MATHEMATICS: PAPER I MARKING GUIDELINES

Transcription

1 NATIONAL SENIOR CERTIFICATE EXAMINATION SUPPLEMENTARY EXAMINATION MARCH 0 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 0

2 NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER I MARKING GUIDELINES SUPPLEMENTARY Page of SECTION A QUESTION (a) () ( 9 )( ) 0 or Check: : is not a solution (6) 9 () 9 (5) () (b) () For A and B: Let y 0 A ( ; 0) B(; 0) For C: let 0 y ()( ) 9 C (0; 9) + For D: D for writing as coordinates y (+ )( ) y D (; ) (6) () or () IEB Copyright 0

3 NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER I MARKING GUIDELINES SUPPLEMENTARY Page of (c) 6 or ( 6)( ) t 9 OR 6 or ( + 6)( + ) t 9 Alternate: Product of roots 8 Numbers are 6 and or 6 and Sum of roots t t OR t 6 9 t 9 or t 9 Alternate: Let the smaller root be a. The other root is a +. The equation is: ( a) ( a+ ) 0 [ ] ( ) ( ) a a a a t a a a+ 8 ( ) a + a 8 0 a+ 6 a 0 a 6 OR a t 9 or t 9 ( )( ) (5) (d) y ( ) ( 5) 0 ( ) y or or 5 y () [9] IEB Copyright 0

4 NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER I MARKING GUIDELINES SUPPLEMENTARY Page of QUESTION (a) ( + i) n P i (5 ) R5 6, () (b) n ( + i) F i ( ) F 5 6, 6 00 F 6 0,9 r A P + 00 t 6 6 A A 6,8 Balance of loan R56 5, 68 nt NB: Discrepancies in rounding ALTERNATE: Balance outstanding 5 6, R56 5, (5) (c) In four years, the farmer paid: 5 6, R 9 89, Balance of loan after four years R56 5,68 Paid towards original loan: ,68 R 98 6, Interest charges were: 9 89, 98 6, R9 08, () [] IEB Copyright 0

5 NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER I MARKING GUIDELINES SUPPLEMENTARY Page 5 of QUESTION (a) () r n 5 n () () () < < () S 5 6,5 () (b) () n S n ( n + 9) n 88 ( n + 9) 9 56 n + 9n 0 n + 9n ± 9 ( )( 956) n 9 ± 59 n n 6 (5) () T S S6 Alternate 6 a T S 6 ( (6) + 9 ) S S S6 8 T 0 d S ( () + 9 ) T + 6( ) 55 S 8 T 55 () IEB Copyright 0

6 NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER I MARKING GUIDELINES SUPPLEMENTARY Page 6 of (c) () Volume of each cylinder Ratio of consecutive volumes Which is a constant The sequence is geometric. πr h 9 9 π r h 0 0 πrh 9 0 () () Volume S a( r ) r 9 π( 8) ( 6) ,5 cm 5 litres () [] QUESTION Quadratic pattern T + + a+ b+ c n an bn c a+ b+ c 9a+ b+ c a+ b 5 and 5a+ b a+ b 5 5a+ b b 5 a Sub into : 5a+ b 5a+ 5 a a a b c Alternate 5 First difference Second difference a, a Tn n + bn + c a+ b 5 and Sub. a () + b 5, b a+ b+ c Sub. a and b + + c c [6] IEB Copyright 0

7 NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER I MARKING GUIDELINES SUPPLEMENTARY Page of QUESTION 5 (a) (b) f '( ) lim Working f( ) f( + h) f( + h) f( + h) y f '( ) f '( ) f '( ) h 0 f '( ) f( + h) f( ) h ( ) + h ( ) + h + h + h + h + h + h lim h 0 h h + h lim h 0 h h + h lim h 0 h (5) y dy + d dy + d (5) [0] 9 marks IEB Copyright 0

8 NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER I MARKING GUIDELINES SUPPLEMENTARY Page 8 of SECTION B QUESTION 6 (a) f( ) 0+ ( ) f( ) 0+ 5 ( 5) + f( ) ( 5) 5 + f( ) ( 5) Alternate b 0 b 5 5 a a ( 0) ( )( ) a 8 a b f( ) a + + a a ( 5) () (b) h ( ) b - q sub.: (0;-) 0 b + q q h ( ) b - sub.: (;-) b b h ( ) () (c) y A shape TP A shape y-int (0; -) y (asymptote) (5; -) (d) Shown () y (e) y log ( + ) () (f) Domain of h Range of h (, ) () [8] (5) IEB Copyright 0

9 NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER I MARKING GUIDELINES SUPPLEMENTARY Page 9 of QUESTION ( ) + + f( ) + + Asymptotes are and y Alternate a + + q + + a+ q + + ( ) ( ) q + a + q + q, a+ q, a f ( ) + Asymptotes are: and y Alternate ( + ) Asymptotes are : and y QUESTION 8 f ( ) + f ( ) + f '( ) f '( ) Alternate + ( + ) + q q q ( ) ; y }Asymptotes [6] Equation of tangent is: y + c 5 ( ) + c c Eq : y IEB Copyright 0 [6]

10 NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER I MARKING GUIDELINES SUPPLEMENTARY Page 0 of QUESTION 9 Eq. AB : y m + 9 sub (;0) 0 m + 9 m length CD + 9 Area ΔCOD y da 9 + d Alternate b a [8] QUESTION 0 (a) f( ) a + b f '( ) a + b f ''( ) 6a + b f ''( ) 6 a( ) + b 0 a+ b b 6a Sub: ( ; 6) in f( ) f( ) a( ) + b( ) 6 a+ 9b 9 0 a+ 9(6 a) 9 0 a 9 a b 6 () (b) () f( ) + f '( ) + f '( ) + and 0 f '( ) 0, when 0 ;0 or 0; () OR [ ) ( ) IEB Copyright 0

11 NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER I MARKING GUIDELINES SUPPLEMENTARY Page of () f is concave up when f ''( ) > 0 f ''( ) 6a + b + + > 0 > () [] QUESTION (a) () Number of arrangements is 5! 0 () () () 6! 60! () 0 () 60 (b) () 0 6 F AF A 6 6 NF A(NF) B 8 0 F BF 0 NF B(NF) P (picking up a fiction book) P ({Shelf A and fiction book} or {Shelf B and fiction book}) P (AF) + P (BF) () () Alternate: () [] IEB Copyright 0

12 NATIONAL SENIOR CERTIFICATE: MATHEMATICS: PAPER I MARKING GUIDELINES SUPPLEMENTARY Page of QUESTION The full sphere touches all eight pieces. Let the side of the cube be a. The radius of the sphere is. The diagonal of the cube is: + +. The diagonal of each face of the cube has length a (by Pythag.) The diagonal of the cube has length: ( ) a + a a (by Pythag.) a Face of cube a But: a a a a a Cut cube diagonally a Volume liquid a π 6,8 π 6,8 8 π 6,8 6,8 8 π, 095, cm a ALTERNATE: Let the side of the cube be a a a π Volume of liquid a 8π 6,8 a a 6,8 a 8π a 6,8, cm 8π Use of volume of cube Indication of volume of spheres Subtracting volumes [] marks IEB Copyright 0 Total: 50 marks