Cambridge IGCSE Mathematics - Additional 0606 — 2023 Oct/Nov Paper 1 · Variant 1

This document has 16 pages. [Turn over Cambridge IGCSE™ * 5 1 7 2 0 6 4 8 8 4 * DC (CE/SG) 316817/2 © UCLES 2023 ADDITIONAL MATHEMATICS 0606/11 Paper 1 October/November 2023 2 hours You must answer on the question paper. No additional materials are needed. INSTRUCTIONS ● Answer all questions. ● Use a black or dark blue pen. You may use an HB pencil for any diagrams or graphs. ● Write your name, centre number and candidate number in the boxes at the top of the page. ● Write your answer to each question in the space provided. ● Do not use an erasable pen or correction fluid. ● Do not write on any bar codes. ● You should use a calculator where appropriate. ● You must show all necessary working clearly; no marks will be given for unsupported answers from a calculator. ● Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless a different level of accuracy is specified in the question. INFORMATION ● The total mark for this paper is 80. ● The number of marks for each question or part question is shown in brackets [ ].

2 0606/11/O/N/23 © UCLES 2023 Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation ax bx c 0 2 + + = , x a b b ac 2 4 2 ! = - - Binomial Theorem ( ) a b a a b a b a b n n n r b 1 2 n n n n n r r n 1 2 2 f f + = + + + + + + - - - e e e o o o where n is a positive integer and ( )! ! ! n r n r r n = - e o Arithmetic series ( ) u a n d 1 n = + - ( ) { ( ) } S n a l n a n d 2 1 2 1 2 1 n = + = + - Geometric series u ar n n 1 = - ( ) ( ) S r a r r 1 1 1 n n ! = - - ( ) S r a r 1 1 1 = - 3 2. TRIGONOMETRY Identities sin cos A A 1 2 2 + = sec tan A A 1 2 2 = + ec cos cot A A 1 2 2 = + Formulae for ∆ABC sin sin sin A a B b C c = = cos a b c bc A 2 2 2 2 = + - sin bc A 2 1 T =

3 0606/11/O/N/23 © UCLES 2023 [Turn over 1 – 720° – 540° – 360° – 180° 0 180° 360° 540° 720° x y 1 – 9 – 8 – 7 – 6 – 5 – 4 – 3 – 2 – 1 The diagram shows part of the graph of cos y a b x c = + b l , where a, b and c are integers. Find the values of a, b and c. [3]

4 0606/11/O/N/23 © UCLES 2023 2 The polynomial ( )x P is such that ( )x ax x bx c 11 P 3 2 = - + + , where a, b and c are integers. ( )x P is divisible by x and has a remainder of 2 3 when divided by x 2 1 + . It is also given that ( ) 2 18 P = l . (a) Find the values of a, b and c. [6] (b) Hence write ( )x P as a product of three linear factors. [2]

5 0606/11/O/N/23 © UCLES 2023 [Turn over 3 The point A has position vector 2 6 - e o. The point B has position vector 6 3 - e o. (a) Find, in vector form, the displacement of B from A. [2] (b) Find the distance AB. [1] The point X is such that AB AX 3 2 = . (c) Find the position vector of X. [2]

6 0606/11/O/N/23 © UCLES 2023 4 x 1 2 3 4 5 y 20 57 104 160 224 The table shows values of the variables x and y, which are related by the equation y Axb = , where A and b are constants. (a) Use the data to draw a straight line graph of lny against lnx. [3] 0 – 1 1 2 3 4 5 6 7 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 ln y ln x

7 0606/11/O/N/23 © UCLES 2023 [Turn over (b) Use your graph to estimate the values of A and b. Give your answers correct to 2 significant figures. [4] (c) Use your graph to estimate the value of y when . x 3 5 = . [2]

8 0606/11/O/N/23 © UCLES 2023 5 (a) A 4-digit code is to be formed from the digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. No digit may be used more than once in any code. A code may start with 0. (i) Find how many codes can be formed. [1] (ii) Find how many codes form an odd number. [1] (iii) Find how many codes form a number greater than 1000. [2] (b) A team of 9 people is to be chosen from a group of 15 people. The group includes a family of 4 people who must not be separated. Find the number of teams that can be chosen. [3]

9 0606/11/O/N/23 © UCLES 2023 [Turn over 6 (a) Write lg lg x 3 2 1 4 2 - + as a single logarithm to base 10. [3] (b) Solve the equation , log log a 2 4 3 5 0 a 4 - - = giving your answers in exact form. [5]

10 0606/11/O/N/23 © UCLES 2023 7 A curve has equation ( ), y x f = where ( ) ( )( ) x x x 2 1 3 2 f 2 = + - . (a) Show that ( )x f l can be written in the form ( )( ) x px q 2 3 2 - + , where p and q are integers. [3] (b) Hence find the coordinates of the stationary points on the curve. [2]

11 0606/11/O/N/23 © UCLES 2023 [Turn over (c) On the axes below, sketch the graph of ( ), y x f = stating the intercepts with the coordinate axes. [3] y x O (d) Find the values of k such that the equation ( )x k f = has 3 distinct solutions. [2]

12 0606/11/O/N/23 © UCLES 2023 8 y x O y x x 4 6 5 2 = - - y x 1 4 = - C A B The diagram shows the line y x 1 4 = - meeting the curve y x x 4 6 5 2 = - - at the points A and B. The tangent to the curve at B meets the horizontal line through A at the point C. Find the x-coordinate of C, giving your answer correct to 2 decimal places. [10]

13 0606/11/O/N/23 © UCLES 2023 [Turn over Additional working space for Question 8.

14 0606/11/O/N/23 © UCLES 2023 9 (a) The first three terms of an arithmetic progression are tan 3 2 i - , tan 2 i - , tan 2 i, where r . 0 2 1 1 i (i) Given that the 12th term of this progression is equal to 19 3 3 , find the exact value of i. [4] (ii) Hence find the exact value of the sum to ten terms of this progression. [2]

15 0606/11/O/N/23 © UCLES 2023 [Turn over (b) The first three terms of a geometric progression are cosec 16 1 4z, cosec 1 4 2z, 1, where r r . 2 2 1 1 z - (i) Given that the sum of the 3rd and 4th terms of this progression is equal to 4, find the possible values of z. [4] (ii) Determine whether or not this progression has a sum to infinity. [2] Question 10 is printed on the next page.

16 0606/11/O/N/23 © UCLES 2023 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. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge Assessment International Education Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at www.cambridgeinternational.org after the live examination series. Cambridge Assessment International Education is part of Cambridge Assessment. Cambridge Assessment is the brand name of the University of Cambridge Local Examinations Syndicate (UCLES), which is a department of the University of Cambridge. 10 (a) Given that y x x 4 3 2 2 = - - , show that x y d d can be written in the form ( ) x x Ax B 4 3 2 2 2 - - + , where A and B are integers to be found. [5] (b) Hence find, in terms of h, the approximate change in y when x increases from 3 to 3 + h, where h is small. [3]

This document consists of 8 printed pages. © UCLES 2023 [Turn over Cambridge IGCSE™ ADDITIONAL MATHEMATICS 0606/11 Paper 1 October/November 2023 MARK SCHEME Maximum Mark: 80 Published 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 International will not enter into discussions about these mark schemes. Cambridge International is publishing the mark schemes for the October/November 2023 series for most Cambridge IGCSE, Cambridge International A and AS Level components, and some Cambridge O Level components.

0606/11 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2023 © UCLES 2023 Page 2 of 8 Generic Marking Principles These general marking principles must be applied by all examiners when marking candidate answers. They should be applied alongside the specific content of the mark scheme or generic level descriptors for a question. Each question paper and mark scheme will also comply with these marking principles. GENERIC MARKING PRINCIPLE 1: Marks must be awarded in line with: • the specific content of the mark scheme or the generic level descriptors for the question • the specific skills defined in the mark scheme or in the generic level descriptors for the question • the standard of response required by a candidate as exemplified by the standardisation scripts. GENERIC MARKING PRINCIPLE 2: Marks awarded are always whole marks (not half marks, or other fractions). GENERIC MARKING PRINCIPLE 3: Marks must be awarded positively: • marks are awarded for correct/valid answers, as defined in the mark scheme. However, credit is given for valid answers which go beyond the scope of the syllabus and mark scheme, referring to your Team Leader as appropriate • marks are awarded when candidates clearly demonstrate what they know and can do • marks are not deducted for errors • marks are not deducted for omissions • answers should only be judged on the quality of spelling, punctuation and grammar when these features are specifically assessed by the question as indicated by the mark scheme. The meaning, however, should be unambiguous. GENERIC MARKING PRINCIPLE 4: Rules must be applied consistently, e.g. in situations where candidates have not followed instructions or in the application of generic level descriptors. GENERIC MARKING PRINCIPLE 5: Marks should be awarded using the full range of marks defined in the mark scheme for the question (however; the use of the full mark range may be limited according to the quality of the candidate responses seen). GENERIC MARKING PRINCIPLE 6: Marks awarded are based solely on the requirements as defined in the mark scheme. Marks should not be awarded with grade thresholds or grade descriptors in mind.

0606/11 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2023 © UCLES 2023 Page 3 of 8 Mathematics-Specific Marking Principles 1 Unless a particular method has been specified in the question, full marks may be awarded for any correct method. However, if a calculation is required then no marks will be awarded for a scale drawing. 2 Unless specified in the question, non-integer answers may be given as fractions, decimals or in standard form. Ignore superfluous zeros, provided that the degree of accuracy is not affected. 3 Allow alternative conventions for notation if used consistently throughout the paper, e.g. commas being used as decimal points. 4 Unless otherwise indicated, marks once gained cannot subsequently be lost, e.g. wrong working following a correct form of answer is ignored (isw). 5 Where a candidate has misread a number or sign in the question and used that value consistently throughout, provided that number does not alter the difficulty or the method required, award all marks earned and deduct just 1 A or B mark for the misread. 6 Recovery within working is allowed, e.g. a notation error in the working where the following line of working makes the candidate’s intent clear. MARK SCHEME NOTES The following notes are intended to aid interpretation of mark schemes in general, but individual mark schemes may include marks awarded for specific reasons outside the scope of these notes. Types of mark M Method marks, awarded for a valid method applied to the problem. A Accuracy mark, awarded for a correct answer or intermediate step correctly obtained. For accuracy marks to be given, the associated Method mark must be 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 in principle independent unless the scheme specifically says otherwise; and similarly where there are several B marks allocated. The notation ‘dep’ is used to indicate that a particular M or B mark is dependent on an earlier mark in the scheme. Abbreviations awrt answers which round to cao correct answer only dep dependent FT follow through after error isw ignore subsequent working nfww not from wrong working oe or equivalent rot rounded or truncated SC Special Case soi seen or implied

0606/11 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2023 © UCLES 2023 Page 4 of 8 Question Answer Marks Guidance 1 4 a = B1 3 b = B1 5 c = − B1 2(a) 0 c = B1 1 P : 4 34 2 a b   − + = −     oe B1 Allow multiples but must be in terms of a, b and one numeric term. ( ) 2 P 3 22 x ax x b  = − + ( ) P 2 12 44 a b  = − + soi M1 For attempt to differentiate and substitute in 2 x = 12 62 a b + = A1 Allow multiples but must be in terms of a, b and one numeric term. 6, 10 a b = = − 2 M1 dep for attempt to solve their simultaneous equations. A1 for both 2(b) ( ) 2 6 11 10 x x x − − M1 For ( ) ( ) 2 11 x their a x x their b − + ( )( ) 3 2 2 5 x x x + − A1 3(a) 5 12 −       B1 5 12 −       B1 3(b) 13 B1 FT on their (a) 3(c) ( ) 2 3 2 2 6 their AB OX   = −  −   M1 Condone ( ) 2 3 2 2 6 their AB OX   = −   −   11 2 12   −         A1

0606/11 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2023 © UCLES 2023 Page 5 of 8 Question Answer Marks Guidance 4(a) ln ln ln y A b x = + soi B1 May be implied by parts (b) and (c) lnx 0 0.69 1.1 1.4 1.6 ln y 3 4 4.6 5.1 5.4 2 M1 for attempt to plot a correct graph, allow one point error on the graph. A1 All points correct on the graph. 4(b) Vertical intercept = ( ) ln 3 A = M1 Dep on a straight line graph 20 A1 Gradient = b M1 Dep on a straight line graph 1.5 b = (allow 1.4 to 1.6) A1 4(c) Reading off graph for ln 1.25 x = to obtain ln y or use of their equation M1 Dep on a straight line graph 120 150 y A1 5(a)(i) 5040 B1 5(a)(ii) 2520 B1 5(a)(iii) There are 504 codes less than 1000 5040 504 4536 − = 2 M1 for their (i) 504 − or 9(a product of 3 relevant numbers) 5(b) With family: 462 B1 Without family: 55 B1 Total: 517 B1 6(a) 3 lg50x 3 B1 for 3 lg x or lg2 or lg100 B1 for 3 lg 2 x

0606/11 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2023 © UCLES 2023 Page 6 of 8 Question Answer Marks Guidance 6(b) 4 1 log log 4 a a = B1 ( ) 2 2 log 4 5log 4 3 0 a a − −= M1 For attempt to obtain a 3-term quadratic equation in log 4 a and an attempt to solve to obtain log 4 ... a = 1 log 4 , log 4 3 2 a a = − = 1 3 1 , 4 16 a a = = oe 3 M1 Dep for dealing with logarithms correctly at least once, to obtain a = … A1 for each correct solution nfww. Alternative: 4 1 log 4 log a a = (B1) ( ) 2 4 4 3 log 5log 2 0 a a + − = (M1) For attempt to obtain a 3-term quadratic equation in log 4 a and an attempt to solve to obtain log 4 ... a = 4 4 log 2, 3log 1 a a = − = 1 3 1 , 4 16 a a = = oe (3) M1 Dep for dealing with logarithms correctly at least once, to obtain a = … A1 for each correct solution nfww. 7(a) ( )( ) ( ) 2 d 2 3 2 1 3 2 2 3 2 d y x x x x =  + − + − 2 M1 for attempt at differentiation of a product, allow one arithmetic error. A1 all correct, allow unsimplified. ( )( ) 2 3 2 9 1 x x − + A1 7(b) 2 , 0 3       B1 Must be from a correct derivative 1 343 , 9 81   −     or 1 , 4.23 9   −     B1 7(c) 3 B1 for a correct cubic shape with a maximum in the second quadrant. B1 for 1 2 − and 2 3 , must have a cubic shape B1 for 4 must have a cubic shape

0606/11 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2023 © UCLES 2023 Page 7 of 8 Question Answer Marks Guidance 7(d) 343 0 81 k   or 0 4.23 k   2 B1 for critical values 0 and their 4.23 or 343 81 8 2 4 6 5 1 4 x x x − − = − oe 2 2 3 0 x x − − = oe 2 M1 for attempt to eliminate y and simplify to a 3-term quadratic equation = 0. A1 for a correct equation. Correct attempt to solve their quadratic equation to obtain 2 values for x or for y M1 3 1, 2 x = − A1 5, 5 y = − A1 d 8 6 d y x x = − When 3 d , 6 2 d y x x = = M1 For finding the value of d d y x using their 3 2 x = Equation of tangent: 6 14 y x = − 2 Dep M1 for attempt to find the equation of the tangent using their 3 2 x = and their 5 y = −. A1 allow unsimplified. 5 6 14 x = − oe M1 For use of their 5 y = in their tangent equation 3.17 x = A1 9(a)(i) 19 3 3tan 11 2tan 2 2 3     − + =     2 M1 for use of 12th term with their common difference. A1 allow unsimplified. 3 tan 2 3 = π 3 = 2 Dep M1 for correct attempt to solve their 3 tan 2 3 = 9(a)(ii) 10 3 3 2 3 9 2 2 3 3       − +                    oe M1 For the use of the sum to 10 terms using their 3 tan 2 3 = 20 3 A1

0606/11 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2023 © UCLES 2023 Page 8 of 8 Question Answer Marks Guidance 9(b)(i) Common ratio = 2 4sin  B1 2 1 4sin 4  + = M1 For use of 1 4 their r + = 3 sin 2 =  π 3 =  2 M1 for a correct attempt to solve their 3 sin 2 =  to obtain at least one solution. 9(b)(ii) Common ratio = 3 soi M1 For attempt to find numerical value of their common ratio. 3 1  so no sum to infinity [as 1 1 r − to have a sum to infinity.] oe A1 Must have correct common ratio. 10(a) ( ) ( ) ( ) ( ) 1 1 2 2 2 2 2 1 4 6 3 2 3 2 2 4 x x x x x −   −  − − −       − 3 B1 for ( ) 1 2 2 1 6 3 2 2 x x −  − , allow unsimplified. M1 for a correct attempt to differentiate a quotient. A1 for all other terms correct. ( ) ( ) ( ) ( ) ( ) 1 2 2 2 2 3 2 3 4 3 2 4 x x x x x − − − − − − M1 For a correct attempt to simplify to obtain the given form. ( ) 2 2 12 2 3 2 4 x x x − + − − A1 10(b) When 3 x = , d 36 2 d 5 y x − + = M1 For attempt using their d d y x 6.8h − 2 Dep M1 for attempt at small changes using their 6.8 − . A1 cao