NATIONAL SENIOR CERTIFICATE GRADE/GRAAD 12
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1 NATIONAL SENIOR CERTIFICATE GRADE/GRAAD MATHEMATICS P/WISKUNDE V FEBRUARY/MARCH/FEBRUARIE/MAART 05 MEMORANDUM MARKS/PUNTE: 50 This memorandum consists of ages. Hierdie memorandum bestaan uit bladsye. Coyright reserved/koiereg voorbehou
2 Mathematics P/Wiskunde V DBE/Feb. Mrt. 05 NOTE: If a candidate answers a question TWICE, only mark the FIRST attemt. If a candidate has crossed out an attemt of a question and not redone the question, mark the crossed out version. Consistent accuracy alies in ALL asects of the marking memorandum. Assuming answers/values in order to solve a roblem is NOT accetable. NOTA: As 'n kandidaat 'n vraag TWEEKEER beantwoord, merk slegs die EERSTE oging. As 'n kandidaat 'n oging om die vraag te beantwoord, doodgetrek het en nie dit oorgedoen het nie, merk die doodgetrekte oging. Volgehoue akkuraatheid word in ALLE asekte van die nasienmemorandum toegeas. Aanvaarding van antwoorde/waardes om 'n robleem o te los, is ONaanvaarbaar. QUESTION/VRAAG. 0 0 Answer only: Full marks 57,6 slegs antw: volunte 57,6 (). ( ;,5 ; 5 ; 7 ; 89) and/ en 89,5 7 5., bo/mond whiskers/ snor (). ositively skewed/ositief skeef skewed to the right/skeef na regs answer/ antwoord ().5 σ 7,7 answer/ antwoord ().6. 57, 6 answer ().6. σ 7,7 answer/ antwoord () [] Coyright reserved/koiereg voorbehou
3 Mathematics P/Wiskunde V DBE/Feb. Mrt. 05 QUESTION/VRAAG. As the temerature increases, the sales of ice-creams increase/soos die temeratuur styg, neem die verkoe toe. reason/rede () As the temerature decreases, the sales of ice-creams decrease/soos die temeratuur daal, neem die verkoe af.. The liveable temerature cannot kee on increasing/die leefbare temeratuur kan nie aanhou styg nie.. a 60,5 b 0,09 y ˆ 0,09 60,5 yˆ 60,5 0, 09 Answer only: Full marks slegs antw: volunte reason/rede reason/rede 60,5 0,09 equation/vgl () (). r 0,96 0,96 ().5 There is a very strong ositive relationshi (correlation)/daar is 'n baie sterk ositiewe verband (korrelasie). very strong/baie sterk () [8] Coyright reserved/koiereg voorbehou
4 Mathematics P/Wiskunde V DBE/Feb. Mrt. 05 QUESTION/VRAAG y P(5 ; ) S(7,5 ; 8) Q( ; 5) O B α R.. PQ ( ) ( y y ) m PQ (5 ) 0 y y 5 5 ( ) 8 6 ( 5) Answer only: Full marks slegs antw: volunte use of distance formula/gebruik afstandformule correct subst into form/korrekte subst in formule 0 () correct subst into gradient formula/ korrekte subst in gradiëntformule gradient/gradiënt () Coyright reserved/koiereg voorbehou
5 Mathematics P/Wiskunde V 5 DBE/Feb. Mrt. 05. Equation of line RS/Vgl van lyn RS: m RS m PQ ( gradients, lines/gradiënte, lyne ) m RS y m c 5 8 c c y y 6 0 y y m( ) 5 y 8 y y 6 0 subst of S(7,5 ; 8) and m into eq /subst van S(7,5 ; 8) en m in vgl value of c /waarde van c or/of st form/st vorm equation/vgl. B is the -intercet of/is die -afsnit van y 0 (0) 6 0 y tan α α 5, OBˆ R (vert o s/regoorst e) O Rˆ B 80 (90 5, ) ( s of / e van ) 6, 87 BS ( ) ( y y ) 5 (8 0) 0 PQ BS and/en PQ BS PQBS arallelogram ( air o sides and / r tos sye en ) midoint of/midt van QS: midoint of/midt van PB: ; ; ; ; PQBS arallelogram (diags bisect each other/hoekl halv mekaar) tan α 5, 6,87 () () correct subst into form/korrekte subst in formule BS 0 BS PQ reason/rede ; ; ; reason/rede Coyright reserved/koiereg voorbehou
6 Mathematics P/Wiskunde V 6 DBE/Feb. Mrt. 05 m m m QB PS QB 5 0 5,5, ,5,5 m PS QB PS PQ BS PQBS arallelogram (both airs o sides /beide r tos sye ) m QB m PS QB PS reason/rede BS ( ) ( y y ) 5 (8 0) 0 QB ( ) ( y y ) PQ BS 5 5 (,5) (5 0) (,5) (5) or 5,59 PS 5 (5 7,5) ( 8) (,5) (5) or 5,59 QB PS PQBS arallelogram (both airs o sides / beide r tos sye ) correct subst into form/korrekte subst in formule PQ 0 QB PS reason/rede [8] Coyright reserved/koiereg voorbehou
7 Mathematics P/Wiskunde V 7 DBE/Feb. Mrt. 05 QUESTION/VRAAG y A E D M( ; ) C( ; ) O B.. Radius ( ) ( ) r Equation of circle/vgl van sirkel: ( ) ( y ) ( ) ( ) ( y ) ( ) r ( ) ( y ).. At/by D: D D D D(5 ; 6) 5 r r and/en y y By insection/deur inseksie: D(5 ; 6) y D D D 8 6 ( ) ( ) or/of ( ) ( y ) () ( ) ( y ) ( ) ( ) value/waarde y - value/waarde - value/waarde y - value/waarde () () () Coyright reserved/koiereg voorbehou
8 Mathematics P/Wiskunde V 8 DBE/Feb. Mrt m MC m m (Tangent radius/raaklyn radius) AB MC m AB y y m( ) y ( ) y.. At/by E: (0 ) ( y ) ( y ) 9 y ± y 7 or y E(0 ; 7) y m c ( ) c y m MC m m m AB AB MC subst m and ( ; ) into eq /subst m en ( ; ) in vgl eq in standard form/ vgl in st vorm (5) 0 simlification/ vereenvoudiging y - values/waardes E(0 ; 7)..5 At/by E: (0 ) ( y ) y 8y 6 y 8y 7 0 ( y 7)( y ) 0 y 7 or y E(0 ; 7) y y mem 7 0 mab EM AB ( m EM mab ) 0 simlification/ vereenvoudiging y - values/waardes E(0 ; 7) m EM reason/rede () Coyright reserved/koiereg voorbehou
9 Mathematics P/Wiskunde V 9 DBE/Feb. Mrt. 05. The centres of the circles are / Die middelunte van die sirkels is P( ; ) and / en Q(5 ; ) QP QP ( 5) ( ( )) 7 8,60 units r M rp 5 8 r M r < QP P The two circles do not intersect/die twee sirkels sny nie both centres/albei Midte QP correct subst into form/korrekte subst in formule distance between centres/afstand tussen midte r r < QP M P (6) [] Coyright reserved/koiereg voorbehou
10 Mathematics P/Wiskunde V 0 DBE/Feb. Mrt. 05 QUESTION/VRAAG 5 5. ² y² ( sin θ)² ( cos θ)² 9 sin² θ 9 cos² θ 9(sin² θ cos² θ) 9() 9 5. sin( 50 ).sin( ) cos(80 ).sin(90 ) sin( 80 ).sin( ) cos(80 ).sin(90 ) (sin )( sin ) ( cos )(cos ) sin cos cos 5.. OT yt sinα OT ² y² r² ² ² ( ) 5.. cos (80 α) cosα (P lies in rd quadrant) (P lies in rd quadrant) siml/vereenv CF/GF 9 answer/antw () sin( 50 ) sin sin( ) sin cos( 80 ) cos sin( 90 ) cos sin cos cos (6) OT sin α y T OT ² y² r² subst cos α answer/antw () () () Coyright reserved/koiereg voorbehou
11 Mathematics P/Wiskunde V DBE/Feb. Mrt. 05 Coyright reserved/koiereg voorbehou 5.. α cos α α cos sin cos α α sin cos α cos α eansion/ uitbreiding squaring each term/kwadreer elke term () eansion/ uitbreiding squaring/kwadrering writing as single fraction/skryf as enkelterm () eansion/ uitbreiding squaring/kwadrering writing as single fraction/skryf as enkelterm ()
12 Mathematics P/Wiskunde V DBE/Feb. Mrt The identity is undefined for/die identiteit is ongedefinieerd as: sin 0 sin 0: 0 ; 80 or/of tan : 90 0 ; 90 ; 80 tan sin 5.. LHS/LK sin sin sin cos cos sin sin sin cos cos sin sin ( cos ) cos sin sin (sin ) cos sin sin cos tan RHS/RK tan sin LHS/LK sin sin sin cos cos cos sin cos sin sin cos sin cos sin ( cos ) sin cos sin.sin sin cos sin cos tan RHS/RK sin cos sin.cos () simlify numerator/ vereenv teller factorising/fakt cos sin simlify to/vereenv sin na cos sin cos sin.cos siml/vereenv factorising/fakt cos sin (6) simlify to /vereenv sin na cos (6) [6] Coyright reserved/koiereg voorbehou
13 Mathematics P/Wiskunde V DBE/Feb. Mrt. 05 QUESTION/VRAAG 6 6. T A 90 K F 6.. In TAK: AK sin KTˆ A KT AK KT.sin sin sin KTˆ A sin KÂT AK KT sin 90 sin AK AK sin correct trig ratio/ korrekte trigverh. answer/antw correct subst into sine rule/korrekte subst in sin-reël answer/antw () () Coyright reserved/koiereg voorbehou
14 Mathematics P/Wiskunde V DBE/Feb. Mrt In AKF: KF AK sinkâf sinafˆk KF AK sin(90 ) sin AK.sin(90 ) KF sin sin.cos sin.cos using sine rule/ gebruik sin-reël correct subst into sine rule/korrekte subst in sin-reël sin(90 ) cos sin.cos (5) In AKF: KF AK sinkâf sinafˆk KF AK sin(90 ) sin AK.sin(90 ) KF sin AT.tan.cos sin.cos sin cos..cos cos sin.cos AT cos AT cos using sine rule/ gebruik sin-reël correct subst into sine rule/korrekte subst in sin-reël sin(90 ) cos sin.cos (5) Coyright reserved/koiereg voorbehou
15 Mathematics P/Wiskunde V 5 DBE/Feb. Mrt A 5 O 0 B 5 C 6.. In BOC: BC² BO² CO².BO. CO.cos 5² 0² 0² (0)(0). cos 00 cos 5 cos 0,5 80 8,8 97,8 Draw a line OD BC: BD DC (line from centre on chord) OBD OCD (90 ; h; s) 7, 5 sin 0 8,59 97,8 using cosine rule/ gebruik cos-reël correct subst/ korrekte subst cos 0,5 97,8 /R correct ratio/ korrekte verh value of/waarde van 97,8 Coyright reserved/koiereg voorbehou
16 Mathematics P/Wiskunde V 6 DBE/Feb. Mrt B ÂC 8, 59 ( at centre at circ/ by midt omt) A Bˆ C BÂC 8, 59 ( 's o equal sides/ e teenoor sye) A ĈB 8,8 (sum of s of /som van e van ) AĈB AÔB ( at centre at circle) ( by midt omt) [60 (97,8 )] 8,8 O ĈB (80 97,8 ) ( 's o equal sides; sum of s of ) ( e teenoor sye; som van e van ), y 8,8 8,8 () () y y y AĈB (, ) 8,8 6.. Area/Oervlakte ABC (BC)(AC)sin AĈB (5)(5)(sin 8,8 ),6 cm 8,8 correct subst into area rule/korrekte subst in o-reël,6 cm () () [6] Coyright reserved/koiereg voorbehou
17 Mathematics P/Wiskunde V 7 DBE/Feb. Mrt. 05 QUESTION/VRAAG 7 A M O B P 7. MB 0 cm answer/antw 7. line from centre to midoint of chord is erendicular to chord/lyn vanaf midt na midt van koord is loodreg o koord answer/antw () () 7. line from centre bisects chord/lyn vanaf midt halveer koord MP 5 OM OP 5 OP 5 OP OP OM OP answer/antw OP OP OP OM OP 5 () () () Coyright reserved/koiereg voorbehou
18 Mathematics P/Wiskunde V 8 DBE/Feb. Mrt OM² MB² OB² subst into/subst Pythagoras ,9 or 5 or 80 answer/antw () [7] Coyright reserved/koiereg voorbehou
19 Mathematics P/Wiskunde V 9 DBE/Feb. Mrt. 05 QUESTION/VRAAG 8 A D B E 0 O C Dˆ Ô 55 ( at centre at circ/ by midt by omt) S R () Â Ô 55 ( at centre at circ/ by midt by omt) S R () Â Dˆ 55 ( s in same segment/ e in dieselfde segment) 8.. Bˆ Dˆ 55 (alternate s/verwiss e; AB DC) Ê Bˆ Â (et of sum of o s /buite v som v tos e) Ê 8. Ê Ô 0 (roven in/bewys in 8..) BEOC is a cyclic quadrilateral (equal s subtended by line/ gelyke e ondersan deur lyn) S R S R R answer/antw S R () () [0] Coyright reserved/koiereg voorbehou
20 Mathematics P/Wiskunde V 0 DBE/Feb. Mrt. 05 QUESTION/VRAAG 9 9. the interior oosite angle/die teenoorstaande binnehoek. answer/antw () 9. S C O V P T R Construction: Draw diameter CT and join CV. Konstruksie: Trek middellyn CT en verbind CV. Vˆ Vˆ 90 in semi-circle/ in halfsirkel R Tˆ Ĉ 90 Tangent diameter/radius/raaklyn middellyn/radius Sum of the angles of triangle/som van die hoeke van 'n driehoek R Ŝ 's same segment/ e in dieselfde segment R V Tˆ R Ŝ (5) Coyright reserved/koiereg voorbehou
21 Mathematics P/Wiskunde V DBE/Feb. Mrt P V T 0 R 50 W S Q Z 9.. Equal chords subtend equal s/gelyke koorde ondersan gelyke e R 9.. Ŵ 0 (tan chord theorem/rkl-koordst) Ŵ 0 9..(a) Rˆ Ŵ 50 (tan chord theorem/rkl-koordst) Ŝ Rˆ (et of /buite v ) Ŵ Ŝ 80 () answer/antw reason/rede answer/antw () R () Rˆ Rˆ Rˆ 0 ( chords subtend s / kde ondersan e) Ŵ 50 (tan chord theorem/rkl-koordst) Ŝ 80 R () Coyright reserved/koiereg voorbehou
22 Mathematics P/Wiskunde V DBE/Feb. Mrt (b) Tˆ Ŝ 80 (et of cyclic quad/buite van koordevh) V Ŵ Tˆ (et of /buite van ) Vˆ In RVW and/en RWS: Rˆ Rˆ 0 (roven/bewys in 9..) Vˆ Ŵ 50 (roven/bewys in 9..) V ŴR Ŝ ( rd in ) RVW RWS ( ) R using the correct s/ gebruik korrekte e R ( rd in ) or ( ) WR RS RV WR WR RV. RS ( RVW RWS) (5) [] Coyright reserved/koiereg voorbehou
23 Mathematics P/Wiskunde V DBE/Feb. Mrt. 05 QUESTION/VRAAG 0 T N R P M 0.. corresonding s/ooreenkomstige e; PN RT answer/antw 0.. ; ; ; answer/antw PM PN ( ΔPNM ΔRTM ) RM RT PN PN PM RP RM RM RN² PN² (RM² NM²) (PM² NM²) RM² PM² RP RP 9 RP RP RP² (Pyth) Use of Pyth. for RN and PN RM RP PM RP 9 RP & () () () RP Coyright reserved/koiereg voorbehou
24 Mathematics P/Wiskunde V DBE/Feb. Mrt. 05 RN PN (RM NM ) (PM NM ) RM PM (PM) PM 8PM (PM) RP RN PN (RM NM ) (PM NM ) RM PM (RP PM) PM RP RP.PM PM PM RP RP. RP RP (Pyth) (Pyth) Use of Pyth. for RN and PN RM RP PM (PM) PM RP PM Use of Pyth. for RN and PN RM RP PM eansion/ uitbreiding PM RP [8] TOTAL/TOTAAL: 50 Coyright reserved/koiereg voorbehou
NATIONAL SENIOR CERTIFICATE GRADE/GRAAD MATHEMATICS P/WISKUNDE V NOVEMBER 0 MEMORANDUM MARKS/PUNTE: 0 This memorandum consists of 3 pages. Hierdie memorandum bestaan uit 3 bladsye. Please turn over/blaai
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