Version.0: 009 hij General Certificate of Education Mathematics 6360 MFP4 Further Pure 4 Mark Scheme 009 examination Januar series
Mark schemes are prepared b the Principal Examiner and considered, together with the relevant questions, b a panel of subject teachers. This mark scheme includes an amendments made at the standardisation meeting attended b all examiners and is the scheme which was used b them in this examination. The standardisation meeting ensures that the mark scheme covers the candidates responses to questions and that ever examiner understands and applies it in the same correct wa. As preparation for the standardisation meeting each examiner analses a number of candidates scripts: alternative answers not alread covered b the mark scheme are discussed at the meeting and legislated for. If, after this meeting, examiners encounter unusual answers which have not been discussed at the meeting the are required to refer these to the Principal Examiner. It must be stressed that a mark scheme is a working document, in man cases further developed and expanded on the basis of candidates reactions to a particular paper. Assumptions about future mark schemes on the basis of one ear 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 to download from the AQA Website: www.aqa.org.uk Copright 009 AQA and its licensors. All rights reserved. COPYRIGHT AQA retains the copright on all its publications. However, registered centres for AQA are permitted to cop material from this booklet for their own internal use, with the following important exception: AQA cannot give permission to centres to photocop an material that is acknowledged to a third part even for internal use within the centre. Set and published b the Assessment and Qualifications Alliance. The Assessment and Qualifications Alliance (AQA) is a compan limited b guarantee registered in England and Wales (compan number 364473) and a registered charit (registered charit number 073334). Registered address: AQA, Devas Street, Manchester 5 6EX Dr Michael Cresswell Director General
MFP4 - AQA GCE Mark Scheme 009 Januar series Ke to mark scheme and abbreviations used in marking M m or dm A B E mark is for method mark is dependent on one or more M marks and is for method mark is dependent on M or m marks and is for accurac mark is independent of M or m marks and is for method and accurac mark is for explanation or ft or F follow through from previous incorrect result MC mis-cop CAO correct answer onl MR mis-read CSO correct solution onl RA required accurac AWFW anthing which falls within FW further work AWRT anthing which rounds to ISW ignore subsequent work ACF an correct form FIW from incorrect work AG answer given BOD given benefit of doubt SC special case WR work replaced b candidate OE or equivalent FB formulae book A, or (or 0) accurac marks NOS not on scheme x EE deduct x marks for each error G graph NMS no method shown c candidate PI possibl implied sf significant figure(s) SCA substantiall correct approach dp decimal place(s) No Method Shown Where the question specificall requires a particular method to be used, we must usuall see evidence of use of this method for an marks to be awarded. However, there are situations in some units where part marks would be appropriate, particularl when similar techniques are involved. Your Principal Examiner will alert ou to these and details will be provided on the mark scheme. Where the answer can be reasonabl obtained without showing working and it is ver unlikel that the correct answer can be obtained b using an incorrect method, we must award full marks. However, the obvious penalt to candidates showing no working is that incorrect answers, however close, earn no marks. Where a question asks the candidate to state or write down a result, no method need be shown for full marks. Where the permitted calculator has functions which reasonabl allow the solution of the question directl, the correct answer without working earns full marks, unless it is given to less than the degree of accurac accepted in the mark scheme, when it gains no marks. Otherwise we require evidence of a correct method for an marks to be awarded. 3
MFP4 - AQA GCE Mark Scheme 009 Januar series MFP4 Q Solution Marks Total Comments (a) 4i + j 3k or equivalent (b)(i) 4 + + 3 = 3 ft From their d.v. 4 3 d.c. s are, and 3 3 3 AF ft (ii) The cosines of the angles between the line and the coordinate axes (c) a = i j + k and b = their d.v. F CAO ft Total 6 (a) det AB = 0 Use of det AB = det A det B det B = AF 3 ft their det AB / 0 (b) C = (AB) T 9 7 = 3 D = [ (BA) T ] T = BA = 4 8 i j k 3(a)(i) x = 3 = 5 7 4 A 3 For reference: A = 3 4, B = 5 3 Total 6 A (ii) (x ) z = 3 5 7 4 8 a = 8 a AF ft or via 8 a (b)(i) A = ½ x = ½ + + =.5 AF ft (ii) (x ) z = 0 a = 8 AF ft or CAO from new start Total 8 4
MFP4 - AQA GCE Mark Scheme 009 Januar series MFP4 (cont) Q Solution Marks Total Comments 4(a) Subst g. λ = into det(m λi) = 0 Or M x = x etc. Solving between x + + z = 0 and x + + z = 0 d Eigenvector(s) α A 3 An non-zero α will suffice 0 (b) Attempt at Char. Eqn. λ 3 5λ 5λ + = 0 A 3 Each coefft. (not the λ 3 ) Use of division/factor theorem etc. With/without (λ + ) factor (λ + )( λ 6λ + ) A Solving remaining quadratic factor λ, 3 = 3± A 8 CAO simplest exact form Total 5(a) D = x + + z x z zx A (b) E.g. b C = C + (C + C 3 ) x + + z z Δ = 0 z x x ( x + + z) + x z+ z = (x + + z) 0 z x x + x z+ A Shown or explained from previous line (c) Working on (R/C-ops) or expanding remaining determinant nd factor = (x + + z x z zx) d Good attempt k = A 3 Total 7 6(a) Use of sin θ or cos θ = (dot product)/(product of moduli) Must be d.v. of line & plane s nml. Num r. = 3 F ft Denom r. = 8 F ft θ = 30 o A 4 CAO (b)(i) λ = 8 noted or found (ii) + λ 0 3 λ = 0 5+ 4λ Attempt at this 3 λ + 5 + 4λ = 0 λ = 4 A Solving a linear eqn. in λ giving Q = (6,, ) F 4 ft (iii) PQ = 4 + 4 + 6 = A Or 4 8, 7.0, 6.97 etc. Sh. Dist. =.sin30 o = 6 F 3 ft previous answer Total 5
MFP4 - AQA GCE Mark Scheme 009 Januar series MFP4 (cont) Q Solution Marks Total Comments 7(a) x = λ x + = 3 3λ At least one correct from setting z = λ Solving for x and in terms of λ x = 7λ 5 and = 4λ A 3 CAO (b) Subst g. x,, z in terms of λ in 5x + k + 7z = 35λ 5 + k(4 ) + 7λ = 0 Factsn. attempt: (4 )(k + 3) = 0 d ( )(k + 3) = 0 A 3 ANSWER GIVEN (c)(i) When k = 3, 5x 3 + 7z = 35λ 5 5λ + 6 + 7λ Subst g. into 3 rd eqn. and demonstrating consistenc The three planes intersect in a line Solns. x = 7λ 5, = 4λ, z = λ F ft (ii) When k 3, λ = ½ Soln. ( ½, 0, ½) F ft Three planes meet at a point 6 Total 8(a)(i) /det 5 Transposed matx. of cofactors (ii) x X ( X + Y ) 5 = A Y = ( Y X) 5 AF ft (b) / 5 / 5 A = 5 / 5 / 5 Enlargement sf 5 (centre O) A Two components in an order + Rotation thro cos (/ 5 ) A 5 or 63.4 o or. rads (c)(i) p = ½, q = 3 noted Or form (ii) 6x + = 3 6 X + 4XY + 4Y 5 ( ) + ( Y XY X ) 4 + 4 = 3 5 0X + 0XY+ 5Y = 75 x Subst g. for x and X + 4XY+ 5Y = 5 A ANSWER GIVEN (iii) It is just an enlarged rotation of E, hence still an ellipse Total 3 TOTAL 75 + = shown 3 6