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22 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 om asseblief

23 Mathematics P/Wiskunde V DBE/November 0 NSC/NSS Memorandum NOTE: If a candidate answers a question TWICE, only mark the FIRST attempt. If a candidate has crossed out an attempt of a question and not redone the question, mark the crossed out version. Consistent accuracy applies in ALL aspects of the marking memorandum. Assuming answers/values in order to solve a problem is NOT acceptable. NOTA: As 'n kandidaat 'n vraag TWEEKEER beantwoord, merk slegs die EERSTE poging. As 'n kandidaat 'n poging om die vraag te beantwoord, doodgetrek het en nie dit oorgedoen het nie, merk die doodgetrekte poging. Volgehoue akkuraatheid word in ALLE aspekte van die nasienmemorandum toegepas. Aanvaarding van antwoorde/waardes om 'n probleem op te los, is ONaanvaarbaar. QUESTION/VRAAG x σ 8, answer/antw.3 (68 8, ; ,) (9,8 ; 86,) 6 candidates had a mark within one standard deviation of the mean/6 kandidate het 'n punt binne een standaardafwyking vanaf die gemiddelde.. a,88...,83 b 0, ,66 y ˆ 0,66x +, 83 yˆ,83+ 0, 66x. y ˆ 0,66x +, 83 y 0,66(60) +,83 6,3...% 6% 6,69% 63% interval answer/antw () () (3) value of a/ waarde van a value of b/ waarde van b equation/vgl (3) subs of 60 into equation answer/antw () answer/antw ().6 (8 ; 6) answer/antw () [] Please turn over/blaai om asseblief

24 Mathematics P/Wiskunde V 3 DBE/November 0 NSC/NSS Memorandum QUESTION/VRAAG. 0 < x 60 0 x < 60 between 0 and 60/tussen 0 en 60.. Class Klas.. Frequency Frekwensie 0 < x < x < x < x < x < x < x < x Cumulative frequency Kumulatiewe frekwensie answer/antw 8 () () grounding at (0 ; 0)/ anker by (0 ; 0) plotting at upper limits/ plot by boonste limiete smooth shape of curve/gladde kurwe Cumulative Frequency/ Kumulatiewe frekwensie Speed in km per hour/ Spoed in km per uur.3 (accept/aanvaar 3 ) motorists/motoriste (accept/aanvaar 0 motorists/motoriste) (3) () [8] Please turn over/blaai om asseblief

3 A(x ; 0) (x ) + (0 ) (x ) + (0 ) x 0x + + 6 (x ) + 6 x 0x + 6 0 (x ) 9 (x 8)(x ) 0 (x ) ± 3 x 8 or/of x x 8 or/of x A( ; 0) A( ;

25 Mathematics P/Wiskunde V DBE/November 0 NSC/NSS Memorandum QUESTION/VRAAG 3 S y K L N M( ; ) O A B x P 3. r MN answer/antw 3. ( x ) + ( y ) equation/vgl 3.3 A(x ; 0) (x ) + (0 ) (x ) + (0 ) x 0x (x ) + 6 x 0x (x ) 9 (x 8)(x ) 0 (x ) ± 3 x 8 or/of x x 8 or/of x A( ; 0) A( ; 0) 3.. m MB () () substitute into eq/ vervang in vgl y 0 standard form/ standaardvorm or perfect square form/kwadr vorm answer/antw (3) subst M and B into form/vervang M and B in form m MB 3 () Please turn over/blaai om asseblief

26 Mathematics P/Wiskunde V DBE/November 0 NSC/NSS Memorandum 3.. m MB mpb (tangent radius/ rkl radius) 3 m PB y 3 x + c y y 3 (x x ) m MB mpb 3 m PB 0 3 (8) + c y 0 3 (x 8) y 3 x 6 y 3 x 6 3. y K y M + r + y At/By L: 3 x 6 9 3x 36 3x 60 x L(0 ; 9) ML L(0 ; 9) ( x (0 ) + 0 x ) + ( y + (9 ) or / of y 0 ) ML ( x () () 0 x ) + () (9 + ) or / of + ( y 3.8 MK KL M K ˆ L 90 (radius tangent/radius rkl) ML is a diameter as it subtends a right angle/ml is middellyn ML 0 r or 7,9 Centre of circle midpoint of ML/Midpt van sirkel midpt v ML x, y 6, Centre/midpt: (, ; 6,) Equation of the circle KLM /Vgl van sirkel KLM: 0 ( x,) + ( y 6,) 6, y 0 ) equation/vgl (3) 9 equation/vgl () equating simultaneously simplification () correct subst into distance formula/ korrekte subst in afstandformule answer in surd form/antw in wortelvorm () S value of/waarde van r x, y 6, answer in correct form/ antw in korrekte vorm () Please turn over/blaai om asseblief

27 Mathematics P/Wiskunde V 6 DBE/November 0 NSC/NSS Memorandum MK KL M K ˆ L 90 (radius tangent/radius rkl) ML is a diameter as it subtends a right angle/ml is middellyn Centre of circle midpoint of ML/Midpt van sirkel midpt v ML x, y 6, Centre/midpt: (, ; 6,) Equation of the circle KLM /Vgl van sirkel KLM: ( x,) + ( y 6,) r subst ( ; ): (,) + ( 6,) r 6, r 0 ( x,) + ( y 6,) 6, By symmetry about LM/deur simmetrie om LM: MK KL M K ˆ L 90 (radius tangent/radius rkl) ML is a diameter as it subtends a right angle/ml is middellyn ML is a diameter /ML is 'n middellyn ML 0 r or /of 7,9 Centre of circle midpoint of ML/Midpt van sirkel midpt v ML x, y 6, Centre/midpt: (, ; 6,) Equation of the circle KLM /Vgl van sirkel KLM: 0 ( x,) + ( y 6,) 6, S x, y 6, value of/waarde van r answer in correct form/antw in korrekte vorm () S value of/waarde van r x, y 6, answer in correct form/antw in korrekte vorm () [] Please turn over/blaai om asseblief

28 Mathematics P/Wiskunde V 7 DBE/November 0 NSC/NSS Memorandum QUESTION/VRAAG y F B( ; ) D M A E O x. y 0: 3x x 3 8 E ; 0 E ; tan DÊO mde 3.3 DÊO 7,6... 7,7 m DÂE 7,6... AB 6,7 tan 6,7 y x + c y y (x x ) () + c y (x ) y x + y x + 9 y-value/waarde x-value/waarde tan DÊO 3 7,6 () 6,7 (3) m AB tan 6, 7 m AB subst of m and ( ; )into formula/ subst m en ( ; ) in formule equation/vgl () Please turn over/blaai om asseblief

x-value/waarde subst/vervang y-value/waarde () x y 9 y 3 (y 9) + 8 y 6 y 7 + 8 y 3 x (3 ) 9 x D( ; 3 ) subst/vervang y

29 Mathematics P/Wiskunde V 8 DBE/November 0 NSC/NSS Memorandum. Solve x y and y 3x + 8 simultaneously: x (3x+8) x 6x x 7 x y 3( ) + 8 y y 3 D( ; 3 ) y 3 subst/vervang x-value/waarde subst/vervang y-value/waarde () x y 9 y 3 (y 9) + 8 y 6 y y 3 x (3 ) 9 x D( ; 3 ) subst/vervang y value/waarde subst/vervang x-value/waarde 3x + 8 x + 6x + 6 x + 9 x 7 x y 3( ) + 8 y ( ) + y 3 y 3 D( ; 3 ) () equating/gelyk stel x value/waarde subst/vervang y-value/waarde () Please turn over/blaai om asseblief

30 Mathematics P/Wiskunde V 9 DBE/November 0 NSC/NSS Memorandum x y 9 () 6x + y 6...() () + (): x 7 x y 9 y 3( ) + 8 adding/optelling x-value/waarde D( y 3 ; 3 ) y 3x + 8.() 6y 3x + 7.() () (): y 9 y 3 3 3x + 8 x D( ; 3 ) y 3 x (3 ) 9 x subst/vervang y-value/waarde () subtracting/aftrekking y-value/waarde subst/vervang x-value/waarde () Please turn over/blaai om asseblief

31 Mathematics P/Wiskunde V 0 DBE/November 0 NSC/NSS Memorandum F 8 3 D( ; ) 3 B( ; ) M(0 ; ) G A( 9 ; 0) 6,7 E( ; 0) C O x. area DMOE area AMO area ADE x A (0) 9 A( 9 ; 0) area AMO area ADE. AO. OM. AE. yd (9)( ). (AO EO). yd 0, 9 3 3,03 area ADE AD.AE.sin D ÂE 9 ( ).6.sin 6,7 3,03 correct method/ korrekte metode x A 9 (9)( ) AE y D 3 9 AD AE 6 3 area DMOE 8, square units/vk eenh answer/antw (6) Please turn over/blaai om asseblief

32 Mathematics P/Wiskunde V DBE/November 0 NSC/NSS Memorandum area DMOE area rectangle DCOG + area DMG + area DEC 7 9 ( 3 ) + ( ) + (3 )( ) 0 8, square units/vk eenh correct method/ korrekte metode 3 0,7 9 answer (6) area DMOE area EDO + area ODM ( EO yd ) + ( OM x D ) or/of 8 or/of 8, square units/vk eenh area DMOE area EOF area DMF ( EO OF) ( OF OM)( x D ) or 8 or 8, square units/vk eenh correct method/ korrekte metode 9 y D or 3 8 EO 3 x D 7 9 OM or answer/antw correct method/ korrekte metode y F 8 EO 3 8 x D 7 FM 3 answer/antw (6) (6) Please turn over/blaai om asseblief

33 Mathematics P/Wiskunde V DBE/November 0 NSC/NSS Memorandum sq units/vk eenh area EOM ( EO OM) ED and DM or,00... or,6.. 0 area EDM ( ED DM sin EDˆ M) sin or, area DMOE area EOM + area EDM 6 +, or/of 8 or/of 8, square units/eenh area EOM 9 0 ED 7 DM 0 area EDM correct method/ korrekte metode answer/antw (6) [9] F 8 3 D( ; ) 3 B( ; ) M(0 ; ) G A( 9 ; 0) 6,7 E( ; 0) C O x Please turn over/blaai om asseblief

34 Mathematics P/Wiskunde V 3 DBE/November 0 NSC/NSS Memorandum QUESTION/VRAAG C A x y 8 P D. sin CÂP CP AP 3 sin x 8 x 60 sin 90 sin x sin x 8 x C PˆA DPˆA 30 ( AP bisects DPˆ C) AD AP 8 + AD,96 + DP 3.AP.DP.cos APˆD (8)() cos (8)()( ),7... correct sine ratio/ korrekte sin-verh 3 () correct sine ratio/ korrekte sin-verh 3 D PˆA 30 () correct subst into cosine rule/ korrekte subst in cos-reël,7...,96 () Please turn over/blaai om asseblief

35 Mathematics P/Wiskunde V DBE/November 0 NSC/NSS Memorandum.3 sin DÂP sin APˆD DP AD sin y sin 30,96 sin 30 sin y,96 0,03... AD 8 cos y y 3,78 AP 8 cos y 0,98... y 3,8 + DP + (,96) + (,96) (8)(,96).AP.DP.cosDÂP (8)(,96).cos y correct subst into sine rule/ korrekte subst in sin-reël sin y subject 3,78 (3) correct subst into cosine rule/ korrekte subst in cos-reël cos y subject 3,8 (3) [9] Please turn over/blaai om asseblief

36 Mathematics P/Wiskunde V DBE/November 0 NSC/NSS Memorandum QUESTION/VRAAG 6 6. cos (80 + x) + tan( x 80 )sin(70 x) cos x ( cos x) + [ ( tan x)]( sin x)(cos x) cos x + cos x sin cos x sin cos x ( sin x)(cos x) x x 6. sin( α β ) cos[ 90 ( α β )] cos[( 90 α ) + β ] cos( 90 α)cos β sin(90 α) sin β sinα cos β cosα sin β sin( α β ) cos[ 90 ( α β )] cos[( 90 + β ) + ( α )] cos( 90 + β )cos( α ) sin(90 + β )sin( α ) ( sin β )cosα cos β ( sinα) sinα cos β cosα sin β 6.3 x y sin 76 cos 76 ( cos 76 sin 76 ) cos (76 ) cos ( cos 8 ) cos ( ) cos 8 ( sin 6 ) cos (90 6 ) sin 6 sin 6 x y sin 76 cos 76 sin 76 sin 76 cos 76 cos 76 sin 76 cos cos 76 sin sin (76 ) sin 6 ( cos x ) or cos x tan x or ( tan x) sin x sin x tan x cos x cos x sin x () rewrite as/herskryf cos[( 90 α ) + β ] expansion/ uitbreiding simpl/vereenv (3) rewrite as/herskryf cos[( 90 + β ) + ( α )] expansion/ uitbreiding simpl/vereenv (3) ( cos 76 sin 76 ) recognition of cos double angle cos cos 8 () cos sin recognition of sine compound angle sin(76 ) x y sin 76 cos 76 cos sin cos ( ) cos 8 sin 6 () cos sin recognition of cos double angle cos 8 () [] Please turn over/blaai om asseblief

37 Mathematics P/Wiskunde V 6 DBE/November 0 NSC/NSS Memorandum QUESTION/VRAAG y or y [0 ; ] critical values/ kritieke waardes notation/notasie () 7. sin x + cos x sin x + sin x sin x + sin x 0 sin x(sin x + ) sin x(sin x + ) 0 sin x 0 or sin x sin x st form/st vorm sin x 0 or sin x () 7. x 0 + k.360 or x 80 + k.360 x k.80, k Z x 0 + k.360 or x k.360, k Z f 0 ; 80 x k.80 0 ; 330 k.360, k Z () y-intercept/afsnit x-intercepts/afsnitte min/max points/ min/maks punte g - 7. f (x) g(x) at/by: x 30 ; 0 ; 80 ; 0 f (x + 30 ) g(x + 30 ) at/by: x 60 ; 30 ; 0 ; Series will converge if/reeks sal konvergeer as: < r < < cos x < < cos x < (3) 30 ; 0 ; 80 ; 0 60 ; 30 ; 0 ; 80 < r < r cos x < cos x < (3) 30 < x < 60 or x (30 ; 60 ) 30 < x < 60 () [9] Please turn over/blaai om asseblief

38 Mathematics P/Wiskunde V 7 DBE/November 0 NSC/NSS Memorandum QUESTION/VRAAG A O C y x B 8.. x 96 ( at centre at circumference/ by midpt by omtrek) Bˆ (sum of s in / som v e in ) Ĉ y Bˆ ( s opp sides/ e teenoor sye) S R S S () () 8. A B O C F 30 D 8.. Fˆ 90 (line from centre to midpt chord/ lyn vanaf midpt na midpt kd) 8.. A Bˆ C 0 (opposite s of cyclic quad/ tos e v koordevh ) S R S R () () Please turn over/blaai om asseblief

39 Mathematics P/Wiskunde V 8 DBE/November 0 NSC/NSS Memorandum 8.3 A x E 3 C x + 7 B 8.3. (a) tangent radius/diameter / raaklyn radius/middellyn R 8.3. (b) tangents from common pt OR tangents from same pt / raaklyne v gemeensk pt OF raaklyne vanaf dies pt 8.3. AB + BC AC x + ( x + 7) 3 (Theorem of/stelling vanpythagoras) x + x + x x + x 0 0 x + 7x 60 0 ( x )( x + ) 0 x ( x ) R () () AB + BC AC x + ( x + 7) 3 standard form answer () [] Please turn over/blaai om asseblief

40 Mathematics P/Wiskunde V 9 DBE/November 0 NSC/NSS Memorandum QUESTION/VRAAG 9 9. A D k h E B C 9.. Same base (DE) and same height (between parallel lines) Dieselfde basis (DE) en dieselfde hoogte (tussen ewewydige lyne) 9.. AD DB AE k EC k But/Maar area DEB area DEC (Same base and same height/dieselfde basis en dieselfde hoogte) area ADE area ADE area DEB area DEC AD DB AE EC same base/dies basis between lines/ tussen lyne () S S S R S () Please turn over/blaai om asseblief

41 Mathematics P/Wiskunde V 0 DBE/November 0 NSC/NSS Memorandum 9. A F D E G M B C 9.. EM FD (Line parallel one side of AM AD OR prop th; EF BD) (Lyn ewewydig aan sy v EM 3 OF eweredigst; EF BD) AM CM AM (diags of parm bisect/hoekl parm halv) CM AM 7 (from 9../vanaf 9..) ME ME h of FDC h of BDC (AD BC) FD. h area FDC area BDC BC. h FD (opp sides of parm ) AD (tos sye v parm ) 3 7 S R answer/antw S R answer/antw AD BC subst into area form/ subst in opp formule S answer/antw (3) (3) () area FDC area ADC FD AD 3 7 (same heights) (dieselfde hoogtes) S R But Area ADC Area BDC (diags of parm bisect area) (hoekl v parm halv opp) area FDC 3 area BDC 7 S answer/antw () [6] Please turn over/blaai om asseblief

42 Mathematics P/Wiskunde V DBE/November 0 NSC/NSS Memorandum QUESTION/VRAAG 0 P S X W x R 3 y T 3 3 Q Y 0.. Tangent chord theorem/raaklyn-koordstelling R 0.. Tangent chord theorem/raaklyn-koordstelling R 0..3 Corresponding angles equal/ooreenkomstige e gelyk R 0.. s subtended by chord PQ OR s in same segment R e onderspan deur dieselfde koord OF e in dieselfde segment 0.. alternate s/verwisselende e ; WT SP R 0. RW RT (Line parallel one side of OR RS RP prop th; WT SP) WR.RP RT (Lyn ewewydig aan sy v OF RS eweredighst: WT SP) S R () () () () () () 0.3 RTW RPS ( ; ; ) RW RT RS RP ( RTW RPS) RW.RP RT RS y Tˆ Rˆ 3 (tan chord theorem/rkl-koordst) y Rˆ 3 Qˆ ( s in same segment/ e in dieselfde segment) S S S R S R () () Please turn over/blaai om asseblief

43 Mathematics P/Wiskunde V DBE/November 0 NSC/NSS Memorandum x P S y X W x y 3 R y T x 3 y y 3 Q Y 0. Qˆ 3 PŜR (ext of cyc quad/buite v kdvh) ŜR Ŵ Qˆ 3 Ŵ P (corresp s/ooreenk e ; WT SP) Qˆ x ( s in same segment/ e in dies segment) Qˆ 3 80 ( x + y) ( s on straight line/ e op reguitlyn) Ŵ Qˆ 3 Ŵ 80 ( x + y) ( s of WRT/ e v WRT ) 0. In RTS and RQP: Rˆ 3 Rˆ y (proven above/hierbo bewys) Ŝ Pˆ ( s in same segment/ e in dies segment) R Tˆ S RQˆ P (3 rd angle of ) RTS RQP ( ; ; ) S R S R S S S S/R S ( ; ; ) (3) (3) (3) Please turn over/blaai om asseblief

44 Mathematics P/Wiskunde V 3 DBE/November 0 NSC/NSS Memorandum 0.6 RT RS ( RTS RQP) RQ RP RS RS RT RS RP RP RQ RP RS RP RT RP RW RS RW RQ RS RQ RS RQ RT RS ( RTS RQP) RQ RP WR.RP But RT (proven in 0./bewys in 0.) RS WR.RP RT RQ WR.RP WR RQ RQ.RS RQ.RS RS RP RS RP (proven in 0./bewys in 0.) S S RS on both RP sides RT RS RP RQ WR.RP RT RS multiplication/ vermenigvuldig (3) (3) RT RQ ( RTS RQP) RS RP RT.RP RQ RS RT.RS and WR (proven in 0./bewys in 0.) RP RT.RS WR RP RQ RT.RP RS RT.RS RS RP RT.RP RS RP S RT.RS WR RP simplification/ vereenvoudiging (3) [0] TOTAL/TOTAAL: 0

GRADE 11 NOVEMBER 2015 MATHEMATICS P2

NATIONAL SENIOR CERTIFICATE GRADE 11 NOVEMBER 2015 MATHEMATICS P2 MARKS: 150 TIME: 3 hours *Imat2* This question paper consists of 14 pages. 2 MATHEMATICS P2 (EC/NOVEMBER 2015) INSTRUCTIONS AND INFORMATION