Cambridge IGCSE Mathematics - Additional 0606 — 2019 May/June Paper 1 · Variant 2

This document consists of 16 printed pages. DC (JM/TP) 165274/2 © UCLES 2019 [Turn over * 0 7 7 3 9 7 3 0 9 1 * ADDITIONAL MATHEMATICS 0606/12 Paper 1 May/June 2019 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 an HB pencil for any diagrams or graphs. Do not use staples, paper clips, 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. Cambridge Assessment International Education Cambridge International General Certificate of Secondary Education

2 0606/12/M/J/19 © UCLES 2019 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 0606/12/M/J/19 © UCLES 2019 [Turn over 1 (a) On the Venn diagrams below, shade the region indicated. A A C B B C A B C + + l l l ( ) A B C , + [2] (b)  { : } x x 0 360 ° ° G G = { : . } cos P x x 2 0 5 = = { : . } sin Q x x 0 5 = = Find P Q + . [3]

4 0606/12/M/J/19 © UCLES 2019 2 Do not use a calculator in this question. Find the coordinates of the points of intersection of the curve ( ) ( ) y x x 2 3 1 2 = + - and the line ( ) y x 3 2 3 = + . [5]

5 0606/12/M/J/19 © UCLES 2019 [Turn over 3 The number, B, of a certain type of bacteria at time t days can be described by B 200 800 e e t t 2 2 = + - . (i) Find the value of B when t 0 = . [1] (ii) At the instant when t B 1200 d d = , show that 3 4 0 e e t t 4 2 - - = . [3] (iii) Using the substitution u e t2 = , or otherwise, solve 3 4 0 e e t t 4 2 - - = . [2]

6 0606/12/M/J/19 © UCLES 2019 4 (a) Given that q pr pr qr 2 2 1 2 2 3 - ` ` j j can be written in the form paqbr c, find the value of each of the constants a, b and c. [3] (b) Solve , . x y x y 3 4 4 3 14 2 1 2 1 2 1 2 1 - = + = - - [3]

7 0606/12/M/J/19 © UCLES 2019 [Turn over 5 9.6 cm 12 cm A B C O rad i The diagram shows the right-angled triangle OAB. The point C lies on the line OB. Angle OAB 2 r = radians and angle AOB = i radians. AC is an arc of the circle, centre O, radius 12 cm and AC has length 9.6 cm. (i) Find the value of i. [2] (ii) Find the area of the shaded region. [4]

8 0606/12/M/J/19 © UCLES 2019 6 (a) Eight books are to be arranged on a shelf. There are 4 mathematics books, 3 geography books and 1 French book. (i) Find the number of different arrangements of the books if there are no restrictions. [1] (ii) Find the number of different arrangements if the mathematics books have to be kept together. [3] (iii) Find the number of different arrangements if the mathematics books have to be kept together and the geography books have to be kept together. [3]

9 0606/12/M/J/19 © UCLES 2019 [Turn over (b) A team of 6 players is to be chosen from 8 men and 4 women. Find the number of different ways this can be done if (i) there are no restrictions, [1] (ii) there is at least one woman in the team. [2]

10 0606/12/M/J/19 © UCLES 2019 7 A pilot wishes to fly his plane from a point A to a point B on a bearing of 055°. There is a wind blowing at 120 km h–1 from the west. The plane can fly at 650 km h–1 in still air. (i) Find the direction in which the pilot must fly his plane in order to reach B. [4] (ii) Given that the distance between A and B is 1250 km, find the time it will take the pilot to fly from A to B. [4]

11 0606/12/M/J/19 © UCLES 2019 [Turn over 8 When e y is plotted against x 1 , a straight line graph passing through the points (2, 20) and (4, 8) is obtained. (i) Find y in terms of x. [5] (ii) Hence find the positive values of x for which y is defined. [1] (iii) Find the exact value of y when x = 3. [1] (iv) Find the exact value of x when y = 2. [2]

12 0606/12/M/J/19 © UCLES 2019 9 y x O P Q y = 5 cos y x 4 2 3 = + The diagram shows the curve cos y x 4 2 3 = + intersecting the line y 5 = at the points P and Q. (i) Find, in terms of r, the x-coordinate of P and of Q. [3]

13 0606/12/M/J/19 © UCLES 2019 [Turn over (ii) Find the exact area of the shaded region. You must show all your working. [6]

14 0606/12/M/J/19 © UCLES 2019 10 x cm 4x cm h cm The diagram shows an open container in the shape of a cuboid of width x cm, length 4x cm and height h cm. The volume of the container is 800 cm3. (i) Show that the external surface area, S cm2, of the open container is such that S x x 4 2000 2 = + . [4]

15 0606/12/M/J/19 © UCLES 2019 [Turn over (ii) Given that x can vary, find the stationary value of S and determine its nature. [5] Question 11 is printed on the next page.

16 0606/12/M/J/19 © UCLES 2019 11 The normal to the curve y x x 2 3 1 3 2 = - + ` `j j at the point where x 3 7 = , meets the y-axis at the point P. Find the exact coordinates of the point P. [7] 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.

This document consists of 11 printed pages. © UCLES 2019 [Turn over Cambridge Assessment International Education Cambridge International General Certificate of Secondary Education ADDITIONAL MATHEMATICS 0606/12 Paper 1 May/June 2019 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 2019 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 May/June 2019 © UCLES 2019 Page 2 of 11 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 May/June 2019 © UCLES 2019 Page 3 of 11 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 Question Answer Marks Guidance 1(a) E B1 E B1

0606/12 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2019 © UCLES 2019 Page 4 of 11 Question Answer Marks Guidance 1(b) o o o o {30 ,150 , 210 , 330 } P = B1 May be seen or implied in a Venn diagram Allow without set notation o o {30 ,150 } Q = B1 May be seen or implied in a Venn diagram Allow without set notation o o {30 ,150 } P Q ∩ = B1 Dep on both previous B marks Must be in set notation 2 Either: ( ) ( ) ( ) 2 2 3 1 3 2 3 + − = + x x x ( )( )( ) 2 2 3 2 6 0 + + − = x x x M1 For attempt to equate line and curve and attempt to simplify to 2 3 + x × a quadratic factor or cancelling 2 3 + x and obtaining a quadratic factor ( )( ) 2 2 3 2 6 0 + + − = x x x ( )( )( ) 2 3 2 3 2 0 + − + = x x x M1 Dep for attempt at 3 linear factors from a linear term and a quadratic term 3 , 0 2   −     B1 3 ,18 2       A1 Dep on first M mark only ( ) 2, 3 − − A1 Dep on first M mark only Or: ( ) ( ) ( ) 2 2 3 1 3 2 3 + − = + x x x ( ) 3 2 4 8 9 18 0 + − − = x x x M1 For attempt to equate line and curve and attempt to simplify to a cubic equation, by collecting like terms ( )( ) 2 2 4 9 + − x x ( )( ) 2 2 3 2 7 6 − + + x x x ( )( ) 2 2 3 2 6 + + − x x x ( )( )( )( ) 2 3 2 3 2 0 + − + = x x x M1 Dep For attempt to find a factor from a 4 term cubic equation (usually 2 + x ), do long division oe to obtain a quadratic factor and factorise this quadratic factor 3 , 0 2   −     A1 3 ,18 2       A1 ( ) 2, 3 − − A1 3(i) 1000 B1

0606/12 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2019 © UCLES 2019 Page 5 of 11 Question Answer Marks Guidance 3(ii) 2 2 d 400e 1600e d − = − t t B t B1 2 2 3 e 4e− = − t t oe M1 For equating an equation of the form 2 2 e e− + t t a b to 1200 and dividing by 400 4 2 e 3e 4 0 − − = t t A1 3(iii) ( )( ) 2 2 e 1 e 4 0 + − = t t M1 For attempt to factorise and solve, dealing with exponential correctly, to obtain 2 e ... = t 1 ln2, ln4 2 = t or awrt 0.693 only isw if appropriate A1 4(a) 5 2 = a B1 3 2 = − b B1 11 2 = c B1 4(b) 1 1 2 2 9 3 12 − − = x y 1 1 2 2 4 3 14 − + = x y M1 For attempt to solve simultaneous equations. Must reach 1 2 ... = kx or 1 2 ... −= ky oe 4 = x A1 1 4 = y A1 5(i) 9.6 12θ = M1 For use of arc length 0.8 θ = A1

0606/12 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2019 © UCLES 2019 Page 6 of 11 Question Answer Marks Guidance 5(ii) Either ( ) tan , 12.36 12 θ = = AB AB Or ( ) 12 17.22 cosθ = = OB OB M1 For attempt to find AB or OB using their θ May be implied by a correct triangle area Allow if using degrees consistently Either Area 1 12 12.36 2 ∆ = × × OAB their Or Area 1 12 17.22 sin 2 θ ∆ = × × × OAB their ( ) 74.1 or 74.2 = M1 Allow if using degrees consistently For attempt to find area of triangle using their θ Area of sector 2 1 12 0.8 2 = × × OAC 57.6 = B1 Allow unsimplified Area of shaded region = 16.5 or 16.6 A1 6(a)(i) 40320 B1 6(a)(ii) No. of ways with maths books as 1 unit = 5! or 5 4! × or 5 5P or 120 B1 No. of ways maths books can be arranged amongst themselves = 4! or 4 4P or 24 B1 Total = ( ) 5! 4! oe 2880 × = B1 6(a)(iii) No. of ways with maths books as 1 unit and geography books as 1 unit = 3! or 3 3P or 3 2! × or 6 B1 No. of ways maths books can be arranged amongst themselves and geography books can be arranged amongst themselves = 4! 3! × or 4 3 4 3 × P P or 144 B1 Total = ( ) 3! 4! 3! oe × × = 864 B1 6(b)(i) 12 6 924 = C B1

0606/12 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2019 © UCLES 2019 Page 7 of 11 Question Answer Marks Guidance 6(b)(ii) Either: 8 6 924 −C M1 For their (i) – the number of teams of just men Total = 896 A1 Or: 5M 1W : 8 4 5 1 × C C ( = 224) 4M 2W : 8 4 4 2 × C C ( = 420) 3M 3W : 8 4 3 3 × C C ( = 224) 2M 4W : 8 4 2 4 × C C ( = 28) M1 For a complete method Total = 896 A1 7(i) 120 β 35 650 α B1 For correct triangle, may be implied by a correct sine rule or cosine rule. ( ) 120 120 650 650 or or sin sin 55 sin35 sin145 α θ = − M1 For use of a correct sine rule to obtain ... α = or ... θ = Or for a correct cosine rule leading to a value for v, followed by a correct sine rule leading to one of the other angles o 6.08 α = or 138.9 β = A1 May be implied by a correct o awrt 49 θ = Bearing is o 048.9 or o 049 A1 7(ii) Either ( ) ( ) 650 120 or sin 145 sin35 sin rv their their α α = − Or 2 2 2 650 120 rv = + − ( ) ( ) 2 650 120 cos 145 theirα × × − M1 For use of sine rule or cosine rule to find resultant velocity Do not allow for a right-angled triangle May be seen in (i) 745 = rv A1 For correct resultant velocity, allow awrt 745 Time taken = 1250 744.7 their M1 For correct attempt at finding time using their v , ≠ 650, 120, 770 or 530 1.68 = hours or I hour 41 mins or 101 mins A1

0606/12 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2019 © UCLES 2019 Page 8 of 11 Question Answer Marks Guidance 8(i) e = + y m c x B1 May be implied by subsequent work Either 20 2 = + m c 8 4 = + m c M1 For at least 1 correct equation M1 Dep For attempt to solve their 2 equations simultaneously to obtain at least one unknown leading to 6, 32 = − = m c A1 For both 6 ln 32   = −     y x A1 Must have correct brackets Mark the final answer given Or: Gradient = ( ) 6 = − m M1 For attempt to find gradient and equate it to m 20 2 = + m c or 8 4 = + m c or 1 e 8 4   − = −     y m x or 1 e 20 2   − = −     y m x M1 For at least 1 correct equation, may be using their m leading to 32 = c and 6 = − m A1 For both 6, 32 = − = m c 6 ln 32   = −     y x A1 8(ii) 3 16 > x oe B1 8(iii) ln30 = y isw B1 8(iv) 6 2 ln 32   = −     x M1 For a correct substitution and attempt to re-arrange using 2, their 32 and their – 6, keeping exactness to obtain x = 2 6 32 e = − x oe A1 Must be exact

0606/12 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2019 © UCLES 2019 Page 9 of 11 Question Answer Marks Guidance 9(i) 5 4 2cos3 = + x M1 For attempt to solve trig equation to obtain one correct solution π 9 A1 π 9 − A1 9(ii) Either: π 9 π 9 4 2cos3 5 d − + − ∫ x x M1 For use of subtraction method π 9 π 9 2 sin3 3 −   −     x x M1 For attempt to integrate to obtain the form sin3 + a x bx B1 For 2sin3 3 x B1 For −x , may be implied by 4 5 − x x 3 π 3 π 3 9 3 9     − −− +             M1 Dep on previous M mark for correct application of their limits in radians from (i) retaining exactness Shaded area = 2 3 2π 3 9 − oe isw A1

0606/12 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2019 © UCLES 2019 Page 10 of 11 Question Answer Marks Guidance 9(ii) Or: Area of rectangle = 2π 5 9 × M1 5 the difference of limits × their in exact radians Area under curve = π 9 π 9 2 4 sin3 3 −   +     x x M1 For attempt to integrate to obtain the form sin3 + a x bx B1 For 2sin3 3 x B1 For 4x 3 4π 3 4π 3 9 3 9     + −− −             2 3 8π 3 9   = +       M1 Dep on previous M mark for correct application of their limits in exact radians from (i) retaining exactness Shaded area = 2 3 2π 3 9 − oe isw A1 10(i) 2 800 4 = x h B1 2 800 4 = h x oe or 800 4 = xh x oe B1 ( ) 2 2 8 4 = + + S hx xh x oe M1 Allow if h is substituted at this point 2 2000 4   = +     S x x A1 Leading to AG, must have S = or surface area = at some point and no errors

0606/12 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2019 © UCLES 2019 Page 11 of 11 Question Answer Marks Guidance 10(ii) 2 d 2000 8 d   = −     S x x x B1 For correct differentiation When 3 d 0, 250 d S x x = = oe (6.30) M1 For equating to zero and attempt to solve, must get as far as x = …, must be using the form 2 + b ax x A1 For correct positive x 476 = S only A1 2 2 3 d 4000 8 d = + S x x 2 2 d d S x > 0 or 24 so minimum B1 For a correct convincing method, with enough detail to reach a correct conclusion of a minimum. Must be using 3 250 = x oe 11 M1 For attempt at differentiating a product B1 For ( ) 1 3 2 3 3 1 3 − × + x ( ) ( ) ( ) 1 2 3 3 d 2 2 3 3 1 3 1 d 3 y x x x x −   = − × × + + +     A1 For all other terms correct 4 3 = y B1 When 7 d 13 , 3 d 3 = = y x x M1 For attempt at normal equation using 1 −their m and their y when 7 3 = x Equation of normal: 4 3 7 3 13 3   − = − −     y x A1 For correct normal equation, may be implied by a correct final answer At y-axis, 73 39 = y 73 0, 39       isw A1