MATHEMATICS: PAPER I MARKING GUIDELINES

Transcription

1 NATIONAL SENIOR CERTIFICATE EXAMINATION NOVEMBER 07 MATHEMATICS: PAPER I MARKING GUIDELINES Time: hours 50 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 07

2 NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER I MARKING GUIDELINES Page of SECTION A QUESTION (a) () () ( ) = ( ) ( ) = ( ) ( ) + ( ) = 0 ( )( + ) = 0 ( )( + ) = 0 = = () = = 5 + = 5 = = () 5 (b) ( + ) < 9 Alternative + + < 9 ( + ) < 9 + 8< 0 + 8< 0 ( + )( ) < 0 Sketch: y = + 8 Critical Values: ; int : = ; = Solution: { : < < } Solution: { : < < } Alternative ( + ) < 9 < + < < < () (c) ( )( + ) = = 0 b = and c = 8 () IEB Copyright 07

3 NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER I MARKING GUIDELINES Page of (d) () () = let = y y = LCD: y y y = y y + y + = 0 () ( y + ) = 0 Alternative y = y + y + = 0 Roots are real and equal. = ()() =0 Roots are real and equal. Alternative = ( ) = ( ) + = + 8 = 0 Roots are real and equal. () [9] IEB Copyright 07

4 NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER I MARKING GUIDELINES Page of QUESTION (a) () () E = { HTT, THT, TTH} P( tails and head) = () 8 (b) () P (A B) = 0 () (c) () () (i) You cannot pick a R and a R5 coin at the same time. () (ii) P(either a R5 or a R) = P(A or B) = P(A) + P(B) mutually eclusive = 0,6 + 0,7 = 0,8 () () () () P(eactly one machine is stamping R5 coins) = 0, + 0, = 0,5 50% () [7] IEB Copyright 07

5 NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER I MARKING GUIDELINES Page 5 of QUESTION (a) ,50 = R () (b) R % = R7 85 () (c) Cost of machinery including import charges = R R7 85 = R 00 5 n A = P + i log 9 00 ( ) n 9, = n = = n 5 50 n = 6,6759 n 6,6 appro. 7 years () (d) () Loan required: R 00 5 R5 50 = R ( + i) n P = i ( ) = 00 = R0 50,76607 = R0 50,77 () IEB Copyright 07

6 NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER I MARKING GUIDELINES Page 6 of () Outstanding Balance = A F A = A = ,57 A , F = 0 50, F = 55 6,77 F 55 6, Outstanding balance = , ,77 = R5 78,67 R5 78,5 NB: If A and F are rounded to the nearest cent, consider Outstanding balance = , , = R5 78, Alternative + 00 Outstanding balance = 0 50, = R5 78,66 R5 78,5 () [5] IEB Copyright 07

7 NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER I MARKING GUIDELINES Page 7 of QUESTION (a) () Constant second difference () () T n = an + bn + c T = a+ b+ c = T = a+ b+ c = T = 9a+ b+ c = 6 a+ b = and 5a+ b = Substitute b = a into 5a+ b = 5 a+ ( a) = a = a = b = and c = 0 Tn = n + n (6) (b) T = 5 cm T 7 = 78 cm T = a+ d = 5 T7 = a+ 6d = 78 d = 6 d = 6 a = 9 cm T = T = cm (5) [] IEB Copyright 07

8 NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER I MARKING GUIDELINES Page 8 of QUESTION 5 (a) () f ( ) = h + h 6h f '( ) = lim h 0 h h(+ h 6) f '( ) = lim h 0 h f '( ) = 6 (5) () f '( ) = ( ) 6 = () (b) y = π + dy d = () [] 7 marks IEB Copyright 07

9 NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER I MARKING GUIDELINES Page 9 of SECTION B QUESTION 6 (a) () Domain = ; () () Range = y y ; y () () (i) 5 units () (ii) 5 units () (b) () y = ab. substitute 0; 0 = ab. a = 9 y = b substitute ; 9 b b = 9 b = ± but b > 0 b = () () f: shape intercept domain f shape intercept range () Range = ; () () () f( ) =. y For f : =. ; y 0 = y y = log ( ) for () (5) See graph in Question 6 (b) () above. () [8] IEB Copyright 07

10 NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER I MARKING GUIDELINES Page 0 of QUESTION 7 (a) f( ) = () + 5 () f( ) = ( + ) T.P.( ; ) () (b) () = = 0 = or = 5 A( 5;0) and B( ; ) () () Horizontal shift: 5< t < () (c) () Length MN ( 5) ( 6 5) = + + Length MN = For ma. length: Let D = 0 7 = 0 7 = 7 7 Ma. length MN = Ma. length MN = units i.e.,5 units (6) () Vertical shift: k > 9 () [7] IEB Copyright 07

11 NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER I MARKING GUIDELINES Page of QUESTION 8 (a) () 9 7 ; ; ;... r = and series is geometric however, series is not convergent since r <. () () = + ( ) = ( )( + ) = = 0 = 9 or () = 7 ; = 5 and = T = S S (b) S S5 S T = T6 = S6 S5 T 6 = 6 T 5 = ar = = ar = 6 T T 6 r T = = 5 a = S n ( n ) = = (7) [] n IEB Copyright 07

12 NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER I MARKING GUIDELINES Page of QUESTION 9 (a) ( ) f = + b + c () () () () f = + b + c = b+ c = 8 f '( ) = + b + c f ''( ) = 6 + b f '' = 6 + b = b = c = 6 (7) (b) For concave up: f ''( ) > > 0 < () [0] IEB Copyright 07

13 NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER I MARKING GUIDELINES Page of QUESTION = + LCD: + ( ) 0( + ) 0 = ( + ) = 0 Therefore Time = =,09 or 6,09 Alternative Let original time taken be represented by y. y = 0... eq. ( + )( y ) = 0... eq. 0 From eq. y = 0 ( + ) = = 0 =,09 or 6,09 Therefore Time = [6] IEB Copyright 07

14 NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER I MARKING GUIDELINES Page of QUESTION (a) () When = and = 0 () () () (b) y = dy d = 5 + substitute = 0 dy d = Equation of tangent: y = + c where c = Equation of tangent: y = + For point of intersection between tangent and line BC, substitute = into y = + y = Pt ; Area of Busi's region = 5 + = 8 units Area of Khanya's region = + = 7 units Therefore Busi's region is larger. (7) [] 76 marks IEB Copyright 07 Total: 50 marks