Cambridge IGCSE Mathematics - International 0607 — 2017 Oct/Nov Paper 6 · Variant 2

* 4 3 6 4 4 2 9 8 7 4 * This document consists of 14 printed pages and 2 blank pages. DC (NF/FC) 134786/2 © UCLES 2017 [Turn over CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/62 Paper 6 (Extended) October/November 2017 1 hour 30 minutes Candidates answer on the Question Paper. Additional Materials: 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 both parts A and B. You must show all relevant working to gain full marks for correct methods, including sketches. In this paper you will also be assessed on your ability to provide full reasons and to communicate your mathematics clearly and precisely. At the end of the examination, fasten all your work securely together. The total number of marks for this paper is 40. Cambridge International Examinations Cambridge International General Certificate of Secondary Education

2 0607/62/O/N/17 © UCLES 2017 Answer both parts A and B. A INVESTIGATION NUMBER WALLS (20 marks) You are advised to spend no more than 45 minutes on this part. This investigation looks at what happens when you place numbers on a Number Wall. You make a Number Wall like this. • Integers are put on the bottom row of bricks. • The number on a brick is the sum of the numbers on the two bricks below. Examples Total = 5 1 4 Total = 2 –5 7 Total = 8 3 1 2 3 5 1 (a) Complete this Number Wall. Total = 1 3 2 (b) In part (a), the number 3 is on the middle brick of the bottom row. In the example, the number 3 is on the end brick of the bottom row. Explain why putting the number 3 on the middle brick of the bottom row increases the total. … …

3 0607/62/O/N/17 © UCLES 2017 [Turn over 2 (a) Complete this Number Wall. Total = 1 2 3 4 (b) Put the numbers 1, 2, 3 and 4 on the bottom row and complete this Number Wall so that the total is bigger than the total in part (a). Total = (c) Complete this Number Wall. You may use negative numbers. Total = 31 4 7 4 10

4 0607/62/O/N/17 © UCLES 2017 3 (a) This Number Wall is 4 bricks high. Complete each brick using expressions in terms of a, b, c and d. Write each expression in its simplest form. a + b a b c d (b) In another wall that is 4 bricks high, the total is 34 and the values of a, b, c and d are all the same. Use the expression for the total you found in part (a) to show that the value of a cannot be an integer.

5 0607/62/O/N/17 © UCLES 2017 [Turn over (c) In this Number Wall that is 5 bricks high, only integers greater than 0 are used. Find both sets of possible values for a, b, c, d and e. a b 43 9 14 6 c d e Set 1 a = … b = … c = … d = … e = … Set 2 a = … b = … c = … d = … e = …

6 0607/62/O/N/17 © UCLES 2017 4 In 1653 a French mathematician, Blaise Pascal, wrote about a triangle of numbers similar to the one shown below. It is made in the same way as Number Walls but • the number on a brick is the sum of the numbers on the two bricks above and • the number on the first and last brick in each row is always 1. 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 Row 1 Row 2 Row 3 Row 4 Row 5 Row 6 6 15 20 15 6 1 (a) The wall in question 3(a) is 4 bricks high. Show clearly how your expression for the total in question 3(a) connects to the numbers in one row of this triangle. Write down which row this is. Row … (b) A wall that is 5 bricks high has a, b, c, d and e, in that order, along the bottom row. Write down an expression in terms of a, b, c, d and e for the total. … (c) Use your expression from part (b) to check that one of the sets of values you found for a, b, c, d and e in question 3(c) gives a total of 43.

7 0607/62/O/N/17 © UCLES 2017 [Turn over 5 (a) Each brick in the bottom row of a Number Wall has the letter a written on it. Complete this table. Height of wall (h bricks high) Total 1 a 2 2a 3 4 h (b) A wall that is 6 bricks high has the same integer written on each brick on the bottom row. The total is 96. Find the integer that has been used. … (c) A wall has the number 5 written on each brick on the bottom row. The total is 20 971 520. Find the height of the wall. …

8 0607/62/O/N/17 © UCLES 2017 6 The bricks on the bottom row of a wall are numbered consecutively using positive integers. For example, 7, 8, 9, … The total for the wall is 80. Find all the possible heights of the wall. …

9 0607/62/O/N/17 © UCLES 2017 [Turn over B MODELLING RANGES (20 marks) You are advised to spend no more than 45 minutes on this part. This task looks at the horizontal distance a ball, fired from a toy cannon on horizontal ground, travels before it hits the ground. The horizontal distance that the ball travels is called the range. The angle the barrel of the toy cannon makes with horizontal ground is called the angle of elevation. angle of elevation the barrel The range depends on the angle of elevation. This diagram shows the path of a ball and its range for three different angles of elevation. Range Range Range

10 0607/62/O/N/17 © UCLES 2017 1 Paul has a toy cannon which fires a ball at a speed of v m/s. He uses a model to calculate the range for the same speed but for different angles of elevation. Angle of elevation (x degrees) 0 10 20 30 40 50 60 70 80 90 Range (r metres) 0 3.5 6.6 8.8 10.0 10.0 8.8 6.6 3.5 0 (a) On the grid, plot these points and draw a graph to show this information. 0 0 1 2 3 4 5 6 7 8 9 10 11 r x 10 20 30 40 50 Angle of elevation (degrees) Range (metres) 60 70 80 90 (b) Paul wants the range to be 8 m. Use the graph to find the angles of elevation he could use. … (c) Paul wants the range to be the maximum. Use the graph to find the angle of elevation he should use. …

11 0607/62/O/N/17 © UCLES 2017 [Turn over (d) A model for the range is r = a sin(2x)°. Find the value of a and write down the model. … (e) Use your model to find r when x = 100. Explain your result. … … … 2 Paul now fires 8 balls from his cannon. Each ball has the same mass and is fired at the same speed of v m/s. He records the angle of elevation and range for each ball. Angle of elevation (x degrees) 10 20 30 40 50 60 70 80 Range (r metres) 3.4 5.9 8.3 9.1 9.1 8.0 5.8 3.2 (a) Paul did not use an angle of elevation of 90°. Give a reason why. … (b) Use the information in the table to draw a graph using the axes on the previous page. (c) How can you tell that the model in question 1 is a good model for these results? … …

12 0607/62/O/N/17 © UCLES 2017 3 Paul now fires balls from his cannon, each with a speed of 20 m/s. (a) Another model for the range is . sin r v x 9 81 2 ° 2 = ^ h . On the axes below, sketch a graph of the expected ranges using this model. 0 0 Range (metres) Angle of elevation (degrees) 80 x r (b) Paul records the actual range when he fires the balls. Angle of elevation (x degrees). 10 20 30 40 50 60 70 80 Range (r metres). 12.0 21.8 29.2 30.7 30.2 26.3 19.5 10.6 On the same axes, sketch the graph for these ranges. Comment on the suitability of using the model to predict the range. … …

13 0607/62/O/N/17 © UCLES 2017 [Turn over 4 Paul changes the speed with which each ball is fired. He records each speed and the maximum range, R metres. Speed (v m/s) 1 5 10 15 20 25 Maximum range (R metres) 0.10 2.49 9.62 19.85 31.03 43.26 (a) On the axes below, sketch the graph of R against v. 0 0 Maximum range (metres) Speed (m/s) 25 v R (b) Paul improves his model in question 3(a) to estimate the maximum range, and to take account of air resistance. The new model is . R v v 9 81 2 981 kv 2 2 # = - where k is a constant. When v = 15 m/s, find the value of k, correct to 2 decimal places. …

14 0607/62/O/N/17 © UCLES 2017 (c) Comment on the suitability of the model as speed increases. … … …

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® IGCSE is a registered trademark. This document consists of 6 printed pages. © UCLES 2017 [Turn over Cambridge Assessment International Education Cambridge International General Certificate of Secondary Education CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/62 Paper 6 (Extended) October/November 2017 MARK SCHEME Maximum Mark: 40 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 International will not enter into discussions about these mark schemes. Cambridge International is publishing the mark schemes for the October/November 2017 series for most Cambridge IGCSE®, Cambridge International A and AS Level components and some Cambridge O Level components.

0607/62 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2017 © UCLES 2017 Page 2 of 6 MARK SCHEME NOTES The following notes are intended to aid interpretation of mark schemes in general, but individual mark schemes may include marks awarded for specific reasons outside the scope of these notes. Types of mark M Method marks, awarded for a valid method applied to the problem. A Accuracy mark, awarded for a correct answer or intermediate step correctly obtained. For accuracy marks to be given, the associated Method mark must be earned or implied. B Mark for a correct result or statement independent of Method marks. When a part of a question has two or more ‘method’ steps, the M marks are in principle independent unless the scheme specifically says otherwise; and similarly where there are several B marks allocated. The notation ‘dep’ is used to indicate that a particular M or B mark is dependent on an earlier mark in the scheme. Abbreviations awrt answers which round to cao correct answer only dep dependent FT follow through after error isw ignore subsequent working nfww not from wrong working oe or equivalent rot rounded or truncated SC Special Case soi seen or implied

0607/62 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2017 © UCLES 2017 Page 3 of 6 Question Answer Marks Partial Marks A INVESTIGATION NUMBER WALLS 1(a) 9 4 5 1 1(b) 3 is added two times oe or 3 [and not 2] is added to 1 oe 1 2(a) 20 8 12 3 5 7 1 2(b) A correct Number wall with total > their 20 1 2(c) 14 17 7 −3 3 1 3(a) a + 3b + 3c + d a + 2b + c b + 2c + d b + c c + d 1 3(b) their (a + 3b + 3c + d) = 34 oe and 4.25 oe or is not an integer 1 C opportunity 3(c) 3 5 1 2 6 and 4 4 2 1 7 2 B1 for each Numbers may be in reverse order C opportunity 4(a) Row 3 gives the coefficients of a, b, c and d 1 4(b) [1]a + 4b + 6c + 4d + [1]e oe 1 4(c) [1 ×]3 + 4 × 5 + 6[× 1] + 4×2 + [1 ×]6 = [43] or [1 ×]4 + 4 × 4 + 6 × 2 + 4[× 1] + [1 ×]7 = [43] 1 Pairs of multiplications may be in any order 5(a) 4a 8a 2(h–1)a oe 1 5(b) 3 1 C opportunity 5(c) 23 2 M1 for their 2(h–1)a = 20 971 520 soi or B1 for 22 C opportunity

0607/62 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2017 © UCLES 2017 Page 4 of 6 Question Answer Marks Partial Marks 6 3 and 5 only 3 Ignore answer of 1 B1 for 3 or 5 only B1 for x, x + 1, x + 2, [...] Allow other letters for x C opportunity Communication: seen in three of the following questions 1 3(b) their 8a = 34 or 34 ÷ 8 or a correct trial for a = 4.25 3(c) Completing all empty bricks with 23, 20, 11, 8, 3, 8 or working with equations 5(b) 26–1a = 96 or better or their expression, with h = 6, = 96 or 32a = 96 or 96 ÷ 32 or wall, height 6, 96 at top, 3 in bottom cells 5(c) (h–1) log2 = log(20 971 520÷5) oe or h – 1 = log2 4194304 oe or T & I using h with more than one trial 6 Construction of wall with consecutive numbers in bottom row with total 70 to 90 or use x, x + 1, [x + 2, …] to set up one correct equation = 80

0607/62 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2017 © UCLES 2017 Page 5 of 6 Question Answer Marks Partial Marks B MODELLING RANGES 1(a) Correct curve 3 Polygon scores 0 B2 for 9 or 10 points correctly plotted or B1 for 7 or 8 points correctly plotted 1(b) 25 to 27 and 63 to 65 1 1(c) 45 1 1(d) [r =] 10.2 [sin2x] 2 B2 for 10.15 to 10.3 or B1 for 10 only seen or M1 for correct substitution of x and r (x and r ≠ 0) into model C opportunity 1(e) –3.52… to –3.42… The shot lands behind him oe 2 B1 for each B1FT if value negative Accept backwards for behind 2(a) It would land on him oe or it would have range 0 oe 1 Ignore extra comments that are not contradictory 2(b) Correct curve 2 B1 for 6, 7 or 8 points correctly plotted Ignore curve for 0 ⩽ x < 10 and 80 < x ⩽ 90 2(c) The ranges are close oe 1 3(a) n shaped curve 1 From (0, 0) to above 80 C opportunity 3(b) n shaped curve and Correct interpretation of two sketches 2 If sketch in part (a) correct then curve should be below sketch B1 for curve from x = 10 to x = 80 always above x-axis B1FT e.g. (for correct sketches), not good for angles from approx. 30 to 60 oe or good at low and high angles oe or the graphs give similar results at the beginning and the end [but not the middle] or the model always overestimates the range oe e.g. (for incorrect sketches), similar interpretation as above but related to their two curves

0607/62 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2017 © UCLES 2017 Page 6 of 6 Question Answer Marks Partial Marks 4(a) Correct sketch 1 Ignore to the left of approx. (1, 0.1) C opportunity 4(b) [0].25 1 Accept [0].249… to [0].250… C opportunity 4(c) Unsuitable for velocities beyond approx 20 because range begins to reduce oe 1 FT 0 ⩽ k ⩽ 0.3 Communication: seen in two of the following questions 1 1(d) Showing rearrangement e.g. 3.5 sin 20 = a or indicating a = maximum on the graph 3(a) Appropriate scale on r axis to allow maximum to be shown 4(a) Appropriate scale on R axis to allow 43.26 to be shown 4(b) Substitution of 15 in three places and 19.85 (i.e. 2 2 15 15 15 2 19.85 9.81 981 × − = k )