AQA---A-level MATHEMATICS 7357/1 Paper 1 Mark scheme June 2022

Version: 1.1 Final Mark Scheme *226A7357/1/MS* MARK SCHEME – A-LEVEL MATHEMATICS – 7357/1 – JUNE 2022 2 Mark schemes are prepared by the Lead Assessment Writer and considered, together with the relevant questions, ...

Version: 1.1 Final Mark Scheme *226A7357/1/MS* MARK SCHEME – A-LEVEL MATHEMATICS – 7357/1 – 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 – 7357/1 – 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 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 – 7357/1 – 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 – 7357/1 – 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 – 7357/1 – JUNE 2022 6 Q Marking instructions AO Marks Typical solution 1 Circles the correct answer 1.2 B1 x y + = 2 2 1 Question 1 Total 1 Q Marking instructions AO Marks Typical solution 2 Circles the correct answer 1.1b B1 2 Question 2 Total 1 Q Marking instructions AO Marks Typical solution 3 Circles the correct answer 2.2a R1 y x = 2log4 Question 3 Total 1 Q Marking instructions AO Marks Typical solution 4 Ticks the correct box 2.2a R1 Question 4 Total 1 MARK SCHEME – A-LEVEL MATHEMATICS – 7357/1 – JUNE 2022 7 Q Marking instructions AO Marks Typical solution 5 Differentiates to obtain a correct derivative either ( ) x − 3 4 2 OE or xxx − +− 3 2 4 24 48 32 PI by −32 obtained with no errors seen in evaluating d d y x 1.1b B1 d ( ) d y x x = − 3 4 2 When x = 0 d d y x = −32 y =16 y x =− + 32 16 Substitutes x = 0 into their d d y x to obtain a numerical value or PI by constant from their d d y x or PI by −32 obtained with no errors seen in evaluating d d y x 1.1a M1 Obtains y x =− + 32 16 ACF Award the mark at the first opportunity and ISW any incorrect rearrangement No errors seen 1.1b A1 Question 5 Total 3 MARK SCHEME – A-LEVEL MATHEMATICS – 7357/1 – JUNE 2022 8 Q Marking instructions AO Marks Typical solution 6(a) Expands to obtain the first two terms Can be unsimplified Condone sign error 1.1a M1 x x x           − ≈+ −      ≈ − 1 2 1 1 1 2 22 1 1 4 Obtains − x 1 1 4 OE Accept if listed as two separate terms. Ignore any extra terms 1.1b A1 Subtotal 2 Q Marking instructions AO Marks Typical solution 6(b) States or uses at least one small angle approximation correctly either sinkx kx ≈ or x x ≈ − 2 cos 1 2 3.1a M1 ( ) x x xx x x x x + ≈ +−   ≈ +−     ≈+ − 2 2 2 sin 4 cos 4 1 2 4 1 4 1 1 4 4 Uses both small angle approximations correctly for sine and cosine sinkx kx ≈ and x x ≈ − 2 cos 1 2 Must have eliminated all trig expressions Inconsistent variables for angles must eventually be consistent to be awarded A1 1.1b A1 Uses their expansion from (a) Must have replaced x with x2 or Applies binomial theorem correctly to   x   −   1 2 2 1 2 ignore any extra terms 3.1a M1 Completes argument to obtain x x   + −     2 4 1 4 or + −x x 1 2 1 4 4 Accept any order of terms Ignore higher powers of x Must be in terms of x Do not ISW 2.1 R1 Subtotal 4 Question 6 Total 6 MARK SCHEME – A-LEVEL MATHEMATICS – 7357/1 – JUNE 2022 9 Q Marking instructions AO Marks Typical solution 7 Sketches one of the sections of the curve shown below Either or Condone translations Do not allow the end points of the curve turning to intersect the asymptote 1.2 B1 Sketches the three branches within the interval from 0 to 2π Condone overlapping branches Asymptotes need not be drawn 1.1a M1 Completes fully correct sketch with asymptotes drawn at approximately the correct positions Labelling not required and can be ignored Ignore anything after 2π or to the left of O 1.1b A1 Question 7 Total 3 MARK SCHEME – A-LEVEL MATHEMATICS – 7357/1 – JUNE 2022 10 Q Marking instructions AO Marks Typical solution 8(a)(i) Obtains correct y-intercept at P (0,5) or y = 5 seen anywhere PI by correct equation for line PQ 3.1a B1 At P ?? = 0 ⇒ ?? = 5 Line PQ 3?? − 5?? = −25 5?? + 3?? = 83 (10, 11) Obtains equation of PQ with correct gradient. For example y xc = + 3 5 or 5y – 3x = k or Forms an equation for the distance or distance squared from (0,5) to a point on L2 For example, d2 = x2 + �− 5 3 ?? + 68 3 � 2 3.1a M1 Obtains correct equation ACF 1.1b A1 Solves simultaneous equations for their PQ and L2 to obtain values for x and y Their PQ must not be a horizontal or vertical line Condone errors in rearrangement of the equation(s) or Minimises their distance or distance squared equation to find one coordinate 3.1a M1 Obtains (10, 11) or x y = = 10, 11 1.1b A1 Subtotal 5 MARK SCHEME – A-LEVEL MATHEMATICS – 7357/1 – JUNE 2022 11 Q Marking instructions AO Marks Typical solution 8(a)(ii) Uses the distance formula to find the value of PQ or 2 PQ or Uses Pythagoras theorem with +2 2 10 6 seen If the coordinates of P and Q are incorrect, differences in x and y must be clearly shown for M1 1.1a M1 = −+− ( ) ( ) = = 2 2 2 10 0 11 5 136 2 34 PQ Completes demonstration to PQ show that k = 2 Must have shown clear use of distance formula Condone not seeing x = 0 substituted in the distance formula Answer of 2 34 and no working shown scores M1 R0 2.1 R1 Subtotal 2 MARK SCHEME – A-LEVEL MATHEMATICS – 7357/1 – JUNE 2022 12 Q Marking instructions AO Marks Typical solution 8(b)(i) Uses a valid method to find a. Evidence could be: Forming the equation of the line mid-way between L1 and L2 5 3 49 x y + = seen or Using (a , -17) as the mid-point of a line segment from L1 to L2 For example: () . () . ( . .) x x x x a +− = = +− = = + = = 5 3 17 15 13 2 5 3 17 83 26 8 26 8 13 2 20 2 or Finding the mid-point of PQ their (5 , 8) and using the gradient of L1 and L2 = − 5 3 For example: ( ) a ( ) + − =− =+ = 8 5 5 17 5 5 3 20 or Substitutes y = 17 and x = a into y x = + 3 5 5 3.1a M1 ( ) x y a a + = +− = = 5 3 49 5 3 17 49 20 Deduces a = 20 2.2a R1 Subtotal 2 Q Marking instructions AO Marks Typical solution 8(b)(ii) Forms expression of the form ( ) ( ) xa y ± +± 2 2 17 using a or their value of a 1.1a M1 ( ) ( ) x y − ++ = 2 2 Obtains correct equation for 20 17 34 their value of a and their radius2 = k 2 17 2 from part (a)(ii) for an integer value of k Condone ( )2 34 1.1b A1F Subtotal 2 Question 8 Total 11 MARK SCHEME – A-LEVEL MATHEMATICS – 7357/1 – JUNE 2022 13 Q Marking instructions AO Marks Typical solution 9(a) Forms an appropriate equation in x only by either using the differences of at least one pair of terms or Using the mean of the first and third term = the second term Condone missing brackets or Forms two simultaneous equations in x and d or Substitutes x = 5 and demonstrates that the three terms obtained, 15, 26 and 37 have a common difference of 11 or Shows that the sum formula for an arithmetic series works when x = 5 The approaches that substitute x = 5 score a maximum of M1 A0 R0 3.1a M1 xx x x ( ) ( ) x x x +− + = + − + −=+ = 5 12 5 6 75 1 34 6 5 Therefore x = 5 is the only solution Obtains a correct equation or Obtains two correct simultaneous equations in x and d Need not be simplified 1.1b A1 Solves to conclude that x = 5 is the only solution Must include the word ‘only’ OE 2.1 R1 Subtotal 3 Q Marking instructions AO Marks Typical solution 9(b)(i) Obtains 15 1.1b B1 a =15 Subtotal 1 Q Marking instructions AO Marks Typical solution 9(b)(ii) Obtains 11 1.1b B1 d = 11 Subtotal 1 MARK SCHEME – A-LEVEL MATHEMATICS – 7357/1 – JUNE 2022 14 Q Marking instructions AO Marks Typical solut