Cambridge IGCSE Mathematics - International 0607 — 2022 Oct/Nov Paper 5 · Variant 1

This document has 12 pages. [Turn over Cambridge IGCSE™ CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/51 Paper 5 Investigation (Core) October/November 2022 1 hour 10 minutes You must answer on the question paper. No additional materials are needed. INSTRUCTIONS ● Answer all questions. ● Use a black or dark blue pen. You may use an HB pencil for any diagrams or graphs. ● Write your name, centre number and candidate number in the boxes at the top of the page. ● Write your answer to each question in the space provided. ● Do not use an erasable pen or correction fluid. ● Do not write on any bar codes. ● You should use a graphic display calculator where appropriate. ● You may use tracing paper. ● You must show all necessary working clearly, including sketches, to gain full marks for correct methods. ● In this paper you will be awarded marks for providing full reasons, examples and steps in your working to communicate your mathematics clearly and precisely. INFORMATION ● The total mark for this paper is 36. ● The number of marks for each question or part question is shown in brackets [ ]. DC (LK/CT) 303200/2 © UCLES 2022 * 6 0 6 6 2 2 7 1 8 1 *

2 0607/51/O/N/22 © UCLES 2022 Answer all the questions. INVESTIGATION ISOSCELES TRAPEZIUMS This investigation looks at the perimeter of isosceles trapeziums drawn on 1 cm isometric grids and the number of unit equilateral triangles in these trapeziums. There is a spare isometric grid on page 12. 1 The diagrams show a sequence of trapeziums. The sloping side length is 1. Both parallel sides increase in length by 1 each time. This is the first trapezium in the sequence. The length of the shorter parallel side is 1. 1 1 2 1 This is the second trapezium in the sequence. The length of the shorter parallel side is 2. 2 3 1 1 This is the third trapezium in the sequence. The length of the shorter parallel side is 3. 3 4 1 1

3 0607/51/O/N/22 © UCLES 2022 [Turn over (a) Draw the next trapezium in the sequence. [1] (b) Complete the table. Shorter parallel side length (x) Longer parallel side length Perimeter 1 2 5 2 3 4 9 4 [2] (c) Write down an expression, in terms of x, for the length of the longer parallel side. … [1] (d) Find an expression, in terms of x, for the perimeter. Give your answer in its simplest form. … [2]

4 0607/51/O/N/22 © UCLES 2022 2 The diagram shows another sequence of trapeziums. Each trapezium has sloping side length 2. Both parallel sides increase in length by 1 each time. x 1 = x 2 = x 3 = (a) Draw the trapezium with sloping side length 2 and x = 5. [1] (b) Complete the table. Shorter parallel side length (x) Longer parallel side length Perimeter 1 3 8 2 4 3 5 4 6 14 5 [2] (c) Write down an expression, in terms of x, for the length of the longer parallel side. … [1] (d) Find an expression, in terms of x, for the perimeter. Give your answer in its simplest form. … [2]

5 0607/51/O/N/22 © UCLES 2022 [Turn over 3 In another sequence each trapezium has sloping side length 3. The length of the shorter parallel side is x. (a) Find an expression, in terms of x, for the length of the longer parallel side of the trapeziums with sloping side length 3. You may use the grid below to help you. … [2] (b) Show that an expression, in terms of x, for the perimeter is x 2 9 + . [1]

6 0607/51/O/N/22 © UCLES 2022 4 y x In each trapezium: • the shorter parallel side length is x • each sloping side length is y. (a) Write down an expression, in terms of x and y, for the length of the longer parallel side. You may use your expressions from Questions 1(c), 2(c) and 3(a) to help you. … [1] (b) Find an expression, in terms of x and y, for the perimeter. Give your answer in its simplest form. … [2]

7 0607/51/O/N/22 © UCLES 2022 [Turn over (c) Show that your expression in part (b) gives the correct result when x 2 = and y 5 = . [3] (d) A trapezium has shorter parallel side length x and sloping side length x. Find an expression, in terms of x, for the perimeter. Give your answer in its simplest form. … [2]

8 0607/51/O/N/22 © UCLES 2022 5 A unit triangle is an equilateral triangle of side length 1. There are 3 unit triangles in this trapezium with x 1 = and y 1 = . There are 12 unit triangles in this trapezium with x 2 = and y 2 = . (a) Work out the number of unit triangles in a trapezium with x 1 = and y 3 = . You may use the grid below to help you. … [2]

9 0607/51/O/N/22 © UCLES 2022 [Turn over (b) Complete the table. You may use the grid below to help you. Shorter parallel side length (x) Sloping side length (y) Number of unit triangles 1 1 3 2 1 3 1 1 2 2 2 12 3 2 1 3 2 3 3 3 27 [3]

10 0607/51/O/N/22 © UCLES 2022 6 In a different sequence of trapeziums the shorter parallel side length, x, is equal to the sloping side length, y. (a) The diagram shows the trapezium with x 4 = and y 4 = . Find the number of unit triangles in this trapezium. … [1] (b) Complete the table. You may use results from Questions 5(b) and 6(a), and patterns to help you. Shorter parallel side length (x) Number of unit triangles 1 3 = 3 # … = 3 # 12 2 … = 3 # 4 = 3 # 22 3 … = 3 # … = 3 # … 4 … = 3 # … = 3 # … [2]

11 0607/51/O/N/22 © UCLES 2022 [Turn over (c) Write down an expression, in terms of x, for the number of unit triangles. … [1] (d) In a trapezium the shorter parallel side length is equal to the sloping side length. There are 675 unit triangles in this trapezium. Find the perimeter of this trapezium. … [4]

12 0607/51/O/N/22 © UCLES 2022 SPARE GRID 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 Assessment International Education Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at www.cambridgeinternational.org after the live examination series. Cambridge Assessment International Education is part of Cambridge Assessment. Cambridge Assessment is the brand name of the University of Cambridge Local Examinations Syndicate (UCLES), which is a department of the University of Cambridge.

This document consists of 5 printed pages. © UCLES 2022 [Turn over Cambridge IGCSE™ CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/51 Paper 5 Core October/November 2022 MARK SCHEME Maximum Mark: 36 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 2022 series for most Cambridge IGCSE™, Cambridge International A and AS Level components and some Cambridge O Level components.

0607/51 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2022 © UCLES 2022 Page 2 of 5 Generic Marking Principles These general marking principles must be applied by all examiners when marking candidate answers. They should be applied alongside the specific content of the mark scheme or generic level descriptors for a question. Each question paper and mark scheme will also comply with these marking principles. GENERIC MARKING PRINCIPLE 1: Marks must be awarded in line with: • the specific content of the mark scheme or the generic level descriptors for the question • the specific skills defined in the mark scheme or in the generic level descriptors for the question • the standard of response required by a candidate as exemplified by the standardisation scripts. GENERIC MARKING PRINCIPLE 2: Marks awarded are always whole marks (not half marks, or other fractions). GENERIC MARKING PRINCIPLE 3: Marks must be awarded positively: • marks are awarded for correct/valid answers, as defined in the mark scheme. However, credit is given for valid answers which go beyond the scope of the syllabus and mark scheme, referring to your Team Leader as appropriate • marks are awarded when candidates clearly demonstrate what they know and can do • marks are not deducted for errors • marks are not deducted for omissions • answers should only be judged on the quality of spelling, punctuation and grammar when these features are specifically assessed by the question as indicated by the mark scheme. The meaning, however, should be unambiguous. GENERIC MARKING PRINCIPLE 4: Rules must be applied consistently, e.g. in situations where candidates have not followed instructions or in the application of generic level descriptors. GENERIC MARKING PRINCIPLE 5: Marks should be awarded using the full range of marks defined in the mark scheme for the question (however; the use of the full mark range may be limited according to the quality of the candidate responses seen). GENERIC MARKING PRINCIPLE 6: Marks awarded are based solely on the requirements as defined in the mark scheme. Marks should not be awarded with grade thresholds or grade descriptors in mind.

0607/51 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2022 © UCLES 2022 Page 3 of 5 Maths-Specific Marking Principles 1 Unless a particular method has been specified in the question, full marks may be awarded for any correct method. However, if a calculation is required then no marks will be awarded for a scale drawing. 2 Unless specified in the question, answers may be given as fractions, decimals or in standard form. Ignore superfluous zeros, provided that the degree of accuracy is not affected. 3 Allow alternative conventions for notation if used consistently throughout the paper, e.g. commas being used as decimal points. 4 Unless otherwise indicated, marks once gained cannot subsequently be lost, e.g. wrong working following a correct form of answer is ignored (isw). 5 Where a candidate has misread a number in the question and used that value consistently throughout, provided that number does not alter the difficulty or the method required, award all marks earned and deduct just 1 mark for the misread. 6 Recovery within working is allowed, e.g. a notation error in the working where the following line of working makes the candidate’s intent clear. 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

0607/51 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2022 © UCLES 2022 Page 4 of 6 Question Answer Marks Partial Marks 1(a) 1 1(b) 3 and 7 5 and 11 2 B1 for 7 or 11 1(c) x + 1 oe 1 1(d) 2x + 3 1 x + their (x + 1) + 1 + 1 or three differences of 2 seen C1 2(a) 1 2(b) 1 3 8 2 4 10 3 5 12 4 6 14 5 7 16 2 B1 for two correct perimeters or for 7 and 16 2(c) x + 2 oe 1 2(d) 2x + 6 or 2(x + 3) 1 x + their (x + 2) + 2  2 or at least three differences of 2 shown on table. C1 3(a) x + 3 oe 1 A trapezium drawn on grid with sloping side length 3 or sketch of a trapezium not on grid with correct lengths of parallel sides marked. C1 3(b) x + x + 3 + 3 + 3 1 4(a) x + y oe 1 4(b) 2x + 3y 1

0607/51 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2022 © UCLES 2022 Page 5 of 5 Question Answer Marks Partial Marks x + y + y + their (x + y) or 3, 6, 9, … 3y soi C1 4(c) Correct trapezium drawn on grid 1 Correct substitution of x = 2 and y = 5 into their (2x + 3y) 1 FT 2 + 5 + 5 + 7 = 19 and 4 +15 = 19 1 4(d) 5x 1 2x + 3x or other unsimplified addition equivalent to 5x or a sketch of a trapezium with dimensions x, x, x and 2x or substitution of y = x into their 4(b) C1 5(a) 15 1 Correct diagram divided into unit triangles or 12 + 3 or 7, 5 and 3 seen corresponding to rows C1 5(b) 5, 7, 8, 16, 15, 21 in final column 3 B2 for 4 correct or B1 for 2 or 3 correct 6(a) 48 1 6(b) 3 = 3  1 = 3  12 12 = 3  4 = 3  22 27 = 3  9 = 3  32 48 = 3  16 = 3  42 2 B1 for 5 entries correct 6(c) 3x2 oe 1 6(d) 75 2 B1 for x = 15 soi 3x2 = 675 seen C1 FT their 3x2 their value of x either substituted into their 5x, or shown by the lengths of each side. C1 FT