Cambridge IGCSE Mathematics - Additional 0606 — 2013 Oct/Nov Paper 2 · Variant 3

This document consists of 19 printed pages and 1 blank page. DC (KN/SW) 67884/4 © UCLES 2013 [Turn over UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education * 7 1 3 2 3 9 8 2 2 2 * ADDITIONAL MATHEMATICS 0606/23 Paper 2 October/November 2013 2 hours Candidates answer on the Question Paper. Additional Materials: Electronic calculator READ THESE INSTRUCTIONS FIRST 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 pencil for any diagrams or graphs. Do not use staples, paper clips, highlighters, glue or correction fluid. DO NOT WRITE IN ANY BARCODES. 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 80. www.theallpapers.com

2 0606/23/O/N/13 © UCLES 2013 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 www.theallpapers.com

3 0606/23/O/N/13 © UCLES 2013 [Turn over For Examiner’s Use 1 Find the coordinates of the stationary points on the curve y = x3 – 6x2 – 36x + 16. [5] www.theallpapers.com

4 0606/23/O/N/13 © UCLES 2013 For Examiner’s Use 2 (i) Find how many different numbers can be formed using 4 of the digits 1, 2, 3, 4, 5, 6 and 7 if no digit is repeated. [1] Find how many of these 4-digit numbers are (ii) odd, [1] (iii) odd and less than 3000. [3] www.theallpapers.com

5 0606/23/O/N/13 © UCLES 2013 [Turn over For Examiner’s Use 3 Find the set of values of k for which the line y = 3x – k does not meet the curve y = kx2 + 11x – 6. [6] www.theallpapers.com

6 0606/23/O/N/13 © UCLES 2013 For Examiner’s Use 4 y x O (0, 3) , 2 3 r c m (a) (i) The diagram shows the graph of y = A + C tan(Bx) passing through the points (0, 3) and , 2 3 r e o. Find the value of A and of B. [2] (ii) Given that the point , 8 7 r e o also lies on the graph, find the value of C. [1] www.theallpapers.com

7 0606/23/O/N/13 © UCLES 2013 [Turn over For Examiner’s Use (b) Given that ( ) 8 5 3 cos x x f = - , state the period and the amplitude of f. [2] period … amplitude … www.theallpapers.com

8 0606/23/O/N/13 © UCLES 2013 For Examiner’s Use 5 (a) (i) In the Venn diagram below shade the region that represents A B j l ^ h . [1] A B (ii) In the Venn diagram below shade the region that represents P Q R k k l. [1] P Q R (b) Express, in set notation, the set represented by the shaded region. [1] S T Answer … www.theallpapers.com

9 0606/23/O/N/13 © UCLES 2013 [Turn over For Examiner’s Use (c) The universal set and the sets V and W are such that n( ) = 40, n(V ) = 18 and n(W) = 14. Given that n( ) V W x k = and n(( ) V W j ) x 3 = l find the value of x. You may use the Venn diagram below to help you. [3] V W www.theallpapers.com

10 0606/23/O/N/13 © UCLES 2013 For Examiner’s Use 6 The expression 2x3 + ax2 + bx + 21 has a factor x + 3 and leaves a remainder of 65 when divided by x – 2. (i) Find the value of a and of b. [5] (ii) Hence find the value of the remainder when the expression is divided by 2x + 1. [2] www.theallpapers.com

11 0606/23/O/N/13 © UCLES 2013 [Turn over For Examiner’s Use 7 The line 4x + y = 16 intersects the curve x y 4 8 1 - = at the points A and B. The x-coordinate of A is less than the x-coordinate of B. Given that the point C lies on the line AB such that AC : CB = 1 : 2, find the coordinates of C. [8] www.theallpapers.com

12 0606/23/O/N/13 © UCLES 2013 For Examiner’s Use 8 Solutions to this question by accurate drawing will not be accepted. x y X D C (10, 4) B (6, –4) A (–4, 6) O The diagram shows a quadrilateral ABCD, with vertices A(− 4, 6), B(6, − 4), C(10, 4) and D. The angle ADC = 90°. The lines BC and AD are extended to intersect at the point X. (i) Given that C is the midpoint of BX, find the coordinates of D. [7] www.theallpapers.com

13 0606/23/O/N/13 © UCLES 2013 [Turn over For Examiner’s Use (ii) Hence calculate the area of the quadrilateral ABCD. [2] www.theallpapers.com

14 0606/23/O/N/13 © UCLES 2013 For Examiner’s Use 9 A particle travels in a straight line so that, t s after passing through a fixed point O, its velocity, v ms–1, is given by sin v t 3 6 2 = + . (i) Find the velocity of the particle when t 4 r = . [1] (ii) Find the acceleration of the particle when t = 2. [3] www.theallpapers.com

15 0606/23/O/N/13 © UCLES 2013 [Turn over For Examiner’s Use The particle first comes to instantaneous rest at the point P. (iii) Find the distance OP. [5] www.theallpapers.com

16 0606/23/O/N/13 © UCLES 2013 For Examiner’s Use 10 120 cm 30 cm h cm The volume of a cone of height H and radius R is R H 3 1 2 r The diagram shows a container in the shape of a cone of height 120 cm and radius 30 cm. Water is poured into the container at a rate of 20 cm s 3 1 r - . (i) At the instant when the depth of water in the cone is h cm the volume of water in the cone is V cm3. Show that V h 48 3 r = . [3] www.theallpapers.com

17 0606/23/O/N/13 © UCLES 2013 [Turn over For Examiner’s Use (ii) Find the rate at which h is increasing when h = 50. [3] (iii) Find the rate at which the circular area of the water’s surface is increasing when h = 50. [4] www.theallpapers.com

18 0606/23/O/N/13 © UCLES 2013 For Examiner’s Use 11 In this question i is a unit vector due east and j is a unit vector due north. At time t = 0 boat A leaves the origin O and travels with velocity (2i + 4j) kmh–1. Also at time t = 0 boat B leaves the point with position vector (–21i + 22j) km and travels with velocity (5i + 3j) kmh–1. (i) Write down the position vectors of boats A and B after t hours. [2] (ii) Show that A and B are 25 km apart when t = 2. [3] www.theallpapers.com

19 0606/23/O/N/13 © UCLES 2013 [Turn over For Examiner’s Use (iii) Find the length of time for which A and B are less than 25 km apart. [5] www.theallpapers.com

20 0606/23/O/N/13 © UCLES 2013 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. BLANK PAGE www.theallpapers.com

CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education MARK SCHEME for the October/November 2013 series 0606 ADDITIONAL MATHEMATICS 0606/23 Paper 2, 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, which would have considered the acceptability of alternative answers. Mark schemes should be read in conjunction with the question paper and the Principal Examiner Report for Teachers. Cambridge will not enter into discussions about these mark schemes. Cambridge is publishing the mark schemes for the October/November 2013 series for most IGCSE, GCE Advanced Level and Advanced Subsidiary Level components and some Ordinary Level components. www.theallpapers.com

Page 2 Mark Scheme Syllabus Paper IGCSE – October/November 2013 0606 23 © Cambridge International Examinations 2013 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. www.theallpapers.com

Page 3 Mark Scheme Syllabus Paper IGCSE – October/November 2013 0606 23 © Cambridge International Examinations 2013 1 x y d d = 3x2 –12x –36 Equate to 0 and solve 3 term quadratic x = –2 and x = 6 y = 56 and y = –200 B2, 1, 0 M1 A1 A1 [5] Allow B1 if 2 terms correct Or one coordinate pair For two y values 2 (a) (i) (ii) (iii) 840 480 Calculates any case(s) correctly Partitions all cases correctly 140 B1 [1] B1 [1] B1 M1 A1 [3] e.g. 1 × 5× 4× 3 = 60, 1 × 5 × 4 × 4 = 80 3 Eliminate x or y Obtain kx2 + 8x + k – 6 (= 0) Use b2 – 4ac*0 Obtain – 4k2 + 24k + 64*0 oe Solve 3 term quadratic (k = 2, 8) k < –2, k > 8 M1* A1 DM1 A1 M1 A1 [1] 4 (a) (i) (ii) (b) A = 3, B = 2 C = 4 120 or 3 2π 5 B1, B1 B1 B1 B1 5 (a) (i) (ii) (b) ' T S ∩ or )' ' ( T S ∪ oe B1 [1] B1 [1] B1 [1] Others will be seen but only accept completely correct set notation www.theallpapers.com

Page 4 Mark Scheme Syllabus Paper IGCSE – October/November 2013 0606 23 © Cambridge International Examinations 2013 (c) 18 – x + x + 14 – x + 3x = 40 x = 4 B1 M1 A1 [3] B1 for any two of x, 3x, 18 – x or 14 – x in correct place (or implied by correct equation) 6 (a) (i) (ii) Equate f(–3) to zero Equate f(2) to 65 – 54 + 9a – 3b + 21 = 0 (9a – 3b = 33) or 16 + 4a + 2b + 21 = 65 (4a + 2b =28) Solve simultaneous equations a = 5, b = 4 Calculate 21 2 4 4 1 2 1 f + − + − =      − b a 20 M1 M1 A1 M1 A1 [5] M1 A1 [2] Or use long division 7 Eliminate x or y Rearrange to quadratic in x or y correctly x2 – 10x + 16 (= 0) or y2 + 8y – 128 (= 0) oe Solve 3 term quadratic x = 2, x = 8 y = 8, y = –16 Correct method for at least one coordinate of C C (4, 0) M1 M1 A1 M1 A1 A1 M1 A1 [8] Or one correct coordinate pair e.g. xc = 3 1 [2 (2) + 1 (8)], OC = OA + 3 1 AB oe 3x 18–x 14–x x www.theallpapers.com

Page 5 Mark Scheme Syllabus Paper IGCSE – October/November 2013 0606 23 © Cambridge International Examinations 2013 8 (a) (i) (ii) X (14, 12) mAX = 3 1 Use m1m2 = –1 for grad CD from grad AX CD is y – 4 = –3(x – 10) or y = –3x + 34 AX is y – 6 = 3 1 (x + 4) or 3y – x = 22 Solve eqn for CD with eqn for AX D (8, 10) Method for area 100 B1 B1 M1 A1√ B1√ M1 A1 [7] M1 A1 [2] √ on grad AX √ on grad AX 9 (a) (i) (ii) (iii) 9 a = k cos 2t 12 cos 2t –7.84 t = 12 7π or awrt 1.8 3t – 3 cos 2t Use limits of 0 and their       12 7π or finds c (≠ 0) and substitutes their       12 7π 11.1 or 3 2 3 3 4 7 + + π B1 [1] M1 A1 A1√ [3] B1 B1, B1 M1 A1 [5] No other functions of t or constants √ on k only Must be negative (if correct) or say “deceleration” Upper limit must be positive www.theallpapers.com

Page 6 Mark Scheme Syllabus Paper IGCSE – October/November 2013 0606 23 © Cambridge International Examinations 2013 10 (a) (i) (ii) (iii) Radius is 4 h Use 3 1 πr2 h      = ×       48 4 3 1 3 2 h h h π π 16 d d 2 h h V π = Use V h t V t h d d d d d d × = with h = 50, π 20 d d = t V 0.128 8 d d 16 2 h h A h A π π = = Use h A t h t A d d d d d d × = with substitution of h = 50, their 0.128 0.8π or 2.51 B1 M1 A1ag [3] B1 M1 A1 [3] B1 M1 M1 A1 [3] On water cone 11 (a) (i) (ii) (iii) (2i + 4j)t (–21i + 22j) + (5i + 3j)t Subtract position vectors ( ) ( ) ( )j i t t − + + − 22 3 21 Substitute t = 2 and use Pythagoras Correctly reach 25 ( ) ( ) 2 2 2 25 22 3 21 = − + + − t t oe t2 – 17t + 30 (= 0) Solve 3 term quadratic t = 15 (and 2) 13 hours B1 B1 [2] M1 M1 A1 [3] M1 A1 M1 A1 A1 [5] Or use t = 2 to find position vectors of A, B 4i + 8j, –11i + 28j Subtract position vectors and use Pythagoras Set expression for distance apart to 25 Not essential to solve quadratic e.g. t1 + t2 = 17 and t1 = 2 www.theallpapers.com