ADVANCED General Certificate of Education Mathematics Assessment Unit F3. assessing. Module FP3: Further Pure Mathematics 3 [AMF31]

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1 ADVACED General Certificate of Education 6 Mathematics Assessment Unit F assessing Module F: Further ure Mathematics [AMF] MDAY 6 UE, AFTER MAR SCHEME 987. F

2 GCE ADVACED/ADVACED SUBSIDIARY (AS) MATHEMATICS Introduction The mark scheme normally provides the most popular solution to each question. ther solutions given by candidates are evaluated and credit given as appropriate; these alternative methods are not usually illustrated in the published mark scheme. The marks awarded for each question are shown in the righthand column and they are prefied by the letters M, W and MW as appropriate. The key to the mark scheme is given below: M W indicates marks for correct method. indicates marks for working. MW indicates marks for combined method and working. The solution to a question gains marks for correct method and marks for an accurate working based on this method. Where the method is not correct no marks can be given. A later part of a question may require a candidate to use an answer obtained from an earlier part of the same question. A candidate who gets the wrong answer to the earlier part and goes on to the later part is naturally unaware that the wrong data is being used and is actually undertaking the solution of a parallel problem from the point at which the error occurred. If such a candidate continues to apply correct method, then the candidate s individual working must be followed through from the error. If no further errors are made, then the candidate is penalised only for the initial error. Solutions containing two or more working or transcription errors are treated in the same way. This process is usually referred to as followthrough marking and allows a candidate to gain credit for that part of a solution which follows a working or transcription error. ositive marking: It is our intention to reward candidates for any demonstration of relevant knowledge, skills or understanding. For this reason we adopt a policy of following through their answers, that is, having penalised a candidate for an error, we mark the succeeding parts of the question using the candidate s value or answers and award marks accordingly. Some common eamples of this occur in the following cases: (a) a numerical error in one entry in a table of values might lead to several answers being incorrect, but these might not be essentially separate errors; (b) readings taken from candidates inaccurate graphs may not agree with the answers epected but might be consistent with the graphs drawn. When the candidate misreads a question in such a way as to make the question easier only a proportion of the marks will be available (based on the professional judgement of the eamining team) F [Turn over

3 Q i j k 5 + QT 6 5 QT. Q. 5 M W AVAIABE MARS Volume of prism units W 5 (i) AB b a AC c a n AB AC (b a) c a) M b c a c b a + a a but a a n a b + b c+ c a M using anticommutativity (ii) Subs a into equation of plane a. (a b + b c + c a) d but a. a b a. c a d a. b c (iii) n a b + b c + c a M W M i j + j ( k) + ( k) i 6k i j W a. b c i. ( i) 6 plane is r F

4 R V (a) (i) y sin tan S W T X dy cos tan ( ) < F ^ h^ h M + cos< tan F ^ h ( ) R V cos tan S W W W T X AVAIABE MARS (b) I : `_ i _ ij M sin + c M sin ( ) + c W F [Turn over

5 (i) I n n cos u n dv cos du n n v sin M AVAIABE MARS I n n sin n n sin W u n du (n ) n dv sin v cos I n n sin n n cos + (n ) n cos n sin + n n cos n (n ) I n W r (ii) V r r 6 I cos I sin + cos I I sin + cos I I sin + c M W V r 6 sin + cos sin cos + R V r r r S + 6 W T X 5 r r + r 6 W r 987. F 55

6 u u 5 (i) HS 7 ^e e ha M AVAIABE MARS u u ^e + e h W u u ^e + e h cosh u RHS (ii) sinh v + + sinh v cosh v cosh vdv integral 8 sinh v cosh v dv W 6 sinh v dv (cosh v ) dv 6 ( 8 sinh v v) + c 56 sinh( sinh ) 6 sinh () + c 987. F 66 [Turn over

7 6 (i) y sech() y AVAIABE MARS y sech () y MW (ii) sech y (cosh y) (cosh y) dy sinh y d dy sech y sech y M M W W 987. F 77

8 (iii) I sech () AVAIABE MARS dv v u sech du I sech + M sech + sin (+ c) Area of blade [ sech + sin ] r a r # + k d sech + n M r sech 6 W (iv) y sech y cosh cosh a b l ln 7 + a A M cosh ln Area r 6 ln W F 88 [Turn over

9 987. F 99 AVAIABE MARS 7 (i) Step Find plane STWV ormal to lines SV and VW with directions and 7 7 n 7 k i j 7 lane STWV r M W Step Find line TW Intersection of planes STWV and RTWU y + z 8 y z 96 z 8 z 7 Also y 56 y + 56 y + y z 7 + η or 7 + η M W

10 (ii) Step Find T as intersection of TW and RT i λ h j 9 + 7λ + h k 7λ 7 λ h T (,, 7) W AVAIABE MARS Step Find angle 8 T direction vector of T is normal to RTWU is cos θ M cos θ θ.8 7 α θ ( s.f.) W T θ α Total F [Turn over

ADVANCED SUBSIDIARY (AS) General Certificate of Education January Mathematics Assessment Unit M1. assessing. Module M1: Mechanics 1 [AMM11]

ADVANCED SUBSIDIARY (AS) General Certificate of Education January 0 Mathematics Assessment Unit assessing Module : Mechanics [AM] FRIDAY 0 JANUARY, AFTERNOON MARK SCHEME 7.0 GCE ADVANCED/ADVANCED SUBSIDIARY