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

This document consists of 15 printed pages and 1 blank page. DC (RW/SG) 106861/1 © UCLES 2016 [Turn over Cambridge International Examinations Cambridge International General Certificate of Secondary Education * 3 9 1 1 9 9 9 3 9 1 * ADDITIONAL MATHEMATICS 0606/21 Paper 2 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.

2 0606/21/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 0606/21/M/J/16 © UCLES 2016 [Turn over 1 Find the values of x for which x x 4 2 7 2 - + ^ ^h h . [3] 2 (a) Illustrate the statements A B f and B C f using the Venn diagram below. [1] (b) It is given that the elements of set  are the letters of the alphabet, the elements of set P are the letters in the word maths, the elements of set Q are the letters in the word exam. (i) Write the following using set notation. The letter h is in the word maths. [1] (ii) Write the following using set notation. The number of letters occurring in both of the words maths and exam is two. [1] (iii) List the elements of the set P Q k l. [1]

4 0606/21/M/J/16 © UCLES 2016 3 Do not use a calculator in this question. (i) Find the value of log p - p2. [1] (ii) Find lg 10 1 n J L KK N P OO. [1] (iii) Show that log lg lg lgy 10 20 4 5 2 - = ^ h , where y is a constant to be found. [2] (iv) Solve log log log x x 2 3 600 r r r + = . [2]

5 0606/21/M/J/16 © UCLES 2016 [Turn over 4 A B C θ rad The diagram shows 3 circles with centres A, B and C, each of radius 5 cm. Each circle touches the other two circles. Angle BAC is θ radians. (i) Write down the value of θ. [1] (ii) Find the area of the shaded region between the circles. [4]

6 0606/21/M/J/16 © UCLES 2016 5 Do not use a calculator in this question. (a) Express 7 8 5 - in the form a b + , where a and b are integers. [3] (b) Given that p q 28 3 2 3 2 + = + ^ h , where p and q are integers, find the values of p and of q. [3]

7 0606/21/M/J/16 © UCLES 2016 [Turn over 6 (i) Express x x 4 8 5 2 + - in the form p x q r 2 + + ^ h , where p, q and r are constants to be found. [3] (ii) State the coordinates of the vertex of y x x 4 8 5 2 = + - . [2] (iii) On the axes below, sketch the graph of y x x 4 8 5 2 = + - , showing the coordinates of the points where the curve meets the axes. [3] O x y

8 0606/21/M/J/16 © UCLES 2016 7 O, P, Q and R are four points such that p OP = , q OQ = and q p OR 3 2 = - . (i) Find, in terms of p and q, (a) PQ, [1] (b) QR. [1] (ii) Justifying your answer, what can be said about the positions of the points P, Q and R? [2] (iii) Given that i j OP 3 = + and that i j OQ 2 = + , find the unit vector in the direction OR. [3]

9 0606/21/M/J/16 © UCLES 2016 [Turn over 8 (a) (i) Use the Binomial Theorem to expand a b 4 + ^ h , giving each term in its simplest form. [2] (ii) Hence find the term independent of x in the expansion of x x 2 5 1 4 + J L KK N P OO . [2] (b) The coefficient of x3 in the expansion of x 1 2 n + J L KK N P OO equals n 12 5 . Find the value of the positive integer n. [3]

10 0606/21/M/J/16 © UCLES 2016 9 (a) Given that tan y a bx c = + has period r 4 radians and passes through the points ,0 2 - ^ h and r , 16 0 J L KK N P OO, find the value of each of the constants a, b and c. [3] a = … b = … c = … (b) (i) On the axes below, draw the graph of cos y x 2 3 1 = + for r r x 3 2 3 2 G G - radians. [3] 3 2r - 2 r - 3 r - 6 r - 6 r 3 r 2 r 3 2r y x –4 4 O (ii) Using your graph, or otherwise, find the exact solutions of cos x 2 3 1 1 2 + = ^ h for r r x 3 2 3 2 G G - radians. [2]

11 0606/21/M/J/16 © UCLES 2016 [Turn over 10 (a) (i) Find how many 5-digit even numbers can be made using each of the digits 1, 2, 3, 4, 5 once only. [2] (ii) Find how many different 3-digit numbers can be made using the digits 1, 2, 3, 4, 5 if each digit can be used once only. [2] (b) A man and two women are to sit in a row of five empty chairs. Calculate the number of ways they can be seated if (i) the two women must sit next to each other, [2] (ii) all three people must sit next to each other. [2]

12 0606/21/M/J/16 © UCLES 2016 11 (i) Find x x x 3 d 2 3 - ^ h y . [2] The diagram shows part of the curve y x x 3 2 3 = - and the lines y x 3 = and y x 2 27 3 = - . The curve and the line y x 3 = meet the x-axis at O and the curve and the line y x 2 27 3 = - meet the x-axis at A. y x x 3 2 3 = - y x 2 27 3 = - y x 3 = y O A x B (ii) Find the coordinates of A. [1] (iii) Verify that the coordinates of B are ,3 9 ^ h. [1]

13 0606/21/M/J/16 © UCLES 2016 [Turn over (iv) Find the area of the shaded region. [4]

14 0606/21/M/J/16 © UCLES 2016 12 A curve has equation y x x x 1 2 5 12 = - - - . (i) Find x y d d . [3] (ii) Find x y d d 2 2 . [2]

15 0606/21/M/J/16 © UCLES 2016 (iii) Find the coordinates of the stationary points of the curve and determine their nature. [5]

16 0606/21/M/J/16 © UCLES 2016 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. BLANK PAGE

® 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 0606/21 Paper 2 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.

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Page 3 Mark Scheme Syllabus Paper Cambridge IGCSE – May/June 2016 0606 21 © Cambridge International Examinations 2016 Question Answer Marks Guidance 4 (i) 3 π isw B1 (ii) [Area triangle ABC =] 2 1 10 sin 2 3 their π   × ×     oe M1 seen or implied by 25 3 or 43.3(0...) [Area 1 sector = ] 2 1 5 2 3 their π × × oe or 2 60 5 360 their π ° × × M1 seen or implied by 25 6 π or 13.0(8...) or 13.09 Complete correct plan M1 e.g. their triangle – 3(their sector) 4.03(1...) or 25 25 3 2 π − isw A1 Units not required 5 (a) ( ) ( ) ( ) 7 5 8 7 5 7 5 + × − + and attempt to multiply M1 56 40 2 + oe A1 not from wrong working 14 10 + A1 (b) 2 4 3 12 q q + + soi B1 2 28 12 q = + oe M1 can be implied by 4 and 16 or –4 and –16 q = 4, −4 p = 16, −16 A1 all values 6 (i) ( ) 2 4 1 9 x + − B3,2, 1,0 one mark for each of p, q, r correct in a correctly formatted expression; allow correct equivalent values; If B0 then SC2 for ( ) 4 1 9 x + − or SC1 for correct 3 values seen in incorrect format e.g. ( ) 4 1 9 x x + − or ( ) 2 4 1 9 x + − or for a correct completed square form of the original expression in a different but correct format. e.g. ( ) 2 2 2 2 9 x + −

Page 4 Mark Scheme Syllabus Paper Cambridge IGCSE – May/June 2016 0606 21 © Cambridge International Examinations 2016 Question Answer Marks Guidance (ii) (−1, 9) B2FT B1FT (–q , –r) r < 0 for each correct coordinate (iii) -1 1 2 3 4 5 6 7 8 9 10 B1 B1 B1 Correct symmetric W shape with cusps on x-axis y-intercept marked at 5 only or coords indicated on graph x-intercepts marked at –2.5 and 0.5 only x-axis or coords indicated on graph or close by 7 (i) (a) q – p B1 (b) 2q – 2p or ( ) 2 − q p B1 (ii) The points are collinear oe PQ


 is a (scalar) multiple of QR


 and they have a point in common. oe B1 B1 Condone PQ


 is parallel toQR


 and … (iii) [OR


 =] 4i – 3j oe soi B1 2 2 4 ( 3) + − (=5) M1 condone 2 2 4 3 + ; may be implied by correct answer or correct FT answer ( ) 1 5 4 – 3 i j oe A1 8 (a) (i) 4 3 2 2 3 4 4 6 4 a a b a b ab b + + + + final answer B2,1,0 −1 each error/omission (ii) ( ) 2 2 1 6 2 5 x x       soi M1 Could be in full expansion 24 25 or 0.96 isw A1 Must be explicitly identified (b) 1 ( 1)( 2) 5 8 6 12 n n n n − −  =     soi leading to a cubic or quadratic ( 2 3 18 0 n n − − = ) M1 Must attempt to expand and remove fractions Solves their quadratic [(n – 6)(n + 3)] M1 must have come from a valid attempt [n =] 6 only, not from wrong working A1 Must be n if labelled −2.5 0.5

Page 5 Mark Scheme Syllabus Paper Cambridge IGCSE – May/June 2016 0606 21 © Cambridge International Examinations 2016 Question Answer Marks Guidance 9 (a) a = 2 b = 4 c = −2 B3 B1 for each correct value (b) (i) -1 1 2 3 B3,2,1, 0 sinusoidal curve symmetrical about y-axis clear intent to have amplitude of 2 2 cycles If not fully correct max B2 (ii) π 2 − , π 6 − , π 6 , π 2 , π 3 − , π 3 cao B2 B1 for any 4 correct 10 (a) (i) 2 × 4! or 2 5! 5 × oe M1 48 A1 (ii) 5 3P or 5! 2! or 5 4 3 × × oe M1 60 A1 (b) (i) 4 × 2[!] × 3oe M1 Correct first step implied by a correct product of two elements 24 A1 (ii) 3! or 3 × 3 seen M1 18 A1 11 (i) ( ) 52 2 3 2 2 5 x x c − + isw B1+B1 (ii) (9, 0) oe B1 Not just x = 9 (iii) Substitute (3, 9) into both lines Or solves simultaneously (6x = 27 – 3x oe) to get x = 3, y = 9 B1 3 × 3 = 9 and 27 3 3 9 2 −× =

Page 6 Mark Scheme Syllabus Paper Cambridge IGCSE – May/June 2016 0606 21 © Cambridge International Examinations 2016 Question Answer Marks Guidance (iv) [ ] 1 Area 9 9 2 AOB = × × oe ( 81 2 or 40.5) M1 Uses their (ii). May split into 2 triangles (13.5 and 27). May integrate. Must be a complete method. [ ] 52 2 3(9) 2(9) 0 2 5 their   − −       (= 24.3) M1 lower limit may be omitted but must be correct if seen 81 243 2 10 their their − M1 must be from genuine attempts at area of triangle and area under curve 16.2 A1 12 (i) ( ) ( ) ( ) 2 2 1 2 5 d d 1 x x y x x − − −   =     − M1A1 Allow slips in d d u x and d d v x but must be explicit. Allow 2 2 ( 1) 2 1 x x x − = − + − 12 isw B1 ALT using 2 12 14 5 1 x x y x − + − = − −24x + 14 B1 2 2 d ( 1)( 24 14) ( 12 14 5) d ( 1) y x x x x x x − − + −− + −   =   −   M1 A1FT FT on their derivative of 3 term quadratic (ii) ( ) 2 3 2 d 1 d y k x x −   = −     M1 No additional terms k = − 6 isw A1

Page 7 Mark Scheme Syllabus Paper Cambridge IGCSE – May/June 2016 0606 21 © Cambridge International Examinations 2016 Question Answer Marks Guidance (iii) ( ) 2 3 12 1 their x    −    −   = 0 and find a value for x M1 12 2 x −24x + 9 = 0 oe (2x − 3)(2x − 1) = 0 oe x = 0.5 and x = 1.5 A1 y = 2 and y = −22 A1 if A0 A0 then A1 for a correct (x, y) pair 3 6 0 ( 0.5) − > − therefore min when x = 0.5 oe B1 or 3 6 48 ( 0.5)   − =   −   therefore min when x = 0.5 oe 3 6 0 (0.5) − < therefore max when x = 1.5 oe B1 or 3 6 48 (0.5)   − = −     therefore max when x = 1.5 oe M1A1 is possible from other methods