GRADE EXAMINATION NOVEMBER ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Time: hours 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. IEB Copyright
GRADE EXAMINATION: ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Page of 5 MODULE CALCULUS AND ALGEBRA QUESTION Let n i LHS i r r r RHS r true for n Assume true for n k Let n k k k k r i r r k k k k k i r k r r r LHS i r r r r k r r by the principle of Mathematical Induction the statement is truen with r [4] QUESTION. xx lg o 5 xx 5 5 x x6 5 x x56 x8 x7 x 8or x 7 but x 8 Thus only solution is x 7 (8). (marks: basic shape for cusp at ( ;-) and for cusp at (:-) x e for x e for vertical asymptote at x () IEB Copyright
GRADE EXAMINATION: ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Page of 5. (a) ( ) n g f x x (4) (b) Domain: x x but also x x Thus the domain is x (c) Range : if x then ln ln As ln x (Graph is not necessary but shown below) x Thus ( ) g f x on its domain in [ x] (4) [] QUESTION. abi bi ( i)( 5i) abi b 55i 6i 65i abi b 87 9i ab 9 and b 87 b 9 and a (8). i i i i i 4 n 4 n i i + = (5). x i x i x i x x x x 4 is a quadratic equation. Any other permissible one will get full marks. x ( i i) x( i)( i) x x 4 [9] IEB Copyright
GRADE EXAMINATION: ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Page 4 of 5 QUESTION 4 f ' x f ( xh) f( x) lim h limh h x h x x xh x xh limh h x x h x x h xhx lim h h x xh x xh lim h x xh x xh x x correct use of limit notation (8) 4. h 4. (a) x π f x f x π x f (b) xπ f' x im lim ππ lim sin π continuous at x lim l f ' x cosx = xπ Thus function is continuous and the gradient from the left is tending to the same value as the gradient from the right at x π thus the function is differentiable at x π. [] QUESTION 5 5. Vertical asymptote x Oblique asymptote x 7x4 xx f xx x Thus the oblique asymptote is y x (8) 5. Using IEB Copyright f x x x x Then f ' x TP : x x x f '' x thus f '' x min and f '' max Co-ords (,7;7,8) Min and (,9;,7) Max ()
GRADE EXAMINATION: ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Page 5 of 5 5. y-intercept: f x-intercept: 4 x 7x 4 7 7 x 4 (or x.78 or x.7) 5.4 Mark allocation: vertical asymptote oblique asymptote basic shape intercepts with axes turning points (8) [4] QUESTION 6 6. Thus : 4: 5 triangle so width is 4 = 8 cm (4) 6. θ cos.97 5 X-sectional area 5 (.97).85 cm Volume 5.85 5sin (.97) 559. cm or (ml) (9) [] IEB Copyright
GRADE EXAMINATION: ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Page 6 of 5 QUESTION 7 7. f ' x sin x x cos x x 7. Require x f x gx h Newton Raphson formula: xn x n n n n x n x sinx x sin x x cos x n n n with x.5 7. x.848 correct to 4DP (4) [6] QUESTION 8 8. (a) x cosxx d cosxx d sin x Let g x then g ' and f ' cosx then f sin sin sin Thus x x x x xcosxdx cosxdx dx c cos x sin x xsin x c g x g' sin x f ' cosx f sin x sin x x cosxdx x.. dx sin x cos x x. c () (b) Let u x then x Thus du 6 dx x sec x d x 6 sec u du tan u 6 c tan( 6 x ) c (8) (c) Let u cot x then Thus du cos ec x.dx cot xcos ec xdx udu u c cot x c (8) 6 IEB Copyright
GRADE EXAMINATION: ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Page 7 of 5 8. Volume 4 πx dx πx dx x 5 x x x 5 x x 5 units () 5 8. Function D is the integral of Function A [4] QUESTION 9 [] Total for Module : marks IEB Copyright
GRADE EXAMINATION: ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Page 8 of 5 MODULE STATISTICS QUESTION. Travelling by car might have the higher mean, but the bus statistic has a high standard deviation. Random samples have a degree of variability. (). Let x be the car commuters and y the bus commuters. H : y x H: y x Rejection Region: Reject H if Z,75 Test statistic: 5, 9,6,8 5, z 5, 55 45 Conclusion: We reject H at the 4% level of significance and suggest sufficient evidence to support the claim. (). Margin of error increases. In other words you could reject a null hypothesis when in fact it is true. () [5] QUESTION. 6! (!)! 6, 96. 7 7 5 4. (a) P(P K ') P(P ' K) (,7)(,4) (,6)(,),46 (b) P(P K) P(P) P(K) P(P K) (,7) (,6) (,7)(,6),88 [4] IEB Copyright
GRADE EXAMINATION: ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Page 9 of 5 QUESTION. (a) r,98 There is a strong negative linear correlation implying as the age of a vehicle increases so its value decreases. () (b) y 59,488 47, 5744x (4) (c) R59 4,88 () (d) Unreliable as a new car would not fit the model for second hand cars. ()., 48, 6(7), 6( x ) x = R 74,7 () [4] QUESTION 4 4. 4. 4. (kx k)dx k( x ) kx 9k kx 9k 9k k 9 k (5) 9 5 ( )d 9 x x () 9 m ( )d,5 9 x x by similar triangles x m 9 x 4 m 9 m 4 m m 9 m 5, or m,8787 Ratio of areas = : Ratio of sides = : m m =,8787 (8) [6] IEB Copyright
GRADE EXAMINATION: ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Page of 5 QUESTION 5 56 7 7 7 5. P(56 x 7) P z 5 5 P(, z,4), 499,554,49 4% (9) 5. 7, 8 x 5 x 65,6 [5] QUESTION 6 5 49 48 47 P 4 untagged.47 6 59 58 57 5 4, 47 6 4 6. 6. Pat least one Pnone 5 4,47 6 4,577,577 (4) 6.,47,577,5 5, [6] Total for Module : marks IEB Copyright
GRADE EXAMINATION: ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Page of 5 MODULE FINANCE AND MODELLING QUESTION. R (). 8% (). R4,5 and R69,5 ().4 4,5 69,5 = 7,6 7, = 7,7 ().5 + 69,5 ( +,8) = 4 8,8 (4) [] QUESTION. 5 = A( +,4) 8 A = 999 667,55 5 999667,75,58 5% (5) 999 667,75. 7 5,87,87 Fv 496 58,, 87 5 + 496 58, = 996 58, 8 7 5,87,87 Pv 496 58,, 87 5 + 496 58, = 996 58, (8) [] IEB Copyright
GRADE EXAMINATION: ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Page of 5 QUESTION. 48 4, 65, 68,65 4 4 4 95 84,76,65 4, 68, 68 4 4 77 996,,68 4 95 84,76 + 77 996, = 7 8,8 49 4, 65, 68 4 4 89,5,65 4, 8, 68 4 4 6 99,,68 4 89,5 + 6 99, = 7 8,8 (4). 4 x, 68, 68 4 4 75 x = 4 56,5 4 6, 68, 68 75 x x x 4 4 x = 4 56,5 () [4] IEB Copyright
GRADE EXAMINATION: ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Page of 5 QUESTION 4 4. r =, () 4. r, K m,47 7,68 7 (4) 4.,95 =, 47.P +, P = 54,4 54 (4) P n 4.4 Pn Pn,Pn, P 7 P 5 = 49,6 5 [6] QUESTION 5 5. 4 5/4 = () 5. kills =, 56 4 = 44 (4) 5.,7,5 4,6 =,756 5.4,6,56 4 = pups Pups = (8) [] QUESTION 6 6. 7 moves () 6. T n =. T n +, T = 7 T T, T 7 n n n T n = T n + n, T = 7 (4) 6. 6 = a. + b.5 and = a.5 + b.7 a = b = T n =.T n.t n T = 7 and T 4 = 5 Each time adding, 4, 6, 8 that is n T n = T n +.T n +, T = 7 and T 4 = 5 (8) [4] Total for Module : marks IEB Copyright
GRADE EXAMINATION: ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Page 4 of 5 MODULE 4 MATRICES AND GRAPH THEY QUESTION. (a) rotation of 8 o about the origin reflection about y = x (4) (b) stretch factor matrix (4) (c) cos 6, sin 6, sin 6, cos 6, 4 4,,9. (a) working sketch (4) (b) working sketch (4) [] QUESTION. (a) () (b) () (c) (). (a) k () (b) k () (c) a.a.a a.a.a... a.a.a 8a.a.a... a.a.a 8k (4) [4] IEB Copyright
GRADE EXAMINATION: ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Page 5 of 5 QUESTION. A 4 4 7 B 4 6 6,5 C 5 (). b b b b b b b b b b = b b = b b =, (8) [8] QUESTION 4 4. Yes, all vertices of the same degree. () 4. No, more than pair of odd vertices. () 4. A B F G H E HD C BA = 6 () 4.4 A E H G F B C D A = 49 8 vertices Hamiltonian circuit girth 5 Several options possible. (8) [] QUESTION 5 5. edges connected correctly edge weight 5. H: matrix is not symmetrical double edge between two vertices a sketch (4) 5. G: G G not connected to other edges a sketch (4) [4] QUESTION 6 6. (a) n () (b) n () (c) (n )/ () 6. vertices 4 edges sketch (4) [] IEB Copyright Total for Module 4: marks Total: marks