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 7 pages./ Hierdie memorandum bestaan uit 7 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. Penalty of only mark for incorrect rounding throughout the paper (Q..) 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). Scatter plot/spreidiagram. no marks: 0 points correctly.. Energy/Energie (in kj) Fat/Vet (in g) plotting 3 5 points correctly plotting 6 9points correctly plotting all 0 points correctly geen punte: 0 punte korrek stip 3 5 pte korrek stip 6 9 pte korrek stip al 0 pte korrek.. y int close to (0 ; 50) one pt close to (5 ; 00) or (0 ; 700) ()
4 Mathematics/P/Wiskunde/V 4 DBE/November 05.. ŷ 54, ,3(8) 54, kj subst answ rounded off correctly/ antw korrek afgerond ().3 (8 ; 300).4 r 0, ,95.5 very strong positive relationship/ baie sterk positiewe verband strong/ 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) () +...() 0 mean/gemiddelde ,73 () T 7 7 median/mediaan 5 + T6 + 7 ().3 SD/SA,64...,6 ().4 (6,73,6 ; 6,73 +,6) (4,47 ; 8,99) lower boundary upper boundary times/keer [9]
6 Mathematics/P/Wiskunde/V 6 DBE/November 05 QUESTION/VRAAG 3 y P (a ; b) S T 45 N(7 ; ) Q( ; 3) M R 3. m PQ tan MN QP [midpt theorem/midpt-stelling] MN y y m( ) y ( 7) y MN PQ m MN y m + c (7) + c 6 c y 6 MN PQ [midpt theorem/midpt-stelling] [midpoint theorem/midp stelling] m tan 45 () OR R m MN subst m and/en N(7 ; ) equation/vgl OR R m MN subst m and/en N(7 ; ) equation/vgl MN 7 4,95 ()
7 Mathematics/P/Wiskunde/V 7 DBE/November QN NS [diag of m/hoekl van m] + + y S 3 S 7 and/en S 6 y S5 QN NS [diag of m/hoekl van m] by inspection/deur inspeksie: S(6 ; 5) 3.6 Equation of/vgl van PQ: y + c 3 + c y a b +...() From distance formula/van afstandsformule: PQ ( ) + ( 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 40 0 (b + 0)(b 4) b 4 (since b > 0) Subst b 4 into (): a P(5 ; 4) Equation of/vgl van PQ: y + c 3 + c y a b +...() From distance formula/van afstandsformule: 7 ( a ( )) + ( b ( 3)) 98 ( a + ) + ( b + 3)...() Subst () into (): 98 ( b + + ) + ( b + 3) 98 ( b + 3) 49 ( b + 3) ± 7 b + 3 ± 7 3 b b 4 (since b > 0) Subst b 4 into (): a P(5 ; 4) method/metode -value/waarde y-value/waarde method/metode -value/waarde y-value/waarde eq of/vgl van PQ subst Q & 7 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 Q & 7 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 + c 3 + c y 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 4 into (): b 5 4 P(5 ; 4) ( a ; b) eq of/vgl van PQ subst Q & 7 into/in distance formula/ afstandsformule subst eq of/vgl v. PQ simplification/ vereenvoudig value of/waarde van a value of/waarde van b (6) 7 7 sin 45 ( ; 3) 45 7 cos 45 a + 7 cos 45 5 b sin 45 4 (6) [7]
9 Mathematics/P/Wiskunde/V 9 DBE/November 05 QUESTION/VRAAG 4 y A P(0 ; 6) Q(5 ; ) θ R B α O S 4. ( 5) (0 5) + ( y ) + (6 ) 4 r ( 5) + ( y ) 4 PQ (0 5) r r r + (6 ) r 4 ( 5) + ( y ) 4 4. ( 0 5) + ( y ) ( y ) y 4y y 4y 0 ( y 6)( y + ) 0 y 6 or / of y S(0 ; ) or y subst (5 ; ) 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 0 st form/st. vorm (neg value)
10 Mathematics/P/Wiskunde/V 0 DBE/November 05 ( 0 5) + ( y ) ( y ) S(0 ; ) ( y ) 4 6 y ± 4 y ± 4 y 6 or / of y Draw/Trek QT PS PT TS [line from centre to chord/ lyn van midpt koord] PT y P yq 6 4 yq ys 4 y S 4 S(0 ; ) P(0 ; 6) 4 T 4 S Q(5 ; ) 0 square form/ kwadraatvorm (neg value) mpq m PQ mapb [tan/raakl radius] 5 APB 4 5 y tan α 4 0 y subst (0 ; 6) & (5 ; ) into grad form/in grad. formule m PQ m APB equation/vgl tan α m APB α 5,34 B(4,8 ; 0) 6 tan α 4,8 α 5,34 6 tan α 4,8 () ()
11 Mathematics/P/Wiskunde/V DBE/November θ B Pˆ S [tan-chord th/raakl-koordst.] 90 α 90 5,34 [ sum in / som van ] 38,66 R 90 α PS 8 PQ SQ 4 PS PQ + SQ.PQ.SQ.cosPQˆ S cosPQˆ S 8 cospqˆ S 8 PQˆ S 77,3 θ PQˆ S 38, Area PQS PS height/hoogte (8)(5) [ at centre circumf] 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 correct subst into cosine rule P Qˆ S 77,3 R 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.. cos 3 sin k cos 3 k 3 reduction/ reduksie answ ito/antw itv k () identity/identiteit cos 3 ito/itv k + ( k ) k k k cos 3 k 5..3 tan ( 3 ) tan 3 sin 3 cos 3 k k k k 3 ( ; k ) k ito/itv k reduction/ reduksie answ ito/antw itv k () 5. tan ( 3 ) tan 3 k k k k 4cos.( sin ) sin(30 + ) 4sin.cos sin 30 4sin.cos 8sin.cos 4(sin.cos ) 4sin reduction/ reduksie answ ito/antw itv k () cos sin sin (α + β) double sine form / dubbel sin form (6)
13 Mathematics/P/Wiskunde/V 3 DBE/November 05 4cos.( sin ) (sin 30 cos cos30 sin )cos + (cos 30 cos + sin 30 sin )sin 4sin.cos 3 3 ( cos sin )cos + ( cos + (sin.cos ) cos + sin (sin.cos ) (cos + sin ) (sin.cos ) () 8cos sin 4(sin cos ) 4sin sin )sin cos sin cos + sin double sine form / dubbel sin form (6)
14 Mathematics/P/Wiskunde/V 4 DBE/November cos 7 cos 3 0 cos 7cos 3 0 cos 7cos 4 0 (cos + )(cos 4) 0 cos or/of cos 4 (no solution) n.360 or/of 40 + n.360 ; n Z ±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 4sin θ 3 3( ) 4( ) epansion/ uitbreiding cos 7 cos 4 0 factors/faktore cos 0 & 40 + n.360 ±0 + n.360 (6) epansion of/ uitbreiding van sin( θ + θ ) epansions of sin θ AND cos θ sin θ subst (5) [4]
15 Mathematics/P/Wiskunde/V 5 DBE/November 05 QUESTION/VRAAG 6 y g f 6. f () cos and/en g() sin( + 30 ) f () cos g() sin( + 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. ( 0 ; 0 ) 0 < < 0 critical values/ kritiese waardes correct interval/ korrekte interval ()
16 Mathematics/P/Wiskunde/V 6 DBE/November The graph of g has to shift 60 to the left and then be reflected about the -ais./die grafiek van g moet 60 na links skuif en dan om die -as gereflekteer word. The graph of g must be reflected about the -ais and then be shifted 60 to the left./die grafiek van g moet om die -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 40 to the left./die grafiek van g moet met 40 na links geskuif word 60 left/links reflection about -ais/refleksie om -as () reflection about -ais/refleksie om -as 60 left/links () 0 right/regs () 40 left/links () [8]
17 Mathematics/P/Wiskunde/V 7 DBE/November 05 QUESTION/VRAAG 7 A θ D B θ θ C 7. CÂD 80 θ [ s sum of / e som van ] 7. sinθ sin(80 θ ) + 3 sinθ sin θ + 3 sinθ sinθ.cosθ + 3 sinθ cosθ ( + 3)sinθ cosθ + 3 correct subst into sine rule/korrekte subst in sin-reël sin θ sinθ. cos θ cos θ as subject/ as onderwerp () AD + 3 [sides opp s/sye to e] AC AD + CD AD.CD.cos θ ( + 3) ( + 3) + () ()( + 3).cosθ 0 4 4( + 3)cosθ 4 cosθ 4( + 3) + 3 A AD + 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 θ C P + 3 D constr/konstr sketch shown/ toon skets
18 Mathematics/P/Wiskunde/V 8 DBE/November cos θ 5 θ 66,4 cos θ 5 size of/grootte v θ In ABC: AB sin θ AC AB sin 33, 5 AB 5 sin 33,,74 θ AB sin 5 θ AB 5 sin correct ratio/ korrekte verh subst correctly/ korrek AB 5 sin θ (5) but/maar: cosθ 5 θ sin 5 θ 3 sin 0 θ 3 sin AB 5,74 0 equation/vgl simplification/ vereenvoudiging value of/waarde v sin θ (5) [0]
19 Mathematics/P/Wiskunde/V 9 DBE/November 05 QUESTION/VRAAG 8 A O D B C 8.. twice or double /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 () Let/Gestel Ô Â [ at centre at circ/midpts omtreks ] Ô 360 [ s in a rev/ e in omw of om 'n pt] Ĉ 80 [ at centre at circ/midpts omtreks ] Â + Ĉ 80
20 Mathematics/P/Wiskunde/V 0 DBE/November B C A D F E 8. Â Ĉ [et 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 ] Bˆ Dˆ [et 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]
21 Mathematics/P/Wiskunde/V DBE/November 05 QUESTION/VRAAG 9 T B H 3 K 3 A S C 9. Kˆ 3 Ĉ [corresp s/ooreenk e ; CA KT] Â 3 [tan-chord th/raakl-koordst] 9. Kˆ 3 Â 3 [proved/bewys in 9.] AKBT is cyc quad [line (BT) subtends equal s/ lyn (BT) onderspan gelyke e] [converse s in same segment/ omgek e in dies segment] 9.3 Kˆ 3 Ĉ [proven in 9.] Bˆ [tan-chord th/raakl-koordst] Kˆ [ s in the same segm/ e in dies segm] TK bisects/halveer A Kˆ B Kˆ Bˆ [ s in the same seg/ e in dies segm] Â 3 [tans from same pt; s opp equal sides/ rkle v dies pt; e to gelyke sye] R R R R R R R ()
22 Mathematics/P/Wiskunde/V DBE/November 05 Kˆ 3 [proven in 9.] TK bisects/halveer A Kˆ B 9.4 Â 3 Kˆ [proven/bewys] TA tangent [converse tan chord theorem OR between line and chord/ omgekeerde raakl-kdst OF tussen lyn en koord] 9.5 B ŜA BKˆ A [A,S,K & B concyclic/konsiklies] ATˆ B 80 [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 ] R (both/beide statements/ bewerings) R () () A, S K and B are concyclic. A, K, B and T are concyclic. A, S, B and T are concyclic. 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 () () [4]
23 Mathematics/P/Wiskunde/V 3 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 CEF CBD 5 4 CF 3, B Dˆ C 90 [ in semi circle/ in halfsirkel] E FˆC B Dˆ C [corresp s/ooreenk e; EF BD] A Bˆ C 90 [tan diameter/raakl middellyn] In BAC and/en FEC: A Bˆ C EFˆC [proven/bewys] Ĉ Ĉ [common/gemeen] BAC FEC [ ] B Dˆ C 90 [ in semi circle/ in halfsirkel] E FˆC B Dˆ C [corresp s/ooreenk e; EF BD] A Bˆ C 90 [tan diameter/raakl middellyn] In BAC and/en FEC: A Bˆ C EFˆC [proven/bewys] Ĉ Ĉ [common/gemeen] using/gebruik Pyth korrek/ correctly /R subst correctly/ korrek /R R R /R R (5)
24 Mathematics/P/Wiskunde/V 4 DBE/November 05 B ÂC FÊC [ sum in / som van ] BAC FEC (5) 0..3 EC 7 4, 5 4 AC BC [ BAC FEC] EC FC AC 7 4,5 3,75 4 AC 9,7 or/of 9 5 CF BC cos Ĉ CE AC 3,75 7 4,5 AC 4 AC 9,7 or/of 9 5 length of/lengte v EC subst correctly/ korrek correct ratios/ korrekte verh's subst correctly/ korrek BCA DBC CB CD. AC BC AC DC ,7 or/of 9 5 Ĉ ABˆ D [tan-chord theorem/rkl-kdstelling] AD tanabˆ D 8 tanĉ AD 5 4 AC 9,7 or/of 9 5 OR Pyth th correct ratio subst correct ratio subst
25 Mathematics/P/Wiskunde/V 5 DBE/November AC is diameter of the circle passing through A, B and C [chord subtends 90 OR converse in semi circle ] AC is middellyn van die sirkel wat deur die punte A, B en C gaan [koord onderspan 90 OF omgek in halfsirkel ] radius 9 9, 7 9,63 or/of 9 or/of AC 30 /R () [7]
26 Mathematics/P/Wiskunde/V 6 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 all 3 statements/ al 3 bewerings 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] all 3 statements/ al 3 bewerings In MNR:,5 + 0,75,565 M Nˆ R 90 [converse Pyth theorem] In PKM:,5,5 + 6,5 Pˆ 90 [converse Pyth theorem],5 3 0,75 3 cos P Kˆ M and cos Rˆ,5 5,5 5 P Kˆ M Rˆ In KPM and RNM P Kˆ M Rˆ [proved] Pˆ MNˆ R [proved] KPM RNM [ ; ; OR 3 rd ] Pˆ MNˆ R P Kˆ M Rˆ [ ; ; OR 3 rd ]
27 Mathematics/P/Wiskunde/V 7 DBE/November 05.. P Kˆ M Rˆ [ KPM RNM] Pˆ is common/gemeen RPQ KPM [ ] RP RQ KP KM [ RPQ KPM] 3,5 RQ,5,5,5 3,5 5 RQ 5,4 or 5,5 NQ 5,4 0,75 4,67 or 4 3 RPQ KPM subst correctly/ korrek RQ 5 5 NQ answ/antw (6) R Nˆ M Pˆ [ KPM RNM] Rˆ is common/gemeen RNM RPQ [ ] RP RQ [ RNM RPQ] RN RM 3,5 RQ 0,75,5 5 RQ 5,4 or 5 NQ 5,4 0,75 4,67 or 4 3 In MNR:,5 + 0,75,565 M Nˆ R 90 [converse Pyth theorem] In PKM:,5,5 + 6,5 Pˆ 90 [converse Pyth theorem] In MNR and QPR R is common M Nˆ R Pˆ 90 MNR QPR [ ] RP RQ [ RNM RPQ] RN RM 3,5 RQ 0,75,5 5 RQ 5,4 or 5 NQ 5,4 0,75 4,67 or 4 3 RNM RPQ subst correctly/ korrek RQ 5 5 NQ answ/antw (6) MNR QPR subst correctly/ korrek RQ 5 5 NQ answ/antw (6) [0] TOTAL/TOTAAL: 49
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