NATIONAL SENIOR CERTIFICATE GRADE /GRAAD MATHEMATICS P/WISKUNDE V NOVEMBER 05 MEMORANDUM MARKS: 50 PUNTE: 50 Tis memoradum cosists of 5 pages. Hierdie memoradum bestaa uit 5 bladsye. Copyrigt reserved/kopiereg voorbeou Please tur over/blaai om asseblief

Matematics/P/Wiskude V DBE/November 05 NSC/NSC Memoradum NOTE: If a cadidate aswers a questio TWICE, oly mark te FIRST attempt. Cosistet accuracy applies i ALL aspects of te markig memoradum. LET WEL: Idie ' kadidaat ' vraag TWEE keer beatwoord, merk slegs die EERSTE pogig. Volgeoue akkuraateid is op ALLE aspekte va die memoradum va toepassig. QUESTION/VRAAG.. 9 0 0.. ( 4)( 5) 0 4 or 5 5 4 0 5 ± (5) () 4()( 4) 5 ± 7 6,6 or 0,59 OR/OF factors 4 5 () stadard form substitutio ito correct formula aswers (4).. 5 5 5 5 6 5 6 5 ( ) 64 OR/OF 5 or 5 6 4 ± 7 6 5 6 7 6 5 ± 6,6 8 or 0,5 7 or 0,59 5 for addig o 6 bot sides 5 ± 6 aswers 7 dividig bot sides by 5 or 6 64 raisig RHS to aswer (4) 5 (4) Copyrigt reserved/kopiereg voorbeou Please tur over/blaai om asseblief

Matematics/P/Wiskude V DBE/November 05 NSC/NSC Memoradum 5 5 64 ( ) 5 OR/OF 5 or 8 or 0,5 dividig bot sides by raisig RHS to 5 aswer 5 (4)..4 5 5 5 64 0,659 0,085 OR/OF 5 0,5 5 log log log log 5 log 0,90 0 0,90 0,5 or ( ) 8 4 4 raisig bot sides to 5 0,659 ad 0,085 dividig bot sides by 0,659 aswer (4) dividig bot sides by logs o bot sides log 0, 90 aswer squarig bot sides (4) 0 ( )( ) OR/OF 0 if, or oly ad factors or oly (4) Copyrigt reserved/kopiereg voorbeou Please tur over/blaai om asseblief

Matematics/P/Wiskude V 4 DBE/November 05 NSC/NSC Memoradum if, OR/OF ( ) ( ) or or 0 oly ad squarig bot sides or 0 or oly (4) 0 ad 0 0 0..5 ad ( 7) < 0 oly 7 < 0-7 0 OR/ OF 7 < < 0 OR /OF -7 ( 7; 0) 0 ad factors iequality or iterval (4) (). Te square of ay umber is always positive or zero So for te sum of two squares to be zero, bot squares must be zero, i.e. Die kwadraat va eige getal is altyd positief of ul. Vir die som va twee kwadrate om ul te wees, moet beide die kwadrate ul wees, d.i. ( y) 0 ad/ e ( 5) 0 y 0 5 0 y 0 ( ) 5 y 0 y 5 ad/ e 5 0 5 5 y 5 (4) Copyrigt reserved/kopiereg voorbeou Please tur over/blaai om asseblief

Matematics/P/Wiskude V 5 DBE/November 05 NSC/NSC Memoradum. b 4 k 0 OR/OF k 4ac < 0 ( )( k) < 0 < 0 4k < 0 k < 4 stadard form < 0 4( )( k) k < 4 (4) for oreal roots OR/OF k 4 k k < 0 4 k < 4 k 4 4 adds 4 to bot sides k 4 k < 0 4 k < 4 (4) Cosider te fuctios y ad y k Beskou die fuksies y e y k y y sketc or eplaatio y k Turig poit of/draaiput va y is ; 4 k does ot ave real roots we te lie y k does ot itersect y. k et gee reële wortels as die ly y sy ie. Terefore k < 4 y k ie met y 4 k < 4 (4) [6] Copyrigt reserved/kopiereg voorbeou Please tur over/blaai om asseblief

Matematics/P/Wiskude V 6 DBE/November 05 NSC/NSC Memoradum QUESTION/VRAAG. T r T T 5 5 0,5 5 8 or 0,65. T 0 OR/OF. For covergece/om te kovergeer < r < Sice/Aagesie r ad/e < < te sequece coverges/die ry kovergeer T 5 4 0 5 or 0,65 8 r aswer () substitutes a 0 ito GP formula substitutes r ito GP formula () < r < sow tat r is < r <.4 a a( r ) S - S r 0 0 0 0 r 0 0 0 0 0 0 0 aswer () (4) OR/OF Copyrigt reserved/kopiereg voorbeou costructig te series Please tur over/blaai om asseblief

Matematics/P/Wiskude V 7 DBE/November 05 NSC/NSC Memoradum Copyrigt reserved/kopiereg voorbeou Please tur over/blaai om asseblief T T T 0 0 4 0 - S S OR/OF ( ) r ar r ar a a r r a r a 0 0 - S S 4 0 aswer (4) r ar a a r ar 0 aswer (4) [0]

Matematics/P/Wiskude V 8 DBE/November 05 NSC/NSC Memoradum QUESTION/VRAAG. d 8 a. T k k ( k ) d ( k )( 8) 8k 8 8k ( 8k ) OR/ OF ( 8( k ) ) k 0 k 0 (8k ) d value aswer () for geeral term lower ad upper values i sigma otatio (). S (4 7) [ a ( ) d] [ ( ) ( )(8) ] [ 6 8 8] [ 8 4] formula substitutio [ 8 4] () 4 7 OR/OF S 4 [ a ( ) d] [ ( ) ( )(8) ] [ 6 8 8] [ 8 4] 7 formula substitutio [ 8 4] () OR/OF S 4 [ a l] [ 8 ] [ 8 4] 7 Copyrigt reserved/kopiereg voorbeou formula substitutio [ 8 4] () Please tur over/blaai om asseblief

Matematics/P/Wiskude V 9 DBE/November 05 NSC/NSC Memoradum OR/OF S S S S 4 5 6 5 8 8 S S 5 5 S4 5 calculates S, S, S ad S 4, S S a a S a 8 4 b c 4 b c 6 b c Hece S 4 7 b c 7...() b c 4...() b 7...() () c 0 a 4 solves simultaeously for b ad c. ().4. Q 6 6 5 9 aswer ().4. Q 9 6 S 8 6 4(8) 7(8) 6 4(8) 7(8) 6464 OR/OF Q Q Q Q 4 6 9 4 9 5 aswer () Q Q Q a 4 a 4 b c 6 6 b c 9 Copyrigt reserved/kopiereg voorbeou b c Hece 4 5 5 Q 9 Q ( ) 5( 9) 4 9 64 64 8 8 8 b c 0...() b c 5...() b 5...() () 5 c 5 a 4 4 5 5 Q aswer () [] Please tur over/blaai om asseblief

Matematics/P/Wiskude V 0 DBE/November 05 NSC/NSC Memoradum QUESTION/VRAAG 4 Give: f ( ) 8 4. y 8 y 8 4. y () 8 6 4 f -8-6 -4-0 4 6 8 - -4 -itercept y-itercept sape asymptote (oly if te grap does ot cut te asymptote) -6-8 y -8 (4) 4. g ( ) 8 OR/OF g ( ) 8 aswer aswer () () [6] Copyrigt reserved/kopiereg voorbeou Please tur over/blaai om asseblief

Matematics/P/Wiskude V DBE/November 05 NSC/NSC Memoradum QUESTION/VRAAG 5 for 4. Give ( ) y Q - O P 4-5. For -itercepts, y 0 0 5. 5.,5 Q(,5 ; 0) : : y ( ) 7 4 : y (4) 5 Domai of : 7 5 OR/OF [ 7; 5] -7 y,5 0-5, 5 y 0 ( ) 7 (4) 5 () 7 5 () y-itercept o a straigt lie lie segmet OR/OF 4 y accurate edpoits ( or y or bot) () -,5 0 - Copyrigt reserved/kopiereg voorbeou Please tur over/blaai om asseblief

Matematics/P/Wiskude V DBE/November 05 NSC/NSC Memoradum 5.4 ( ) For te iverse of, y y ( ) ( ) ( ) 4 6 OR/OF y () ( ) ad ( ) itersect we y OR/OF ( ) For te iverse of, y y ( ) ( ) y () () Copyrigt reserved/kopiereg voorbeou Please tur over/blaai om asseblief

Matematics/P/Wiskude V DBE/November 05 NSC/NSC Memoradum 5.5 OP ( 0) ( y 0) ( ) 4 9 5 9 For OP to be at its miimum, OP as to be a miimum Vir OP om miimum te wees, moet OP ' miimum wees d( OP ) b 0 OR / OF d a 0 0 5 6 6 Miimum legt of OP 5 9 5 5 OR/OF For miimum distace OP te lie m (give) 5 6 OP, (, ) 6 5 mop OP as equatio y 4 6 y P P (, 0) ( 0,6 0),4 or 0,6,8 uits 9 5 or ( ) 5 or,4 uits OP y substitute y 5 9 -value aswer m OP equatio of OP (5) -value aswer (5) Copyrigt reserved/kopiereg voorbeou Please tur over/blaai om asseblief

Matematics/P/Wiskude V 4 DBE/November 05 NSC/NSC Memoradum OR/OF For miimum distace OP te lie O ( 0;0) P( ; ) Q ; 0 OP 9 9 4 9 4 4 0 7 8 0 ( 0) ( 0) ( 0) 4 9 6 Hece, at P 5 OP OP 6 5 6 5 9 5,4 ( ) 6 5 9 5 PQ ( 5 6)( ) OQ 0 6 5 or (pytag) OP y substitute y 0 7 8 -value aswer (5) OR/OF For miimum distace OP te lie ta Qˆ ˆ Q 6,4 OP si 6,4,5 OP,4 ta Q ˆ Q ˆ 6,4 si 6,4 OP, 5 aswer (5) Copyrigt reserved/kopiereg voorbeou Please tur over/blaai om asseblief

Matematics/P/Wiskude V 5 DBE/November 05 NSC/NSC Memoradum OR/OF OP ( 0) ( y 0) ( 0) ( 0) 4 9 5 9 By usig te cai rule (wic is ot i te CAPS): dop ( ) 5 9 ( 0 ) d 0 ( 5 9) ( 0 ) 0 ( 0 ) 0 5 6 6 5 5 OP 5 6,4 6 9 5 ( 0) ( 0 ) OP y substitute y 5 9 -value aswer (5) OR/OF For miimum distace OP te lie Let te y-itercept be R OR uits O,5 Q OQ uits RQ 5 (Pytagoras) R P RQ 5 Area OQR base eigt.or.oq. 5.OP... 5.OP OP,4 5. 5. OP.. equatig aswer (5) Copyrigt reserved/kopiereg voorbeou Please tur over/blaai om asseblief

Matematics/P/Wiskude V 6 DBE/November 05 NSC/NSC Memoradum 5.6. f ( ) Turig poit at f ( ) > 0 or f > 0 f as a local miimum at f et ' lokale miimum by OR/OF Turig poit at f ( ) > 0 () ( ;, ) ( ) f ( ) < 0 for 5 f is decreasig o te left of Q / f is daled liks va Q. ( ) f ( ) > 0 for (,5 ; 4) f is icreasig o te rigt of Q / f is styged regs va Q. f () as a local miimum we / f () et lokael miimum by decreasig left of Q icreasig rigt of Q () OR/OF f ( ) c 5.6. ( ) 5 f ( ) c f as a miimum value sice a > 0 f et ' miimum waarde omdat a > 0 eplaatio () m f ( 4) 4 aswer () [9] Copyrigt reserved/kopiereg voorbeou Please tur over/blaai om asseblief

Matematics/P/Wiskude V 7 DBE/November 05 NSC/NSC Memoradum QUESTION/VRAAG 6 6.. T (0;8) ( 0;8) 6.. 8 0 0 ( )( ) Q(; 0) y 0 factors () () OR/OF 9 Q(; 0) 8 0 y 0 9 () 6.. -coordiate of S is 4,5/-koördiaat va S is 4,5 By symmetry about te lie 4,5/Deur simmetrie om die ly 4,5: R (4,5 4,5 ; 0) (6 ; 0) 6..4 For all R OR/OF ( ) 6. If C ( ; y) is te cetre of te yperbola/as C ( y) die iperbool y 6 ad y 6 4 y 6 y 0 () ; aswer () ; die middelput is va asymptote y 4 y 4 asymptote - 0 sape (icreasig yperbolic fuctio) (4) [] Copyrigt reserved/kopiereg voorbeou Please tur over/blaai om asseblief

Matematics/P/Wiskude V 8 DBE/November 05 NSC/NSC Memoradum QUESTION/VRAAG 7 7. R450 000 aswer 7. A P( i) f ( ) 450000( i) 4 76,90 450000( i) 4 substitutio of 450 000 ito correct formula () substitutio of ( 4 ; 4 76,90) ito correct formula i 4 i 0,4 4 76,90 450000 makig i te subject 7. 7.4 Te rate of depreciatio is4,% p.a. Die waardevermiderigskoers is 4,% p.j. At T : A P ( i) g( ) 450000( i) a 450000 ( 0,08) R64490,66 Future Value R64 490, 66 R4 76,90 R70 75,76 Let be te value of motly paymet [ ( i) ] Fv i 6 0.06 7075,76 0.06 R997, 4 aswer (4) i 0,08 & 4 correct substitutio ito formula aswer R70 75,76 0,06 i 6 () substitutio ito correct formula aswer (5) [] Copyrigt reserved/kopiereg voorbeou Please tur over/blaai om asseblief

Matematics/P/Wiskude V 9 DBE/November 05 NSC/NSC Memoradum Copyrigt reserved/kopiereg voorbeou Please tur over/blaai om asseblief QUESTION/VRAAG 8 8. ( ) lim ) ( lim lim ) ( ) ( lim ) ( ) ( ) ( ) ( ) ( ) ( ) ( 0 0 0 0 f f f f f f OR/OF ) lim( ) ( lim lim lim ) ( ) ( ) ( lim ) ( ) ( lim ) ( 0 0 0 0 0 0 f f f fidig f() formula factorisatio aswer (5) formula fidig f() factorisatio aswer (5) 8.. 5 4 4 4 4 4 4 d dy y y OR/OF 4 4 4 4 5 ()

Matematics/P/Wiskude V 0 DBE/November 05 NSC/NSC Memoradum Copyrigt reserved/kopiereg voorbeou Please tur over/blaai om asseblief By usig te cai rule (wic is ot part of CAPS): ( ) ( )( ) ( ) ( ) 5 5 5 4 4 d dy y ( )( ) () 8.. [ ] ) )( ( D D OR/OF By usig te quotiet rule (wit is ot part of CAPS): ( ) ( ) ( ) D factorisatio () ( ) ( ) ( ) () []

Matematics/P/Wiskude V DBE/November 05 NSC/NSC Memoradum QUESTION/VRAAG 9 9. Substitute Q(; 0) ito ( ) a b 9. lie a( ) b() 0 8 4a b 0 () a b 9 At Q : ( ) () 0 a() b 0 4a b 0 4a b lie : a a Substitute i lie : b 6 f...lie ( ) ( ) ( ) 6( ),5 a b...lie Average gradiet/gemiddelde gradiët f ( Q ) f ( P ) Q P substitute Q ito fidig derivative () equatig derivative to 0 solvig simultaeously for a ad b f ( ),5 formula (5) Average gradiet/ Gemiddelde gradiët 0 (,5) ( ) 4,5 substitutio aswer (4) Copyrigt reserved/kopiereg voorbeou Please tur over/blaai om asseblief

Matematics/P/Wiskude V DBE/November 05 NSC/NSC Memoradum 9. ( ) 6 ( ) 6 ( ) ( ) 6 ( ) 6 For Vir <, is cocave up ad for <,is kokaaf a bo e vir cocavity cages at / kokwiteit verader by 9.4 Te grap of as a poit of iflectio at Die grafiek va et ' buigput by OR/OF >, is cocave dow >,is kokaaf a oder. / Te grap of cages from cocave up to cocave dow at / Die grafiek va verader by va kokaaf op a kokaaf af eplaatio // usig ( ) aswer aswer () () () 9.5 Gradiet of g is /Gradiët va g is Gradiet of taget is/gradiët va die raakly is: ( ) 6 ( ) 6 8 0 6 0 ( )( ) 0 oly ( ) 6 ( ) factors selectio of -value (4) [7] Copyrigt reserved/kopiereg voorbeou Please tur over/blaai om asseblief

Matematics/P/Wiskude V DBE/November 05 NSC/NSC Memoradum QUESTION/VRAAG 0 0. 0. 0 0 ta 60 ta 60 r r r 0 ta 60 r aswer V coe π r π π 9 dv π d dv π d 9 7π ( 9) or 84,8 cm /cm formula () substitutio of te value of r i terms of simplified volume aswer derivative aswer (5) [7] Copyrigt reserved/kopiereg voorbeou Please tur over/blaai om asseblief

Matematics/P/Wiskude V 4 DBE/November 05 NSC/NSC Memoradum QUESTION/VRAAG. P(A) P(B) 0, 0,6 0,6 i.e. P(A) P(B) P(A ad B) Terefore A ad B are idepedet/dus is A e B oafaklik 0, 0,6 P(A) P(B)P(A ad B) coclusio ().. 7 7 8 54 7 7 ().. 7!5040 7!.. Tere are vowels optios for first positio Tere are 4 cosoats 4 optios for last positio 4 Te remaiig 5 letters ca be arraged i 5 4 ways 5 4 ( 5 4 ) 4 440 aswer () Daar is klikers opsies vir die eerste posisie Daar is 4 kosoate 4 opsies vir die laaste posisie Die oorblywede 5 letters ka as volg geragskik word 5 4 ways/maiere ( 5 4 ) 4 440 (4) Copyrigt reserved/kopiereg voorbeou Please tur over/blaai om asseblief

Matematics/P/Wiskude V 5 DBE/November 05 NSC/NSC Memoradum. Orage OO t t t Orage Yellow t t t t t Yellow Orage OY YO P(O) P(Y) t t t t Yellow YY 5 P(Orage,Orage) P(Yellow,Yellow) 00 t t 5 t t t t 00 t t 4t 4 t t 5 t 4 ( t 4) ( t 4t 4) t 0 ( t 4)( t ) 5 0 t Tere are orage balls i te bag/daar is oraje balle i die sak 4 4 TOTAL/TOTAAL: P(O,O) t t P(Y,Y) t t t t t t t t (o ca) 5 00 (6) [7] 50 marks Copyrigt reserved/kopiereg voorbeou