ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES

Transcription

1 GRADE EXAMINATION NOVEMBER 009 ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Time: hours 00 marks These marking guidelines were used as the basis for the official IEB marking session. They were prepared for use by examiners and sub-examiners, all of whom were required to attend a rigorous standardisation meeting to ensure that the guidelines were consistently and fairly interpreted and applied in the marking of candidates' scripts. At standardisation meetings, decisions are taken regarding the allocation of marks in the interests of fairness to all candidates in the context of an entirely summative assessment. 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, and different interpretations of the application thereof. Hence, the specific mark allocations have been omitted. Please note that learners who provided alternate correct responses to those given in the marking guidelines will have been given full credit. IEB Copyright 009 PLEASE TURN OVER

2 GRADE EXAMINATION: ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Page of 9 MODULE CALCULUS AND ALGEBRA QUESTION Let n : 9 9 is divisible by Statement is true for n Assume statement true for n k : k k k is divisible by (or p, p0z) Let n k : k k k. k. k. k ( k k k ) which is divisible by. p If the statement is true for n k, it is also true for n k. Statement true for n0n marks IEB Copyright 009

3 GRADE EXAMINATION: ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Page of 9 QUESTION y. (a) x e x y e y l n x y l n x f ( x) l n x (). (b) x int : l n x form 6, f x e x e 6, y int : ln (9). (a) Squaring: x iy a abi i b a abi b x a b and y ab (5). (b) x iy 5i x ( ) 5 9 and y (5)( ) 0 (). A B x ( - x) ( x) x x (Any method) A ( x) B ( x) x : A A x : B B ( x ) ( x) or x ( x )( x ) A B x x ( x ) B( x ) A x A A x B B (9) ( x ) ( x ) marks IEB Copyright 009 PLEASE TURN OVER

4 GRADE EXAMINATION: ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Page of 9 QUESTION. ( x i) ( x i) factor ( x ) (i x ) x i x x x x 5 () ( x x 5) (or any other method.) x i or x i () 5 marks QUESTION. ( ; 6) j(x) h(x) (6). x x x 6 x 5x 6 0 ( x 6) ( x ) 0 x y 7 ( ; 7) (7) marks IEB Copyright 009

5 GRADE EXAMINATION: ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Page 5 of 9 QUESTION 5 lim h( x) 5. x lim ( x x ) x lim h ( x) x lim x k x ( ) ( ) k ( ) k 5 (6) lim h ( x) 5. x lim ( 6x x) x ( ) ( ) 6 lim h ( x) lim ( ) x x lim h ( x) lim h ( x) x x Not diff.b. (8) marks QUESTION 6 sin Max if (Dist) 0 cos x (x ) Dist : x ( x ) 6 cos x x (6) 6. Let D (x) x cos x D (x) 8 sin x xn cos x xn xn 8sin xn x 0 0,5 x 0,5550 x 0,6 x 0,66 x 0,66 x 0,6 n (9) 5 marks IEB Copyright 009 PLEASE TURN OVER

6 GRADE EXAMINATION: ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Page 6 of 9 QUESTION 7 f ( x) ( x ) ( x ) ( x 7) ( x ) (6) 6 marks QUESTION 8 8. ( x) cos ( tan (x)).sec (x). f () g ( x) x ( x x) x x OR 5 g ( x) x x 5 g ( x) x x () cos x xsin x. h ( x) () (cos x) 8. f ( π ) g (), 58 or h π g () ; f ( π ) ; h () π marks IEB Copyright 009

7 GRADE EXAMINATION: ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Page 7 of 9 QUESTION 9 dy dy 9. x y x. 0 0 dx dx dy dy. x y x dx dx dy ( x) y x dx dy y x (9) dx x dy 9. 0 dx y - x 0 y x Subst: (y) (y). y y 8 0 y 8y y 8 0 y y 8 0 y y 0 (y ) (y ) 0 y or y x or x (;) or ( ; ) (9) 8 marks QUESTION 0 0. Damon used fg fg and chose f g g '( x) x and f ( x) cos x () 0. Choose g ( x) cos x and f ( x) x () sin x sin x 0. x cos x dx x.. dx x sin x cos x C (6) marks IEB Copyright 009 PLEASE TURN OVER

8 GRADE EXAMINATION: ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Page 8 of 9 QUESTION ( 5x). (a) ( 5x ) dx C. 5 (5) cos9x 9. (b) ( sin 9x sin ( x) ) dx cos( x). (a) LHS ( sin x)( sin x) sin x sin x sin cos x x C (6) sec x R.H.S. (6). (b) 0 a sec tan x x dx ] a 0 tan a tan 0 tan a a π 6 (7) marks IEB Copyright 009

9 GRADE EXAMINATION: ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Page 9 of 9 QUESTION. 0 m x x dx ( x ) m m 0 m x 0 OR Let u x dy x dx du x dx and if x m, u m m 0 U U du m 0 m (9). Area. 9 () b a. V π ( f ( x) ) dx m π x o x dx () 5 marks IEB Copyright 009 PLEASE TURN OVER

10 GRADE EXAMINATION: ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Page 0 of 9 QUESTION. f ( a) h ( a) g ( a) g ( a) g ( a) () f ' ( a). g ( a) f. h' ( a) ( g ( a)) ( a). g' ( a)..6 6 ( ) (7) 9 0 marks Total for Module : 00 marks IEB Copyright 009

11 GRADE EXAMINATION: ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Page of 9 MODULE STATISTICS QUESTION. Sample mean x 69,5% 0, 695 Population mean μ 70 % 0, 7 Standard deviation σ 0, 0 H 0 : μ 70 H : μ 70 i.e. two sided test 0,695 0,7 z, 0,0 8 Two sided critical value ±, 96 Accept alternative hypothesis i.e. reject claim 0,0. 99% CI: 0,695 ±,576 8 ( 0,695 ± 0,009) ) (68,%: 70,6%) (0) (8) 8 marks QUESTION Using a calculator the following should be determined A,676 B 0,9806 r 0,88. (a) r 0,88 (6) (b) relatively strong positive (). (a) F,676 0,9806I (5) (b) Prashail,676 0, ,676 () Shosho,676 0, ,08 () (c) Prashail is interpolation whilst Shosho is extrapolation () Prashail more reliable (). (a) Use calculators i 86, 7 and σ 5, 57 () (b) Student 7 () (c) If it is removed the strength of the linear relationship changes quite dramatically (from r0.88 to r0.67) and gets weaker. y 0,9806(70), which is very close to the 56 given. (5) marks IEB Copyright 009 PLEASE TURN OVER

12 GRADE EXAMINATION: ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Page of 9 QUESTION. (a) Defective balls 0, () (b) Let X the number of defective balls P ( x 0) 0, P ( x ) 0, ,87 0,77 Thus a 8,% chance of being accepted (0). X has a binomial distribution X Bin (0;0,5) 0 6 P ( x 6) C6 0,5 ( 0,5) 0,057 Or,5 % chance of needing 6 bats to be replaced (0). X has a normal distribution X N (80; 5) P ( x > 90) P z > P ( z >,58) 5 0,5 0,9506 0,009 thus 0,9 % chance of an arm length of more than 90 cm. (8) 0 marks IEB Copyright 009

13 GRADE EXAMINATION: ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Page of 9 QUESTION P (A B). P (B A) P (A) P (A) P (A B) (5) P (A B). P (A B) P (B) P (B) P (A B) () 5. P (A) P (B) 5 P (B) 5 P (A) 8 P (A) 5 P (A) 0 8 P (A) P (A) (9) 5. P (A B) P (A) 0, () 5 0 marks Total for Module : 00 marks IEB Copyright 009 PLEASE TURN OVER

14 GRADE EXAMINATION: ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Page of 9 MODULE FINANCE AND MODELLING QUESTION. ( i ) (,)(,087)(,065)(,08) i. 906 i 8,59% (7) 5 ( i) 0,. (a) x where i i x R069,7 (7) (b) Balance outstanding (loaninterest)-(paymentsinterest) ( i) 5 000( i) 069,7 i 6889,7 8 6,98 R 78 65,76 Or Balance outstanding present value of outstanding payments 5 ( i) 069,7 i R 78 65,90 (8) marks QUESTION. n i x R ( i) ) FV (7) i 8 ( i). FV 000 i 5 FV ( i) R 09 (0). A 0, 9 97 R 65 99, 0 () IEB Copyright 009

15 GRADE EXAMINATION: ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Page 5 of 9. PV ,0 R85 979, 70 0,08 i n ( i) ,70 x i x R 6 59,5 (8) 6 marks QUESTION. 0,09 k T k T k 000, 0 (8). 0,09 T ,0 50 0,09 T ,0 8,6 0,09 T 8,6 000,0 506, (5) marks IEB Copyright 009 PLEASE TURN OVER

16 GRADE EXAMINATION: ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Page 6 of 9 QUESTION. a 0, 5 intrinsic growth rate of krill p 500 carrying capacity of krill (5). c is the death rate of whales dies every year thus c 0, (). Krill decreases rapidly to this point as its population is above the carrying capacity. At A it reaches sustainable population but there are some whales thus population continues to decrease but very slowly. (). ± 00 tons/ acre ().5 Stable population's ± 00 tons krill/ acre ± whales () K n.6 K n K n 0, K n bk nwn 500 W W fbw K cw n W n n n 7 ( 500, 0 0,0) 0 n K n n K n 9,0 tons/ acre K 7 And 0, n K n, 0 K nwn W 89 9,5 whales () n 7 marks QUESTION rabbits () 5. Population A more rapid growth () 5. From 0 to 5 months population doubles, i.e. from 0 to 0 rabbits ( i ) 5 i,87% 5% per month (7) 5. Grows to rapidly so overshoots stable population therefore starts to decline again. Keeps oscillating until population stabilises. () marks Total for Module : 00 marks IEB Copyright 009

17 GRADE EXAMINATION: ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Page 7 of 9 MODULE GRAPH THEORY AND MATRICES QUESTION. Fifth degree rotational symmetry cos7 sin 7,6 sin 7 cos7 0,80 (6). (a) Stretch by factor,5 parallel to x-axis, invariant y-axis () (b), Therefore the new coordinates are (0 ; 0) (). No. Petals will not all be same shape.,5 0 0 cos7 sin 7 sin 7 cos7 cos7 sin 7 sin 7 cos7,5 0 0 () 5 marks QUESTION,5 0. Stretch parallel to x-axis 0 0. Reflection about x-axis 0. Need to reflect then multiply by inverse of stretch. 0, 0 0 0, () () (8) marks IEB Copyright 009 PLEASE TURN OVER

18 GRADE EXAMINATION: ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Page 8 of 9 QUESTION. det P a a a a a ( 8 a ) a ( a 9a) ( a ) 7a a But det P 0 7a a 0 ( 7 a )( a ) 0 a since a is an integer. (0) 0 marks QUESTION z 7 And y z 9 y And x y z 0 x (8). (a) A. adja. adja det A ( 7) ( ) 8 Cofactors matrix ( ) A 7 (7) 8 7 (b) X A. Y () 7 marks IEB Copyright 009

19 GRADE EXAMINATION: ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Page 9 of 9 QUESTION 5 5. AHCGF ABDEF ABCGF < x 7 < x < 5 thus maximum value of x (0) 5. (a) H; C; D and G () (b) Pairs Weight HC and DG HG and DC HD and CG 5 9 Need to duplicate HC and GE and DE (7) 0 marks QUESTION 6 6. Kruskal's: remove JHB Select shortest edges to form MST (NB no circuits) Select: PE EL : 650 EL DBN : 950 CT PE : 00 CT BLM : 00 DBN NEL : 00 BLM WIND : 700 Then include edges: JHB BLM : 700 JHB DBN : 800 Lower bound R (9) 6. (a) No. Same edges selected () (b) Yes. The lower bound would be R 8850 () 6. JHB BLM : 700 BLM CT : 00 CT PE : 00 PE EL : 650 EL DBN : 950 DBN NEL : 00 NEL WIND : 600 Back to JHB : 00 TOTAL: R 00 (9) IEB Copyright 009 marks Total for Module : 00 marks Total: 00 marks