ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES

Transcription

1 GRADE EXAMINATION NOVEMBER 0 ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Time: hours 00 marks These marking guidelines are prepared for use by examiners and sub-examiners, 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.

2 GRADE EXAMINATION: ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Page of 8 MODULE CALCULUS AND ALGEBRA QUESTION n RTP: 9 8n = 8 p n ; n Step : For n = true because 9 6 = 64 = 8 k Step : Assume true for n= k, i.e. 9 8k = 8q where q Step : for n= k+ ( k + ) LHS = 9 8( k + ) k = 9.9 8k 8 k k = 9 8k k = 8q + 89 ( ) k ( q q ) = 8 + Statement true n ; n by the principle of MI [4] QUESTION. logx = log + log ( x + 5) log ( x ) ( x + 5) logx = log x x x = x+ 0 ( ) = x x 0 = 0 ( x 5)( x+ ) = 0 x = 5 but x. (a) ( x) e is increasing f ( x) = x + e is decreasing () (b) The range of f( x ) is (0; ) (4) (c) The inverse function: x y + e y + e = x y e = x y = ln( ) x ( ) ln( x f x = ) or ln x x (4)

3 GRADE EXAMINATION: ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Page of 8. (a) (b) Domain of g( f ( x )) is ( ) (4) ; 0 [; ) (4) [5] QUESTION. ( )( ) + i a + bi + i = 8i + 6 ( ) ( ) a b + b+ 6+ a i = 6+ 8i 8i+ 6 i a b = 6 and b+ 6 + a= 8 a+ ib ( + ) = + i i a b= 9 solve simultaneously 6i 8i + 6i a+ b= = 4 i 5b = 5 and a = i b = = 5 a+ ib ( + ) = 4 + i a = 4 and b= (7). + = and x= + i x i = x 4x = x 4x+ x 5x kx 0 ( ) ( ) Thus ( x x )( x ) 4 + = 0 x x x = 0 Thus k = 7 (8) [5]

4 GRADE EXAMINATION: ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Page 4 of 8 QUESTION 4 4. (a) (-)For no labels of graphs h(x) g(x) (b) sinx 0 x but sin x since there exist values of x 0 where sinx< 0 () px if x < π π f x = cosx if π x 5π π x if x> π For f to be continuous at x = π we need pπ = cosπ Thus pπ = p = (4) π π π (ii) f = cos = 0 But π 5π lim f ( x) = π π = x π π f not continuous at x = (5) 4. (a) (i) ( ) (8)

5 GRADE EXAMINATION: ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Page 5 of 8 (b) (i) the gradient of f as x π is the gradient of f as x π + π sinπ = 0 is ( ) thus the gradient from left gradient from right not differentiable Alternatively: which is constant (straight line) Clearly see the gradient from right gradient from left at π not differentiable (5) π lim ' = sin π = x π 5π lim + f '( x) = π = x π (ii) f ( x ) (iii) No, as although they have the same gradient the function is discontinuous at π x =. () []

6 GRADE EXAMINATION: ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Page 6 of 8 QUESTION 5 5. (a) You can't take the log of a negative number. () (b) lnx = 0 0 x= e = () (c). lnx. x d x x lnx = = dx lnx lnx lnx ( ) ( ) Thus turning point when lnx = 0 which is when x = e which gives e y = = e lne Thus coordinates of TP are (;) ee (8) 5. (a) Oblique asymptote: x x ( x )( x ) Thus: x x = = x 4 + so the oblique asymptote is 4 x+ x+ y = x (b) 9 No. For them to touch x 4+ = x 4 x x + = which is not possible. () [] QUESTION 6 i Width = and xi = n n n i Approximate area = +. i= n n n 8i = + i= n n n n 8 = i + n i= n i= 8 n n n = n n 6 n = n + 6n area = lim n = n 6n [4]

7 GRADE EXAMINATION: ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Page 7 of 8 QUESTION 7 7. ( ) r = r r = 6 r =,75 cm 7. 5 sin θ =, 75 θ sin; cos or tan or cosine rule could = 0,76 be used. θ =,5 So the arclength ACB =,75,5 =,04 cm (5) 7. Area of segment ( ) ( ) = 0.5, 75,5 0.5, 75 sin,5 =, 7566 cm (5) [6] QUESTION 8 8. LHS: ( ) ( ).. + = 8 6 = = RHS () 8. dy dy + + = dx dx dy ( 6xy + y ) = x + y dx dy x y = dx xy y x y x. y. y 0 (8) 8. at ( ; ) ( ) dy 4 = = dx ()( ) ( ) 5 y = x+ c 5 =. + c 5 c = 5 y = x+ (5) 5 5 [5]

8 GRADE EXAMINATION: ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Page 8 of 8 QUESTION 9 9. x 4 x x sec dx = x tan x c + x 9. (cos x) ( sin ) x dx Let cosx= u du du Then: sin x. or sinxdx dx = = Thus: cos u du = u + c = x + c (7) 9 9 cos θ θdθ sin θ θ dθ cos9θ =..cosθ + c sin5 = 9 + sin ( ) 9.4 y y + dy du Let u = y + then = or du = dy dy Thus ( ) y y + dy = u. u du = u u du 5 5 = u u + c= ( y+ ) ( y+ ) + c 5 5 Let f( y) = y f '( y) = g'( y) = y+ g( y) = ( y+ ) fg ' dy = fg f ' gdy y = ( y + ) ( y + ) dy 5 y. = ( y + ) ( y + ) + c.5 5 y 4 = ( y + ) ( y + ) + c (9) 5 (- for no c ) [8]

9 GRADE EXAMINATION: ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Page 9 of 8 QUESTION 0 0. a a π Vol = π dx = π = + πunits x x a (8) Vol a = π units () 0. ( ) [0] QUESTION 8k S = + x k ( 5 x) ds = = 0 dx Thus kx k ( x) ( ) ( ) 8 5 x x = 0 ( ( 5 x) x) 4( 5 x) + x( 5 x) + x = 0 or ( ) 8 5 x = x ( ) 0 x= 0 or x= 0 km from A ( 5 x) = x x = 0 (Not necessary for the way question was stated: ds 4 4 Check for min: = 48kx + 6k ( 5 x) > 0 for x = 0 thus minimum.) dx [0] Total for Module : 00 marks

10 GRADE EXAMINATION: ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Page 0 of 8 MODULE STATISTICS QUESTION. X ~ bin (0 ;7) 0 7 P( x= 7) = ( 0, ) ( 0,8) = 0, (a) = (b) 6 6 = 5 (c) 6 5 = = 80! P at least one = P none.4 ( ) ( ) () () (4) (5) or (7) [6] QUESTION X ~ N ( 00;5 ) P < x< = P < z < 5 5 = P(, < z <, 9) = 0, 47 0, 408 = 0,065 6,5% (0) P X < a = or a 00,0 = 5 a = 69,5 a 70 less than 70% [6]. ( 0 9). ( ) 0,0

11 GRADE EXAMINATION: ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Page of 8 QUESTION. x = 57,8% y = 60,7% ().. ( ) ( )( ) n x ( x) ( ) ( )( ) 0( 878) ( 578) n xy x y b = = = 0,80 sub ( 57,8 ;60,7) ( ) or 60, 7 = a + 0,80 57,8 a = 4,44 y = 4,44 + 0,80x 0( 9070) ( 578)( 607) r = = 0,8546 () 0( 878) ( 578) 0( 4) ( 607).4 Allie, y = 4,44 + 0,80( 85) = 8,5 ().5 This is a reliable estimate as a strong correlation exists. () [6] QUESTION 4 4. (a) 00: x = 6 σ = 49, 45 0: y = 8, 4 σ = 96,8055 (4) (b) 4. Rejection Region: Reject H0 if Z >, 8 Test statistic: ( 8, 4 6 ) (0) ( 96,8055) ( 49, 45) z = = 0, Conclusion: We fail to reject H 0 at the 0% l.o.s and suggest insufficient evidence to support the claim. (0) (0, 5)(0, 75),88 < 0, 04 n (0, 5)(0, 75) < n 47 6n < 09 44, < n n = 45 [0]

12 GRADE EXAMINATION: ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Page of 8 QUESTION 5 5. P( FT ) = P( FT ') = 0,9 PT ( ) = 0,6 P( F T) P( FT ) = PT ( ) P( FT ') = 0,6 P( F T) ( T' ) PT ( ') P F = ( 0,9)( 0, 64) = P( F T ' ) 0,576 = P( F T ' ) 5. (a) P( F ) = 0, ,6 = 0,96 () (b) PF ( ' T') = 0,96 = 0,064 () [] (0) QUESTION Alicia: z = =, 8 Rae: z = = 0,89 8 () 6. x 60 Alicia:, = or x = 87,99 Rae: x 60 0,89 = or x = 49, 6. Standard deviation was lowered to reduce the wide spread of values allowing more consistency. () [9] Total for Module : 00 marks

13 GRADE EXAMINATION: ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Page of 8 MODULE FINANCE AND MODELLING QUESTION 0 0, = 7 860, , = 60,70 (4). 0 0,8 0 0, = x x = 45 0, , 0684 i + = + i = 0, , Fv = = 9 94,9 (0) 0, Using i = 0,069: x = 9 90,07 Using i = 0,06879: x = 9 94,9 Using i = 0,07: x = 9 954,68 [0] QUESTION ( 0,) ( 0,05) ( i) = ( 0,) 6 i = 0,0847 = 8,47% T 6 = A( 0,) 6 = 0,5 44A T = A( 0,) ( 0,05) = 0,69 088A 0,5 44A = 0,69 088A ( x) x = 8,47% (8) [8]

14 GRADE EXAMINATION: ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Page 4 of 8 QUESTION. 80 0, , 0856 OB = , 0856 = , , 95 = , , , 0856 OB = , 70 0, = (8). 6 0, , ,70 8,6 = 674 6,08 5 = 8,6 T 7 = , = 8 9,50 T 65 = 8 9,50 + 0,0856 = 7 074,60 T 80 = 7 074,60 + 0, , , , , , ,0856 = 674 6,08 (8)

15 GRADE EXAMINATION: ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Page 5 of , , , 0856 = 089 9, ,4 = 5,04 0, ,04 + = 557,4 59 0, , , , 0856 = 04 89,56 0 8, = 557,4 0, , 0856 = , = 40,77 40, , = 557, , , 0856 = + y 0, y = 557,4 (0) [6] 60

16 GRADE EXAMINATION: ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Page 6 of 8 QUESTION ,6 0,8 /7 = 5,67 4. H n + = 6,6H n () 4. H 0 = 5 H = 65 H = 089 () 5 r 4.4 ( + 5,6) = + 5 r =,9 (annual rate, compounded weekly) H = 5( +,9/5) = 5,95 0,95 hog per week ( + 5,6) = ( + r) 5 r = 0,0695 (weekly rate) H = 5( + 0,07) = 5,95 0,95 hog per week [6] QUESTION 5 5. (a) antelope: leopards: () (b) 40 < leopards < 5 () (c) antelope = () (d) 40 = b b = 0,00 (5) (e) f 0, = 4 f = 0,00988 (5) 5. (a) Both populations initially increase; formerly only leopards increase. () (b) Range of both populations larger in second plot. () (c) Equilibrium point reached over longer period of time in second plot. () [] QUESTION 6 6. T n + =, T n, T = 7 (4) 6. T 4 = 6,495 n 4 7 > n = 4 = 6, 495 (4) [8] Total for Module : 00 marks

17 GRADE EXAMINATION: ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Page 7 of 8 MODULE 4 MATRICES AND GRAPH THEY QUESTION. M N = = (4). not a square matrix. (). one row is a multiple of another determinant is zero. () (4) [] QUESTION. 0 7 = tan θ = θ = 7,565 o cos (7,565) sin (7,565) 0,8 0, 6 = sin (7,565) cos (7,565) 0, 6 0, AND 0 0 (4).4 cosθ sinθ 5 45 = sinθ cosθ cos θ + 75sin θ = 45 AND 5sin θ 75cos θ = 65 sin θ = 7/5 AND cos θ = 4/5 θ = 96,6 o (8) [4] QUESTION AB = = BA = = (8). a b c a d g a + b + c ad + be + cf ag + bh + ci d e f b e h = ad + be + cf d + e + f dg + eh + fi g h i c f i ag + bh + ci dg + eh + fi g + h + i (0) [8]

18 GRADE EXAMINATION: ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Page 8 of 8 QUESTION 4 4. Every vertex is used once and once only before returning to start. () 4. A B J B C D E C E F C G B H G F G H J A Doubled edges BJ + FG + GH + CE = 5 BJ + GC + GH + FE = 5 Circuit with all nine vertices and edges Total weight = 48 (original graph) + 5 (doubled edges) = 99 m () = 48 (existing edges) + 5 (BH + EF) + CJ CJ = 7 m (0) [4] QUESTION edges () 5. Kruskal : shortest edges in graph are being chosen () 5. 6 () 5.4 JE : will complete a mini-circuit () 5.5 DE, AB, IJ, DI EF / JG HG / CI BI = 60 (8) [6] QUESTION n = 5 () 6. Eulerian Circuit needs even degrees at vertices n odd amount of vertices. n () [6] Total for Module 4: 00 marks Total: 00 marks