AQA A-level MATHEMATICS 7357/3 Paper 3 Mark scheme June 2022 Version: 1.1 Final Mark Scheme

A-level MATHEMATICS 7357/3 Paper 3 Mark scheme June 2022 *226A7357/3/MS* MARK SCHEME – A-LEVEL MATHEMATICS – 7357/3 – JUNE 2022 2 Mark schemes are prepared by the Lead Assessment Writer and considered, together ...

A-level MATHEMATICS 7357/3 Paper 3 Mark scheme June 2022 *226A7357/3/MS* MARK SCHEME – A-LEVEL MATHEMATICS – 7357/3 – JUNE 2022 2 Mark schemes are prepared by the Lead Assessment Writer and considered, together with the relevant questions, by a panel of subject teachers. This mark scheme includes any amendments made at the standardisation events which all associates participate in and is the scheme which was used by them in this examination. The standardisation process ensures that the mark scheme covers the students’ responses to questions and that every associate understands and applies it in the same correct way. As preparation for standardisation each associate analyses a number of students’ scripts. Alternative answers not already covered by the mark scheme are discussed and legislated for. If, after the standardisation process, associates encounter unusual answers which have not been raised they are required to refer these to the Lead Examiner. It must be stressed that a mark scheme is a working document, in many cases further developed and expanded on the basis of students’ reactions to a particular paper. Assumptions about future mark schemes on the basis of one year’s document should be avoided; whilst the guiding principles of assessment remain constant, details will change, depending on the content of a particular examination paper. Further copies of this mark scheme are available from aqa.org.uk Copyright information AQA retains the copyright on all its publications. However, registered schools/colleges for AQA are permitted to copy material from this booklet for their own internal use, with the following important exception: AQA cannot give permission to schools/colleges to photocopy any material that is acknowledged to a third party even for internal use within the centre. Copyright © 2022 AQA and its licensors. All rights reserved. MARK SCHEME – A-LEVEL MATHEMATICS – 7357/3 – JUNE 2022 3 Mark scheme instructions to examiners General The mark scheme for each question shows: • the marks available for each part of the question • the total marks available for the question • marking instructions that indicate when marks should be awarded or withheld including the principle on which each mark is awarded. Information is included to help the examiner make his or her judgement and to delineate what is creditworthy from that not worthy of credit • a typical solution. This response is one we expect to see frequently. However credit must be given on the basis of the marking instructions. If a student uses a method which is not explicitly covered by the marking instructions the same principles of marking should be applied. Credit should be given to any valid methods. Examiners should seek advice from their senior examiner if in any doubt. Key to mark types M mark is for method R mark is for reasoning A mark is dependent on M marks and is for accuracy B mark is independent of M marks and is for method and accuracy E mark is for explanation F follow through from previous incorrect result Key to mark scheme abbreviations CAO correct answer only CSO correct solution only ft follow through from previous incorrect result ‘their’ Indicates that credit can be given from previous incorrect result ISW Ignore subsequent working AWFW anything which falls within AWRT anything which rounds to ACF any correct form AG answer given SC special case OE or equivalent NMS no method shown PI possibly implied sf significant figure(s) dp decimal place(s) MARK SCHEME – A-LEVEL MATHEMATICS – 7357/3 – JUNE 2022 4 AS/A-level Maths/Further Maths assessment objectives AO Description AO1 AO1.1a Select routine procedures AO1.1b Correctly carry out routine procedures AO1.2 Accurately recall facts, terminology and definitions AO2 AO2.1 Construct rigorous mathematical arguments (including proofs) AO2.2a Make deductions AO2.2b Make inferences AO2.3 Assess the validity of mathematical arguments AO2.4 Explain their reasoning AO2.5 Use mathematical language and notation correctly AO3 AO3.1a Translate problems in mathematical contexts into mathematical processes AO3.1b Translate problems in non-mathematical contexts into mathematical processes AO3.2a Interpret solutions to problems in their original context AO3.2b Where appropriate, evaluate the accuracy and limitations of solutions to problems AO3.3 Translate situations in context into mathematical models AO3.4 Use mathematical models AO3.5a Evaluate the outcomes of modelling in context AO3.5b Recognise the limitations of models AO3.5c Where appropriate, explain how to refine models MARK SCHEME – A-LEVEL MATHEMATICS – 7357/3 – JUNE 2022 5 Examiners should consistently apply the following general marking principles No Method Shown Where the question specifically requires a particular method to be used, we must usually see evidence of use of this method for any marks to be awarded. Where the answer can be reasonably obtained without showing working and it is very unlikely that the correct answer can be obtained by using an incorrect method, we must award full marks. However, the obvious penalty to students showing no working is that incorrect answers, however close, earn no marks. Where a question asks the student to state or write down a result, no method need be shown for full marks. Where the permitted calculator has functions which reasonably allow the solution of the question directly, the correct answer without working earns full marks, unless it is given to less than the degree of accuracy accepted in the mark scheme, when it gains no marks. Otherwise we require evidence of a correct method for any marks to be awarded. Diagrams Diagrams that have working on them should be treated like normal responses. If a diagram has been written on but the correct response is within the answer space, the work within the answer space should be marked. Working on diagrams that contradicts work within the answer space is not to be considered as choice but as working, and is not, therefore, penalised. Work erased or crossed out Erased or crossed out work that is still legible and has not been replaced should be marked. Erased or crossed out work that has been replaced can be ignored. Choice When a choice of answers and/or methods is given and the student has not clearly indicated which answer they want to be marked, mark positively, awarding marks for all of the student's best attempts. Withhold marks for final accuracy and conclusions if there are conflicting complete answers or when an incorrect solution (or part thereof) is referred to in the final answer. MARK SCHEME – A-LEVEL MATHEMATICS – 7357/3 – JUNE 2022 6 Q Marking instructions AO Marks Typical solution 1 Circles correct answer 1.1b B1 x < 4 Question 1 Total 1 Q Marking instructions AO Marks Typical solution 2 Ticks correct box 3.1a B1 ( )d xx x − ∫ 5 2 0 5 Question 2 Total 1 Q Marking instructions AO Marks Typical solution 3 Circles correct answer 3.1a B1 x = –1 Question 3 Total 1 Q Marking instructions AO Marks Typical solution x x + + c 3 3 2 2 3 3 4 Integrates one x term correctly Accept unsimplified terms 1.1a M1 Obtains correct answer ACF Must include + c Accept unsimplified terms Do not ISW 1.1b A1 Question 4 Total 2 MARK SCHEME – A-LEVEL MATHEMATICS – 7357/3 – JUNE 2022 7 Q Marking instructions AO Marks Typical solution 5(a) Sketches sine wave with correct orientation through the origin to at least one period Ignore any numbers on y-axis 1.1a M1 Sketches y = sin 2x through the correct axes intersections Condone only slight difference in amplitudes Ignore any numbers on y-axis Ignore any sections of the graph outside of 0 360 ≤ ≤x   1.1b A1 Subtotal 2 Q Marking instructions AO Marks Typical solution 5(b) Deduces correct values of A CAO 2.2a B1 ±1 Subtotal 1 Question 5 Total 3 MARK SCHEME – A-LEVEL MATHEMATICS – 7357/3 – JUNE 2022 8 Q Marking instructions AO Marks Typical solution t x = ⇒ =− × . . 2 95 2 95 = −180.5 Length = 180.5 cm 6(a) Substitutes t = 9.5 into 2 x t = −2 or 2t 2 3.4 M1 Obtains 180.5 Condone incorrect or missing units ISW 1.1b A1 Subtotal 2 Q Marking instructions AO Marks Typical solution 6(b)(i) Obtains 9 − 1.4t or −4t OE Ignore labels 1.1b B1 d . d y t t = − 9 14 d d x t t = −4 d dd d dd y yt x tx = × d . d y t x t − = − 9 14 4 Uses chain rule to obtain d d y x Condone sign error 3.1a M1 Obtains a correct expression Do not ISW 1.1b A1 Subtotal 3 MARK SCHEME – A-LEVEL MATHEMATICS – 7357/3 – JUNE 2022 9 Q Marking instructions AO Marks Typical solution . t t − = − 9 14 4 0 ( ) . . .. . t y = = =× − × = 2 45 6 43 7 9 6 43 0 7 6 43 28 9 Width of surfboard = 58 cm 180.5 ÷ 3 = 60.2 ≈ 58 Hence the width is approximately one third of the length 6(b)(ii) Equates their d d y x or their d dt y or their numerator of their d d y x to 0 PI by correct t from correct d d y x 3.1a M1 Obtains correct value for t ACF eg t = 6.4 or 9 1.4 Must come from correct d d y x 1.1b A1 Substitutes their value for t into the model for y and obtains a value for y provided 0 < t < 9.5 3.4 M1 Compares correct width and correct length and 1 3 or 3 with a correct concluding statement OE CSO Allow 180 for length 3.2a R1 Subtotal 4 Question 6 Total 9 MARK SCHEME – A-LEVEL MATHEMATICS – 7357/3 – JUNE 2022 10 Q Marking instructions AO Marks Typical solution 7(a)(i) Forms a correct expression for the gradient or sets up two correct simultaneous equations PI by a = −0.7 or b = 1.5 Ignore missing labels 1.1a M1 . . . . . − = − 4 49 1 94 1 5 3 46 1 76 T d ( ) T d −= − =− + 10 10 10 10 log 1.94 1.5 log 1.76 log 0.7 1.5log Obtains a = −0.7 or b = 1.5 OE Ignore missing labels 1.1b A1 Obtains a = −0.7 and b = 1.5 or seen in the logarithmic equation ISW 1.1b A1 Subtotal 3 Q Marking instructions AO Marks Typical solution 7(a)(ii) Uses one law of logarithm correctly Allow use of original equation without values for a and b If values are used, a ≠ 0 3.3 M1 . . . . . . . . log log log T d T d T d T d − − − =−     = −   = = × 1 5 10 10 10 1 5 0 7 1 5 07 15 0 7 0 7 10 1