NATIONAL/NASIONALE SENIOR CERTIFICATE/SERTIFIKAAT GRADE/GRAAD 12

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1 NATIONAL/NASIONALE SENIOR CERTIFICATE/SERTIFIKAAT GRADE/GRAAD MATHEMATICS P/WISKUNDE V NOVEMBER 06 MEMORANDUM MARKS/PUNTE: 0 This memorandum consists of 6 pages. Hierdie memorandum bestaan uit 6 bladsye.

2 Mathematics P/Wiskunde V DBE/November 06 NOTE: If a candidate answered a question TWICE, mark only the FIRST attempt. If a candidate has crossed out an attempt to answer a question and did not redo it, mark the crossed-out version. Consistent accuracy applies in ALL aspects of the marking memorandum. Stop marking at the second calculation error. Assuming answers/values in order to solve a problem is NOT acceptable. LET WEL: Indien 'n kandidaat 'n vraag TWEE keer beantwoord het, sien slegs die EERSTE poging na. As 'n kandidaat 'n poging om 'n vraag te beantwoord, doodgetrek en nie oorgedoen het nie, sien die doodgetrekte poging na. Volgehoue akkuraatheid is op ALLE aspekte van die memorandum van toepassing. Staak nasien by die tweede berekeningsfout. Om antwoorde/waardes om 'n probleem op te los, te veronderstel, word NIE toegelaat NIE.

3 Mathematics P/Wiskunde V DBE/November 06 QUESTION/VRAAG Distance from the store in km Afstand vanaf die winkel in km Average number of times shopped per week Gemiddelde aantal keer wat kopers die winkel per week besoek Average number of times shopped per week SCATTER PLOT/SPREIDIAGRAM Distance from the store in km. Strong/Sterk. 0,9 ( 0,946..). a,7 (,7 ) b, (,76 ) y ˆ, +,7.4 y ˆ,(6) +, 7 times value of a value of b equation/vgl substitition. On scatter plot/op spreidiagram A line close to any of the following points: ( ; 6) or (0 ; ) or (6 ; ) or (0 ;,7) () () () () [9]

4 Mathematics P/Wiskunde V 4 DBE/November 06 QUESTION/VRAAG. Positively skewed OR skewed to the right/positief skeef OF skeef na regs (). Range/Omvang,,9 0,8 m subtract values answer (). Intervals Cumulative frequency Klasse Kumulatiewe frekwensie 9,,, <, 4 6, <,7 9,7 <,9 60,9 <, 6, <, 60 ().4 upper OGIVE/OGIEF limits / 70 6 boonste 60 limiete 0 cum f/ 4 kum f 40 shape/ 0 vorm 0 grounded 0 geanker Number of learners Heights (m). method (using 80 to determine the height),6 (accept any value between,6 and,69).6. The mean would change by 0, m Die gemiddelde sal met 0, m verander.6. No influence/change as there is no difference in variation of data./geen invloed /verandering aangesien daar geen verskil in die variasie van die data is nie. Q (4) method () () () []

5 Mathematics P/Wiskunde V DBE/November 06 QUESTION/VRAAG D y y + 6 A( 7 ; ) M O < α F(p ; 0) C(6 ; ) G. M Midpt of AC [diags of rectangle bisect/ hoekl v reghoek halveer] M ; M ;. 0 mbc 6 p 6 p 0 m BC p 6 p 6. m AD m BC [AD BC] m BC 6 p p p 4 y y ( ) C(6 ;) y ( 6) y 9 but y 0 4 p B -value of M y-value of M answer answer m BC equating answer m BC substituting (6 ; ) answer () () ()

6 Mathematics P/Wiskunde V 6 DBE/November 06 y + c + c 9 c y p 9.4 DB AC [diag of rectangle / hoekl v reghoek ] AC ( ) + ( y y AC AC (6 + 7) + + ( ) AC 70 DB 70 or,04. tanα mbc α 6,4 ).6 In quadrilateral OFBG: O FˆB 6,4 [vert opp s/regoorst e] F ÔG GBˆ F 90 O ĜB 60 [ ,4 ] [sum s quad/som e vierh 60 ] O ĜB 6, 7 m AB 90 + O ĜA,4 O ĜA 6,4 O ĜB 80 6,4 6,7 F ÔG GBˆ F 90 GOFB is cyc quad O ĜB 80 6,4 [ s of cyc quad 80 ] 6,7 OFˆB 6,4 XÔG FBˆ G 90 OGBF is a cyclic quad OĜB 80 6,4 OĜB 6,7 m BC substituting answer substitution length of AC AC BD tanα mbc α 6,4 () size of m AB O Fˆ B

7 Mathematics P/Wiskunde V 7 DBE/November 06.7 M ; is the centre/is die middelpunt 70 r radius [BD is diameter/middellyn] y 4,.8 C Bˆ M BÂM 4 [diag of square bisect s/hoekl v vierk halv e] BC will be a tangent [converse tan chord th/omgekeerde raakl-koordst] A Mˆ B 90 [diag of square bisect ] AB is diameter BC AB BC is tangent A [ line radius or converse tan-chord th] M B C M is centre r 70 equation S R S R () () [9]

8 Mathematics P/Wiskunde V 8 DBE/November 06 QUESTION/VRAAG 4 y L T( ; 7) M R S( ; ) O K(a ; b) 4. in semi circle/ at centre on circle in halfsirkel / by middelpt op sirkel R () mts mts mrs mrs y + c () + c c 0 y [TS SR] substitution m TS m RS substitution m and ( ; ) equation ()

9 Mathematics P/Wiskunde V 9 DBE/November m TS m y m RS y y ( ) y ( ) RS r 6 4 TR. r 6 4 [TS SR] 4 TM ( 9) TR. r M 9 ; 6 R + yr and 6 R( ; 6) M 9 ; 6 R ; 6 R( ; 6) 4 Answer only: full marks Answer only: only coordinate correct ( mark) m RS substitution m and ( ; ) equation r substitution M coordinate y coordinate M coordinate y coordinate () ()

10 Mathematics P/Wiskunde V 0 DBE/November m TM TM : SR : y y () 6 ST + c...() y ( ) + ( y ) ST y ( ) + ( 7) ST TS 9 sin R or TR 4 TS 9 SR 9 or ( 9)( 9) area of TSR 9 ( 4)( 9) sin R sin R or 7 6 m TR 9 mtr mktl mktl y y ( ) y 7 ( ) y 9 substitute K( a; b) : b a 9 + or ,4 [ r tangent]...() m TR 7 6 equating coordinate y coordinate substitution ratio area rule ratio m TR m KTL y 9

11 Mathematics P/Wiskunde V DBE/November m TR 9 mtr mktl b 7 a b 7 ( a ) b a 9 [ r tangent] m TR m KTL substitution ( ; 7) & (a ; b) 4.4. KR TR + TK ( a ) + ( b 6) ( ) + (6 7) + ( a ) + ( b 7) 0a + b a + 9 4b + 49 b 4a 8 b a 9 TK TR ( a - ) + ( b 7) 4 ( a - ) + ( b 7) 4 Substitute b a 9 [from 4.4.4] ( a - ) + (a 9 7) 4 a ( a - ) + (a 6) 4 6a a 864a a 870a ± (870) 4(4)(60) a 90 a or a 4 b () 9 or b (4) 9 9 K( ; ) subst into Pyth multiplication simplification substitution into distance formula substitution of b a 9 standard form subst into formula or factorise values of a value of b (6)

12 Mathematics P/Wiskunde V DBE/November 06 ( a ) + ( b 7) TK TR 4 ( a ) + ( b 7) 4 Substitute b a 9 [from 4.4.4] ( a ) ( a ) ( a ) a or 4 b () 9 or b (4) 9 9 K( ; ) + (a 9 7) + (a 6) + 44( a ) ( a ) a ± substitution into distance formula substitution of b a 9 ( a ) ± values of a value of b (6) KR ( a ) ( a ) ( a ) a 4a a TR + ( b 6) + (a 9 6) + (a ) 0a a 870a a ( a )( a 4) 0 a or a 4 + TK a + 90 substitution substitution of b a 9 standard form factors values of a b () K( ; ) 9 or b (4) value of b (6) []

13 Mathematics P/Wiskunde V DBE/November 06 QUESTION/VRAAG.... sin96 sin6 p cos6 sin p 6 reduction answer statement answer () () + p p 6 ( ; p) in terms of p p cos6 p. sin(a + B) cos[90 (A + B)] cos[(90 A) B] cos(90 A)cosB + sin(90 A)sinB sin AcosB + cos AsinB. cos A cos( A).cos(90 + A) answer () co-ratio correct form epansion sin A cosa.( sina) sin A cosa.( sina) sin AcosA cosa.( sina) sin A cosa sina sinacosa () cos A cos( A)cos(90 + A) (cos A ) cosa. sina cos A cosa sina 4 (4cos A 4cos A + ) 4 4cos A 4cos A cosa. sina cosa. sina 4cos A( cos A) 4cos Asin A cosa. sina cosa. sina cosa.sina cosa. sina identity ()

14 Mathematics P/Wiskunde V 4 DBE/November 06 ( sin A) sin A cosa sina.4. cosa. sina ( 4sin A + 4sin A) cosa. sina 4sin A( sin A) cosa. sina sina cos A cosa. sina cosb cos B 4 cos B 4 cos B or or [ 0 B 90 ] identity identity value of B cos () cos B + cos B + value of B cos B + cos.4. sin B cos B sin B or sin B () () + y ( ) 4 + y y y B ( ; y) y sin B or ()

15 Mathematics P/Wiskunde V DBE/November cosb sin B sin B sin B or cos(b + 4 ) cosb.cos4 sinb.sin4 0 or sin B epansion & answer () (4) cos(b + 4 ) cosb.cos4 sinb.sin4 or 0 0 epansion & answer (4) []

16 Mathematics P/Wiskunde V 6 DBE/November 06 QUESTION/VRAAG g - intercepts/ afsnitte y- intercept/ afsnit turning pts/ draaipte f 6. f ( ) sin maimum value 6. sin cos tan ref 6,7,4 + k.80 ; k Z 76,7 + k.90 ; k Z or.8 + k.90, k Z () tan, 4 or 6,6 o 76,7 or,8 k. 90 ; k Z (4) sin cos tan ref 6,7,4 + k.60 76,7 + k.80 or,4 + k.60, k Z or 66,7 + k.80 ; k Z tan, 4 &,4 76,7 & 66,7 k. 80 ; k Z (4) 6.4 ( 0,8 ;,8 ) 0,8 < <,8 values notation values notation []

17 Mathematics P/Wiskunde V 7 DBE/November 06 QUESTION/VRAAG 7 E Volume of pyramid (area of base) ( height) D C O 7. DB + [Theorem of Pyth] 8 DB 8 OB DB OB sin 4 OB sin 4 OB or OF/OR OB cos 4 OB OB A 8 or or, or or, θ or or, B substitution into Pyth value of DB correct ratio OB as subject correct ratio special angle

18 Mathematics P/Wiskunde V 8 DBE/November AÔB 90 (diagonals bisect ) OB OA OB OA AB AB OB AO OB + BO [pyth] OB or or, BE EO + OB (Pyth) cosθ cosθ AB EB 9 BE + 9 BE + AE AB + EB AB.EBcosθ AB + EB AE AB cosθ AB.EB AB.EB + [EB AE] Pyth substitution into Pyth length of BE correct cosine rule cos θ as subject simplification () BE EO + OB (Pyth) BE + 9 BE + AE AB + EB AB.EBcosθ cosθ cosθ substitution into Pyth length of BE correct cosine rule substituting cos θ as subject s ()

19 Mathematics P/Wiskunde V 9 DBE/November 06 BE EO + OB BE + 9 BE + cosθ (Pyth) A E F + 9 θ B substitution into Pyth length of BE sketch with values substitution () Eˆ 80 θ sin E sin θ E Ê 80 θ sin E sin θ sin θ sinθ 9 + sinθ cosθ sinθ θ A 9 + cosθ cosθ 9 + Volume (area of base) ( height) (9) cosθ 9 + θ 7, θ B E subst into sine rule diagram sinθ cosθ substitution -value substitution () (4) []

20 Mathematics P/Wiskunde V 0 DBE/November 06 QUESTION/VRAAG 8 8. S P 70 T R Q 8.. Alternate angles / verwiss hoeke, PQ SR R 8..(a) Tˆ 70 [ s opp sides/ e teenoor sye] Qˆ 80 (70 ) [ s/e 80 ] 40 R 8..(b) Pˆ 40 [tangent chord th/raakl-koordst] R () ()

21 Mathematics P/Wiskunde V DBE/November C S O A T B 8.. AT 0 [line from centre to chord/lyn vanaf midpt koord] S 8.. AO OS + AS [Pyth : AOS] OT + AT OS + AS [Pyth : ΔAOT] But AS 4 [line from centre to chord/lyn vanaf midpt koord] 7 OT OT OT OT OT equating AS 4 substitution 7 OS OT OT () AO Let OS 7, then OT In AOT: AO AO In AOS: AO AO OA radius testing in AOT testing in AOS conclusion () ()

22 Mathematics P/Wiskunde V DBE/November 06 equating AO OS + AS [Pyth : AOS] OT + AT OS + AS [Pyth : ΔAOT] Let OT. Then OS 7 But AS 4 [line from centre to chord/lyn vanaf midpt koord] () (7) AO AS 4 [line from centre to chord/lyn vanaf midpt koord] AO OS + AS [Pyth : AOS] 7 OT 49 AO (AO AO 9 AO 6 AO + AS 0 ) + 4 [Pyth : AOT] AS 4 substitution radius AS 4 substitution 7 OS OT equating subst Pyth radius () () []

23 Mathematics P/Wiskunde V DBE/November 06 QUESTION/VRAAG 9 F E D A 4 B y C 9.. tangent chord theorem/raaklyn-koordstelling R 9.. corresponding/ooreenkomstige s/e; FB DC R 9. Ê BĈD BCDE cyclicquad [converse et cyc quad/omgek: buite kdvh] 9. R Dˆ Ê [ s in the same segment/ e in dies segment] Dˆ FBˆ D [alt s, BF CD/ verwiss e,bf CD] 9.4 Bˆ y OR Bˆ Ĉ [ s in the same segment/ e in dies segment] Bˆ y OR Bˆ + Bˆ [from 9. and 9.4] Ĉ y [from 9. and 9.] Bˆ Ĉ () () () () In BFE and BEC Ê Ê [ ] Fˆ Bˆ + Bˆ 4 [tan - chord theorem] BFE/// CBE [,, ] Bˆ Ĉ identifying s [9]

24 Mathematics P/Wiskunde V 4 DBE/November 06 QUESTION/VRAAG 0 0. P S h h T Q R 0. Constr : Join S to R and T to Q and draw h from T PS/ Verbind SR entq en trek h van S van T PS] from S PT and h PT en h constr/konstruksie Proof : area PST area QST area area area But area area area PS SQ PS h SQ h PS SQ PT h PST PT STR TR TR h PST area PST QST area STR PST area ΔPST QST area ΔSTR PT TR equal altitudes equal altitudes [common] [same base, height; ST QR] area PST area QST PS h SQ h area PST PT area STR TR R (6)

25 Mathematics P/Wiskunde V DBE/November M y K L H G F 0.. Corresponding/Ooreenkomstige s/e; GF LK R 0..(a) GL LM FK KM GH y GH y OR GL y [prop theorem/ eweredighst; GF LK] [LH HG] S R () GL GH

26 Mathematics P/Wiskunde V 6 DBE/November (b) 0..(c) 0.. K GFˆM [corresponding/ ooreenkomst s; GF LK] LKˆ M or K MĤF [et cyclicquad/ buite koordevh] M ĤF G Fˆ M In MFH and MGF: Mˆ Mˆ [common/gemeen] M ĤF G Fˆ M [proven/bewys] MFH MGF [ ] OR/OR K GFˆM [corresponding/ ooreenkomst s; GF LK] LKˆ M or K MĤF [et cyclicquad/ buite koordevh] M ĤF GFˆ M In MFH and MGF: Mˆ Mˆ [common/gemeen] M ĤF G Fˆ M [proven/bewys] Fˆ Ĝ [ s of 80 ] MFH MGF GF MF [ s] FH MH y MF MH y y y y MG MF 9 6 [ s] [from 0..( c)] S R R S R S R () () () substitution simplificatio n [0] TOTAL MARKS 0

NATIONAL SENIOR CERTIFICATE/ NASIONALE SENIOR SERTIFIKAAT GRADE/GRAAD 10

NATIONAL SENIOR CERTIFICATE/ NASIONALE SENIOR SERTIFIKAAT GRADE/GRAAD 0 MATHEMATICS P/WISKUNDE V NOVEMBER 05 MEMORANDUM MARKS: 50 PUNTE: 50 This memorandum consists of pages./ Hierdie memorandum bestaan