Cambridge IGCSE Mathematics (with coursework) 0581 — 2014 May/June Paper 4 · Variant 2

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 fl uid. DO NOT WRITE IN ANY BARCODES. Answer all questions. If working is needed for any question it must be shown below that question. Electronic calculators should be used. If the degree of accuracy is not specifi ed in the question, and if the answer is not exact, give the answer to three signifi cant fi gures. Give answers in degrees to one decimal place. For π, use either your calculator value or 3.142. 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 of the marks for this paper is 130. MATHEMATICS 0581/42 Paper 4 (Extended) May/June 2014 2 hours 30 minutes Candidates answer on the Question Paper. Additional Materials: Electronic calculator Geometrical instruments Tracing paper (optional) This document consists of 16 printed pages. [Turn over IB14 06_0581_42/FP © UCLES 2014 *6688308894* Cambridge International Examinations Cambridge International General Certifi cate of Secondary Education PAPA CAMBRIDGE

2 0581/42/M/J/14 © UCLES 2014 1 Jane and Kate share $240 in the ratio 5 : 7 . (a) Show that Kate receives $140. Answer(a) [2] (b) Jane and Kate each spend $20. Find the new ratio Jane’s remaining money : Kate’s remaining money. Give your answer in its simplest form. Answer(b) … : … [2] (c) Kate invests $120 for 5 years at 4% per year simple interest. Calculate the total amount Kate has after 5 years. Answer(c) $ … [3] (d) Jane invests $80 for 3 years at 4% per year compound interest. Calculate the total amount Jane has after 3 years. Give your answer correct to the nearest cent. Answer(d) $ … [3] (e) An investment of $200 for 2 years at 4% per year compound interest is the same as an investment of $200 for 2 years at r % per year simple interest. Find the value of r. Answer(e) r = … [3] __________________________________________________________________________________________

3 0581/42/M/J/14 © UCLES 2014 [Turn over 2 f(x) = 21 x – 2x , x ≠ 0 (a) Complete the table of values for f(x). x –3 –2.5 –2 –1.5 –1 –0.5 0.4 0.5 1 1.5 2 f(x) 6.1 5.2 4.3 3.4 5 5.5 –2.6 –3.8 [3] (b) On the grid, draw the graph of y = f(x) for –3 Y x Y – 0.5 and 0.4 Y x Y 2 . y x 7 6 5 4 3 2 1 –1 –2 –3 –4 0 –1 –2 –3 2 1 [5] (c) Solve the equation f(x) = 2 . Answer(c) x = … [1] (d) Solve the equation f(x) = 2x + 3 . Answer(d) x = … [3] (e) (i) Draw the tangent to the graph of y = f(x) at the point where x = –1.5 . [1] (ii) Use the tangent to estimate the gradient of the graph of y = f(x) where x = –1.5 . Answer(e)(ii) … [2] __________________________________________________________________________________________

4 0581/42/M/J/14 © UCLES 2014 3 80 m 90 m 95 m 49° 55° D A B C NOT TO SCALE The diagram shows a quadrilateral ABCD. Angle BAD = 49° and angle ABD = 55°. BD = 80 m, BC = 95 m and CD = 90 m. (a) Use the sine rule to calculate the length of AD. Answer(a) AD = … m [3] (b) Use the cosine rule to calculate angle BCD. Answer(b) Angle BCD = … [4]

5 0581/42/M/J/14 © UCLES 2014 [Turn over (c) Calculate the area of the quadrilateral ABCD. Answer(c) … m2 [3] (d) The quadrilateral represents a fi eld. Corn seeds are sown across the whole fi eld at a cost of $3250 per hectare. Calculate the cost of the corn seeds used. 1 hectare = 10 000 m2 Answer(d) $ … [3] __________________________________________________________________________________________

6 0581/42/M/J/14 © UCLES 2014 4 Q y x 8 7 6 5 4 3 2 1 –1 –2 –3 –4 –5 –6 –7 –8 0 –1 1 2 3 4 5 6 7 8 –2 –3 –4 –5 –6 –7 –8 (a) Draw the refl ection of shape Q in the line x = –1 . [2] (b) (i) Draw the enlargement of shape Q, centre (0, 0), scale factor –2 . [2] (ii) Find the 2 × 2 matrix that represents an enlargement, centre (0, 0), scale factor –2 . Answer(b)(ii) f p [2]

7 0581/42/M/J/14 © UCLES 2014 [Turn over (c) (i) Draw the stretch of shape Q, factor 2, x-axis invariant. [2] (ii) Find the 2 × 2 matrix that represents a stretch, factor 2, x-axis invariant. Answer(c)(ii) f p [2] (iii) Find the inverse of the matrix in part (c)(ii). Answer(c)(iii) f p [2] (iv) Describe fully the single transformation represented by the matrix in part (c)(iii). Answer(c)(iv) … … [3] __________________________________________________________________________________________

8 0581/42/M/J/14 © UCLES 2014 5 12 cm 10 cm 4 cm 8 cm NOT TO SCALE The diagram shows a cylinder with radius 8 cm and height 12 cm which is full of water. A pipe connects the cylinder to a cone. The cone has radius 4 cm and height 10 cm. (a) (i) Calculate the volume of water in the cylinder. Show that it rounds to 2410 cm3 correct to 3 signifi cant fi gures. Answer(a)(i) [2] (ii) Change 2410 cm3 into litres. Answer(a)(ii) … litres [1]

9 0581/42/M/J/14 © UCLES 2014 [Turn over (b) Water fl ows from the cylinder along the pipe into the cone at a rate of 2 cm3 per second. Calculate the time taken to fi ll the empty cone. Give your answer in minutes and seconds correct to the nearest second. [The volume, V, of a cone with radius r and height h is V = 3 1 πr 2h.] Answer(b) … min … s [4] (c) Find the number of empty cones which can be fi lled completely from the full cylinder. Answer(c) … [3] __________________________________________________________________________________________

10 0581/42/M/J/14 © UCLES 2014 6 21° 117° y° x° S P Q R T NOT TO SCALE (a) The chords PR and SQ of the circle intersect at T. Angle RST = 21° and angle STR = 117°. (i) Find the values of x and y. Answer(a)(i) x = … y = … [2] (ii) SR = 8.23 cm, RT = 3.31 cm and PQ = 9.43 cm. Calculate the length of TQ. Answer(a)(ii) TQ = … cm [2]

11 0581/42/M/J/14 © UCLES 2014 [Turn over (b) EFGH is a cyclic quadrilateral. EF is a diameter of the circle. KE is the tangent to the circle at E. GH is parallel to FE and angle KEG = 115°. Calculate angle GEH. Answer(b) Angle GEH = … [4] (c) A, B, C and D are points on the circle centre O. Angle AOB = 140° and angle OAC = 14°. AD = DC. Calculate angle ACD. Answer(c) Angle ACD = … [5] __________________________________________________________________________________________ 115° G F H E K NOT TO SCALE 140° 14° O B A D C NOT TO SCALE

12 0581/42/M/J/14 © UCLES 2014 7 (a) 1.0 0.8 0.6 0.4 0.2 0 20 40 60 Mass (grams) 80 100 10 30 50 70 90 Frequency density m The histogram shows some information about the masses (m grams) of 39 apples. (i) Show that there are 12 apples in the interval 70 < m Y 100 . Answer(a)(i) [1] (ii) Calculate an estimate of the mean mass of the 39 apples. Answer(a)(ii) … g [5] (b) The mean mass of 20 oranges is 70 g. One orange is eaten. The mean mass of the remaining oranges is 70.5 g. Find the mass of the orange that was eaten. Answer(b) … g [3] __________________________________________________________________________________________

13 0581/42/M/J/14 © UCLES 2014 [Turn over 8 The distance a train travels on a journey is 600 km. (a) Write down an expression, in terms of x, for the average speed of the train when (i) the journey takes x hours, Answer(a)(i) … km/h [1] (ii) the journey takes (x + 1) hours. Answer(a)(ii) … km/h [1] (b) The difference between the average speeds in part(a)(i) and part(a)(ii) is 20 km/h. (i) Show that x 2 + x – 30 = 0 . Answer(b)(i) [3] (ii) Find the average speed of the train for the journey in part(a)(ii). Show all your working. Answer(b)(ii) … km/h [4] __________________________________________________________________________________________

14 0581/42/M/J/14 © UCLES 2014 9 If the weather is fi ne the probability that Carlos is late arriving at school is 10 1 . If the weather is not fi ne the probability that he is late arriving at school is 3 1 . The probability that the weather is fi ne on any day is 4 3 . (a) Complete the tree diagram to show this information. Fine Not fine Late Weather Arriving at school Not late Not late Late … … … … 3 4 1 10 [3] (b) In a school term of 60 days, fi nd the number of days the weather is expected to be fi ne. Answer(b) … [1] (c) Find the probability that the weather is fi ne and Carlos is late arriving at school. Answer(c) … [2] (d) Find the probability that Carlos is not late arriving at school. Answer(d) … [3] (e) Find the probability that the weather is not fi ne on at least one day in a school week of 5 days. Answer(e) … [2] __________________________________________________________________________________________

15 0581/42/M/J/14 © UCLES 2014 [Turn over 10 f(x) = x 1 , x ≠ 0 g(x) = 1 – x h(x) = x 2 + 1 (a) Find fg 2 1 ` j. Answer(a) … [2] (b) Find g–1(x), the inverse of g(x). Answer(b) g–1(x) = … [1] (c) Find hg(x), giving your answer in its simplest form. Answer(c) hg(x) = … [3] (d) Find the value of x when g(x) = 7 . Answer(d) x = … [1] (e) Solve the equation h(x) = 3x. Show your working and give your answers correct to 2 decimal places. Answer(e) x = … or x = … [4] (f) A function k(x) is its own inverse when k –1(x) = k(x). For which of the functions f(x) , g(x) and h(x) is this true? Answer(f) … [1] __________________________________________________________________________________________ Question 11 is printed on the next page.

16 0581/42/M/J/14 © UCLES 2014 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. 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. 11 The total area of each of the following shapes is X. The area of the shaded part of each shape is kX. For each shape, fi nd the value of k and write your answer below each diagram. A B C D NOT TO SCALE NOT TO SCALE 72° O J K NOT TO SCALE F E G I H AB = BC = CD k = … Angle JOK = 72° k = … EF = FG and EI = IH k = … NOT TO SCALE NOT TO SCALE A O B The shape is a regular hexagon. k = … The diagram shows a sector of a circle centre O. Angle AOB = 90° k = … [10]

CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education MARK SCHEME for the May/June 2014 series 0581 MATHEMATICS 0581/42 Paper 4 (Extended), maximum raw mark 130 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 2014 series for most IGCSE, GCE Advanced Level and Advanced Subsidiary Level components and some Ordinary Level components. PAPA CAMBRIDGE

Page 2 Mark Scheme Syllabus Paper IGCSE – May/June 2014 0581 42 © Cambridge International Examinations 2014 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 Qu Answers Mark Part Marks 1 (a) 240 7 ) 7 5 ( × + ÷ [=140] oe M2 M1 for 240 ÷ (5 + 7) or 240 × 7 (b) 2 : 3 final answer 2 B1 for ratio of form 2x : 3x seen or SC1 for 3 : 2 (c) 144 3 M2 for 120 + 100 5 4 120 × × oe or M1 for 100 5 4 120 × × (d) 89.99 cao mark final answer 3 B2 for 89.9[8…] shown but not spoiled or answer 90[ .0..] nfww or M1 for 80 × 3 100 104       oe If M1 spoiled by adding 80 or subtracting 80 then SC1 for answers 169.99 or 9.99 (e) 4.08 3 M2 for 100 2 200 × × r = 200 × 1.042 – 200 oe or M1 for 200 × 1.042 [216.3[2]] oe or 100 2 200 × × r oe

Page 3 Mark Scheme Syllabus Paper IGCSE – May/June 2014 0581 42 © Cambridge International Examinations 2014 Qu Answers Mark Part Marks 2 (a) 3, 3, – 1 3 B1 B1 B1 (b) Complete correct curve 5 B3FT 11 points or B2FT for 9 or 10 points or B1FT for 7 or 8 points And B1indep two separate branches not touching or crossing y-axis (c) 0.5 to 0.6 1 (d) Correct line and 0.4 to 0.5 or no line and 0.4 to 0.5 nfww 3 Must check line - not if wrong line B2 for y = 2x + 3 ruled correctly or SC1 for correct freehand line or ruled line with either gradient 2 or y-intercept 3 but not y = 3 (e) (i) Tangent at x = – 1.5 1 No daylight at x = –1.5. Consider point of contact as midpoint between two vertices of daylight, the midpoint must be between x = – 1.7 and –1.3 (ii) – 2 to – 1 2 Dependent on tangent mark awarded Allow integer/integer if in range Or M1 for rise/run also dep on any tangent drawn or close attempt at tangent at any point Must see correct or implied calculation from a drawn tangent 2 1 0 -1 -2 3 6 4 2 0 -2 -4 2 1 0 -1 -2 3 6 4 2 0 -2 -4 2 1 0 -1 -2 3 6 4 2 0 -2 -4 2 1 0 -1 -2 3 6 4 2 0 -2 -4

Page 4 Mark Scheme Syllabus Paper IGCSE – May/June 2014 0581 42 © Cambridge International Examinations 2014 Qu Answers Mark Part Marks 3 (a) 86.8 or 86.83…. 3 M2 for 49 sin 55 sin 80 or M1 for 55 sin 49 sin 80 x = oe (b) 51.2 or 51.15 to 51.16 4 M2 for [cos =] 90 . 95 .2 80 90 95 2 2 2 − + oe or M1 for BCD cos . 95 . 90 .2 90 95 80 2 2 2 − + = A1 for 100 17 725 10 or 228 143 etc. or 0.627….. (c) 6700 or 6698 to 6703 3 M2 for 0.5 × 80 × their(a) × sin(180-55-49) oe [3368 – 3370…] [If AB used then AB= 102.8 to 103] + 0.5 × 90 × 95 × sin(their(b)) oe [3329 – 3332] or M1 for one of these triangle area methods oe (d) 2180 or 2176 to 2179 3FT FT their (c) × 0.325 correctly evaluated to 3 sf or better M2 for their (c) × 000 10 3250 or SC1 FT for figs 218 or figs 2176 to 2179

Page 5 Mark Scheme Syllabus Paper IGCSE – May/June 2014 0581 42 © Cambridge International Examinations 2014 Qu Answers Mark Part Marks 4 (a) Image at (–3, 2), (–5, 2), (–5, 4), (–3, 3) 2 SC1 reflection in y = – 1 or x = k or 4 correct points not joined (b) (i) Image at (–2, –4), (–6, –4), (–6, –8), (–2, –6) 2 SC1 other enlargement of scale factor -2, correct size and correct orientation or 4 correct points not joined (ii)       − − 2 0 0 2 2 SC1 for       k k 0 0 , k may be algebraic or numeric but not 0 or 1 (c) (i) Image at (1, 4), (3, 4), (3, 8), (1, 6) 2 SC1 for trapezium with vertices at (1, 6) and (3, 8) or correct stretch with y-axis invariant or 4 correct points not joined (ii)       2 0 0 1 2 SC1 for       k 0 0 1 k may be algebraic or numeric but not 0 or 1 or for       1 0 0 2 (iii)       1 0 0 2 2 1 oe isw 2FT FT inverse of their (c)(ii) (algebraic or numeric) B1FT their (c)(ii) for       d b c a 2 1 or       1 0 0 2 p ie FT their correct fraction or their transposed matrix FT for 2 and 1 mark dependent on det 0 ≠ (iv) Stretch, [factor ] 2 1 , invariant [line] x-axis oe 3 B1 B1 B1 each independent cao

Page 6 Mark Scheme Syllabus Paper IGCSE – May/June 2014 0581 42 © Cambridge International Examinations 2014 Qu Answers Mark Part Marks 5 (a) (i) 2412 to 2413.… B2 Must be at least 4 figures shown M1 for 12 8 π 2 × × oe (ii) 2.41[0] B1 (b) 1 min 24 s 4 B3 for 83.76 to 83.8[0] or 84 or 1.396 to 1.397 or 1.4 or 1 min 23.76 to 1 min 23.8 seen or M2 for 2 10 4 π 2 3 1 ÷ × × [ 80/3π ] or M1 for 10 4 π 2 3 1 × × [160/3π or 167.5 to 167.6] (c) 14 3 M1 for 10 4 π 2410 2 3 1 × × or ) ( 2410 b part from vol cone their A1 for 14.3 to 14.4…. 6 (a) (i) [x =] 21, [y =] 42 2 B1 B1 (ii) 3.79 or 3.8[0] or 3.792 to 3.802 2 M1 for 43 .9 23 .8 31 .3 = TQ oe or 43 .9 117 sin sin 21 sin = TQ x their or oe (b) 40 4 B3 for angle between HE and tangent = 25 or GFH = 40 or EGH = 25 and angle EHG = 115 (accept 90 and 25 at H for 115) B2 for angle EGH = 25 or angle EHG = 115 (accept 90 and 25 at H for 115) B1 for angle FEG = 25 or angle EFG = 65 (c) 38 5 B4 for angle ADC = 104 or M4 for x + 14 + 20 + x + 70 =180 or better or B3 for angle OBA = 20 and angle OBC = 56 or angle CBA = 76 or reflex angle AOC = 208 or B2 for angle OAB or OBA = 20 and angle ACB = 70 or obtuse angle AOC = 152 or angle BOC = 68 or B1 for angle OAB or OBA = 20 or angle ACB = 70

Page 7 Mark Scheme Syllabus Paper IGCSE – May/June 2014 0581 42 © Cambridge International Examinations 2014 Qu Answers Mark Part Marks 7 (a) (i) (100 – 70) × 0.4 [= 12] or better 1 Accept 39 78 24 × oe (ii) 60.9 or 60.89… nfww 5 B1 for 3 or 4 correct extra frequencies 3, 6, 10, 8 soi M1 for at least 4 of mid-interval values 15, 40, 55, 65, 85 soi M1 for fx Σ where x is any value in each interval allow their frequencies provided integers and they must be shown [3 × 15 + 6 × 40 + 10 × 55 + 8 × 65 + 12 × 85] [2375] M1 (dependent on second M1) for ÷ 39 or ÷ (3 + 6 + 10 + 8 +12) (b) 60.5 3 M2 for 20 × 70 – 19 × 70.5 oe or M1 for either 20 × 70 or 19 × 70.5 8 (a) (i) x 600 1 Not x x 600 = (ii) 1 600 + x 1 Not 1 600 + = x x (b) (i) x 600 – 1 600 + x = 20 oe )1 ( 20 600 )1 ( 600 + = − + x x x x or better x x x x 20 20 600 600 600 2 + = − + 600 20 20 0 2 − + = x x 0 30 2 = − + x x M1FT A1 A1 FT their (a)(i) – their (a)(ii) = 20 oe If M0, SC1FT for their(a)(ii) – their (a)(i) = 20 oe May still be over common denominator and can be implied by third line. Allow recovery if bracket omitted Dep on M1A1 and conclusion reached with at least one of the interim lines and without any errors or omissions

Page 8 Mark Scheme Syllabus Paper IGCSE – May/June 2014 0581 42 © Cambridge International Examinations 2014 Qu Answers Mark Part Marks (ii) x = 5 100 B3 B1FT B2 for ) 5 )( 6 ( − + x x [= 0] oe or SC1 for ) )( ( b x a x + + where ab = – 30 or a + b = 1 or B2 for 1.2 30 .1.4 1 1 2 − − − + − or or 2 1 2 1 30 2 −       + or B1 for 1.2 1 q or − + − or 30 1.4 12 − − or 2 2 1       + x FT 600 ÷ (their x + 1) if 0 > x correctly evaluated 9 (a) 3 2 , 3 1 , 10 9 , 4 1 3 B1 for 4 1 B1 for 10 9 B1 for 3 1 and 3 2 (b) 45 1 (c) 40 3 oe 2 M1 for 10 1 4 3 × oe (d) 120 101 oe 3 M2 for 3 2 4 1 10 9 4 3 × + × only or 3 1 4 1 (c) 1 × − −their only or M1 for 10 9 4 3 × or 3 2 4 1 × or their (c) + 3 1 4 1 × (e) 1024 781 oe 2 M1 for 5 4 3 1       − oe

Page 9 Mark Scheme Syllabus Paper IGCSE – May/June 2014 0581 42 © Cambridge International Examinations 2014 Qu Answers Mark Part Marks 10 (a) 2 2 B1 for g 2 1 2 1 =       soi or [fg=] x − 1 1 (b) 1 – x 1 Accept equivalents e.g. –(x – 1) (c) 2 2 2 + −x x 3 M1 for 1 ) 1( 2 + −x B1 for ( ) [ ] 2 2 1 1 x x x x + − − = − or better (d) – 6 1 (e) )1 )( 1( 4 ) 3 ( 2 − − or better ) 3 (− − = p and 1 2× = r oe 0.38, 2.62 B1 B1 B1B1 or for 2 2 3       − x Must see r q p + or r q p − or both or for 1 2 3 2 3 2 −       − + or SC1 for answers 0.4 and 2.6 or 0.3819 to 0.3820 and 2.618… or 0.38 and 2.62 seen in working or for –0.38 and –2.62 as final ans (f) f(x) and g(x) 1 Accept f and g or 1/x and 1 – x

Page 10 Mark Scheme Syllabus Paper IGCSE – May/June 2014 0581 42 © Cambridge International Examinations 2014 Qu Answers Mark Part Marks 11 3 1 1 Allow equivalent decimal throughout (3sf or better where necessary) 360 72 oe 1 4 1 2 M1 for 2 2 1       or (2)2 or 12 : 22 or 22 : 12 oe seen 6 1 2 M1 for [X = 6 × ] 0.5 × l2 × sin60 or [X = 6 × ] 0.5 × l2 × sin120 Or recognition that the area of the obtuse- angled triangle shaded is equal to the area of one of the 6 equilateral triangles from the centre π 2 π − or π 2 1− or 0.363 or 0.3630 to 0.3635 4 If fraction given as answer, check if it falls into range B1 for [sector=] 2 π 4 1 r oe B1 for [triangle =] 2 2 1 r oe M1dep for sector their triangle their sector their − dep on B1B1 earned