NATIONAL SENIOR CERTIFICATE/ NASIONALE SENIOR SERTIFIKAAT GRADE/GRAAD 12

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

2 Mathematics/P/Wiskunde/V DBE/November 05 NOTE: If a candidate answers a question TWICE, mark only the FIRST attempt. If a candidate crossed out an attempt of a question and did not redo the question, 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, sien slegs die EERSTE poging na. Indien 'n kandidaat 'n antwoord doodgetrek het en nie oorgedoen het nie, sien die doodgetrekte poging na. Volgehoue akkuraatheid word in ALLE aspekte van die memorandum toegepas. Hou op 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 3 DBE/November 05 QUESTION/VRAAG Fat/Vet (in g) Energy/Energie (in kj)... Energy/Energie (in kj) Scatter plot/spreidiagram Fat/Vet (in g). plotting 3 points correctly plotting 6 points correctly plotting all points correctly stip 3 punte korrek stip 6 punte korrek stip alle punte korrek.. any of/ enige van ±(;000) ±(9;60) ±(8;53) ()

4 Mathematics/P/Wiskunde/V DBE/November 05.. ŷ 5, ,3(8) 5,9 500 kj subst.3 (8 ; 300). r 0, ,95.5 very strong positive relationship/ baie sterk positiewe verband very strong/ baie sterk () () () () []

5 Mathematics/P/Wiskunde/V 5 DBE/November 05 QUESTION/VRAAG Sum of the values on uppermost faces/ Som van die waardes op boonste vlakke Frequency/ Frekwensie (0) () +...() mean/gemiddelde 30 6,73 T T6 median/mediaan () ().3 SD/SA,6...,6 (). (6,73,6 ; 6,73 +,6) (,7 ; 8,99) times/keer interval [9]

6 Mathematics/P/Wiskunde/V 6 DBE/November 05 QUESTION/VRAAG 3 y P S T 5 N(7 ; ) x Q( ; 3) M R 3. m PQ tan 5 3. MN QP [midpt theorem/midpt-stelling] MN y y m( x x ) y (x 7) y x 6 MN PQ [midpt theorem/midpt-stelling] m MN y mx + c (7) + c 6 c y x MN RS 7 MN [midpoint theorem/midp stelling] m tan 5 () m MN subst m and/en (7 ; ) equation/vgl m MN subst m and/en (7 ; ) equation/vgl (),95 RS 7 9,899 9,90 ()

7 Mathematics/P/Wiskunde/V 7 DBE/November QN NS [diag of m/hoekl van m] + x + y S 3 S 7 and/en x S 6 y S 5 QN NS [diag of m/hoekl van m] by inspection/deur inspeksie: S(6 ; 5) 3.6 Equation of/vgl van PQ: y x + c 3 + c y x + a b +...() From distance formula/van afstandsformule: PQ ( x x ) + ( y y ) 7 ( a ( )) + ( b ( 3)) 98 ( a + ) + ( b + 3)...() Subst () into (): 98 ( b + + ) + ( b + 3) 98 b + 6b b + 6b b + b 80 0 b + 6b 0 0 (b + 0)(b ) b (since b > 0) Subst b into (): a + 5 P(5 ; ) Equation of/vgl van PQ: y x + c 3 + c y x + a b +...() From distance formula/van afstandsformule: 7 ( a ( )) + ( b ( 3)) 98 ( a + ) + ( b + 3)...() Subst () into (): 98 ( b + + ) + ( b + 3) 98 ( b + 3) 9 ( b + 3) ± 7 b + 3 ± 7 3 b b (since b > 0) Subst b into (): a + 5 P(5 ; ) method/metode x-value/waarde y-value/waarde method/metode x-value/waarde y-value/waarde eq of/vgl van PQ subst into/in distance formula/ afstandsformule subst eq of/vgl v. PQ st form/st vorm value of/waarde van b value of/waarde van a (6) eq of/vgl van PQ subst into/in distance formula/ afstandsformule subst eq of/vgl v. PQ simplification/ vereenvoudig value of/waarde van b value of/waarde van a (6)

8 Mathematics/P/Wiskunde/V 8 DBE/November 05 Equation of/vgl van PQ: y x + c 3 + c y x + a b +...() From distance formula/van afstandsformule: 7 ( a ( )) + ( b ( 3)) 98 (a + ) + (a + 3) (a + ) a + 7 (since/aangesien a > 0) a 5 Subst a into (): b 5 P(5 ; ) eq of/vgl van PQ subst into/in distance formula/ afstandsformule subst eq of/vgl v. PQ simplification/ vereenvoudig value of/waarde van a value of/waarde van b (6) [8]

9 Mathematics/P/Wiskunde/V 9 DBE/November 05 QUESTION/VRAAG y A P(0 ; 6) Q(5 ; ) θ R B α O x S. ( x 5) (0 5) + ( y ) + (6 ) r ( x 5) + ( y ) PQ (0 5) r r r + (6 ) r ( x 5) + ( y ). ( 0 5) + ( y ) 5 + ( y ) 5 + y y + y y 0 ( y 6)( y + ) 0 y 6 or / of y S(0 ; ) or y subst (5 ; ) & (0 ; 6) into circle eq/in sirkelvgl value of/waarde van r equation/vgl subst (5 ; ) & (0 ; 6) into dist. form/in afst. form value of/waarde van r equation/vgl st form/st. vorm factors/faktore

10 Mathematics/P/Wiskunde/V 0 DBE/November 05 ( 0 5) + ( y ) 5 + ( y ) S(0 ; ) ( y ) 6 y ± y ± y 6 or / of y square form/ kwadraatvorm simplify/vereenv.3. Draw/Trek QT PS PT TS [line from centre to chord/ lyn van midpt koord] PT y P yq 6 yq ys y S S(0 ; ) or/of y 6 mpq m PQ mapb [tan/raakl radius] 5 APB 5 y x tan α T P(0 ; 6) S Q(5 ; ) PT subst (0 ; 6) & (5 ; ) into grad form/in grad. formule m PQ m APB equation/vgl tan α m APB α 5,3 B(,8 ; 0) 6 tan α,8 α 5,3 6 tan α,8 () ()

11 Mathematics/P/Wiskunde/V DBE/November 05.5 B Pˆ S 90 5,3 [ sum in / som van ] 38,66 θ 38,66 [tan-chord th/raakl-koordst.] 90 α size of/grootte v B PˆS size of/grootte v θ R P Qˆ S θ [ at centre at circ/ midpts omtreks ] P Qˆ T θ [ PQT SQT] sin θ tan θ 5 θ 38,66 θ 38,66 P(0 ; 6) T S θ θ Q(5 ; ) R any correct ratio/ enige korrekte verhouding.6 Area PQS PS height/hoogte (8)(5) 0 sq units/vk eenh P Qˆ S 38,66 [ at centre at circum/ midpts omtreks ] 77,3 Area PQS PQ.QS.sin P Qˆ S...sin 77,3 0 sq units/vk eenh area formula/e: PQS PS 8 h 5 size of/grootte v P Qˆ S area rule/reël: PQS subst correctly/ subst korrek [0]

12 Mathematics/P/Wiskunde/V DBE/November 05 QUESTION/VRAAG sin 03 sin 3 k 5.. x + ( k ) x k x k cos 3 k k 3 (x ; k ) reduction/ reduksie answ ito/antw itv k sketch/skets x ito/itv k () cos 3 sin k cos 3 k tan ( 3 ) tan 3 k k k k identity/identiteit cos 3 ito/itv k reduction/ reduksie answ ito/antw itv k () 5. tan ( 3 ) tan 3 sin 3 cos 3 k k k k cos x.( sin x) sin(30 x + x) sin x.cos x sin 30 (sin x.cos x) sin x reduction/ reduksie answ ito/antw itv k cos x sin x sin (α + β) () double sine form / dubbel sin form (6)

13 Mathematics/P/Wiskunde/V 3 DBE/November cos x 7 cos x 3 0 cos x 7cos x 3 0 cos x 7cos x 0 (cos x + )(cos x ) 0 cos x or/of cos x (no solution) 5. x 0 + n.360 or/of x 0 + n.360 ; n Z sin 3θ sin(θ + θ ) sin θ cosθ + cos θ sinθ sinθ cosθ cosθ + ( sin θ )sinθ 3 sinθ ( sin θ ) + sinθ sin θ 3 3sinθ sin θ 3 3( ) ( ) expansion/ uitbreiding factors/faktore no sol/geen opl cos x 0 & 0 + n.360 ; n Z (6) expansion/ uitbreiding double cos form/ dubbel cos form identity/identiteit subst (5) []

14 Mathematics/P/Wiskunde/V DBE/November 05 QUESTION/VRAAG 6 y g x f 6. f (x) cos x and/en g(x) sin(x + 30 ) f (x) cos x g(x) sin(x + 30 ) p 30 and/en q value of/waarde v p value of/waarde v q sin (60 + p) and/en cos 0 + q sin (60 + p) cos 0 + q p 30 q value of/waarde v p value of/waarde v q 6. x ( 0 ; 0 ) 0 < x < 0 critical values/ kritiese waardes notation/notasie ()

15 Mathematics/P/Wiskunde/V 5 DBE/November The graph of g has to shift 60 to the left and then be reflected about the x-axis./die grafiek van g moet 60 na links skuif en dan om die x-as gereflekteer word. The graph of g must be reflected about the x-axis and then be shifted 60 to the left./die grafiek van g moet om die x-as gereflekteer word en dan met 60 na links geskuif word. The graph of g has to shift 0 to the right./die grafiek van g moet 0 na regs geskuif word. The graph of g has to shift 0 to the left./die grafiek van g moet met 0 na links geskuif word 60 left/links reflection about x-axis/refleksie om x-as () reflection about x-axis/refleksie om x-as 60 left/links () 0 right/regs () 0 left/links () [8]

16 Mathematics/P/Wiskunde/V 6 DBE/November 05 QUESTION/VRAAG 7 A x + 3 x + 3 θ D B θ θ x C 7. CÂD 80 θ [ s sum of / e som van ] 7. sinθ sin(80 θ ) x + 3 x sinθ sin θ x + 3 x sinθ sinθ.cosθ x + 3 x xsinθ cosθ ( x + 3)sinθ x cosθ x + 3 correct subst into sine rule/korrekte subst in sin-reël sin θ sinθ. cos θ cos θ as subject/ as onderwerp () AD x + 3 [sides opp s/sye to e] AC AD + CD AD.CD.cos θ ( x + 3) ( x + 3) + (x) (x)( x + 3).cosθ 0 x x( x + 3)cosθ x cosθ x( x + 3) x x + 3 A AD x + 3 correct subst into cosine rule/korrekte subst in cos-reël simplification/ vereenvoudiging cos θ as subject/ as onderwerp Draw/Trek AP CD cos θ x x + 3 x + 3 θ C x P x + 3 x D constr/konstr sketch shown/ toon skets

17 Mathematics/P/Wiskunde/V 7 DBE/November cos θ 5 θ 66, size of/grootte v θ In ABC: AB sin θ AC AB sin 33, 5 AB 5 sin 33,,7 θ AB sin 5 θ AB 5 sin correct ratio/ korrekte verh subst correctly/ korrek AC 5 AB 5 sin θ (5) but/maar: cosθ 5 θ sin 5 θ 3 sin 0 θ 3 sin AB 5,7 0 equation/vgl simplification/ vereenvoudiging value of/waarde v sin θ (5) [0]

18 Mathematics/P/Wiskunde/V 8 DBE/November 05 QUESTION/VRAAG 8 A O D B C 8.. twice/twee keer of dubbel R 8.. Ô Â [ at centre at circ/midpts omtreks ] Ô Ĉ [ at centre at circ/midpts omtreks ] Ô + Ô 360 [ s in a rev/ e in omw of om 'n pt] Â + Ĉ 360 Â + Ĉ 80 ()

19 Mathematics/P/Wiskunde/V 9 DBE/November B C A D F E Ĉ Â [ext of cyclic quad/buite v kdvh] Ê 80 Ĉ [opp s of cyclic quad/tos e v kdvh] Ê 80 Â EF AB [co-interior s 80 /ko-binne e 80 ] Dˆ Bˆ [ext of cyclic quad/buite v kdvh] Fˆ 80 Dˆ [opp s of cyclic quad/tos e v kdvh] Fˆ 80 Bˆ EF AB [co-interior s 80 /ko-binne e 80 ] R R R R R R (5) (5) [9]

20 Mathematics/P/Wiskunde/V 0 DBE/November 05 QUESTION/VRAAG 9 T B H 3 K 3 A x S C 9. Kˆ 3 Ĉ [corresp s/ooreenk e ; CA KT] Ĉ x [tan-chord th/raakl-koordst] Kˆ x 3 9. Kˆ 3 x Â 3 [proved/bewys in 9.] AKBT is cyc quad [line (BT) subtends equal s/ lyn (BT) onderspan gelyke e] 9.3 Bˆ Ĉ x [tan-chord th/raakl-koordst] Bˆ Kˆ x [ s in the same segm/ e in dies segm] Kˆ 3 Kˆ x TK bisects/halveer A Kˆ B Bˆ Â3 x [tans for same pt; s opp equal sides/ rkle v dies pt; e to gelyke sye] Bˆ Kˆ x [ s in the same seg/ e in dies segm] Kˆ 3 Kˆ x TK bisects/halveer A Kˆ B R R R R R R R ()

21 Mathematics/P/Wiskunde/V DBE/November 05 TA TB [tans for same pt/rkle v dies pt] KATB is cyc quad Kˆ 3 Kˆ x [equal chords subtend equal angles/ gelyke koorde onderspan gelyke hoeke] TK bisects/halveer A Kˆ B 9. Â 3 Kˆ x [proven/bewys] TA tangent [converse tan chord theorem/ omgekeerde raakl-kdst] 9.5 The circle passing through points A, K and B contains the point S on the circumference (A, S, K and B concyclic)./die sirkel deur punt A, K en B bevat die punt S op die omtrek (A, S, K en B konsiklies). The circle passing through A, K and B contains the point T on the circumference (proven in 9.)./Die sirkel deur punt A, K en B bevat die punt T op die omtrek (bewys in 9.). points A, S, B and T are also concyclic/punte A, S, B en T is konsiklies R R R () () B ŜA BKˆ A x [A,S,K & B concyclic/konsiklies] ATˆ B 80 x [A,T,B & K concyclic/konsiklies] points A, S, B and T are also concyclic/punte A, S, B en T is ook konsiklies [opp s of quad 80 /tos e van vierhoek80 ] (both/beide statements/ bewerings) R () []

22 Mathematics/P/Wiskunde/V DBE/November 05 QUESTION/VRAAG 0 B 7 8 E F C A D 0. B Dˆ C 90 [ in semi circle/ in halfsirkel] DC 7 8 [Th of/stelling v Pythagoras] DC 5 CF CE [line one side of /lyn een sy van ] CD CB CF 5 CF 3, In BAC and/en FEC: A Bˆ C 90 [tan diameter/raakl middellyn] E FˆC 90 [corresp s/ooreenk e; EF BD] Ĉ Ĉ [common/gemeen] BAC FEC [ ] using/gebruik Pyth korrek/ correctly /R subst correctly/ korrek R /R R (5) In BAC and FEC: A Bˆ C 90 [tan diameter/raakl middellyn] E FˆC 90 [corresp s/ooreenk e; EF BD] Ĉ Ĉ [common/gemeen] B ÂC FÊC [ sum in / som van ] BAC FEC R /R (5)

23 Mathematics/P/Wiskunde/V 3 DBE/November 05 ABEF cyc quad/ kdvh [opp s of quad supp/tos e v vh suppl)] F ÊC Â [ext of cyc quad/buite v kdvh] In BAC and/en FEC: F ÊC Â [proven/bewys] Ĉ Ĉ [common/gemeen] B ÂC FÊC [ sum in / som van ] BAC FEC ABEF cyc quad/ kdvh [opp s of quad supp/tos e v vh suppl)] F ÊC Â [ext of cyc quad/buite v kdvh] In BAC and FEC: F ÊC Â [proven/bewys] Ĉ Ĉ [common/gemeen] BAC FEC [ ] R /R R /R R (5) (5) 0..3 EC 7, 5 AC BC [ s/e] EC FC AC 7,5 3,75 AC 9,7 or/of 9 5 length of/lengte v EC subst correctly/ korrek CF BC cos Ĉ CE AC 3,75 7,5 AC AC 9,7 or/of AC is diameter of the circle [chord subtends 90 ] AC is middellyn van die sirkel [koord onderspan 90 ] radius 9 9, 7 9,63 or/of 9 30 correct ratios/ korrekte verh's subst correctly/ korrek /R () [7]

24 Mathematics/P/Wiskunde/V DBE/November 05 QUESTION/VRAAG. equiangular or similar/gelykhoekig of gelykvormig K,5 P,5 M,5 0,75 N R ().. Q KP,5 PM KM,5 ; ; RN 0,75 NM RM.5 KP PM KM RN NM RM KPM RNM [Sides of in prop/sye v eweredig] RN 0,75 NM RM,5 KP,5 ; ; PM KM,5 RN NM RM KP PM KM KPM RNM [Sides of in prop/sye v eweredig].. P Kˆ M Rˆ Pˆ is common/gemeen RPQ KPM [ ] RP RQ [ s] KP KM 3,5 RQ,5,5,5 3,5 5 RQ 5, or 5,5 NQ 5, 0,75,67 or 3 all 3 statements/ al 3 bewerings R all 3 statements/ al 3 bewerings R RPQ KPM subst correctly/ korrek RQ 5 5 NQ answ/antw (6) [0] TOTAL/TOTAAL: 50

NATIONAL SENIOR CERTIFICATE/ NASIONALE SENIOR SERTIFIKAAT GRADE/GRAAD 12

NATIONAL SENIOR CERTIFICATE/ NASIONALE SENIOR SERTIFIKAAT GRADE/GRAAD MATHEMATICS P/WISKUNDE V NOVEMBER 05 MEMORANDUM MARKS: 50 PUNTE: 50 This memorandum consists of 7 pages./ Hierdie memorandum bestaan