 for attempt to get a direction vector Answer given M1 for use of their velocity vector with 2i + 3j. Follow through on their velocity vector M1 for a difference of velocities M1 for attempt to get relative displacement or other valid method. M1 for equating like vectors and attempt to get t

Page 7 Mark Scheme Syllabus Paper IGCSE – May/June 2008 0606 01 © UCLES 2008 11 (i) 5 tan 3 x = − o o 121.0 ,301.0 x = (ii) 2 3sec sec 4 0 y y − − = ( )( ) o o o 3sec 4 sec 1 0 3 cos , 1 4 41.4 ,318.6 ,180 y y y y − + = = − = (iii) 2 0.6 0.9273,2.2143 z − = z = 0.764, 1.407 (allow 1.41) M1 A1, √A1 [3] M1 M1 M1 B1, A1 [5] M1 M1 A1, A1 [4] M1 for use of tan and attempt at one solution A1 for each, √ on first solution for x M1 for use of correct identity and formation of a 3 term quadratic in one variable. M1 for factorising a 3 term quadratic M1 for all terms in terms of cos B1 for 0 180 , A1 for the other pair M1 for correct order of operations M1 for a valid attempt at a second solution A1 for each 12 EITHER (i) ( 3,0) ± allow (ii) ( ) 2 d 3 e e 2 d x x y x x x − − = − − + ( ) 2 e 2 3 x x x − = − + d d y x = 0 , 2 2 3 0 x x − − = leading to x = 3, –1 and y = 6e–3( 0.299), –2e (5.44) (iii) ( ) ( ) 2 2 2 d e 2 2 e 2 3 d x x y x x x x − − = − − − + When x = 3, 2 2 d d y x is –ve, max When x = –1, 2 2 d d y x is +ve, min B1, B1 [2] M1, A1 M1 A1 A1 [5] M1 B1 B1 [3] M1 for a correct attempt to differentiate a product or a quotient A1 allow unsimplified M1 for attempting to solve d d y x = 0 A1 for each pair M1 for attempt at second differential or use of gradient method B1 for each

Page 8 Mark Scheme Syllabus Paper IGCSE – May/June 2008 0606 01 © UCLES 2008 12 OR (i) 0 1 , 1 1 v v t = = + (ii) ( ) 1 1 2 2 1 v t t = − − + 4 4 1 1 1 ; 4 5 20 v v = − = ( 0.05) (iii) ( ) ( ) 4 2 2 1 1 17 ; 200 2 2 1 a a t t = − + = − − + (–0.085) (iv) ( ) 1 1 2 2 1 t t − − + = 0, t = 5 (v) 3 ln 4 s = (1.386) 4 16 2 ln 5 s = (1.509) In 4th sec, 4 2 ln 5 s = (0.123) (allow 0.124) M1, A1 [2] M1 A1 [2] M1, A1 [2] DM1, A1 [2] M1 A1 [2] M1 for attempt to differentiate M1 for attempt to differentiate M1 for attempt to differentiate DM1 for equating v to zero M1 for attempt to find s3 and s4