Version.0 klm General Certificate of Education June 00 Mathematics MPC4 Pure Core 4 Mark Scheme
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 alrea 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 00 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 6447) and a registered charity (registered charity number 074). Registered address: AQA, Devas Street, Manchester M5 6EX
MPC4 - AQA GCE Mark Scheme 00 June series 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, or (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.
MPC4 - AQA GCE Mark Scheme 00 June series MPC4 (a) f( 4) = 8 + 6 4 64 6 4 M Use x = in evaluation 4 = 4 A NMS /; no ISW (b)(i) ( ) g = number(s) + d = 0 Use factor theorem to find d 4 M See some processing d = A NMS / (ii) g( x) ( 4x )( x bx ) = + BF a= c= ; F on d ( c = d) x 6 = 4b or x 4 = b M Any appropriate method; PI b = A NMS / Total 7 Alternatives: (a) x + x 4x 8x + 6x 4x 8x x 8x 4x 8x x x x + 4 (M) (A) () Complete division with integer remainder Remainder = 4 stated (b)(i) Division as for (a) d last line (M) Candidate s d = (A) () (a) = = 6t B Both derivatives correct; PI dt dt 6t = M Correct use of chain rule = t A CSO (b) t = mt = mn = M AF Substitute t= F on gradient; m ± T m N = m T Attempt at equation of normal using x, y =, ( ) ( ) Normal has equation y ( x ) M Condone one error = + A 4 CSO; ACF (c) x y t = or t = x y = + M Correct expression for t in terms of x or y A ACF Total 9 4
MPC4 - AQA GCE Mark Scheme 00 June series (a)(i) 7x = A( x ) + B( x+ ) M x= x= Substitute two values of x and solve for A m and B 7= A+ B A = B = A Or solve } condone one error = A+ B (ii) 7x = ( x+ )( x ) pln( x ) qln( x ) ( ) ( ) + + M Condone missing brackets = ln x + + ln x ( + c) AF F on A and B; constant not required (b) + + + + 6x x 6x x 4x = x x+ x x+ Alternatives: 4x x x + = + M B A Total 8 P = Q= 4 and R= (a)(i) By cover up rule 7 x= A= 5 4 x= B = x = and x = 5 (M) and attempt to find A and B A = B = (A,A) () SC NMS A and B both correct / One of A or B correct / (b) x x+ 6 x + x+ 6x x+ 4x or 6x + x+ = P x x+ + Qx+ R ( ) (M) (B) (A) () Complete division, with ax + b remainder P = stated Q = 4 and R = stated or written as expression Multiply across and equate coefficients or = Px + ( Q P) x+ P+ R (M) use numerical values of x P = (B) P = stated Q P= P+ R= Q= 4 and R= Q = 4 and R = stated or written as (A) () expression 5
MPC4 - AQA GCE Mark Scheme 00 June series 4(a)(i) ( ) + x = + x+ kx M = + x + x A (ii) ( x) 8 9 6 + 9 = 6 + x B 6 9 9 = k( + x+ ( x) ) M x replaced by 9 x or start binomial again 6 8 6 6 Condone missing brackets 4 = 64 + 54x + x A Accept 7.5975x (b) x = M Use x = 46 + 7 A 46 seen with a = 7 b =, or Alternative: Total 7 k 7 k (a)(ii) ( 6 9x) + = 6 6 9 6 (9 ) + x + x (M) Use ( a+ bx) n from FB. Allow one error. Condone missing brackets. 4 = 64 + 54x + x (A) () Accept 7.5975x 5(a)(i) cos x = sin x B ACF in terms of sin (PI later) ( sin x) + sin x+ = 0 Substitute candidate s cosx in terms of M sin x (at least terms) 6sin x+ sin x+ 4 = 0 sin x sin x = 0 A AG Factorise correctly or use formula sin + sin = 0 M correctly sin x = sin x = A Both; condone 0.67 or 0.66 or better (ii) ( x )( x ) (b)(i) R = B Accept.6 or better tan α = α =.7 MA OE; accept α =.69(0) (ii) x α = cos R M Candidate s R. Or cos( x α) = x α = 06., 5.9 x = 69.9, 4.8 A A Total One correct answer Both correct, no extras in range R 6
MPC4 - AQA GCE Mark Scheme 00 June series 6(a) x + cos π = 7 x = 7 M Or x = 7 cosπ x = A CSO d (b) ( ) d xy = xy + x M A cos πy = πsin πy B ( ) ( ) d d At (, ) 4 y y 8 πsin π 0 M = A 5 CSO; OE Total 7 terms added, one with d x Substitute candidate s x from (a) and y = with 0 on RHS and both derivatives attempted and no extra derivatives 7
MPC4 - AQA GCE Mark Scheme 00 June series 7(a) OB = B PI 4 AB = = M Use ± ( OB OA) A 0 (b)(i) 4+ λ = + μ 4+ λ + μ = μ = μ M λ 4 μ + + λ = 4 μ or set up equations 6 = μ μ = m Solve for λ and μ λ = 4 λ = A Both correct 4+ λ = 4 = + μ = + = A 4 (ii) P is (,, ) B Independent check with conclusion: minimum intersect (c) OC = OA + AC Or = OA + PB 4 OC = + C is (,, ) A OC = OB + BC = OB + PA M OA + PB in components or OC = OB + BC = OB + AP 4 OC = + ( ) is,, C A 4 Total M OB + AP in components 8
MPC4 - AQA GCE Mark Scheme 00 June series Alternative: 7(c) AP= BC AP = BC = ( 4) + ( ) + ( ) = 5 (M) BC = k 0 BC = k 5 so k =± (A*) For k = and k= OC = OB + k 0 = + 0 or 0 = or *If k = or k = (ie only one k), one correct point gets /4 (M) Either (A) (4) Both 9
MPC4 - AQA GCE Mark Scheme 00 June series 8(a) dt = d t Correct separation; or = 5( x + ) B x + 5 Condone missing integral signs 5 ( ) x + = t + C BB Correct integrals; condone x = 80 t = 0 C = 8 M Use ( ) x= 9 t 0 0, 80 to find a constant C x + = 8 AF F on integrals if in form x + = qt + c A 6 OE; CSO; x = correct expression in t (b) t = 60 x= f ( 60) M Evaluate ( ) (c)(i) ( 9 ) = 8 A CSO f 60, ie x = (C not required) d A da ka A dt = M = product involving A; k required dt Condone terms in t A 9 4.5 = M Condone one slip in denominator + 4e t (ii) 0.09 e 0.09t 4 = A ( ) 0.09t = ln m Take ln correctly ( 4 ) ln t = 0.09 4 = 5.4 ( hours) A 4 Total 4 TOTAL 75 CAO; condone more than sf if correct 5.407068 Allow 5h 4m 0