Cambridge IGCSE Mathematics - Additional 0606 — 2020 Oct/Nov Paper 1 · Variant 2

Cambridge IGCSE™ * 7 3 9 4 7 0 3 4 3 7 * DC (PQ/FC) 187996/3 © UCLES 2020 [Turn over This document has 16 pages. Blank pages are indicated. ADDITIONAL MATHEMATICS 0606/12 Paper 1 October/November 2020 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 0606/12/O/N/20 © UCLES 2020 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 0606/12/O/N/20 © UCLES 2020 [Turn over 1 The curve y x k 2 4 2 = + + intersects the straight line ( ) y k x 4 = + at two distinct points. Find the possible values of k. [4] 2 - 5 - 4 - 3 - 2 - 1 0 1 2 5 y x The diagram shows the graph of ( ) y x f = , where ( )x f is a cubic polynomial. (a) Find ( )x f . [3] (b) Write down the values of x such that ( )x 0 f 1 . [2]

4 0606/12/O/N/20 © UCLES 2020 3 (a) Write down the amplitude of cos x 2 3 1 - . [1] (b) Write down the period of cos x 2 3 1 - . [1] (c) On the axes below, sketch the graph of cos y x 2 3 1 = - for x 3 G G r r - radians. - r 0 r 2r 3r 1 - 1 - 2 - 3 2 3 y x [3]

5 0606/12/O/N/20 © UCLES 2020 [Turn over 4 The 7th and 10th terms of an arithmetic progression are 158 and 149 respectively. (a) Find the common difference and the first term of the progression. [3] (b) Find the least number of terms of the progression for their sum to be negative. [3]

6 0606/12/O/N/20 © UCLES 2020 5 Find the coefficient of x2 in the expansion of x x x x 3 2 5 - + b bl l . [5]

7 0606/12/O/N/20 © UCLES 2020 [Turn over 6 ( )x x x 2 3 f 2 = + - for x 1 H- (a) Given that the minimum value of x x 2 3 2 + - occurs when x 1 =- , explain why ( )x f has an inverse. [1] (b) On the axes below, sketch the graph of ( ) y x f = and the graph of ( ) y x f 1 = - . Label each graph and state the intercepts on the coordinate axes. O y x [4]

8 0606/12/O/N/20 © UCLES 2020 7 A curve has equation ln y x x 2 1 3 5 2 = + - ` j for x 3 5 2 2 . (a) Find the equation of the normal to the curve at the point where x 2 = . [6] (b) Find the approximate change in y as x increases from 2 to h 2 + , where h is small. [1]

9 0606/12/O/N/20 © UCLES 2020 [Turn over 8 (a) Find the number of ways in which 12 people can be put into 3 groups containing 3, 4 and 5 people respectively. [3] (b) 4-digit numbers are to be formed using four of the digits 2, 3, 7, 8 and 9. Each digit may be used once only in any 4-digit number. Find how many 4-digit numbers can be formed if (i) there are no restrictions, [1] (ii) the number is even, [1] (iii) the number is greater than 7000 and odd. [3]

10 0606/12/O/N/20 © UCLES 2020 9 A curve has equation ( ) y x x 2 1 4 3 = - + . (a) Show that ( ) x y x Ax B 4 3 4 d d = + + , where A and B are constants. [5] (b) Hence write down the x-coordinate of the stationary point of the curve. [1] (c) Determine the nature of this stationary point. [2]

11 0606/12/O/N/20 © UCLES 2020 [Turn over 10 The polynomial ( )x x ax bx 6 2 p 3 2 = + + + , where a and b are integers, has a factor of x 2 - . (a) Given that ( ) ( ) 1 2 0 p p =- , find the value of a and of b. [4] (b) Using your values of a and b, (i) find the remainder when ( )x p is divided by x 2 1 - , [2] (ii) factorise ( )x p . [2]

12 0606/12/O/N/20 © UCLES 2020 11 In this question all lengths are in centimetres and all angles are in radians. A B C O r r r r D E F The diagram shows the rectangle ADEF, where AF DE r = = . The points B and C lie on AD such that AB CD r = = . The curve BC is an arc of the circle, centre O, radius r and has a length of 1.5r. (a) Show that the perimeter of the shaded region is ( . . ) sin r 7 5 2 0 75 + . [5]

13 0606/12/O/N/20 © UCLES 2020 [Turn over (b) Find the area of the shaded region, giving your answer in the form kr2, where k is a constant correct to 2 decimal places. [4]

14 0606/12/O/N/20 © UCLES 2020 12 (a) 0 v ms–1 t s 20 30 V 60 90 The diagram shows the velocity–time graph of a particle P that travels 2775 m in 90 s, reaching a final velocity of V ms 1 - . (i) Find the value of V. [3] (ii) Write down the acceleration of P when t 40 = . [1]

15 0606/12/O/N/20 © UCLES 2020 (b) The acceleration, a ms 2 - , of a particle Q travelling in a straight line, is given by cos a t 6 2 = at time t s. When t 0 = the particle is at point O and is travelling with a velocity of 10ms 1 - . (i) Find the velocity of Q at time t. [3] (ii) Find the displacement of Q from O at time t. [3]

16 0606/12/O/N/20 © UCLES 2020 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 the Cambridge Assessment Group. Cambridge Assessment is the brand name of the University of Cambridge Local Examinations Syndicate (UCLES), which itself is a department of the University of Cambridge. BLANK PAGE

This document consists of 9 printed pages. © UCLES 2020 [Turn over Cambridge IGCSE™ ADDITIONAL MATHEMATICS 0606/12 Paper 1 October/November 2020 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 2020 series for most Cambridge IGCSE, Cambridge International A and AS Level and Cambridge Pre-U components, and some Cambridge O Level components.

0606/12 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2020 © UCLES 2020 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.

0606/12 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2020 © UCLES 2020 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

0606/12 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2020 © UCLES 2020 Page 4 of 9 Question Answer Marks Guidance 1 ( ) ( ) ( ) 2 2 4 4 0 x k x k − + + + = ( ) ( ) ( ) 2 2 4 4 0 x k x k + −− + + = B1 Discriminant:( ) ( ) ( ) 2 4 4 2 4 k k + − × × + M1 Use of discriminant to obtain 2 critical values using their 3 term quadratic 4 ± A1 For critical values 4 k < − 4 k > A1 2(a) ( )( )( ) 1 5 1 2 2 y x x x = − + + − 3 B1 for negative soi B1 for 1 2 soi B1 for ( )( )( ) 5 1 2 x x x + + − or 3 2 4 7 10 x x x + − − 2(b) 5 1 x −< < − B1 2 x > B1 3(a) 2 B1 3(b) 6π or o 1080 B1 3(c) 3 B1 for passing through ( ) π, 0 − and ( ) 3π, 3 − – must be a curve B1 for correct shape with max on y-axis and a min at 3π x = B1 for passing through ( ) 0, 1 and ( ) π, 0 only on the positive x-axis 4(a) 6 158 a d + = 9 149 a d + = B1 For both equations, may be implied by a correct a and d 3, d = − B1 176 a = B1

0606/12 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2020 © UCLES 2020 Page 5 of 9 Question Answer Marks Guidance 4(b) ( )( ) ( ) ( ) 352 1 3 0 2 n n + − − < M1 For correct attempt at sum formula with their a and their d, 355 3 or 118.3 oe A1 119 A1 5 5 3 10 40 ... x x x + + + 3 M1 for attempt to expand 5 2 x x   +     , with at least 2 correct terms A1 for 3 10x A1 for 40x Term in 2 : x ( ) ( ) 1 40 3 10 × − × M1 For ( ) ( ) 1 40 3 10 their their × ± × 10 A1 6(a) It is a one-one function because of the given restricted domain or because x ⩾ –1 B1 6(b) 4 B1 for f( ) y x = for 1 x > − only B1 for 1 on x-axis and –3 on y-axis for f( ) y x = B1 for 1 f ( ) y x − = as a reflection of f( ) y x = in the line y x = , maybe implied by intercepts with axes B1 for 1 on y-axis and –3 on x-axis for 1 f ( ) y x − =

0606/12 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2020 © UCLES 2020 Page 6 of 9 Question Answer Marks Guidance 7(a) ( ) ( ) ( ) 2 2 2 6 2 1 2ln 3 5 d 3 5 d 2 1 x x x y x x x + − − − = + or ( ) ( ) ( ) 1 2 2 2 d 6 2 1 2 2 1 ln 3 5 d 3 5 y x x x x x x − − = + − + − − 3 B1 for 2 6 3 5 x x − M1 for attempt at a quotient or equivalent product A1 for all terms other than 2 6 3 5 x x − correct When 2, 0 x y = = B1 May be implied When 2, x = d 6 2 d 2 2 1 y x = + or 24 6 2 7 − or 2.22 oe Normal: ( )( ) 2 2 1 2 6 2 y x + = − − oe or ( ) 7 2 24 6 2 y x = − − − oe or ( ) 1 2 2.22 y x = − − oe or ( ) 4 2 2 12 y x + = − − oe or ( ) 9 4 2 2 24 6 2 y x + = − − + oe 0.451 0.638 y x = − + 2 M1 for attempt at normal using their y and their perp gradient A1 Allow equivalent surd forms 7(b) 6 2 2 2 1 h       +   or 24 6 2 7 h − or other equivalent surd forms, or 2.22h B1 FT on their d d y x from (a) 8(a) 12 9 3 4 220 126 C C × = × or 12 7 5 4 792 35 C C × = × or 12 8 4 5 495 56 C C × = × or other equivalents 27720 3 B1 for one correct combination in a product of 2 or 3 combinations Must be numeric B1 for a second appropriate combination in the same product Must be numeric 8(b)(i) 120 B1 8(b)(ii) 48 B1

0606/12 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2020 © UCLES 2020 Page 7 of 9 Question Answer Marks Guidance 8(b)(iii) Starts with 7 or 9 24 B1 May be implied by 12 and 12 Starts with 8 18 B1 42 B1 Alternative Ends with 3 18 (B1) Ends with 7 or 9 24 (B1) May be implied by 12 and 12 42 (B1) 9(a) ( ) ( ) ( ) 1 1 2 2 d 1 2 1 4 4 3 2 4 3 d 2 y x x x x − = − × × + + + 3 B1 for ( ) 1 2 1 4 4 3 2 x − × + oe M1 for a correct attempt at a product A1 for all other terms correct ( ) ( ) 1 2 d 2 4 3 2 1 4 3 d y x x x x − = + −+ + or equivalent M1 For attempt to simplify to the given form ( ) ( ) 1 2 4 3 1 d d 4 3 x y x x + = + A1 9(b) 1 3 − B1 FT on their 3 1 0 x + = 9(c) For a complete method using 2nd derivative, or gradient or y values either side or one side of their stationary point e.g. x < 1 3 − 1 3 − > 1 3 − d d y x – 0 + x < 1 3 − 1 3 − > 1 3 − y < 2.15 − 2.15 − > 2.15 − M1 Must be using values of 3 4 x > − Minimum A1 Must be from correct work

0606/12 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2020 © UCLES 2020 Page 8 of 9 Question Answer Marks Guidance 10(a) ( ) p 2 : 48 4 2 2 0 a b + + + = 2 25 0 a b + + = B1 For 2 25 0 a b + + = or multiple ( ) ( ) p 1 2p 0 = − 12 0 a b + + = B1 For 12 0 a b + + = 13, 1 a b = − = 2 M1 for attempt to solve their equations in a and b leading to 2 values A1 for both 10(b)(i) 1 6 13 1 p 2 2 8 4 2  = − + +     M1 For attempt to find 1 p 2       using their a and b 0 A1 10(b)(ii) ( )( )( ) 2 2 1 3 1 x x x − − + 2 M1 for realising that 2 factors are known and 3rd factor can be got by observation or algebraic long division, or for making use of 2 x − or 2 1 x − in order to obtain a quadratic factor A1 Must see all factors together 11(a) 1.5 BOC ∠ = rad B1 2 sin0.75 BC r = M1 For a complete attempt to find BC – must be using a right-angled triangle to get required result – Given answer 2 sin 0.75 BC r = A1 Perimeter = 2 2 sin 0.75 4 1.5 r r r r + + + M1 Dep on first M mark for attempt at perimeter ( ) 7.5 2sin0.75 r + A1 Given answer 11(b) Area = ( ) ( ) 2 1 2 2 sin0.75 1.5 sin1.5 2 r r r r + − − Area = 2 2 2 3.36 0.75 0.4987 r r r − + 3 M1 for a correct plan M1 for ( ) 2 2 sin0.75 r r r + , using their 2 sin 0.75 r B1 for segment ( ) 2 2 1 1.5 sin1.5 0.251 2 r r − = Area = 2 3.11r A1

0606/12 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2020 © UCLES 2020 Page 9 of 9 Question Answer Marks Guidance 12(a)(i) Area under graph: ( ) ( ) ( ) 1 1 60 40 30 30 30 2775 2 2 V + × + + × = or ( ) ( ) ( ) 1 1 20 30 40 30 30 30 2 2 V × + + + + × 2 M1 for attempt to find the area under the graph Dep M1 on previous M mark for attempt to equate to 2775 and simplify in order to find V or 30 V − 55 A1 12(a)(ii) 0 B1 12(b)(i) ( ) 3sin 2 v t c = + M1 Must have 3sin 2t ± 10 c = M1 Dep for attempt to find c + , 3sin 2 10 v t = + A1 12(b)(ii) 3 cos2 10 2 s t t d = − + + M1 For attempt to integrate their v, must have 3 cos2 2 t ± 3 2 d = M1 Dep on previous M mark for attempt to find d. 3 3 cos2 10 2 2 s t t = − + + A1