Cambridge IGCSE Mathematics - International 0607 — 2015 Oct/Nov Paper 4 · Variant 1

This document consists of 20 printed pages. DC (LK/CGW) 103946/2 © UCLES 2015 [Turn over Cambridge International Examinations Cambridge International General Certificate of Secondary Education * 1 5 4 5 7 5 4 0 2 1 * CambrIdgE InTErnaTIonal maThEmaTICs 0607/41 Paper 4 (Extended) october/november 2015 2 hours 15 minutes Candidates answer on the Question Paper. Additional Materials: Geometric Instruments Graphics 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. Do not use staples, paper clips, glue or correction fluid. You may use an HB pencil for any diagrams or graphs. DO noT WRITE IN ANY BARCODES. Answer all the questions. Unless instructed otherwise, give your answers exactly or correct to three significant figures as appropriate. Answers in degrees should be given to one decimal place. For π, use your calculator value. You must show all the relevant working to gain full marks and you will be given marks for correct methods, including sketches, even if your answer is incorrect. 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 120.

2 0607/41/O/N/15 © UCLES 2015 Formula List For the equation ax2 + bx + c = 0 x = a b b ac 2 4 2 ! - - Curved surface area, A, of cylinder of radius r, height h. A = 2πrh Curved surface area, A, of cone of radius r, sloping edge l. A = πrl Curved surface area, A, of sphere of radius r. A = 4πr2 Volume, V, of pyramid, base area A, height h. V= Ah 3 1 Volume, V, of cylinder of radius r, height h. V = πr2h Volume, V, of cone of radius r, height h. V = 3 1 πr2h Volume, V, of sphere of radius r. V = 3 4 πr3 sin sin sin A a B b C c = = A C B c b a a2 = b2 + c2 – 2bc cos A Area = sin bc A 2 1

3 0607/41/O/N/15 © UCLES 2015 [Turn over Answer all the questions. 1 Sunil has $80 and Asha has $75. (a) Write the ratio 80 : 75 in its simplest form. Answer(a)… : … [1] (b) (i) Sunil spends $24. Work out $24 as a percentage of $80. Answer(b)(i) …% [1] (ii) Sunil invests $50 at a rate of 2% per year compound interest. Calculate the interest Sunil has after 20 years. Answer(b)(ii) $ … [4] (c) During each month, Asha spends 5 1 of the money that she had at the beginning of the month. (i) Work out how much of the $75 Asha has at the end of the 2nd month. Answer(c)(i) $ … [2] (ii) Calculate the number of whole months it takes for Asha to have less than $5. Answer(c)(ii) … [3]

4 0607/41/O/N/15 © UCLES 2015 2 (a) p 2 3 = J L KK N P OO q 14 8 = J L KK N P OO (i) Find 2p + 3q . Answer(a)(i) J L K KK N P O OO [2] (ii) Find q p - . Answer(a)(ii)  [3] (b) The graph of y = f(x) is mapped onto the graph of y = f(x + 2) by a translation with vector u v J L KK N P OO. Find the value of u and the value of v. Answer(b) u = … v = … [2]

5 0607/41/O/N/15 © UCLES 2015 [Turn over (c) 7 D 8 y x 6 5 4 3 2 1 –1 –2 –3 –4 –5 –6 0 1 2 3 4 5 6 –6 –5 –4 –3 –2 –1 T E (i) Draw the image of triangle T under a rotation of 90° clockwise about the point ( , ) 1 1 - - . [3] (ii) Describe fully the single transformation that maps triangle T onto triangle D. … … [2] (iii) Describe fully the single transformation that maps triangle T onto triangle E. … … [3]

6 0607/41/O/N/15 © UCLES 2015 3 y x 5 –5 –5 5 0 ( ) f x x 1 2 = - , x 0 ! (a) On the diagram, sketch the graph of ( ) f y x = , for values between x 5 =- and x 5 = . [2] (b) Solve the inequality ( ) f x 0 1  Answer(b) … [2]

7 0607/41/O/N/15 © UCLES 2015 [Turn over (c) Find ( ) f x 1 - . Answer(c) … [3] (d) On the diagram, sketch the graph of ( ) f y x 1 = - , for values between x 5 =- and x 5 = . [2] (e) Describe fully the single transformation that maps the graph of ( ) f y x = onto the graph of ( ) f y x 1 = - . … … [2]

8 0607/41/O/N/15 © UCLES 2015 4 (a) 9 cm 3 cm 4.5 cm X B A C D NOT TO SCALE 4 cm AB and CD are parallel. AD and CB intersect at X. CD = 9 cm, AB = 4 cm, AX = 4.5 cm and BX = 3 cm. Calculate the length of CX. Answer(a) … cm [2] (b) S P Y Q R 4 cm 7 cm 8 cm 6 cm NOT TO SCALE P, Q, R and S lie on a circle. PR and QS intersect at Y. QR = 6 cm, PS = 8 cm, PY = 7 cm and YS = 4 cm. Calculate the length of RY. Answer(b) … cm [2]

9 0607/41/O/N/15 © UCLES 2015 [Turn over (c) 12 cm NOT TO SCALE E F w cm The two shapes are mathematically similar. The area of E is 90 cm2 and the area of F is 45 cm2. Find the value of w. Answer(c) w = … [3]

10 0607/41/O/N/15 © UCLES 2015 5 B C D A 12 cm 11 cm 10 cm 35° NOT TO SCALE Calculate (a) BC, Answer(a) … cm [2] (b) angle CAD, Answer(b) … [3] (c) the area of the quadrilateral ABCD. Answer(c) … cm2 [3]

11 0607/41/O/N/15 © UCLES 2015 [Turn over 6 120 students estimate the mass, m kg, of a bag of oranges. The frequency table shows the results. Mass (m kg) . m 0 5 1 1 G . m 1 2 1 1 G . . m 1 2 1 4 1 G . . m 1 4 1 6 1 G . m 1 6 2 1 G Frequency 4 36 40 32 8 (a) Calculate an estimate of the mean. Answer(a) … kg [2] (b) Complete the histogram to show the information in the table. 20 0 0.2 0.4 0.6 0.8 1.0 Mass (kilograms) 1.2 1.4 1.6 1.8 2.0 60 40 m 80 120 100 Frequency density 140 160 200 180 [3]

12 0607/41/O/N/15 © UCLES 2015 7 (a) A solid metal cuboid measures 20 cm by 8 cm by 2 cm. 1 cm3 of the metal has a mass of 7.85 g. (i) Calculate the mass of the cuboid. Answer(a)(i) … g [2] (ii) The surface of the cuboid is painted at a cost of 8 cents per cm2. Calculate the cost of painting the cuboid. Give your answer in dollars. Answer(a)(ii) $ … [3] (b) Another cuboid measures 16 cm by 6 cm by 4 cm. It is cut into cubes, each of side 2 cm. Calculate the number of cubes. Answer(b) … [2] (c) Another solid metal cuboid measures 20 cm by 12 cm by 4 cm. It is melted down and made into spheres of radius 1.5 cm. Calculate (i) the largest number of spheres of radius 1.5 cm that can be made, Answer(c)(i) … [3]

13 0607/41/O/N/15 © UCLES 2015 [Turn over (ii) the volume of metal remaining after the spheres have been made, Answer(c)(ii) … cm3 [2] (iii) the radius of the sphere that can be made using all the remaining metal. Answer(c)(iii) … cm [2] (d) A plastic cone has radius r cm and perpendicular height 3r cm. 1 cm3 of the plastic has a mass of 0.9 g. A wooden hemisphere has a radius of 2r cm. 1 cm3 of the wood has a mass of 0.45 g. Find the mass of the cone as a fraction of the mass of the hemisphere. Give your answer in its lowest terms. Answer(d) … [4]

14 0607/41/O/N/15 © UCLES 2015 8 0 2 –1 4 y x ( ) f x x 3 2 = - ( ) g x xx = (a) On the diagram, sketch the graphs of ( ) f y x = and ( ) g y x = for values between x 0 = and x 2 = . [4] (b) Solve the equation x x 3 x 2 - = for x 0 2 G G . Answer(b) x = … [1]

15 0607/41/O/N/15 © UCLES 2015 [Turn over (c) Solve the equation x 3 2 - = 0 for x 0 2 G G . Answer(c) x = … [1] (d) (i) Find the co-ordinates of the local minimum point on the graph of y = g(x). Answer(d)(i) ( … , … ) [2] (ii) Find the range of g(x) for the domain x 0 2 1 G . Answer(d)(ii) … [2] (e) (i) Find the values of the following. g(0.1) = … g(0.01) = … g(0.001) = … [3] (ii) Complete the statement. Starting from x = 0.1, as x gets closer and closer to 0, g(x) gets closer and closer to the value … [1]

16 0607/41/O/N/15 © UCLES 2015 9 0 A 1 1 1 B 2 3 The diagram shows two unbiased dice, A and B. The numbers on die A are 0, 1, 1, 1, 2, 3. The numbers on die B are 1, 2, 2, 3, 3, 3. When a die is rolled, the number shown on the top face is recorded. (a) Both dice are rolled. Find the probability that (i) both dice show 3, Answer(a)(i) … [2] (ii) the numbers showing on the two dice add up to 2. Answer(a)(ii) … [3]

17 0607/41/O/N/15 © UCLES 2015 [Turn over (b) Die B is rolled until it shows 2. Find the probability that this occurs when the die is rolled for the 4th time. Answer(b) … [2] (c) Die A is rolled until it shows 3. The probability that this occurs when the die is rolled for the nth time is 46656 3125 . Find the value of n. Answer(c) … [2]

18 0607/41/O/N/15 © UCLES 2015 10 f(x) = 2x + 3 g(x) = x – 1 h(x) = log (x + 1) (a) Find f(h(9)). Answer(a) … [2] (b) Find g(f(x)) in its simplest form. Answer(b) … [2] (c) Find ( ) ( ) f g x x 1 1 + in terms of x. Give your answer as a single fraction. Answer(c) … [3]

19 0607/41/O/N/15 © UCLES 2015 [Turn over (d) Solve the equation. ( ) h x 1 =- Answer(d) x = … [2] (e) Solve the equation. ( ( )) g x 5 2 = Give exact answers. Answer(e) x … or x = … [3] Question 11 is printed on the next page.

20 0607/41/O/N/15 © UCLES 2015 11 (a) Cakes cost x cents each and drinks cost y cents each. 2 cakes and 1 drink cost $1.57 . 1 cake and 3 drinks cost $2.96 . Find the total cost of 3 cakes and 2 drinks. Give your answer in dollars. Answer(a) $ … [6] (b) A child’s train ticket costs $x. An adult’s train ticket costs $(x + 5). Claudia buys 11 tickets. She spends $24 on children’s tickets and $24 on adults’ tickets. Write down an equation in x and solve it to find the cost of a child’s ticket. Answer(b) $ … [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. CAMBRIDGE INTERNATIONAL EXAMINATIONS Cambridge International General Certificate of Secondary Education MARK SCHEME for the October/November 2015 series 0607 CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/41 Paper 4 (Extended), maximum raw mark 120 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 October/November 2015 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 – October/November 2015 0607 41 © Cambridge International Examinations 2015 Abbreviations cao correct answer only dep dependent FT follow through after error isw ignore subsequent working oe or equivalent SC Special Case nfww not from wrong working soi seen or implied Question Answer Mark Part Marks 1 (a) 16 : 15 1 (b) (i) 30 1 (ii) 24.3[0] or 24.29 to 24.30 4 M3 for 50 × 1.0220 – 50 or M2 for 50 × 1.0220 or M1 for 50 ×1.02n , n ⩾2oe (c) (i) 48 2 M1 for 75 × (0.8)2 oe (ii) 13 3 B2 for answer 12.13 to 12.14 or M1 for 75 × 0.8n = 5 oe 2 (a) (i)       30 46 2 B1 each component SC1 for       30 46 (ii) 13 3 M2 for 2 2 5 12 + or M1 14 – 2 and 8 – 3 or better seen (b) [u = ] –2 [v = ] 0 1 1 (c) (i) Image at (1, –3), (1, –6), (2, –6) 3 SC2 for rotation about (–1, –1) 90˚ anti- clockwise or SC1 for rotation 90˚ clockwise with different centre (ii) Reflection x = –1 1 1 (iii) Stretch [factor] 4 invariant x-axis oe 1 1 1

P 3 4 5 Pag Q 5 ge ues (a (b (c (e (a (b (c (a (b (c 3 stio a),(d b) c) e) a) b) c) a) b) c) on d) 0 1 R y 6. 3 8. 9. 59 85 0 < x x − 2 Refl = x .75 .49 .83 9.2 5.3 C < x x o lect x 5 9 or or 2 or or Cam 2 < oe tion r 8.4 r 9. r 59 r 85 mb oe n 484 829 9.16 5.4 brid A e 4 to 9 to 6 to or dge © An o 8 o 9 o 5 85 e IG Ca nsw .48 .83 9.1 .34 GC amb wer 85… 30 17 4 to M CSE brid … o 85 Mar E – dge 5.37 k S – O e In 7 Sch Oct nter hem ob rnat me ber/ tion e /No nal M ove Ex Mar 2 2 2 3 1 1 2 2 3 2 3 3 em xam rk mbe mina M b M b B M r M M M I M M M M o o M M o M M er 2 atio M1 but M1 but B1 M1 rear M1 M1 M1 If a M2 M1 M1 M2 or M or    M1 M2 or M M1 M1 20 ons fo wi fo wi for fo rran fo fo fo answ 2 fo fo 2 fo M1 12   w fo 2 fo M1 fo fo 15 s 20 or in ith or in ith r 0 or c nge or c or c or c we or 9 4 or 6 8 or 1 w fo 2 2    w or B or [c fo or 0 or 0 015 nac cor nac cor < x orr eme orr orr orr r in 9 4 = 6 8 = 12 = w or 2 = 12 BC cos or 1 5.0 5.0 5 ccur rrec ccur rrec x oe rect ent rect rect rect nco CX 3 = R 4 = = 45 90 45 90 = C s=] 21 1 × 1 × P rate ct o rate ct o e o t an t t m t ch t di orr CX 3 Y 4 45 90 5 0 5 0 o cos 12 1 = 2× 2× Part e re orie e re orie or x nd a mult han ivis rect oe oe 5 0 or oe 35 s 2 2 2 + × 2 12 ×the 10 × S t M ecta enta ecta enta x < app tipl ngin sion t m e or e or oe 5 o 10 12 + × 1 + eir s × Syl 0 Mar ang atio ang atio < 2 prop ica ng x n max r be r be or 90 45 or b 2 0 2 1 − × 2 10 BC sin( llab 060 rks gul on. gul on. oe pri atio x an xim ette ette bet 0 5 o bett 11 10 − 2 − × C (the bu 07 s ar h ar h e ate on nd mum er er tter oe s ter 21 2× sin eir s hyp hyp e y m p r soi 12 35 n CA P per per poss 1 × 5 o ) AD Pa 4 bol bol sib 0co oe ) o pe 41 la la ble i osθ oe er is θ

Page 4 Mark Scheme Syllabus Paper Cambridge IGCSE – October/November 2015 0607 41 © Cambridge International Examinations 2015 Question Answer Mark Part Marks 6 (a) 1.31 or 1.308… 2 M1 for mid-values soi (b) Fully correct histogram 3 B1 for correct widths 0.2, 0.2, 0.2, 0.4 B2 for 4 correct heights or B1 for 2 or 3 correct heights or for 180, 200, 160, 20 seen 7 (a) (i) 2512 final answer 2 M1 for 20 × 8 × 2 × 7.85 (ii) 34.56 cao 3 M2 for 2(20 × 8 + 20 × 2 + 8 × 2) × 0.08 oe or M1 for 2(20 × 8 + 20 × 2 + 8 × 2) oe or for area × 0.08 (b) 48 2 M1 for 8 × 3 × 2 oe (c) (i) 67 3 M2 for (20 × 12 × 4) ÷       × 3 5.1 3 4 π or M1 for       × 3 5.1 3 4 π (ii) 12.6 or 12.7 or 12.8 or 12.62 to 12.82... 2 M1 for (20 × 12 × 4) – their 67 × their       × 3 5.1 3 4 π oe must be positive (iii) 1.44 or 1.45 or 1.444 to 1.452 cao 2 M1 for ( ) 3 4 3 r their π = ii oe or better (d) 8 3 4 B3 for equivalent fraction or 0.375 SC2 for answer 16 3 or M1 for 9.0 3 3 1 2 × × r r π and M1 for 45 .0 ) 2 ( 3 4 2 1 3 × ×     r π 8 (a) Correct graphs 2 2 B1 for inaccurate parabola B1 for correct shape but inaccurate (b) 1.28 or 1.277 to 1.278 1 2 1.5 1 0.5 0 4 3 2 1 0 -1 2 1.5 1 0.5 0 4 3 2 1 0 -1 2 1.5 1 0.5 0 4 3 2 1 0 -1 2 1.5 1 0.5 0 4 3 2 1 0 -1

Page 5 Mark Scheme Syllabus Paper Cambridge IGCSE – October/November 2015 0607 41 © Cambridge International Examinations 2015 Question Answer Mark Part Marks (c) 1.73 or 1.732… 1 (d) (i) 0.368 or 0.3678 to 0.3679 0.692 or 0.6922… 1 1 SC1 for (0.37, 0.69) (ii) 0.692 ⩽ y ⩽ 4 oe 2FT B1FT for 0.692 ⩽ y oe or B1 for y ⩽ 4 oe or B1FT for 0.692 and 4 SC1FT for 0.692 < y < 4 (e) (i) 0.794 or 0.7943… 0.955 or 0.9549 to 0.9550 0.993 or 0.9931… 1 1 1 (ii) 1 1 9 (a) (i) 36 3 oe 2 M1 for 6 3 6 1 × oe (ii) 36 5 oe 3 M2 for 6 2 6 1 × + 6 1 6 3 × oe or M1 for 6 2 6 1 × or 6 1 6 3 × oe or for list or space diagram showing at least 3 combinations (b) 1296 128 oe 2 M1 for 6 2 6 4 6 4 6 4 × × × oe (c) 6 2 M1 for 6 1 6 5 ×       k or SC1 for answer 5 10 (a) 5 2 M1 for [h(9)=] 1 or for 3 )) 1 (log( 2 + + x (b) 2x + 2 or 2(x + 1) 2 M1 for 2x + 3 – 1 (c) )1 )( 3 2 ( 2 3 − + + x x x oe final answer 3 B1 for denominator (2x + 3)(x – 1) oe B1 for x – 1 + 2x + 3 as numerator (d) –10 9 oe 2 M1 for 1 10 1 − = + x oe or appropriate sketch

Page 6 Mark Scheme Syllabus Paper Cambridge IGCSE – October/November 2015 0607 41 © Cambridge International Examinations 2015 Question Answer Mark Part Marks (e) 5 1± final answer cao 3 B2 for 5 1= − x or 5 1 − = − x or 2 20 2 ± oe or for –1.24 (or –1.236...) and 3.24 (or 3.236...) or M1 for 5 )1 ( 2 = − x oe 11 (a) 2.79 cao nfww 6 6 B1 for 2x + y = 157 oe B1 for x + 3y = 296 oe M1 for eliminating one variable or appropriate sketch B1 for [x =] 35 oe B1 for [y =] 87 oe or M1 for x and y working in one equation (b) 3 nfww 4 4 M2 for 11 5 24 24 = + + x x oe or B1 for 5 24 or 24 + x x M1 for full method e.g. sketch of graph showing solution or quadratic with correct factors or 11 2 ) 120 )( 11 ( 4 7 7 2 × − − ± − Note that x + x + 5 = 11 is wrong working