Version 1.0. General Certificate of Education (A-level) June Mathematics MM1B. (Specification 6360) Mechanics 1B. Final.

Transcription

1 Version.0 General Certificate of Education (A-level) June 0 Mathematics MB (Specification 6360) Mechanics B Final Mark Scheme

2 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 events which all examiners participate in and is the scheme which was used by them in this examination. The standardisation process 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 standardisation each examiner analyses a number of candidates scripts: alternative answers not already covered by the mark scheme are discussed and legislated for. If, after the standardisation process, examiners encounter unusual answers which have not been raised 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 from: aqa.org.uk Copyright 0 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 ) and a registered charity (registered charity number ). Registered address: AQA, Devas Street, Manchester 5 6EX.

3 Key to mark scheme abbreviations M mark is for method m or dm mark is dependent on one or more M marks and is for method 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 or ft or F follow through from previous incorrect result CAO correct answer only CSO correct solution only AWFW anything which falls within AWRT anything which rounds to ACF any correct form AG answer given SC special case OE or equivalent A, or (or 0) accuracy marks x EE deduct x marks for each error NMS no method shown PI possibly implied SCA substantially correct approach c candidate sf significant figure(s) 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. 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.

4 MB (a)(i) 0.6 = 0 + a 0.9 A : Correct use of constant acceleration 0.6 equation with u = 0 to find a. a = = 0. ms AG A 3 A: Correct equation..8 A: Correct a but some intermediate working must be seen. Note that 0 = a 0.9 Scores M0A0A0 Verification methods require a conclusion for full marks to be awarded. Condone seeing just the second line of working. (a)(ii) 0.9 = ( ) t 0.9 t = = 3 seconds = t 0.6 t = = 3 seconds = 0.t t = 3 seconds A () () : Correct use of constant acceleration equation with u = 0 (and a = 0. if needed) to find t. A: Correct time. Note: Do not penalise 0.9 = ( ) t in the first method. Note: 0= t scores M0A0 in the second method. (b) T = T = = 8000 N A A 3 Total 8 : Three term equation of motion. Must have these three terms but can have incorrect signs. Must use a = 0. A: Correct equation with correct signs. (Allow 800g) A: Correct tension. Accept 8008 or 800 from use of g = 9.8.

5 MB (cont) (a) B: Diagram with four forces showing R or N or 4g or 39. or 39.4 arrow heads and labelled. Ignore negative signs in labels. 30 B Note: Award mark if forces drawn on the diagram in the question. F or µr or 0.3R Note: Do not accept 4kg for the weight. Note Accept μ R for F. mg or 4g or W or 39. or 39.4 (b) ( R = = ) 39. N B B: Correct normal reaction. Accept 4g (c) ( F = ) =.76 =.8 N (to 3sf) A : Use of ( F = ) μr A: Correct friction. Accept.g or.7 or.76 N. Do not condone further work after the value for friction has been obtained. (d) 4a = a = = 4.56 ms 4 AF AF 3 Total 7 : Three term equation of motion. AF: Correct equation. AF: Correct acceleration. FT candidates F from part (c). Accept 4.55 from.8.

6 MB (cont) 3(a) s = 3.5 = 400 m B B: Correct distance. (b) 600 = (3 + 8) t 600 t = = 64 seconds 5 d A 3 : Seeing 000 candidate s answer to part (a) calculated d: Use of constant acceleration equation(s) to find t, with u = 3 and v = 8 A: Correct time. Accept only 64 (c) v (ms ) t (s) B B BF 3 B: Shape of the graph. B: Correct velocities (ie 8 and 3) on vertical axis. BF: Correct times (ie.5 and 76.5) on the horizontal axis. (Follow through incorrect answers to part (b)). Award marks for graph if seen in earlier parts. (d) Average Speed 000 = = 6. ms AF Total 9 : Use of 000 over candidate s total time (not 64 or.5). AF: Correct speed. AWRT 6.. FT candidate s answer to part (b) or (c). 4 (a) 6(5i+ 8) j + m(i 5) j = 6(8) i + m( Vj) 6 5+ m = m = 48 A m = = 9 A 3 : Conservation of momentum, with addition of terms, as either 4 term vector equation (seen either in part (a) or part (b) three term equation for i component. Allow one error, for example switching masses. A: Correct equation for i components. A: Correct m. (b) = 9V = 9V V = = 7 9 AF AF 3 Total 6 : Conservation of momentum for j component with correct signs. Allow one error, for example switching masses. Note: omitting any mass scores M0. AF: Correct equation. Allow m instead of 9 at this stage. AF: Correct velocity. Condone 7j FT candidate s mass from part (a). Only award FT marks if mass positive. 08 Note V = 5 m

7 MB (cont) 5 (a) 5g T = 5a A : Three term equation of motion with T 3g = 3a A 5g or 49, 5a (not 5ga) and T. A: Correct equation. g = 8a : Three term equation of motion with g a = =.45 ms 3g or 9.4, 3a (not 3ga) and T. AG 8 A 5 A: Correct equation. A: Correct acceleration from correct working. Note: Do not penalise candidates who consistently use signs in the opposite direction throughout, provided they then give their final answer as.45. If the final answer is.45 don t award the final A mark. Special Case: Whole String Method g = 8a and g a = =.45 OE AA. 8 (b) T = = = 36.8 N (to 3 sf) A : Substitution of a =.45 into a three term equation of motion to find the tension. Contains T, mg and ma where m = 3 or 5 A: Correct tension. Accept or 36.7 (c) Light and Inextensible BB B: Light B: Inextensible (Allow inelastic or not stretchy) Ignore irrelevant non-contradictory assumptions. (d)(i) 0.96 =.45 t 0.96 t = = 0.4 seconds.45 (ii) v = v = 0.98 v = = 0.98 ms 0.96 = (0 + v) 0.4 v = 0.98 m s A A 3 A (A) () Total 4 : Use of constant acceleration equation with s = 0.96, u = 0 and a =.45 to find t. A: Correct equation. A: Correct t : Use of constant acceleration equation with s = 0.96, a =.45,u = 0 and candidate s time (as needed) to find v. A: Correct v.

8 MB (cont) 6 (a) 000 = V 4 : Equation for horizontal motion to V = 50 ms- A find V. Must not contain g. Could contain cos0 or equivalent. A: Correct V. (b) ( h = ) = 78.4 metres to 3sf (c) ( vy ) or ( vy ) ( v ) = = 39. ms = = 39. ms = = 53 ms A A da 4 : Vertical equation to find height with u = 0 and a = ±9.8. A: Correct height. Accept 78.4 : Calculation of vertical component of velocity with u = 0 and a = ±9.8. A: Correct vertical component. d: Calculation of speed. A: Correct speed. (d) tan α = or tanα = α = 8.9 AF A 3 : Using tan to find angle with opposite and adjacent sides. Can be inverted as shown in brackets. AF: Correct trig expression. A: Correct angle sin α = or sinα = α = 8.9 (AF) : Using sin to find angle with hypotenuse and one other side. Can be changed as shown in brackets. AF: Correct trig expression. A: Correct angle. 50 cos or cos 39. α = α = α = 8.9 ( ) (AF) : Using cos to find angle with hypotenuse and one other side. Can be changed as shown in brackets. AF: Correct trig expression. A: Correct angle. Accept 8.83 from this method. Note: Accept 8.98 from 53. Accept negative angles Total Note: FT value of V from (a) and speed from (c) if needed. Do not FT 39. from (c) in place of 53. Note: Accept energy methods if used correctly in part (c).

9 MB(cont) 7(a) v= (0.5i j) 0( = 0i+ 7.5j) A : Calculating velocity with u= 0i+ 0j and t = 0. v = =.5 ms da 4 A: Correct expression for velocity. d: Calculating speed. A: Correct speed. (b) tan θ = or or tan θ = or AF θ = 053 A cos θ = or or cosθ = (AF) θ = sin θ = or or sin θ = (AF) θ = : Using trig to find angle. Can be inverted as shown in brackets. AF: Correct trig expression with any correct equivalent fraction. A: Correct angle to the nearest degree. Accept 53. Note: For 37 award A0A0 But for = 53 award AA. For 7, award AA0 Note: 53. as final answer scores AA0 Condone finding angle from acceleration or position vector. (c) ( ) ( r = ) i+ j t ( = t i+ t j) ( t ) ( t ) 500 = t 500 = 4 = seconds A da A 5 : Finding an expression for position vector in terms of t. A: Correct position vector. d: Using distance to form an equation for t. A: Correct equation. A: Correct time. a = = 0.65t t = 40 (A) (da) : Finding magnitude of acceleration. A: Correct acceleration d: Using distance to form an equation for t. A: Correct equation. A: Correct time t = 600 t = 40 = t or 300 = 0.375t (A) (d) : Working with one component. A: Correct distance (300 or 400) A: Correct equation. d: Solving for t. A: Correct t. Total Note: 500.5=40 is not acceptable and scores 0

10 MB (cont) 8(a) Pcos80 Qcos80 = 50a Psin80 + Qsin80 = 50g A B : Horizontal equation of motion in the form Pcos80 ± Qcos80 = 50a 50a or Psin80 ± Qsin80 = 50a P Q= cos80 A: Correct horizontal equation. B: Correct vertical equation. 50g P+ Q= AG Note: the above marks could be awarded sin80 for a correct vector equation. 50a 50g P = + d: Solving for P with an attempt to d cos80 sin 80 eliminate Q. a g A: Correct result from correct working. P = 5 + A 5 Must see an expression for P or cos80 sin80 P sin80 cos80 (b) Pcos80 = 50a Psin80 = 50g a = tan80 g g a = =.73 tan80 d A 3 Total 8 TOTAL 75 : Using Q = 0 into correct original equation(s) or resolving without Q. d: Eliminating P A: Correct a. Note: use of P = ±Q scores M0dM0A0 Note: use of P = 0 can lead to ±.73 but scores M0dM0A0 unless fully justified by a symmetry argument.