NATIONAL SENIOR CERTIFICATE GRADE 12

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1 NAIONAL SENI CERIFICAE GRADE MAHEMAICS P NOVEMBER 008 MEMANDUM MARKS: 0 his memoradum cosists of pages. Copright reserved Please tur over

2 Mathematics/P DoE/November 008 NSC Memoradum Cotiued Accurac will appl as a geeral rule. If a cadidate does a questio twice ad does ot delete either, ol mark the FIRS attempt. If a cadidate does a questio, crosses it out ad does ot re-do it, mark the deleted attempt. QUESION.. 0 ( )( ) 0 or for ot equal to zero i this questio ol. If 0 appears oce i this questio the full marks stadard form 0 factorisatio both aswers ().. ( ) B the formula stadard form 0 substitutio both aswers simplificatio 0 ± ( ) () ()( ) ±,79 or 0, 79 for iaccurate roudig off for both aswers. stadard form substitutio ito formula aswers () ( ) simplificatio 0 ± () ( ) ( )() ±,79 or 0, 79 for iaccurate roudig off for both aswers. stadard form substitutio ito formula aswers If egative discrimiat: ma / () Copright reserved Please tur over

3 Mathematics/P DoE/November 008 NSC Memoradum.. < < 0 > 0 ( )( ) > 0 < or > ( ; ) (; ) stadard form factorisatio / < > () < 0 < 0 < ( )( ) < or > / Iaccurate iequalit i the begiig / If fial aswer does ot have iequalit sigs (ie. questio has bee chaged to a equatio) / If the cadidate has used AND or istead of or If Aswer is ( )( ) > 0 the: / < < stadard form factorisatio / < > () Copright reserved Please tur over

4 Mathematics/P DoE/November 008 NSC Memoradum Copright reserved Please tur over. 0 ) )( ( ) ( ) ( or substitutio simplificatio of ) ( stadard form factorisatio both values values (8) ) )( ( or 0 ) )( ( 0 ) ( ) )( ( 0 or substitutio simplificatio of stadard form factorisatio both values values (8) commo factor commo bracket both -values -values (8)

5 Mathematics/P DoE/November 008 NSC Memoradum 0 ( )( ) ( ) 0 ( )( ) 0 or. ( )( ) ( ) commo factor commo bracket both -values -values factorisatio simplificatio (8). herefore whe , the value is ( )( ) ( ) > sice 0 > ca ever be equal to ½ aswer () If cadidate has substituted directl, 0/ (aswer would be 0 b substitutio) Aswer ol: / Correct aswer but icorrect mathematics 0 / iequalit coclusio () Which has o real solutio sice > 0 for all R {0} equatio coclusio () Copright reserved Please tur over

6 Mathematics/P 6 DoE/November 008 NSC Memoradum calculatio b ac 0 ()() 8 < 0 or 8 < 0 () o real roots equatio coclusio QUESION Which has o real solutio sice 0 for all R () [6].... aswers ; 6 formula for...to terms ( to terms) 8 geometric series a ( r ) r [ a ( ) d ] 0, S 0 00,00 [ () () ] S terms of st sequece terms of d sequece 0 S0... to terms 8 S0 [ () () ] S 0, S ,00 ( to terms) aswer for geometric series () formula for liear series [ () () ] 000 aswer (7) If used 0 terms i each series: ma /7 (aswer the is 876) Aswer ol: 6 / 7 Write out series ad the correct aswer: full marks Copright reserved Write out both series ad ot add them: 6 / 7 Please tur over

7 Mathematics/P 7 DoE/November 008 NSC Memoradum.. 60 ; 78 aswers () a a 0 a substitutio b c 8 b c 7 b c (i) 8 b c b c (ii) (ii) (i): b c 7 b c 7 b c 0 7 solvig simultaeousl b 7 c 0 geeral term (6) 8 0 th term of sequece with a 8 ad d Add both sides to terms [6 ( )] ( 7) 8 0 Add both sides sequece substitutio (6) Copright reserved Please tur over

8 Mathematics/P 8 DoE/November 008 NSC Memoradum a(0) b(0) c 0 c 0 costat secod differece a b 8 b 7 7 ( 7) fidig 0 c 0 secod differece a substitutio b 7 (6) [(0) ( )()] 8 0( ) ( )( ) 8 [ ( first first differece) ( )(secod differece) ] ( )( ) ( ) ( ) d differece ( )( ) ( )(8) ( )(8) formula substitutio simplificatio aswer (6) formula substitutio simplificatio aswer (6) Copright reserved Please tur over

9 Mathematics/P 9 DoE/November 008 NSC Memoradum ( )( ) ( )( ) ( )( ) formula substitutio simplificatio ( 6)(8) ( )(8) ( )(0) aswer (6) observatio aswer (6) B trial ad error: 6 / 6 Aswer ol: 6 / 6.. ( 7) ( )( ) 0 or th term is 0. / if did ot reject Aswer ol: / B trial ad error ad the write : / / if just equate that the foud If liear ad valid aswer : / substitutio stadard form factorisatio aswer () [] Copright reserved Please tur over

10 Mathematics/P 0 DoE/November 008 NSC Memoradum QUESION.. ( 8 ) 8 6 ratio. aswer ratio () < < < < iequalit aswer. S S S S a r substitutio ito formula for S substitutio of aswer () () S 8 S S 7 series substitutio aswer Formula Icorrect: 0 / () [7] Copright reserved Please tur over

11 Mathematics/P DoE/November 008 NSC Memoradum QUESION. p q a a a. c substitute ( ; ) c c 6 aswer p aswer q substitutio of (; ) aswer () Aswer for p mark Aswer for q mark Aswer for a marks correct poit ( ; ) substitutio aswer raslatio of the lie uits up ad uits right ( ) 6 () substitutio of addig aswer () Substitutio of ( ; ): 0 / Aswer ol: / [7] Copright reserved Please tur over

12 Mathematics/P DoE/November 008 NSC Memoradum QUESION. & h f g EXPONENIAL shape (must be icreasig above -ais) -it PARABOLA shape turig poit -itercept -itercepts (8) INVERSE/LOG -it shape (must be icreasig o the right of the - ais) () ( ; 8) If -itercepts ot show but correct o graph / for - itercepts. Calculatio of -itercepts of parabola 0 ( ) 8 0 ( ) 8 8 ( ) 0 ( ) 8 ( ) 0 6 or 0 or 0 ( )( ) or. ( ) () () Copright reserved Please tur over

13 Mathematics/P DoE/November 008 NSC Memoradum. h substitutio. h ( ) h( ) (. ). ( ) h( ). ( ) substitutio ( ) ( ) If umerical eamples are used : / () () [] Copright reserved Please tur over

14 Mathematics/P DoE/November 008 NSC Memoradum QUESION 6 6. aswer aswer If correct umbers but ot writig as a equatio / () 6. h( ) ta( ) h( ) ta( ) f ( ) h is the reflectio of f about the -ais h is the reflectio of f about the lie 0 If uits left out: / reflectio about -ais () reflectio about 0 If calculatio ol: / If aswer is: Reflectio ol: 0 / () If do calculatio ad sa reflectio: / Ol h( ) ta( ) h( ) ta( ) f ( ) / 6. si () [6] Copright reserved Please tur over

15 Mathematics/P DoE/November 008 NSC Memoradum QUESION 7 Pealise ONCE i questio 7 for earl roudig off. 7. A P( i) (.) (.) log(,), log,6968..,69 ears or ears or ears 8 moths or,7 ears (, ) formula substitutio appl log fuctio aswer () A P( i) ( (.) log(,) ) 00, log,6968..,69 ears or ears or ears 8 moths or,7 ears Accept ears : / Icorrect Formula: 0/ (, ) formula substitutio of 00 appl log fuctio aswer () A P( i) (.08) R76,6 R76,6 R R976,6 Some calculators give R 97 6,0 [( i) ] F i 60 [,0] 976,6 0,0 976, [,0] R 9,00 substitutio R 7 6,6 R 97 6,6 () Icorrect Formula: 0/ F R976,6 or aswer i i,0 formula simplificatio aswer (6) Copright reserved Please tur over

16 Mathematics/P 6 DoE/November 008 NSC Memoradum 60 [,0] 976,6 0,0 976,6 8, R 9,00 F R976,6 60 i,0 formula simplificatio aswer Cotiued Accurac applies. (6) Service [000(,0) 000(,0) 000(,0) 000(,0) Copright reserved 97,77 60 [,0] 976,6 Service 0,0 976,6 8, ,77 R 8, [,0 ] Service,0 97,77 R8, 60 [,0] 976,6 Service 0,0 976,6 8, ,77 Preset Value pamet of R (,0) (,0) (,0) { (,0) (,0) } 60 (,0) 000(,0) (,0) R 7 7, Preset Value of the sikig fud ,6(,0) R 6 9,9 otal Value of sikig fud R 7 7, R 6 9,9 R 666,9 60 (,0) 666,9 0,0 R 8, 000] 97,77 settig up of correct equatio aswer () 97,77 settig up of correct equatio aswer () 77, 666,9 settig up of correct equatio aswer () Please tur over

17 Mathematics/P 7 DoE/November 008 NSC Memoradum ( i i eff eff ) ( 0,0) 0, P( i) (,680) 000 0,680 97,77 60 [,0] 976,6 97,77 0,0 976,6 8, ,77 R 8, [(,0) ) 000 0, ,0,0 9, So mothl deposit must be icreased b R 9, New mothl deposit R 9 R 9, R 8, substitutio ito formula 97,77 settig up of correct equatio aswer R 8, () substitutio ito formula 9, settig up of correct equatio aswer R 8, () [7] Copright reserved Please tur over

18 Mathematics/P 8 DoE/November 008 NSC Memoradum QUESION 8 8. f ( h) f ( ) f ( ) lim h 0 h ( h) ( ) lim h 0 h 6h h lim h 0 h 6h h lim h 0 h h( 6 h) lim h 0 h lim( 6 h) h 0 6 defiitio f ( h) f ( ) lim h 0 h ( h) substitutio of correct aswer () Pealt for icorrect otatio If a cadidate has used the rules ol: 0/ d d 6 d d d d If removed coefficiets, or moved the umbers from the deomiator to the umerator: Cotiued accurac applies for each correct derivative Ma / d If leave out pealise mark. d Simplificatio or 6 () [8] Copright reserved Please tur over

19 Mathematics/P 9 DoE/November 008 NSC Memoradum QUESION 9 9. ( )( ) 0 or AB, uits ; aswer () ( )( ) 0 or AB (, ( ) (0 0) AB, uits ; aswer () 9. g ( ) ( )( ) 0 or at : 9. g ( ) 6 6 g ( ) 6( ) 6( ) g ( ) 8 g ( ) a q ( ) q q 6 6 g ( ) 6 6 g ( ) 6( ) 6( ) g ( ) 8 g ( ) ( ) 7 6 g ( ) 0 g ( ) 6 6 factorisatio aswer g () () method of settig up straight lie equatio substitutio of poit ( ; ) aswer i equatio form () g () formula substitutio of poit ( ; ) aswer i equatio form () Copright reserved Please tur over

20 Mathematics/P 0 DoE/November 008 NSC Memoradum 9. -coordiate of is -coordiate of (7) g() () () () 0 7 ( ; 7) 0 < k < 7 aswer () 0 k 0 k 0 7 < 0 k < 0 7 < k < 0 0 < k < 7 9. g ( ) 6 6 g ( ) g ( ) < 0 g ( ) > 0 g () chages sig at poit of iflectio at urig poits A(-;0); (;7) Now co-ordiate of poit of iflectio is 7 < 0 k < 0 aswer Aswer Ol: / 0 k 7 : / k > 0: / k < 7: / 6 0 poits () () () [8] Copright reserved Please tur over

21 Mathematics/P DoE/November 008 NSC Memoradum QUESION V 00 π r h 00 h π r Surface Area π rh π r 00 S( r) π r.π r π r 00 S( r) π r r S( r) π r 00r ds π r 00r dr ds At miimum : 0 dr 00 π r 0 r π r r π r,99 cm π r h formula substitutio formula substitutio epoets correct ds dr ds 0 dr π r 00r 00 r π r,99 or r 00 π If did ot put 0, pealise mark () () () d If otatio is, igore d otatio. [9] Copright reserved Please tur over

22 Mathematics/P DoE/November 008 NSC Memoradum QUESION , N 0. See attached graph () &. See attached graph () aswer aswer aswer (). feasible regio () If shadig ol, ad did ot state feasible regio /. P 00 0 aswer () P.6 P 0 Maimum at (0 ; 7) m Sice the gradiet of the ew profit fuctio is equal to the gradiet of the costrait 60, there are poits other tha (0 ; 7) that give a optimal solutio. () gradiet search lie aswer () Read correctl from the cadidate s graph for the poit for maimum profit. If used vertices method: / for accurate aswer. m more poits i optimal solutio (more tha oe solutio) () If just aswer Yes 0 / If just aswer No 0 / [6] Copright reserved Please tur over

23 Mathematics/P DoE/November 008 NSC Memoradum (0 ; 7) FEASIBLE REGION , N ACUNA Search Lies MAAA Copright reserved

NATIONAL SENIOR CERTIFICATE GRADE 12

NATIONAL SENIOR CERTIFICATE GRADE 12 NAIONAL SENI CERIFICAE GRADE MAHEMAICS P NOVEMBER 00 MEMANDUM MARKS: 50 his memoradum cosists of 7 pages. Mathematics/P DoE/November 00 QUESION.. x ( x 4) 5.... x 4x 5 0 ( x 5)( x + ) 0 x 5 or x 4x 0x