Cambridge IGCSE Mathematics - International 0607 — 2015 Oct/Nov Paper 4 · Variant 3

This document consists of 16 printed pages. DC (ST/SW) 103703/4 © UCLES 2015 [Turn over Cambridge International Examinations Cambridge International General Certificate of Secondary Education * 3 4 2 3 1 7 5 2 2 9 * CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/43 Paper 4 (Extended) October/November 2015 2 hours 15 minutes Candidates answer on the Question Paper. Additional Materials: Geometrical 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 0607/43/O/N/15 © UCLES 2015 Formula List For the equation ax bx c 0 2 + + = x a b b ac 2 4 2 ! = - - Curved surface area, A, of cylinder of radius r, height h. r A rh 2 = Curved surface area, A, of cone of radius r, sloping edge l. r A rl = Curved surface area, A, of sphere of radius r. r A r 4 2 = Volume, V, of pyramid, base area A, height h. V Ah 3 1 = Volume, V, of cylinder of radius r, height h. r V r h 2 = Volume, V, of cone of radius r, height h. r V r h 3 1 2 = Volume, V, of sphere of radius r. r V r 3 4 3 = sin sin sin A a B b C c = = cos a b c bc A 2 2 2 2 = + - sin bc A 2 1 Area = A C B c b a

3 0607/43/O/N/15 © UCLES 2015 [Turn over Answer all the questions. 1 C D B 41° 15 cm A NOT TO SCALE 17 cm (a) Calculate the length of BD. Answer(a) … cm [2] (b) Calculate the area of triangle ACD. Answer(b) … cm2 [2] (c) Use the cosine rule to find the length of AD. Answer(c) … cm [3]

4 0607/43/O/N/15 © UCLES 2015 2 (a) Jay buys a bicycle for $220. He later sells it for $160. Calculate his percentage loss. Answer(a) … % [3] (b) A television has a sale price of $216 after a reduction of 10%. Calculate the original price of the television. Answer(b) $ … [3] (c) The population of a village is 2180. The population decreases by 3% each year. (i) Calculate the population in 20 years time. Answer(c)(i) … [3] (ii) Calculate the number of whole years it takes for the population to decrease from 2180 to less than 1000. Answer(c)(ii) … [2]

5 0607/43/O/N/15 © UCLES 2015 [Turn over 3 (a) The speeds, v km/h, of 120 cars passing under a bridge are measured. The table shows the results. Speed (v km/h) v 30 50 1 G v 0 0 5 6 1 G v 0 0 6 7 1 G v 0 75 7 1 G v 75 90 1 G Frequency 2 25 46 41 6 (i) Write down the interval that contains the lower quartile. Answer(a)(i) … [1] (ii) Calculate an estimate of the mean. Answer(a)(ii) … km/h [2] (iii) Complete the table of frequency densities. Speed (v km/h) v 30 50 1 G v 0 0 5 6 1 G v 0 0 6 7 1 G v 0 75 7 1 G v 75 90 1 G Frequency density [3] (b) The table below shows the monthly rainfall and the average midday temperatures of a city. Month Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Rainfall (r mm) 15 20 20 35 70 90 75 70 50 30 12 8 Temperature (t ° C) 35 25 22 15 10 10 15 20 27 30 38 36 Find the equation of the line of regression, giving t in terms of r. Answer(b) t = … [2]

6 0607/43/O/N/15 © UCLES 2015 4 (a) (i) Shade in one more square so that the diagram has one line of symmetry. [1] (ii) Shade in two more squares so that the diagram has rotational symmetry of order 2 and no lines of symmetry. [1] (b) C B A P Q R NOT TO SCALE Triangle ABC and triangle PQR are mathematically similar. AB : PQ = 3 : 2 . (i) CB = 10.5 cm. Calculate the length of RQ. Answer(b)(i) … cm [2] (ii) The area of triangle ABC is 45 cm2. Calculate the area of triangle PQR. Answer(b)(ii) … cm2 [2]

7 0607/43/O/N/15 © UCLES 2015 [Turn over 5 1 0 1 2 3 4 5 6 7 8 9 10 x y 2 3 4 5 6 7 8 U T (a) (i) Describe fully the single transformation that maps triangle T onto triangle U. Answer(a)(i) … … [3] (ii) Describe fully the inverse of the transformation in part(a)(i). Answer(a)(ii) … … [2] (b) (i) Draw the image of triangle T under a reflection in the line y = x. [2] (ii) Draw the image of triangle T under a rotation of 90° anti-clockwise about the point (6, 8). [2] (c) Describe fully the single transformation equivalent to a rotation 90° clockwise about (0, 0) followed by a reflection in the line y x =- . You may use the grid to help you. 0 x y Answer(c) … … [3]

8 0607/43/O/N/15 © UCLES 2015 6 The diagram shows a solid cone inside a cylinder. The base radius of the cone and the radius of the cylinder are both 10 cm. The height of both the cone and the cylinder is 30 cm. 30 cm 10 cm NOT TO SCALE (a) Find the volume of the cylinder not occupied by the cone. Answer(a) …cm3 [3] (b) Water is poured into the cylinder until it reaches a depth of 15 cm. 15 cm NOT TO SCALE (i) Calculate the volume of the part of the cone that is below the water level and show that it rounds to 2749 cm3, correct to the nearest cubic centimetre. [4] (ii) Calculate the amount of water that has been poured into the cylinder. Give your answer in litres. Answer(b)(ii) … litres [3]

9 0607/43/O/N/15 © UCLES 2015 [Turn over 7 (a) Kim walks 10 km at 4 km/h and then a further 6 km at 3 km/h. Calculate Kim’s average speed. Answer(a) … km/h [3] (b) Chung runs at x km/h for 45 minutes and then at (x – 2) km/h for 30 minutes. Find an expression, in terms of x, for Chung’s average speed in km/h. Give your answer in its simplest form. Answer(b) … km/h [4]

10 0607/43/O/N/15 © UCLES 2015 8 (a) (i) Solve the inequality. ( ) ( ) x x 2 3 5 3 1 - + Answer(a)(i) … [3] (ii) Show your answer to part(a)(i) on the number line. –10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10 x [1] (b) Solve the equation. ( ) ( ) x x 3 1 25 2 2 + + + = Give your answers correct to 2 decimal places. Answer(b) x = … or x = … [6]

11 0607/43/O/N/15 © UCLES 2015 [Turn over (c) Solve the equations. (i) logx x 5 = - Answer(c)(i) x = … [3] (ii) logx x 5 = - Answer(c)(ii) x = … or x = … [2] (d) Simplify, giving your answer as a single fraction. x x x 1 1 2 - - + Answer(d) … [3]

12 0607/43/O/N/15 © UCLES 2015 9 (a) 124° 25° D A B C NOT TO SCALE In the quadrilateral ABCD, DA = AB and DA is parallel to CB. Angle DAB = 124° and angle BDC = 25°. Calculate angle BCD. Answer(a) … [3] (b) Nine of the angles of a 10-sided polygon are each 142°. Calculate the other angle. Answer(b) … [3]

13 0607/43/O/N/15 © UCLES 2015 [Turn over (c) D F X 20° O 25° E A B C NOT TO SCALE A, B, C and D lie on the circle, centre O. BD is a diameter and EDF is a tangent at D. AC and BD intersect at X. Angle BCA = 25° and angle BDC = 20°. Calculate (i) angle ADE, Answer(c)(i) … [2] (ii) angle DAC, Answer(c)(ii) … [2] (iii) angle AXD. Answer(c)(iii) … [1]

14 0607/43/O/N/15 © UCLES 2015 10 In this question, the weather is only considered to be either wet or dry. When the weather is dry the probability that Sara will go walking is 5 3 . When the weather is wet the probability that Sara will go walking is 10 1 . The probability of a dry day is 3 2 . (a) Complete the tree diagram. 2 3 Weather 3 5 … … Walking dry yes no … … yes no wet [3] (b) Show that the probability that Sara goes walking is 30 13 . [2] (c) The probability that Sara does not go walking when the weather is wet is 30 9 . Complete this tree diagram. 13 30 Weather 1 13 … … Walking yes dry wet … … dry wet no [3]

15 0607/43/O/N/15 © UCLES 2015 [Turn over 11 ( ) f x x 16 2 = - ( ) , g x x x 1 2 1 ! = + - ( ) h x 2x = (a) Find h(3). Answer(a) … [1] (b) Find the range of g(x) for the domain {0, 1}. Answer(b) … [1] (c) f(x – 2) can be written as ( )( ) x a x b + + . Find the value of a and the value of b. Answer(c) a = … b = … [4] (d) Find the inverse of (i) g(x), Answer(d)(i) … [3] (ii) h(x). Answer(d)(ii) … [2] (e) Describe fully the single transformation that maps the graph of ( ) f y x = onto the graph of y x 2 32 2 = - . … … [2] Question 12 is printed on the next page

16 0607/43/O/N/15 © UCLES 2015 12 –6 –10 10 y x 4 0 (a) On the diagram, sketch the graphs of ( ) y x 2 12 = + and y 2 5 x = - for values of x between x = – 6 and x = 4. [4] (b) Write down the equation of each asymptote of the graph of (i) y x 2 12 = + , Answer(b)(i) … … [2] (ii) y 2 5 x = - . Answer(b)(ii) … [1] (c) Solve the inequality. x x 2 5 2 12 0 for x 2 2 - + . Answer(c) … [2] 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 0607/43 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 43 © 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) 9.84 or 9.840 to 9.841 2 M1 for sin41 = 15 BD oe or better (b) 83.6 or 83.64 to 83.65 2 M1 for 0.5 × 17 × their (a) oe (c) 41 cos 15 17 2 15 17 2 2 × × − + 129 or 129.0 to 129.1 11.4 or 11.36… M1 A1 A1 If 0 scored SC2 for 11.4 or 11.36… 2 (a) 27.3 or 27.27… 3 M2 for 100 220 160 220 × − oe or M1 for 220 160 220 − oe or 100 220 160 × oe (b) 240 3 M2 for 216 ÷ 0.9 oe or M1 for 216 = 90% (c) (i) 1190 or 1186 or 1185. … 3 M2 for 2180 × 0.97 20 oe or M1 for 2180 × 0.97 k k integer > 1 oe (ii) 26 2 M1 for 2180 × 0.97 n = 1000 oe If 0 scored, SC1 for answer 25 3 (a) (i) 60 70 v < ≤ 1 (ii) 65.9 or 65.93 to 65.94 2 M1 for at least 3 correct mid-values seen (iii) 0.1, 2.5, 4.6, 8.2, 0.4 oe 3 B2 for 3 or 4 correct or B1 for 2 correct (b) –0.286r + 35.4 or (–0.2861…)r + (35.38 to 35.39) 2 B1 for (–0.286 or –0.2861…)r + k or for kr + (35.4 or 35.38 to 35.39) or SC1 for –0.29r + 35

Page 3 Mark Scheme Syllabus Paper Cambridge IGCSE – October/November 2015 0607 43 © Cambridge International Examinations 2015 Question Answer Mark Part Marks 4 (a) (i) or 1 (ii) 1 (b) (i) 7 2 M1 for RQ 5. 10 2 3 = oe or better (ii) 20 2 M1 for 2 2 3 2 or 2 3             oe 5 (a) (i) Enlargement [factor] 0.5 oe [centre] (0, 8) 1 1 1 (ii) Enlargement [factor] 2 and [centre] (0, 8) 1 1FT FT scale factor and centre (b) (i) Image at (4, 4), (8, 4), (8, 6) 2 M1 for y = x drawn (ii) Image at (6, 8), (6, 6), (10, 6) 2 SC1 for 90 ˚ anti-clockwise but different centre (c) Reflection, x-axis oe 3 M2 for full method seen i.e. diagram or unit vectors. or M1 for one of transformations correctly carried out If 0 scored, SC1 for any reflection in answer

Page 4 Mark Scheme Syllabus Paper Cambridge IGCSE – October/November 2015 0607 43 © Cambridge International Examinations 2015 Question Answer Mark Part Marks 6 (a) 6280 or 6283 to 6284 3 M2 for 30 10 3 2 2 × × ×π oe or M1 for 30 10 3 1 2 × ×     π ( ) π 1000 (b) (i) − × × × 30 10 3 1 2 π 15 5 3 1 2 × × ×π oe 2748.8 to 2749.3 M3 A1 Allow use of their volume of cone from (a) or 8 7 × 30 10 3 1 2 × × ×π or 8 7 their volume of cone from (a) M2 for 15 5 3 1 2 × × ×π oe or B1 for radius of small cone = 5 not 2749 alone (ii) 1.96 or 1.963 to 1.964 3 B2 for 1960 or 1963 to 1964 or M1 for 2 10 15 2749 π × × − M1 for correctly converting their volume in cc to litres. 7 (a) 3.56 or 3.555 to 3.556 3 M2 for 3 6 4 10 6 10 + + or M1 for 4 10 or 3 6 (b) 5 4 5 − x or x – 0.8 or x – 5 4 or 0.2(5x – 4) final answer nfww 4 M3 for oe 60 30 60 45 oe 60 30 ) 2 ( 60 45 + × − + × x x or M2 for ] 60 [ 30 ) 2 ( ] 60 [ 45 × − + × x x oe or M1 for one of these products or evidence of total distance ÷ total time

Page 5 Mark Scheme Syllabus Paper Cambridge IGCSE – October/November 2015 0607 43 © Cambridge International Examinations 2015 Question Answer Mark Part Marks 8 (a) (i) x > –7 oe 3 M2 for 2x – 5x < 15 + 6 or better or B1 for 2x – 6 or 5x + 15 (ii) Line with empty circle at –7 and arrow to right 1FT Strict FT, must be from an inequality. (b) Sketch of y = (x + 3)² + (x + 1)² – 25 oe or 0 15 8 2 2 = − + x x –5.39 and 1.39 M2 or B2 B4 M1 for sketch of (x + 3)² + (x + 1)² B1 for 9 3 3 2 + + + x x x or 1 2 + + + x x x oe B3 for –5.391... and 1.391... or B2 for –5.39 or 1.39 or B1 for –5.391... or 1.391... or M1 for sketch of parabola or correct substitution in formula or reaching ( ) 2 2 2 23 x + − oe (c) (i) Appropriate sketch which could lead to answer 4.36 or 4.360… M2 B1 M1 for correct sketch of logx or other equation containing logx (ii) 4.36 or 4.360… 5.76 or 5.760… B1FT B1 (d) )1 )( 1 ( 2 2 + − + − x x x x oe final answer 3 B1 for )1 ( 2 )1 ( − − + x x x oe seen B1 for denominator )1 )( 1 ( + − x x oe 9 (a) 127 3 M1 for angle ADB or ABD ( ) 0.5 180 124 = − implied by 28 in diagram M1 for angle DBC = angle ADB. (b) 162 3 M2 for (10 – 2) × 180 – 9 × 142 or M1 for (10 – 2) × 180 (c) (i) 65 2 B1 for angle ADB = 25 or angle ACD = 65 (ii) 70 2 B1 for angle BAC = 20 or angle FDC = 70 (iii) 85 1

Page 6 Mark Scheme Syllabus Paper Cambridge IGCSE – October/November 2015 0607 43 © Cambridge International Examinations 2015 Question Answer Mark Part Marks 10 (a) 3 1 5 2 10 1 and 10 9 1 1 1 (b) 10 1 3 1 5 3 3 2 × + × M2 M1 for one of these FT from (a) (c) 30 17 and 13 12 17 8 and 17 9 1 2 M1 for 30 9 30 17 = × x oe 11 (a) 8 1 (b) 2, 1 1 (c) –6 and 2 4 B3 for ) 2 )( 6 ( + − x x or SC3 for 6 and –2 or B2 for 16 4 2 2 2 − + − − x x x or better or M1 for 16 ) 2 ( 2 − − x or for ab bx ax x + + + 2 (d) (i) x x − 2 oe final answer 3 M1 for interchanging x and y M1 for a correct multiplication M1 for a correct rearrangement and a correct division If answer incorrect maximum possible is M2 (ii) x 2 log or 2 log log x 2 M1 for 2 log log x y = or x y = 2 log oe or y x 2 = (e) Stretch [factor] 2 and x-axis invariant 1 1

Page 7 Mark Scheme Syllabus Paper Cambridge IGCSE – October/November 2015 0607 43 © Cambridge International Examinations 2015 Question Answer Mark Part Marks 12 (a) Fully correct sketches 2 2 B1 for rectangular hyperbola with correct orientation but inaccurate Correct curve crossing positive x-axis and negative y-axis B1 for exponential curve with correct orientation but inaccurate (b) (i) x = –2 y = 0 1 1 (ii) y = –5 1 (c) x > 2.9[0] or 2.897… 2 B1 for 2.9[0] or 2.897… seen 6 4 2 0 -2 -4 6 10 5 0 -5 -10 6 4 2 0 -2 -4 6 10 5 0 -5 -10 6 4 2 0 -2 -4 6 10 5 0 -5 -10 6 4 2 0 -2 -4 6 10 5 0 -5 -10