NATIONAL SENIOR CERTIFICATE GRADE 12

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1 NAIONAL SENI CERIFICAE GRADE MAHEMAICS P NOVEMBER 00 MEMANDUM MARKS: 50 his memoradum cosists of 7 pages.

2 Mathematics/P DoE/November 00 QUESION.. x ( x 4) x 4x 5 0 ( x 5)( x + ) 0 x 5 or x 4x 0x + 0 ( 0) ± x x x 0 ± 8 4,5 84 or ( 0) (4) x 0,05 4(4) y x x x 6 + ( x ) x x 0 ( x )( x + ) 0 x or x y 0 or y 4 Solutios are (x ; y) ( ; 0) or ( ; 4) x y + ( y + ) y y + 6y + y 6 + y + 4y 0 y( y + 4) 0 y 0 ( y + ) 6 + y or y 4 x or x Solutios are (x ; y) ( ; 0) or ( ; 4) y x y x y ( x )( x + ) x + x x 6 y y( x + ) or or y 0 x y 4 Solutios are (x ; y) ( ; 0) or ( ; 4) stadard form factors both aswers ito formula 84 s y x factors s x y + simplify factors aswers y x factorise equal aswers (4) (6) (6) (6)

3 Mathematics/P DoE/November 00. m m + m m + m m + 7 m ( m + ) 7 squarig both sides equatig umbers ad surds squarig both sides i m + 7 m + m + 4 m + 4 m 4 5 m m + m m + m m + 7 ad m are possible solutios m 7 7 m (7 ) m(7 m) ( 4)( ) 0 or 7m + 0 ( m 4)( m ) 0 4 or m 4 or m m or m 4 or 4 m squarig both sides of m + 7 equatig umbers ad surds solvig simultaeously for m ad m m m + m m + m By ispectio (m ; ) (4 ; ) or ( ; 4) m squarig both sides of m + 7 s for m ad [8]

4 Mathematics/P 4 DoE/November 00 QUESION. 0 ; 4 ; 8 ; ; ( ) O the st day she will cycle 00 km. sequece 0 days O the st day she will cycle 00 km. (Aswer oly full marks) ( ). S [a + ( ) d] 4 S4 [(0) + (4 )4] 644 km. No. It will ot be humaly possible to just keep o icreasig the distace covered idefiitely. For example: 4 06 km i oe day. 000 formula o reaso () [8]

5 Mathematics/P 5 DoE/November 00 QUESION. 45. a + b + c Secod differece of terms is. a a + b 7 + b 7 b 4 a + b + c c 5 c a + b + c Secod differece of terms is. a 0 c 0 5 b 4 + b b + 4 value of a value of b value of c value of a value of c value of b If a + b + c 5 a + b + c a a + b 7 4a + b + c b 4 5a + b a + b + c c 0 settig up equatios simultaeous equatios aswers [6]

6 Mathematics/P 6 DoE/November 00 QUESION S rs S rs a ar S( r) a( r ) ( r ) a S r 5 ; 5 ; 5 ; a + ar + ar ar settig up of S rs ar + ar ar + ar 5 r 5 he series coverges because < r < S S S S S S subtractio commo factors divisio value of r explaatio (6) () a 5 ito correct formula aswer both sums both sums () ()

7 Mathematics/P 7 DoE/November a S S S () r + S S + + NOE: + 4 S a 4 secod term a 4 develop patter formula simplificatio [7] If + the a 4 4( ) S 4( ) maximum of marks +

8 Mathematics/P 8 DoE/November 00 QUESION 5 5. (0 ) + b b 5. g ( x) ( x ) + urig poit of g: ( ; ) f ( ; ) lies o f. ( ; ) lies o both f ad g D( ; ) + b (0 ; ) simplificatio () 5. log x 5.4 y y (4 ; ) ( ; ) P ito f ( ; ) lies o both f ad g. () y-itercept oe other poit shape icreasig f - ( ; ) x h( x) ( x + ) x + traslatio () Shift oe uit to the left ad shift two uits dow to give y x 5.6 x 0 x ( Max. value of ) occurs at max. value of 4 ( x ) x at ( ; ) () []

9 Mathematics/P DoE/November 00 QUESION 6 6. p q 6. y [0 ; ] () 0 y 6.. cos x (si x ) si x cos x cos x si x cos x + his would be solved by fidig the x-values of the poits of itersectio of the graphs of f ad g maipulatio () 80 about,5 [6] QUESION f (0) 0 + f (0) y-itercept (0 ; ) x-itercepts x + 0 x + K K ( x ) x x-itercept ( ; 0) x 0 y 0 (4)

10 Mathematics/P 0 DoE/November x + + x + x + x + x + LCD simplificatio () x + x + ( x + ) + x + x + + x + x + + x + split the fractio simplificatio () 7. Vertical asymptote: x Horizotal asymptote: y () 7.4 y asymptotes 5 shape itercepts 4 (4) x NOE: If the graph does ot represet a fuctio, cadidates do ot get the mark for shape x x + x ( ; ] ( ; ) x or x > maipulatio NOE: 0 marks for < x [5]

11 Mathematics/P DoE/November 00 QUESION 8 8. Depreciatio value 7 00( 0,5) R 07,50 [ ] (,05) (,05) (,05) 0, log 0, log,05 0, or ( ) ,05 (,05) 54,65 Number of paymets Balace outstadig R 0, Amout paid i last moth 0.8 0,50 + R 78, formula formula use of logs use of logs settig up equatio ito correct formula 8..4 otal repaid ( ) + 78,6 R77 78,6 (6) () [7]

12 Mathematics/P DoE/November 00 QUESION. f ( x + h) f ( x) f ( x) lim h 0 h 5( x + h) + ( 5x + ) lim h 0 h 5( x + xh + h ) + + 5x lim h 0 h 0xh 5h lim h 0 h h( 0x 5h) lim h 0 h lim( 0x 5h) h 0 0x f ( x + h) 5( x + h) 5x + 0xh 5h + method simplificatio factorisig f ( x + h) h f ( x) 5x 0xh 5h h 0xh 5h h 0x 5h + + 5x f ( x) lim( 0x 5h) 0x h 0. D [( x )( x + ) ].... x D x x + [ x + x 8] Depth after days Rate of decrease i depth h (t) t 4 () 4 Rate of decrease i depth after days 4,5 metres/day Rate of decrease i depth,5 metres per day 6,75 m simplificatio aswer h (t) derivative of t aswer (,5) uits (metres per day) [4]

13 Mathematics/P DoE/November 00 QUESION 0 0. x ad/or x 0. Whe x <, f ( x) > 0 ad so f is icreasig f ( x) > 0 Whe < x <, f ( x) < 0 ad so f is decreasig f ( x) < 0 Whe x >, f ( x) > 0 ad so f is icreasig () At x : At x : local maximum local miimum aswer (4) f ( x) ax + bx + c is miimum-valued a > 0 f has a shape f (x) miimum-valued a > 0 At x : local maximum At x : local miimum f (x) x umber lie (4) 0. At x : At x : + x local maximum local miimum,5 (4)

14 Mathematics/P 4 DoE/November shape x-values of turig poits correct () []

15 Mathematics/P 5 DoE/November 00 QUESION. r x Area rectagle 8x Radius small circle r A( x) area rectagle area circles. A( x) 8x πr + π r A( x) 8x π (x) π (x) 6 A( x) 8x 4πx πx 5π A( x) 8x x r x area rectagle 8x radius small circle r formula 6 π x A ( x) 8 πx A ( x) 8 πx 04 A ( x) πx 7 x 04π x π x 0, x 0, m. Area of circles 5π x 5π (0,) 0,88 m Area of circles 5π x 5π π 6 m π () [0]

16 Mathematics/P 6 DoE/November 00 QUESION.. ad. x + y 7 000x y x + y x + 4y 4 x + y x, y 0 Number of hectares of sweet potatoes y costrait costrait costrait ( correct expressio, correct iequalities) (6) lie graphs feasible regio Number of hectares of mealies.4 P 000x + 800y ().5 Maximum at (4 ; 4) search lie 4 hectares of mealies ad 4 hectares of sweet potatoes () x.6 Profit per hectare sweet potatoes P 0 000x + 000y 5 P y x Maximises at (5 ; ) P () Profit R per hectare gradiet of or P 0 000x + 000y [7] OAL: 50

17 Mathematics/P 7 DoE/November 00 QUESION. AND. 4 y Number of hectares of sweet potatoes x Number of hectares of mealies

NATIONAL SENIOR CERTIFICATE GRADE 12

NATIONAL SENIOR CERTIFICATE GRADE 12 NAIONAL SENI CERIFICAE GRADE MAHEMAICS P NOVEMBER 008 MEMANDUM MARKS: 0 his memoradum cosists of pages. Copright reserved Please tur over Mathematics/P DoE/November 008 NSC Memoradum Cotiued Accurac will