ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES

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1 GRADE EXAMINATION NOVEMBER 4 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 4

2 GRADE EXAMINATION: ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Page of 8 MODULE CALCULUS AND ALGEBRA QUESTION Step : Let n = LHS = = RHS Step : Assume true for n = k then (k ) = k Step : For n = k (k ) + (k + ) = k + k + = ( k + ) by the P.M.I. the statement is true for n [] QUESTION. (a) (b) 4e x > 7 x e >.75 ln.75 < x.56 < x x <.44 IEB Copyright 4

3 GRADE EXAMINATION: ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Page of 8. (a) 7 ; (b) For gx ( ) f( x) require k (4) (note change of inequality) [6] IEB Copyright 4

4 GRADE EXAMINATION: ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Page 4 of 8 QUESTION. 5 ( i) i( 6i ) = 4i 6i + i (note change of mark allocations) = 6 i. x A B Ax + A + Bx B = + = (note change of mark allocations) ( x )( x + ) x x + ( x )( x + ) x: A + B = Const: +A B = A = 9 B = 7 7 x = + x x + x x thus ( )( ) (). lim 4Σk Σ n n ( + ) lim 4n n = n n n lim = = n n [4] QUESTION 4 4. Since f '(x) > f is always increasing (a) Greatest value of f(x) is at x = F (4) (b) f"(x) = points of inflection at (and f(x) ) (note change of mark allocations) x = B; C; E 4. No. We have no indication of what the y-values are for the graph. (No given points) (4) 4. f '( x ) thus f( x ) has no turning points. (4) [8] IEB Copyright 4

5 GRADE EXAMINATION: ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Page 5 of 8 QUESTION 5 5. ( )( ) y = sin x x (note change of mark allocations) dy 4 = cosx( x) + ( x).sin x dx sin x OR y = ( x) cos ( ) ( ).sin dy x x + x x = 6 dx ( x) 5. (a) + = + 8 4= 5 point ( ; ) lies on curve. () (b) dy dy x + y y + y x = dx dx dy dy x + y y xy dx dx = dy ( y xy) = y x dx.. dy y x = dx y xy () dy 4 (c) at ( ; ) = = dx 4 8 y = x+ c 8 = 8 + c 7 c = 8 7 tangent y = x+ (4) 8 8 IEB Copyright 4

6 GRADE EXAMINATION: ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Page 6 of 8 5. (a) ( ) y = bx+ c dy = +. = + dx d y ( bx c) b b( bx c) ( ) ( ) = b bx + c. b = b bx + c dx d y ( ) ( ) 4 4 = 6 b bx + c. b = 6b bx + c dx n d y = n dx n! b bx + c [4] n n (b) ( ) ( ) ( n+ ) QUESTION 6 6. SA rh rαh r α = + + but r = h A= r + r α + r α (note change of mark allocations) = r ( α + ) 6. Vol = r αh If r = h and total volume cm Then = r α (note change of mark allocations) Thus α = (4) r 4 6. A= r + r = + r r dsa 4 4r dr = r = for minimum SA 4r 4 = r = and thus α = [] IEB Copyright 4

7 GRADE EXAMINATION: ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Page 7 of 8 QUESTION 7 7. (a) Vertical asymptotes: ( x )( x ) = (b) ( )( ) x + 6x 5x 9 = x + 6x+ 5 x 6 + x 9 x = 5 or x = (4) x + x x = x 6+ x (this step not necessary) x + 6x+ 5 x + 6x+ 5 oblique asymptote = y = x 6 (note change in mark allocation) f x > = Implies concave up 5< x < (4) (c) ( ) 7. (a) f ( x) = cos x, 5x initial estimate f ( x) = sin x, 5 cos x, 5x xn+ = xn + sin x +, 5 x =, (b) If f ( x) = i.e. sin x =, 5 or x = sin 4 Algebraically by OR Graphically will not cut x-axis () [6] QUESTION 8 du 8. Let u = x = dx cos ( cos ) udu = + u du sin u = u+ + c 6 x sin 6x = + + c IEB Copyright 4

8 GRADE EXAMINATION: ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Page 8 of 8 8. cos θ sin 5θdθ = sin(5 θ + θ ) + sin(5 θ θ ) dθ = sin 7 θ + sin θdθ cos7θ cosθ = c + 7 cos7θ cosθ = + c Let u = x then x= u du = dx 5u 5 du = u u du u u 5u = + + c (note changes of sign in last steps) = u + u + c = ( x) + ( x) + c [] QUESTION ( x 4 4 ) π 4 π ( x 4) π 4 64 π dx ( ) 4 6 x dx π V = = 4 = = = units 9. No The line y = x, not an axis of symmetry rotation would be different. Or any other valid reason. (4) 9. a ( x + 4) π dx 6 4 ( x 4) 5 a π = = π 6 5 π 5 5 = ( a 4) ( 4) ) = π 8 ( a 4) 5 = 8 4 a 4 =,955 a =, 485 [8] Total for Module : marks IEB Copyright 4

9 GRADE EXAMINATION: ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Page 9 of 8 MODULE STATISTICS QUESTION. (a) X ~ N ( 8, 4 ; 4,6 ) 8, 4 4 8, 4 P( < x< 4) = P < z < 4,6 4,6 = P,8 < z <,5 ( ) =, ,68 =, 6 P x> k = (b) ( ),9 z =, 8 k 8, 4, 8 = 4.6 k =,5. (a) (b) (,)(,68),, n, 76,49 n n 9 A 98% CI for p is, ±, (,)(,68) 9 (,4 ;,496) (4) [6] QUESTION. (a) False (b) True (c) False (d) True (e) True (f) False IEB Copyright 4

10 GRADE EXAMINATION: ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Page of 8. H : µ = 8 H : µ < 8 Rejection Region: Reject H if z <, 75 Test Statistic: 7,9 8 z = =,9, Conclusion: Since z <, 75 we reject H at the 4% level of significance and suggest sufficient evidence to support the claim. () [6] QUESTION. (a) A and B are independent. () (b) P( A B) = A and B are mutually exclusive. (). P(A B) = P(A B) P(B) = P(A B) P(B) P(B) = P(A B). (a) (b) = =, = 4 (4) or 5 4 = 5 =, > = + = P( x ) ( ) ( ) ( ) ( ) 4,6, 4,6, 4, [] IEB Copyright 4

11 GRADE EXAMINATION: ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Page of 8 QUESTION 4 4. (a) x P( x ) = P X = x is a probability mass function. (4) ( ) (b) x = = (5) 4. (a) ( ) P correct = (, 65) + (,5) 5 =,7 (4) (,5) P guesses correct = 5, 65 + (,5) 5 7 = =, 97 (4) 7 [7] (b) ( / ) QUESTION 5 5. (a) Sub ( ) ; y into y = 7x+ 6 y = 7( ) + 6 = 6 () (b) y = 7(5) + 6 = 8 () 5. r = =,49 7 A correlation coefficient is negative and considered very weak. () 5. The estimate is unreliable as the correlation is too weak. () [9] Total for Module : marks IEB Copyright 4

12 GRADE EXAMINATION: ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Page of 8 MODULE FINANCES AND MODELLING QUESTION,9. Tn+ = Tn +, T = 5 (5) 4. 4, 5 = ( c ) c = 5,9%, = (,59) n n =, n = 4 years (9) [4] QUESTION., ,75 + OB = = 97 9,48, , 674 x +, =, 674 x = 9 69,8. 8 9, ,8 48 = ,6 [] QUESTION.. 4,58 i + = + 4 i + =, i = 5,77% per annum, compound monthly 85, =, 577 n,499 =,48 n n =7,8 8 7 = 8 months earlier () [8] IEB Copyright 4

13 GRADE EXAMINATION: ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Page of 8 QUESTION 4 4.,95 = +,6 x x =,65 5 = 5 + y y = 4. F n + =,95.F n + 5, F = 5 F = 8 7 or 8 8 (rounding) Population increasing, but at slower rate as time goes on. (5) 4. F n 5[,95 ] =,95.5 +,95 (k) (n) (c, K) = 9 97 (5) [6] QUESTION 5 5. (a) prey: accept 8 - predator: accept 7 - () (b) < prey < 94 (accept ;9 94 ) () (c) quadrant () (d) rate of deadly interactions between predator and prey () (e) Could lead to extinction of prey at some time points in cycles prey population is already very low ( ) and now has a higher rate of fatalities per cycle OR prey decreases more rapidly per cycle double amount of kills per cycle and hence potentially more predators OR Equilibrium point of prey half the original R = c/(bf) so denom doubled, hence value halved (4) IEB Copyright 4

14 GRADE EXAMINATION: ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Page 4 of 8 POPULATIONS TIME PERIODS initial points discrete plot accuracy of plots [] QUESTION 6 6. Q n + =,P n +,9Q n + 4 (4) 6. For equilibrium, P n + = P n :,P +,6Q = 6. For equilibrium, Q n + = Q n :,P,Q = 4. P = , Q = 875 [] Total for Module : marks IEB Copyright 4

15 GRADE EXAMINATION: ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Page 5 of 8 MODULE 4: MATRICES AND GRAPH THEORY QUESTION. (a) 4a ( ) = a = ½ (4) (b) 6 4 (4) (c) A B = a b a 4 = 7 a = 4 b = 4a = b = 8. (a) q x p () (b) p x r () (c) r x r () [] QUESTION. (a) () (b) (shear) (factor) (direction) =. 6 o 8 = 45 o cos 45 sin 45, = sin 45 cos 45, 77. cos A sin A,8,6 = sin A cos A,6,8 cos A =,8 AND sin A =,6 A = 8 + 6,87 = 6,87 so A = 8,45 tan A = y = x () [4] IEB Copyright 4

16 GRADE EXAMINATION: ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Page 6 of 8 QUESTION. det = 6 (). 4 6 (). a = 6= 8 b = 4 6 = c = 4 ( ) = [] QUESTION 4 4. Dijkstra () 4. Prim () 4. Fleury () 4.4 Nearest Neighbour () [8] QUESTION 5 5. A B C D E F G H J K L A B 8 C 6 D 6 E 4 F 8 G 6 H 7 4 J 8 K 4 L 6 IEB Copyright 4

17 GRADE EXAMINATION: ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Page 7 of 8 5. C K L F E 4 G 7 J 5 x G 8 x H J A E 8 x L 9 B 8 C 4 D H 7 x A F E G H J L = 9 OR A B,C,D,E,F,G,H,J,K,L AB C,D,E,F,G,H,J,K,L ABF () C,D,E,G,H,J,K,L ABFC or ABFCE (4) D,G,H,J,K,L ABFCEG (7) D,H,J,K,L ABFCEGD () H,J,K,L ABFCEGDH or ABFCEGDHK () J,L ABFCEGDHKJ () L ABFCEGDHKJL (9) A F E G H J L = 9 () 5. EG HJ (4) 5.4 (a) vertices, therefore each vertex must be connected to 5 other vertices. True only of G. () (b) converse not true OR Theorem gives one case of when there will be HC; it is not exhaustive () [6] IEB Copyright 4

18 GRADE EXAMINATION: ADVANCED PROGRAMME MATHEMATICS MARKING GUIDELINES Page 8 of 8 QUESTION 6 6. (a) m + n () (b) m.n () 6. x = or x 5 = 7 (4) 6. vertices edges (4) [] Total for Module 4: marks Total: marks IEB Copyright 4