Version 1.0: 0110 klm General Certificate of Education Mathematics 660 MFP Further Pure Mark Scheme 010 examination - January series
Mark schemes are prepared by the Principal Examiner and considered, together with the relevant questions, by a panel of subject teachers. This mark scheme includes any amendments made at the standardisation meeting attended by all examiners and is the scheme which was used by them in this examination. The standardisation meeting ensures that the mark scheme covers the candidates responses to questions and that every examiner understands and applies it in the same correct way. As preparation for the standardisation meeting each examiner analyses a number of candidates scripts: alternative answers not already covered by 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 they are required to refer these to the Principal Examiner. It must be stressed that a mark scheme is a working document, in many 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 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 to download from the AQA Website: www.aqa.org.uk Copyright 010 AQA and its licensors. All rights reserved. COPYRIGHT AQA retains the copyright on all its publications. However, registered centres 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 centres to photocopy any material that is acknowledged to a third party even for internal use within the centre. Set and published by the Assessment and Qualifications Alliance. The Assessment and Qualifications Alliance (AQA) is a company limited by guarantee registered in England and Wales (company number 67) and a registered charity (registered charity number 107). Registered address: AQA, Devas Street, Manchester 6EX Dr Michael Cresswell Director General
Key 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 accuracy mark is independent of M or m marks and is for method and accuracy mark is for explanation or ft or F follow through from previous incorrect result MC mis-copy CAO correct answer only MR mis-read CSO correct solution only RA required accuracy AWFW anything which falls within FW further work AWRT anything which rounds to ISW ignore subsequent work ACF any correct form FIW from incorrect work AG answer given BOD given benefit of doubt SC special case WR work replaced by candidate OE or equivalent FB formulae book A,1 or 1 (or 0) accuracy marks NOS not on scheme x EE deduct x marks for each error G graph NMS no method shown c candidate PI possibly implied sf significant figure(s) SCA substantially correct approach dp decimal place(s) 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. However, there are situations in some units where part marks would be appropriate, particularly when similar techniques are involved. Your Principal Examiner will alert you to these and details will be provided on the mark scheme. 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 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 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.
1(a) 1 (b) 1 (c) 1 (d) 9 Total (a) 8 AB AD = 0 7 1 = 1 8 Attempt at vector product of any two suitable vectors 1 + 1 + 8 Magnitude of this attempted = 7 ft on minus signs only AG (b) 1 AB AD AE = 1 8 1 6 = Attempt at scalar triple product (or ) on any suitable vectors ft (c) Distance = (b) / 7 = 6 ft; must be deduced Total 8 (a) t 0 0 Decent attempt at AB 98 t t t1 correct 8 t 0 all correct t = 7 Allow this ft if only 1 or elements of AB incorrect AB = I Must be from AB completely correct (b) A 1 = 1 1 1 1 17 ft 1/det if related to B CAO; must be deduced NB 1 B scores only Total 7
MFP (cont) (a) 1 k k + 1 0 = k Attempt at det. of coefft. mtx. k Correct 1 k 1 k =, 1 Setting det. = 0 and solving for k (b) k = = x y+ z 8 = x+ y = x + y + z ft Eliminating one variable twice 8x + 9y = / 1y 8z = 1 / x + z = 11 ; Correct eqn. once; twice k = 1 = x yz 1 = y = x + y z x z = 1 / x z = 10 OR y = 1 and y = 7 found (a) 1 x x1 = x+ 1 x+ / x z = 11 8 Total 1 ft Eliminating one variable twice Inconsistency correctly shown x Use of x + 1 ; either term correct giving y = 1 x CSO (b) Finding det(b) (= 1) and mult g. by. =. cm ft (c) 1 1 = 1 0 1 This is a shear parallel to the x-axis AB attempted mapping (eg) (1, 1) to (, 1) Any point not on Ox and its image Total 10
MFP (cont) 6(a)(i) 6 p + 1 p = 0 1 Equating dot product to zero p + 6 p + 6 + = 0 Solving a linear eqn. in t p = CAO (ii) 6 p + 1 m p = 1 m =, p = 1, 6 p + 1 ALT p = 0 1 p = 1 1 p 8p 18p 9 () () (b)(i) 6x y + z =, 17x + y + z = 7, (ii) eg. 1: 7[x + y = 8] Eliminating one variable; correct x+ y = =λ 1 Parametrisation attempt Substituting back to find rd variable m1 z = 9λ + 7 0 1 1 r = 8, 1 or 0 + λ 6 ft (any pt. on l) 9 0 7 9 1 OR method for finding the d.v. 9 (M) Method for finding any pt. on l () Putting them together as a line eqn. () ft (c) r = a + u d 1 + v d where a is an pt. on plane ft their previous a; or (0, 7, 0) d 1 = any d.v. ft their previous d.v. 0 d = 7 any a in plane 0 ft if is clear where it has come from Total 16
MFP (cont) 7(a)(i) R 1 = R 1 + R or R = R + R 1 leading to R 1/ = ( q q 0) (ii) 1 1 0 Δ= ( q) 1 1q 7 6 6 10 q 0 1 0 = ( q) q11 1q 7 0 6 10 q 0 1 0 = ( q)( q11) 1 1q 7 0 6 10 q By C 1 = C 1 C (eg) nd linear factor correct = ( q)(q 11) Full attempt at rd factor = ( q)(q 11)(q 10) (b)(i) 16 7 8 1 1 7 = 0 6 6 10 7 8 = so that λ = 7 (ii) λ = 10, 11 noted or used ft λ = 10 0 = 6x + y+ 7 z 0 = 1x 11y7z 0 = 6x+ 6 y Using y = x to get z x / y Eigenvector ( 7, 7, 1) Any non-zero multiple will do λ = 11 0 = x + y+ 7 z 0 = 1x 1y7z 0 = 6x+ 6 yz Either y = x or z = 0 Eigenvector (1, 1, 0) 7 Any non-zero multiple will do (c) x y z x y = =, = = z 7 7 7 Any line eqns. or x = y, z = 0 any one ft correct Total 18 TOTAL 7