AQA A-level MATHEMATICS PAPER 2 2020 .

AQA A-level MATHEMATICS PAPER 2 2020 MS. 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 or m marks and is for accuracy B mark is independent of M or 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 SCA substantially correct approach sf significant figure(s) dp decimal place(s) MARK SCHEME – A-LEVEL MATHEMATICS – 7357/2 – JUNE 2020 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 – 7357/2 – JUNE 2020 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 – 7357/2 – JUNE 2020 6 Q Marking instructions AO Marks Typical solution 1 Circles correct answer 2.2a B1 ( ) 1 f x x e − = − Total 1 Q Marking instructions AO Marks Typical solution 2 Ticks correct box 2.2a B1 sec x = 0 Total 1 Q Marking instructions AO Marks Typical solution 3 Uses the product of ( ) 5 2x and 3 3 x    ±    terms condone sign error and/or omission of nCr term Or Obtains any two correct terms (unsimplified) term 3.1a M1 ( ) 3 5 8 2 3 3 2 56 32 27 coefficient is 48384 C x x x   × −  = × × −   ∴ − Multiplies their ( ) 5 2x and 3 3 x    −    by 8C3 or 8C5 or 56 OE For this mark condone( ) 3 2x and 5 3 x    −    1.1a M1 Obtains correct coefficient of x 2 −48384 condone inclusion x 2 1.1b A1 Total 3 MARK SCHEME – A-LEVEL MATHEMATICS – 7357/2 – JUNE 2020 7 Q Marking instructions AO Marks Typical solution 4 Uses or states small angle approximation for tan 5x ≈5x 1.1b B1 ( ) 2 2 2 tan 5 5 cos 4 1 4 1 1 2 5 8 5 8 x x x x x x x x × ≈ − − − ≈ − ≈ − Uses or states small angle approximation for ( ) 2 4 cos 4 1 2 x x ≈ − Condone omission of bracket 1.1b B1 Substitutes their expressions Of the form tan 5x≈mx and 2 cos 4 1 2 nx x ≈ − into tan 5 cos 4 1 x x x − Condone correct extra terms 1.1b M1 Deduces 5 8 A = − from a reasoned argument CSO 2.2a R1 Total 4 MARK SCHEME – A-LEVEL MATHEMATICS – 7357/2 – JUNE 2020 8 Q Marking instructions AO Marks Typical solution 5 Uses a suitable substitution u = 4x +1 oru = 4x +1 OE 3.1a M1 6 25 1 0 4 25 3 1 2 2 1 25 5 3 2 2 0 5 3 2 2 d 4 1 4 d 1 0 4 6 25 1 4 1 1 4 1 d d 4 4 1 d 16 1 2 2 16 5 3 1 25 25 8 5 3 875 12 u u x x x u x u u x u x x x u u u u u u u − = + ⇒ = = − ⇒= = ⇒= − = − + = = −     = −           = −       = ∫ ∫ ∫ Differentiates their substitution correctly 1.1b A1 Completes substitution to obtain correct integrand for their suitable substitution. Can be unsimplified. 1.1a M1 Correctly integrates their simplified integrand provided it is of the form 3 1 A u 2 2 u    −    or ( ) 4 2 B u −u 1.1a A1 Substitutes correct limits for their substitution or 6 and -1/4 for x 1.1a M1 Completes rigorous argument to show the required result. AG 2.1 A1 Total 6 MARK SCHEME – A-LEVEL MATHEMATICS – 7357/2 – JUNE 2020 9 Q Marking instructions AO Marks Typical solution 6(a) Begins a valid method to find the coordinates Uses gradient of L to find gradient of perpendicular radius Or Forms equation of circle with unknown radius and solves simultaneously with equation of L Or differentiates equation of circle implicitly 3.1a M1 ( ) 5 12 298 12 298 5 5 5 9 7 12 12 5 73 y x y x y x y x + = − = + − = − − = 19 14 x y = = (19,14) Uses (7, 9) to find the equation of the radius Or Uses (7, 9) correctly in their equation of circle Or Uses 12 5 − after their implicit differentiation 1.1a M1 Obtains 12y −5x = 73OE Or Correctly eliminates a variable to obtain a quadratic in x or y for example obtain a quadratic in x or y ( ) ( ) 2 2 2 2 298 12 7 9 5 253 12 7 5 x x k x x k  −  − +  −  =    −  ⇒ − +   =   ( ) ( ) 2 2 2 2 298 5 7 9 12 214 5 9 12 y y k y y k  −   −  + − =    −  ⇒   + − =   1.1b A1 Equates discriminant to zero PI By correct answer or Solves their simultaneous equations of tangent and radius PI by correct answer 3.1a M1 Obtains correct values for x and y (19,14) 1.1b A1 Subtotal 5 MARK SCHEME – A-LEVEL MATHEMATICS – 7357/2 – JUNE 2020 10 6(b) Obtains ( ) ( ) 2 2 2 x −7 + y −9 = r PI if 2 r is replaced with correct value using their point from (a) 1.1a M1 ( ) ( ) 2 2 2 x −7 + y −9 = r ( ) ( ) 2 2 2 2 2 19 7 14 9 12 5 169 − + − = r + = ( ) ( ) 2 2 Uses their point to find radius or x −7 + y −9 =169 radius squared. Must have obtained a point in part (a) 1.1a M1 Obtains correct equation of circle CSO ACF 1.1b A1 Subtotal 3 Question Total 8 MARK SCHEME – A-LEVEL MATHEMATICS – 7357/2 – JUNE 2020 11 Q Marking instructions AO Mark Typical solution 7(a)(i) Identifies the error lies in step 1 without contradiction. 2.3 E1 Mistake is 1