Cambridge IGCSE Mathematics - Additional 0606 — 2022 May/June Paper 1 · Variant 3

[Turn over DC (NF/FC) 303987/4 © UCLES 2022 This document has 16 pages. Any blank pages are indicated. * 7 3 3 8 0 7 0 3 6 9 * ADDITIONAL MATHEMATICS 0606/13 Paper 1 May/June 2022 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 [ ]. Cambridge IGCSE™

2 0606/13/M/J/22 © UCLES 2022 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 0606/13/M/J/22 © UCLES 2022 [Turn over 1 (a) Find the rational numbers a, b and c, such that the first three terms, in descending powers of x, in the expansion of x x 3 9 1 2 5 - e o can be written in the form . ax bx cx 10 7 4 + + [3] (b) Hence find the coefficient of x4 in the expansion of . x x x 3 9 1 1 1 2 5 3 2 - + e e o o [3]

4 0606/13/M/J/22 © UCLES 2022 2 In this question, all lengths are in centimetres and all angles are in radians. B A C O 8 10 The diagram shows a circle, centre O, radius 8. The points A, B and C lie on the circumference of the circle. The chord AB has length 10. (a) Show that angle BOA is 1.35 correct to 2 decimal places. [2] (b) Given that the minor arc BC has a length of 18, find angle BOC. [2] (c) Find the area of the minor sector AOC. [3]

5 0606/13/M/J/22 © UCLES 2022 [Turn over 3 (a) Find the exact solution of the equation . 2 3 5 0 e e x x 6 3 - - = [3] (b) Solve the following simultaneous equations. 1 e e e x x y 4 7 5 7 2 ' = - + xy 18 0 + = [5]

6 0606/13/M/J/22 © UCLES 2022 4 Variables x and y are such that when e y 4 is plotted against x, a straight line of gradient 5 2, passing through (10, 2), is obtained. (a) Find y in terms of x. [3] (b) Find the value of y when x 45 = , giving your answer in the form lnp. [2] (c) Find the values of x for which y can be defined. [1]

7 0606/13/M/J/22 © UCLES 2022 [Turn over 5 The velocity, , vms 1 - of a particle moving in a straight line, t seconds after passing through a fixed point O, is given by . sin v t 6 3 = (a) Find the time at which the acceleration of the particle is first equal to . 9ms 2 - - [4] (b) Find the displacement of the particle from O when . . t 5 6 = [4]

8 0606/13/M/J/22 © UCLES 2022 6 (a) It is given that f x x 2 2 " | for , x 0 H g x x 2 1 " | + for . x 0 H Each of the expressions in the table can be written as one of the following. f f g g fg gf f g f g 2 2 1 1 - - l m l m Complete the table. The first row has been completed for you. [5] Expression Function notation 2 gl 0 x 4 x x 8 8 2 2 + + x 4 3 + x 2 1 -

9 0606/13/M/J/22 © UCLES 2022 [Turn over (b) It is given that ( ) ( ) x x 3 1 h 2 = + - for . x a H The value of a is as small as possible such that h 1 - exists. (i) Write down the value of a. [1] (ii) Write down the range of h. [1] (iii) Find ( )x h 1 - and state its domain. [3]

10 0606/13/M/J/22 © UCLES 2022 7 A curve has equation ( ) y x x 5 2 1 2 3 = + + for . x 0 H (a) Show that ( ) ( ) ( ), x y x x Ax B 5 2 1 d d 2 2 1 = + + + where A and B are integers to be found. [4] (b) Show that there are no stationary points on this curve. [1]

11 0606/13/M/J/22 © UCLES 2022 [Turn over (c) Find the approximate change in y when x increases from 1 to ,p 1+ where p is small. [2] (d) Given that when x 1 = the rate of change in x is 2.5 units per second, find the corresponding rate of change in y. [2]

12 0606/13/M/J/22 © UCLES 2022 8 (a) A 6-digit number is formed from the digits 0, 1, 2, 5, 6, 7, 8, 9. A number cannot start with 0 and each digit can be used at most once in any 6-digit number. (i) Find how many 6-digit numbers can be formed if there are no further restrictions. [1] (ii) Find how many of these 6-digit numbers are divisible by 5. [3] (iii) Find how many of these 6-digit numbers are greater than 850 000. [3]

13 0606/13/M/J/22 © UCLES 2022 [Turn over (b) A team of 8 people is to be chosen from 12 people. Three of the people are brothers who must not be separated. Find the number of different teams that can be chosen. [3]

14 0606/13/M/J/22 © UCLES 2022 9 (a) Solve the equation , cosec 3 2 3 4 2 r z- = b l for . 0 1 1 r z Give your solutions in terms of r. [4] (b) Given that cosec x 2 1 2i - = and , tan y 1 2i + = find y in terms of x. [4]

15 0606/13/M/J/22 © UCLES 2022 10 (a) Show that ( ) x 1 + 6 4 2 2 + - x x 2 3 1 + + can be written as . ( )( ) x x x 2 3 1 14 10 2 + + + [2] (b) Hence find the exact value of ( )( ) . x x x x 2 3 1 14 10 d 2 0 2 + + + y Give your answer in the form , ln p q + where p and q are rational numbers. [6]

16 0606/13/M/J/22 © UCLES 2022 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. BLANK PAGE

This document consists of 9 printed pages. © UCLES 2022 [Turn over Cambridge IGCSE™ ADDITIONAL MATHEMATICS 0606/13 Paper 1 May/June 2022 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 May/June 2022 series for most Cambridge IGCSE, Cambridge International A and AS Level and Cambridge Pre-U components, and some Cambridge O Level components.

0606/13 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2022 © UCLES 2022 Page 2 of 9 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.

0606/13 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2022 © UCLES 2022 Page 3 of 9 Maths-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, 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 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 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

0606/13 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2022 © UCLES 2022 Page 4 of 9 Question Answer Marks Guidance 1(a) 10 7 4 10 243 45 3   x x x 3 B1 for 10 243x B1 for 7 45  x B1 for 4 10 3 x 1(b) 2 3 3 6 1 2 1 1 1          x x x oe B1 Coefficient of term in 4x =   10 243 2 45 3    M1 For their 243+   2 45   their + 10 3 their Must have 3 terms 469 3 oe A1 2(a) 7 cos 32  BOA or 5 sin 2 8  BOA M1   1.350 263...  BOA 1.35  BOA (correct to 2 dp) A1 Must see detail of extra decimal places to justify 2 dp answer 2(b) 8 18  M1 2.25  A1 2(c) 2π 2.25 1.35...     AOC   2.683 M1 For use of 2π ( ) 1.35...   their b Area =   1 64 2.683.. 2 their M1 For use of sector area formula 85.9 or 85.8 A1 Allow awrt 85.9 Alternative Area = 1 1 64π 64 1.35... 64 2.683 2 2                   (2) M1 for a correct plan M1 for one correct use of sector area formula 85.9 or 85.8 (A1) Allow awrt 85.9 or 85.8

0606/13 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2022 © UCLES 2022 Page 5 of 9 Question Answer Marks Guidance 3(a)    3 3 2e 5 e 1 0    x x M1 For attempt to solve a 3-term quadratic equation in 3e x , or using an appropriate substitution. May also be implied by correct use of quadratic formula 1 5 ln 3 2  x 2 Dep M1 for a correct attempt to obtain x = … A1 cao with negative root discounted. 3(b) 7 7 2 e e    x y M1 For correct attempt to deal with powers of e 7 5   x y A1 2 5 126 0    x x or 2 7 5 18 0    y y M1 Dep for attempt to obtain a 3-term quadratic equation equated to zero in either x or y 14, 9   x x A1 For both 9 , 2 7   y y A1 For both 4(a) Intercept = –2 soi B1 4 2 e 2 5   y x M1 For attempt at straight line equation with their intercept 1 2 ln 2 4 5         y x oe A1 4(b) 1 ln16 4  y M1 Dep on M1 in part (a) ln 2  y A1 4(c) 5  x 1 5(a) Acceleration 18cos3  t 2 B1 for cos3 k t , 2, 0   k k 1 cos3 2  t oe M1 For attempt to solve their 1 cos3 2  t to obtain a value for t 2π 9  t or 0.698 A1

0606/13 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2022 © UCLES 2022 Page 6 of 9 Question Answer Marks Guidance 5(b)   2cos3   t c 2 B1 for cos3 k t , 18, 0   k k Displacement 2 2cos3   t M1 For attempt to find value of c 2.92 A1 Alternative 2cos3  t (2) B1 for cos3 k t , 18, 0   k k   5.6 0 2cos3  t =    2cos16.8 2   (M1) For correct application of limits using their cos3 k t , 18, 0   k k 2.92 (A1) 6(a) Expression Function notation 0 g 4x f 2 8 8 2   x x fg 2 g 1 2  x 1 g 5 B1 for each one correct 6(b)(i) 1  a B1 6(b)(ii)  h 3 x  B1 6(b)(iii)   2 1 3    x y 1 3   y x M1 For a correct attempt to find the inverse, allow one sign error  1 h 1 3    x x A1 Must be using correct notation 3 x  B1 Must be using correct notation 4 3 x 

0606/13 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2022 © UCLES 2022 Page 7 of 9 Question Answer Marks Guidance 7(a)         1 3 2 2 2 3 5 2 2 1 2 1 d 2 d 5               x x x y x x 3 B1 for   1 2 3 2 2 1 2   x oe M1 for attempt to differentiate a quotient or product A1 for all terms other than   1 2 3 2 2 1 2   x correct       1 2 2 2 1 14 5    x x x A1 7(b) When d 0 d  y x , 14  x and 0.5  x but 0 x so no stationary point B1 FT on their   14  x 7(c) 5 3 12 p , 15 3 36 p or 0.722p 2 M1 for calculation of d d y x when 1  x and multiplication by p 7(d) 25 3 24 or 1.80 oe 2 M1 for multiplication of their d d y x calculated in (c) by 2.5, must be numeric 8(a)(i) 17640 B1 8(a)(ii) Ends in a 5: 2160 or 6 6 1 4 P P  B1 Ends in a 0 : 2520 or 7 6 1 4 P P  B1 4680 B1 8(a)(iii) Starts with 85 : 360 Starts with 86 : 360 Starts with 87 : 360 Starts with 89: 360 oe 1440 or 6 4 4 P  B1 Starts with 9 : 2520 or 7 5P B1 3960 B1

0606/13 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2022 © UCLES 2022 Page 8 of 9 Question Answer Marks Guidance 8(b) With brothers : 126 or 9 5 C B1 Without brothers : 9 or 9 8 C B1 135 B1 9(a) 2 π 3 sin 2 3 4          soi B1 π π 5π , , 3 2 6  3 M1 for a correct method of solution, may be implied by one correct solution. A1 for a second correct solution A1 for a third correct solution and no extra solutions within the range 9(b) 2 1 cot 1   y B1 Attempt to use 2 2 cot 1 cosec    M1 For attempt to eliminate  1 1 2 2    y x oe 2 Dep M1 for attempt to rearrange to obtain the required form Alternative 2 1 2 1 sin   x and 2 2 sin 1 cos    y (B1) 1 2 1 1 1 1 2 1     x y x oe (M1) For attempt to eliminate  1 1 2 2    y x oe (2) Dep M1 for attempt to rearrange to obtain the required form 10(a)           2 2 6 1 4 2 3 2 2 3 1 2 3 1         x x x x x x M1 For dealing with the fractions, allow an extra  1  x in each of the terms in the numerator and in the denominator Allow one sign error      2 14 10 2 3 1    x x x A1 AG - Must have sufficient evidence of expansion and simplification to obtain the given answer

0606/13 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2022 © UCLES 2022 Page 9 of 9 Question Answer Marks Guidance 10(b)       4 2ln 2 3 2ln 1 1            x x x 3 B1 for each term, must have the correct signs with each term Must be using part (a)   4 2ln8 2ln3 2ln 2 4 3           M1 Dep on at least one log term in their integral, for use of limits 8 16 ln 3 9  2 A1 for 16 ln 9 A1 for 8 3