ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES

Transcription

1 GRADE EXAMINATION NOVEMBER 008 ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Time: hours 00 marks These marking guidelines are prepared for use by eaminers and sub-eaminers, all of whom are required to attend a standardisation meeting to ensure that the guidelines are consistently interpreted and applied in the marking of candidates' scripts. The IEB will not enter into any discussions or correspondence about any marking guidelines. It is acknowledged that there may be different views about some matters of emphasis or detail in the guidelines. It is also recognised that, without the benefit of attendance at a standardisation meeting, there may be different interpretations of the application of the marking guidelines. IEB Copyright 008 PLEASE TURN OVER

2 GRADE EXAMINATION: ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Page of 8 MODULE CALCULUS AND ALGEBRA QUESTION. (a) ( ) ( ) + + () (b) (5) or ± 4 + i or. ( )( + ) or ( ) ( ) is a factor. By inspection: i ( ) ( ) ( )( 5 i )( + 5i ) 6 marks (0) QUESTION. (a) log log( 0) log (4) (b) e 5e 5 4 e 5 e,5 ln,5 0,... (4) (c) Let k ( k 6 )( k + ) 0 6 (6) ± 6 IEB Copyright 008

3 GRADE EXAMINATION: ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Page of 8 L. 07 0log ,7 log L log0 5, log L 5, L 0 5,0 0-6 (6) 0 marks QUESTION Prove:. + + k y k is divisible by y k k. y. y ( + from Step k k ( p y) y ). y. y ( + k k ( p y) y ). y. y + k p( y) y ( y) + k ( y) ( p y ) So we have shown that provided that k k y is divisible by y then so is + + k y k. But since the statement is true for n, then by the argument above it is true for n, and hence n and so on for all natural values of n. marks IEB Copyright 008 PLEASE TURN OVER

4 GRADE EXAMINATION: ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Page 4 of 8 QUESTION 4 e ( e ) 4. (a) f ( ) e f ( ) e () (b) y O (4) 4. At Q, log ( + ) At P, log ( + ) (6) marks QUESTION 5 5. a() 4() + 5 a (5) 5. f ( ) if if < lim h 0 f '( ) 4 lim f '( ) 4() 4 + h 0 method Therefore differentiable at. (9) 4 marks IEB Copyright 008

5 GRADE EXAMINATION: ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Page 5 of 8 QUESTION g( ) 6. ( )( ) or ( )( ) 0 ( )( ) or or () 6. ( + )( ) Vertical: and lim 5 + Horizontal: y (7) 6. g'( ) ( )( 4 5) ( 5+ )( 4 ) ( ) (8) ( ) 6.4 Δ 44 4(8)(7) Therefore no local ma or min turning point. () 6.5 y y (7) 7 marks IEB Copyright 008 PLEASE TURN OVER

6 GRADE EXAMINATION: ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Page 6 of 8 QUESTION f (6) 7. '( ) ( ) 7. cos( a ).cos + sin ( sin( a )( ) ) cos( a ) cos + sin( a ) sin cos( a ) cos( a ) (7) marks QUESTION 8 dy d 8. t dt dt dy t t d (4) 8. t t + 0 () y 9 (0;9) (5) 9 marks QUESTION 9 9. solutions () 9. 0,5 () r+ r ,607 (6 dp) (0) marks IEB Copyright 008

7 GRADE EXAMINATION: ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Page 7 of 8 QUESTION 0 0. y' a + b + c y'' 6a + b 0 6a() + b (5) b 6a 0. a(0) + b(0) + c c Passes through (0;0) d 0 Passes through (; -) 8a + 4b 6 b a 4 6a a 4 a b 6 y 6 (9) 4 marks QUESTION. π P () 6,8 cm. A Δ ABC + segments π π sin + sin,8 cm or or A Sector + segments π + π sin π A sectors triangles π sin π (8) marks IEB Copyright 008 PLEASE TURN OVER

8 GRADE EXAMINATION: ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Page 8 of 8 QUESTION. ( ) d or Let u ( ) d ( ) 8, 0 du d d du units d 5. (a) 0, 4 5,4 h ln,4 Area 0,4895,8 h ln,8 Area 0,540 4, h ln 4, Area 0, ,6 h4 ln 4,6 Area4 0,604 5 h ln 5 Area 0, TOTAL 8 8 du u u 8,0 (0),85 units (0) ln 5.ln 5 5 ln +,75units (6) (b) [ ] 5 QUESTION V ( 4 4 ) d π ( )d π V π ( ) d 6 marks 4 π 4 π π [ ] 4 4 π π + )] π[( ) ( + 4π units 4π units marks IEB Copyright 008

9 GRADE EXAMINATION: ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Page 9 of 8 MODULE STATISTICS QUESTION. H H 0 8 (4) 8. Testing for change therefore involves two rejection regions. 5% level of significance means each region is,5%. The critical value that corresponds is,96. μ Z σ n 45 8 Z 850,5 40 Z, ,96 This value of Z falls within the acceptance region, therefore not enough evidence to reject H 0. Accept H0 (7). 8,96 850,5 40 8, 6g (4) 5 marks IEB Copyright 008 PLEASE TURN OVER

10 GRADE EXAMINATION: ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Page 0 of 8 QUESTION. As increases y decreases. (). (a) r 0, 98 (4) (b) very strong correlation (). b n y ( ) n y n (9,88) 5(78,5) b (89490) (5) b 0,007 y y b( ) Now for a: y 6,54 0,007( 4,58) (0) y 0, ,85.4 (a) y 6,85 0,007(95) y 6,7 () (b) unreliable, outside of range, etrapolation () marks IEB Copyright 008

11 GRADE EXAMINATION: ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Page of 8 QUESTION 0. Proportion not in favour Proportion in favour % confidence interval z,645 p( p) p ±,645 n ±, ,6099; 0,674 (0) [ ] 05,89 + 0,. (a) 08 () (b) 05,89 08,96 σ 700 (6) σ 8, (0,55) (0,45) (0,45) 0 0,995 (9) 7 marks IEB Copyright 008 PLEASE TURN OVER

12 GRADE EXAMINATION: ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Page of 8 QUESTION ! 7!! 040 () 4. (a) 0, 0,5 5 () (b) 0,4 4 0,5 5 () (c) P (A). P (B) 0,4 0,5 0, and P (A B) 0, not equal not independent () 4. (a) 4 8 P( X ) for 0,,, 0 elsewhere formula (6) 4 8 (b) 55 (5) (a) ( 0, 0,0) d ( 4 ) ,48 Probability delay would be between and 6 hours is 0,48 (8) (b) m ,5 m m 0, m 400m (9) m,9 or m 7,07( N / A) 6 marks IEB Copyright 008

13 GRADE EXAMINATION: ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Page of 8 MODULE FINANCE AND MODELLING QUESTION. P 0, 0,4 0, , i i 6,% 5 ( ) ( ) P( i) 7 ( i) 7 (6). A Paying more than the required amount in order to pay back more quickly. B Cash flow problem - Paying only sufficient to cover interest. C Interest rates rise. Paying less than the required amount. D Gets promotion, bigger salary. Able to pay more per month and finish the loan in the right time. (8) 4 marks QUESTION. 0, , ,095 n n 7,5 months So 76 months in total years (0). Amount remaining 0, , ,095 R ,5 R 5 89, ,6 () marks IEB Copyright 008 PLEASE TURN OVER

14 GRADE EXAMINATION: ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Page 4 of 8 QUESTION ,6 0,6 0, R6860, 4 (8). 58 0, ,4 0,6 R890,08 (9) 58 7 marks QUESTION ( 6) q q 5 ( 6) p 0 5 p 0 (8) 4. n 0,4 Tn+ 00, + + Tn m (7) 5 marks IEB Copyright 008

15 GRADE EXAMINATION: ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Page 5 of 8 QUESTION 5 5. (a) Rabbits 0, Foes 5 () (b) No. The number of rabbits and foes are tending to a limit. (4) (c) Rabbits 55. Foes 8 () (d) 0 < r < 55 4 < f < 8 (4) 5. (a) In the pred-prey model one must include the influence of the contact between rabbits and foes. The term represents the number of rabbit deaths and is a function of the product of the number of rabbits and foes. () (b) R c fb 0,048 0, 0, a c F b 0,64 0, ,048 0, 0, (8) marks QUESTION 6 6. A Logistic model since there is a limit to the number of ants over time. () 6. (a) 500 () (b) r 500 0,5 9r 0 r 5 (5) 9 9 marks IEB Copyright 008 PLEASE TURN OVER

16 GRADE EXAMINATION: ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Page 6 of 8 MODULE 4 MATRICES AND GRAPHS QUESTION 0. (a) Enlargement scale factor (4) 0 0 (b) Shear factor, invariant y-ais: (5). (a) Rotation 90 0 and stretch factor, parallel to -ais. (5) (b) (6) cos60 sin 60 sin 60 cos (6) 6 marks QUESTION cofactors : adj inverse , marks (0) (6) IEB Copyright 008

17 GRADE EXAMINATION: ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Page 7 of 8 QUESTION () Two planes parallel. () det, so unique solution () det 0, planes not parallel, so this is the one. 0 marks QUESTION , since 9 edges () 4. Both the vertices of order five must go to all other vertices hence the minimum order of any other verte must be () 4. One with order 5 four with an order of one with an order of 0 marks (5) QUESTION 5 5. B, D, H, I () 5. (Chinese Postman Problem) Recognises the need to travel a route twice (Routes B E D and I H repeated is not a minimum route) Routes B E I and D G H are repeated A solution that starts and finishes at A Every edge has been travelled at least once Solution for the shortest inspection route: e.g. A B C D E B E G D G F H G H I E I G C A (4) 5. Summing all the edges + B E I repeat + D G H repeat 5,4 + 5,6 + 4, 6, km (4) 5.4 Yes BD would remove two of the vertices with odd order and HI,4 Total travelled would be shorter than before at 5,4 + 6,4 +,4 6, km (6) 6 marks IEB Copyright 008 PLEASE TURN OVER

18 GRADE EXAMINATION: ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Page 8 of 8 QUESTION 6 6. Students must demonstrate that they have used Prim's algorithm, rather than just an intuitive approach or Kruskal's algorithm students lose 50% of the mark if they do not demonstrate use of the correct algorithm. E Office (488 m). E E4 (587 m). E4 E (5 m) 4. Office E (54 m) (0) m () marks Total: 00 marks IEB Copyright 008