Cambridge IGCSE Mathematics - Additional 0606 — 2016 May/June Paper 1 · Variant 2

ADDITIONAL MATHEMATICS 0606/12 Paper 1 May/June 2016 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. This document consists of 16 printed pages. DC (ST/SW) 106862/3 © UCLES 2016 [Turn over Cambridge International Examinations Cambridge International General Certificate of Secondary Education * 0 9 9 3 8 1 1 3 1 0 *

2 0606/12/M/J/16 © UCLES 2016 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/16 © UCLES 2016 [Turn over 1 (a) The universal set  is the set of real numbers and sets X, Y and Z are such that X = {integer multiples of 5}, Y = {integer multiples of 10}, Z = {r, 2, e}. Use set notation to complete the two statements below. Y … X Y  Z = … [2] (b) On each of the Venn diagrams below, shade the region indicated. A B (A B) C A (B C ) C A B C [2]

4 0606/12/M/J/16 © UCLES 2016 2 (i) The first 3 terms in the expansion of x 2 4 1 5 - c m are a b x c 2 + + x . Find the value of each of the integers a, b and c. [3] (ii) Hence find the term independent of x in the expansion of x x 2 4 1 3 4 5 - + c ^ m h. [2]

5 0606/12/M/J/16 © UCLES 2016 [Turn over 3 Vectors a, b and c are such that a = y 2 J L KK N P OO, b = 1 3 J L KK N P OO and c = 5 5 - J L KK N P OO. (i) Given that a b c = - , find the possible values of y. [3] (ii) Given that b c b c b c 4 2 n m + + - = - ^ ^ ^ h h h, find the value of n and of m. [3]

6 0606/12/M/J/16 © UCLES 2016 4 Do not use a calculator in this question. Find the positive value of x for which x x 4 5 2 5 1 0 2 + + - - = ^ ^ h h , giving your answer in the form b a 5 + , where a and b are integers. [6]

7 0606/12/M/J/16 © UCLES 2016 [Turn over 5 (i) Show that cos sec sin tan 1 1 i i i i - + = ^ ^h h . [4] (ii) Hence solve the equation cos sec sin 1 1 i i i - + = ^ ^h h for 0 G G i r radians. [3]

8 0606/12/M/J/16 © UCLES 2016 6 Show that x x 4 1 d d e x 3 + ^ h can be written in the form x px q 4 1 e x 3 + + ^ h, where p and q are integers to be found. [5]

9 0606/12/M/J/16 © UCLES 2016 [Turn over 7 A O B y = 1 – 2cos3x y x The diagram shows part of the graph of y x 1 3 2cos = - , which crosses the x-axis at the point A and has a maximum at the point B. (i) Find the coordinates of A. [2] (ii) Find the coordinates of B. [2] (iii) Showing all your working, find the area of the shaded region bounded by the curve, the x-axis and the perpendicular from B to the x-axis. [4]

10 0606/12/M/J/16 © UCLES 2016 8 0.2 1 2 3 4 x2 0.4 0.6 lg y 0 0.7 0.8 0.9 1.0 0.1 0.3 0.5 0.0 (1, 0.73) (4, 0.10) Variables x and y are such that when lgy is plotted against x2, the straight line graph shown above is obtained. (i) Given that y Abx2 = , find the value of A and of b. [4] (ii) Find the value of y when x = 1.5. [2] (iii) Find the positive value of x when y = 2. [2]

11 0606/12/M/J/16 © UCLES 2016 [Turn over 9 A curve passes through the point ,2 3 4 - c m and is such that x x y 3 10 d d 2 1 = + - ^ h . (i) Find the equation of the curve. [4] The normal to the curve, at the point where x 5 = , meets the line y 3 5 =- at the point P. (ii) Find the x-coordinate of P. [6]

12 0606/12/M/J/16 © UCLES 2016 10 x cm y cm B C A D The diagram shows a badge, made of thin sheet metal, consisting of two semi-circular pieces, centres B and C, each of radius x cm. They are attached to each other by a rectangular piece of thin sheet metal, ABCD, such that AB and CD are the radii of the semi-circular pieces and AD = BC = y cm. (i) Given that the area of the badge is 20 cm2, show that the perimeter, P cm, of the badge is given by P x x 2 40 = + . [4]

13 0606/12/M/J/16 © UCLES 2016 [Turn over (ii) Given that x can vary, find the minimum value of P, justifying that this value is a minimum. [5]

14 0606/12/M/J/16 © UCLES 2016 11 (a) 10 10 20 v ms–1 30 20 30 40 50 60 t s O The diagram shows the velocity-time graph of a particle P moving in a straight line with velocity v ms–1 at time t s after leaving a fixed point. (i) Find the distance travelled by the particle P. [2] (ii) Write down the deceleration of the particle when t = 30. [1]

15 0606/12/M/J/16 © UCLES 2016 [Turn over (b) The diagram shows a velocity-time graph of a particle Q moving in a straight line with velocity v ms–1, at time t s after leaving a fixed point. 5 10 15 20 3 6 v ms–1 t s O The displacement of Q at time t s is s m. On the axes below, draw the corresponding displacement-time graph for Q. [2] 5 10 15 20 30 60 90 s m t s O Question 11(c) is printed on the next page.

16 0606/12/M/J/16 © UCLES 2016 (c) The velocity, v ms–1, of a particle R moving in a straight line, t s after passing through a fixed point O, is given by v 4 6 e t2 = + . (i) Explain why the particle is never at rest. [1] (ii) Find the exact value of t for which the acceleration of R is 12 ms–2. [2] (iii) Showing all your working, find the distance travelled by R in the interval between . t 0 4 = and . t 0 5 = . [4] 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 International Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at www.cie.org.uk after the live examination series. 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.

® IGCSE is the registered trademark of Cambridge International Examinations. This document consists of 7 printed pages. © UCLES 2016 [Turn over Cambridge International Examinations Cambridge International General Certificate of Secondary Education ADDITIONAL MATHEMATICS 0606/12 Paper 1 May/June 2016 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 will not enter into discussions about these mark schemes. Cambridge is publishing the mark schemes for the May/June 2016 series for most Cambridge IGCSE®, Cambridge International A and AS Level components and some Cambridge O Level components.

Page 2 Mark Scheme Syllabus Paper Cambridge IGCSE – May/June 2016 0606 12 © Cambridge International Examinations 2016 Abbreviations awrt answers which round to cao correct answer only dep dependent FT follow through after error isw ignore subsequent working oe or equivalent rot rounded or truncated SC Special Case soi seen or implied www without wrong working Question Answer Marks Guidance 1 (a) Y X ⊂ or Y X ⊆ only Y Z ∩ = ∅ or { } only B1 B1 (b) (i) (ii) B1 B1 2 (i) 2 20 5 32 x x − + B3 B1 for each correct term – must be integers (ii) ( ) 20 3 32 4 16 x x   × + − × =     Accept o 16x M1 A1 for ( ) ( ) 20 3 32 4 their their x x  −  × + ×     3 (i) 6 2   − =   −   b c 2 4 36 4 y + = + 6 y = ± B1 M1 A1 may be implied by further correct working for one correct attempt at using the modulus (ii) 4 2 µ λ + = or 4 24 7 µ λ − + = 4 µ λ − = − or 8 8 µ λ − = leading to 4 8 , 3 3 µ λ = = oe allow 1.33 and 2.67 or better B1 B1 DB1 for one correct equation in µ and λ for a second correct equation in µ and λ for both, must have both previous B marks

Page 3 Mark Scheme Syllabus Paper Cambridge IGCSE – May/June 2016 0606 12 © Cambridge International Examinations 2016 Question Answer Marks Guidance 4 ( ) ( ) 2 4 5 2 5 1 0 x x + + − −= ( ) ( ) ( )( ) ( ) 2 2 5 2 5 4 4 5 1 2 4 5 x − − ± − − + − = + ( ) ( ) 2 5 9 4 5 16 4 5 2 4 5 x − − ± − + + = + ( ) ( ) 2 5 5 2 4 5 − − + = + ( ) 3 5 2 4 5 + = + ( )( ) ( )( ) 3 5 4 5 2 4 5 4 5 + − = + − 7 5 22 + = M1 A1 DM1 A1 M1 A1 You must be convinced that a calculator is not being used. for use of quadratic formula (allow one sign error), allow 2 9 4 5 b = − all correct for attempt to simplify the discriminant (minimum of 4 terms must be seen in discriminant, 2 terms involving √5 and 2 constant terms) for ( ) 3 5 2 4 5 + + or 3 5 8 2 5 + + , ignore negative solution if included for attempt to rationalise an expression of the form 5 5 a b c d ± ± as part of their solution of the quadratic Must obtain an integer denominator Final A1 can only be awarded if all previous marks have been obtained 5 (i) ( )( ) 1 cos 1 sec 1 cos 1 cos cos cos θ θ θ θ θ θ − + = − + − sec cos θ θ = − 1 cos cos θ θ = − 2 1 cos cos θ θ − = 2 sin cos θ θ = sin tan θ θ = www M1 DM1 A1 A1 M1 for expansion and use of 1 sec cos θ θ = consistently, allow one sign error for attempt at a single fraction, dependent on first M1

Page 4 Mark Scheme Syllabus Paper Cambridge IGCSE – May/June 2016 0606 12 © Cambridge International Examinations 2016 Question Answer Marks Guidance Alternative method: ( ) cos 1 1 cos cos θ θ θ +   −     2 1 cos cos θ θ − = 2 sin cos θ θ = sin tan θ θ = www M1 DM1 A1 A1 for attempt at a single fraction for second factor and use of 1 sec cos θ θ = for expansion (ii) sin tan sin θ θ θ = ( ) sin tan 1 0 θ θ − = π tan 1, , 4 θ θ = = allow 0.785 or better sin 0, 0, π θ θ = = or 3.14 or better B1 B1 B1 for π 4 θ = from tan 1 θ = for 0 θ = from sin 0 θ = for π θ = from sin 0 θ = 6 ( ) 1 3 2 d e 4 1 d x x x   +     ( ) ( ) 1 1 3 3 2 2 1 e 4 4 1 3e 4 1 2 x x x x − = × + + + ( ) ( ) 3 1 3 2 1 2 2e 3e 4 1 4 1 x x x x = + + + ( ) ( ) 3 1 2 e 2 12 3 4 1 x x x = + + + ( ) ( ) 3 1 2 e 12 5 4 1 x x x = + + B1 B1 B1 DM1 A1 for ( ) 1 3 2 e 4 1 x r x − + must be part of a sum, 1 1 or 2 or 4 2 2 r = × for ( ) 1 3 2 e 4 1 x s x + must be part of a sum, s is 1 or 3 for all correct, allow unsimplified for ( ) ( ) 3 1 2 e 4 1 x a bx x + + , dependent on first 2 B marks , must be using a correct method, collecting terms in the numerator correctly 7 (i) 1 π cos3 , 2 9 x x = = or 0.349, o 20 , allow 0.35 M1 A1 for correct attempt to solve the trigonometric equation (ii) π , 3 3 B      or ( ) 1.05, 3 , ( ) o 60 , 3 B1B1 B1 for each, must be in correct position or in terms of x = and y =

Page 5 Mark Scheme Syllabus Paper Cambridge IGCSE – May/June 2016 0606 12 © Cambridge International Examinations 2016 Question Answer Marks Guidance (iii) π π 3 3 π π 9 9 2 1 2cos3 d sin3 3 x x x x   − = −     ∫ π π 2 3 3 9 3 2     = − − ×             = 2π 3 9 3 + oe or 1.28 M1 A1 DM1 A1 for sin3 x a x ± attempt to integrate at least one term for correct integration for correct use of limits from (i) and (ii), must be in radians 8 (i) 2 lg lg lg y x b A = + lg 0.21 b = ± 0.617 b = allow 0.62, 0.21 10− lg 0.94 A = allow 0.93 to 0.95 8.71 A = allow awrt 8.5 to 8.9 B1 B1 B1 B1 for lg 0.21 b = ± may be implied Alternative method 0.73 5.37 or 10 Ab = 0.1 4 1.259 or 10 Ab = 3 0.63 10 b − = 0.617 b = allow 0.62, 0.21 10− 8.71 A = allow awrt 8.5 to 8.9 B1 B1 B1 B1 for both equations, allow correct to 2 sf (ii) 2 1.5, 2.25 x x = = 2.93 y = , allow awrt 2.9 or 3.0 M1 A1 for correct use of graph 2 1.5 y theirA theirb = × or ( ) 2 lg lg 1.5 lg y theirA theirb = + (iii) lg 0.301, y = or ( ) 2 2 '8.71 0.617 ' x = 1.74 x = , allow 3 or awrt 1.7, 1.8 M1 A1 for correct use of graph to read off 2 x ( ) 2 2 x theirA theirb = or ( ) ( ) 2 lg2 lg lg theirb x theirA = + 9 (i) ( ) ( ) 1 2 2 3 10 3 y x c = + + passes through 4 2, 3   −     , so 4 c = − ( ) 1 2 2 3 10 4 3 y x = + − oe B1 B1 M1 A1 for p( ) 1 2 3 10 x + where p is a constant for ( ) 1 2 2 3 10 3 x + oe unsimplified for attempt to find c, must have attempt to integrate, must have the first B1

Page 6 Mark Scheme Syllabus Paper Cambridge IGCSE – May/June 2016 0606 12 © Cambridge International Examinations 2016 Question Answer Marks Guidance (ii) When 5 x = , 2 3 y = − perpendicular gradient 5 = − Equation of normal: ( ) 2 5 5 3 y x + = − − When 5, 3 y = − 5.2 x = oe B1 B1 M1 A1 DM1 A1 for attempt at the normal using their perpendicular gradient and their y value (but not 4 5 or 3 3 y = − − ). for use of 5 3 y = − in their normal equation to get as far as ... x = 10 (i) Area: 20 2 πx xy = + 2 20 πx y x − = 2π 2 2 P x x y = + + 20 2π 2 2 π x x x x   = + + −     40 2x x = + B1 B1 M1 A1 for attempt to use perimeter and obtain in terms of x only all steps seen, www AG Alternative method: 20 2 πx xy = + 2π 2 2 P x y x = + + ( ) 2 2 π 2 x xy x x = + + ( ) 2 20 x = +2x 40 2x x = + B1 M1 B1 A1 for attempt to use perimeter and write in 2 πx xy x + for replacing 2 πx xy + with 20 all steps seen, www AG

Page 7 Mark Scheme Syllabus Paper Cambridge IGCSE – May/June 2016 0606 12 © Cambridge International Examinations 2016 Question Answer Marks Guidance (ii) 2 d 40 2 d P x x = − When d 0 d P x = , 2 5 x = allow 4.47, 20 leading to 8 5 P = , allow 17.9 2 2 3 d 80 d P x x = , always positive so a minimum M1 DM1 A1 A1 A1 for attempt to differentiate for equating to zero and attempt to solve at least as far as 2 x = for this statement or use of gradient inspection either side of correct x 11 (a) (i) Distance = area under graph = 1275 M1 A1 for attempt to find the area, one correct area seen ( triangle, rectangle or trapezium) as part of a sum. (ii) deceleration is 1.5 oe B1 (b) B1 B1FT for a straight line between ( ) 0,0 and ( ) 10,60 FT a straight line between ( ) 10, 60 and ( ) 20, 90 , a displacement vector 10 30       from their ( ) 10, 60 their (c) (i) 2 e t is always positive or oe B1 (ii) 2 8e t a = 2 3 e 2 t = 1 3 ln 2 2 t = , 3 ln 2 or 1 ln1.5 2 M1 A1 for attempt to differentiate, must be of the form 2e t p , equate to 12 and solve. Allow fractions equivalent to 3 2 (iii) 0.5 2 0.4 2e 6 t s t   = +   ( ) ( ) 0.8 2e 3 2e 2.4 = + − + ( ) 8.436 6.851 = − 1.59 = , allow 1.58 M1 A1 DM1 A1 for attempt to integrate to get 2e 6 t q t + all correct for correct use of limits or considering distances separately, ignore attempts at c 5 10 15 20 10 20 30 40 50 60 70 80 90 100 x y