NATIONAL/NASIONALE SENIOR CERTIFICATE/SERTIFIKAAT GRADE/GRAAD 12

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1 NATIONAL/NASIONALE SENIOR CERTIFICATE/SERTIFIKAAT GRADE/GRAAD MATHEMATICS P/WISKUNDE V FEBRUARY/MARCH/FEBRUARIE/MAART 07 MEMORANDUM MARKS / PUNTE: 50 This memorandum consists of pages. Hierdie memorandum bestaan uit bladsye.

2 Mathematics P/Wiskunde V DBE/Feb. Mar./Feb. Mrt. 07 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 Cumulative frequency/kumulatiewe frekwensie Mathematics P/Wiskunde V DBE/Feb. Mar./Feb. Mrt. 07 QUESTION/VRAAG Ogive/Ogief (50, 6) (60, 65) (40, 45) 0 0 (0, 5) 0 (0, ) 0 (0, 0) Money spent/geld gespandeer (R) Amount of money/ Bedrag geld (in R) Frequency Frekwensie 0 x 0 0 x 0 0 x x x 60 a 0 b learners/leerders answer. Modal class/modale klas: 0 x 40 answer. a = b = 6 45 = 6.4 No. of learners/aantal leerders = = = 0 Answer only: full marks answer answer 54 or 55 or 0 () () () () [6]

4 Mathematics P/Wiskunde V 4 DBE/Feb. Mar./Feb. Mrt. 07 QUESTION/VRAAG. Class/Klas A Class/Klas B IQR of Class B/IKV van Klas B = Q Q = 7 5 = marks/punte.. Although the boxes contain the same number of data points, the marks for Class A are more widely spread./alhoewel die monde dieselfde aantal datapunte bevat, is die punte van Klas A meer verspreid. Although the boxes contain the same number of data points, the marks for Class B are more clustered./alhoewel die monde dieselfde aantal datapunte bevat, is die punte van Klas B nader aan mekaar... Medians are the same/mediane is dieselfde Ranges are the same OR Maximum and minimum values are the same/variasiewydtes is dieselfde OF die maksimum en minimum waarde is dieselfde 75% of both classes obtained 5 and above/75% van albei klasse behaal 5 en meer. 7 and 5 only () Class A is more widely spread () Class B is more clustered () any TWO of the reasons mentioned (). COUPLE/PAAR JUDGE / BEOORDELAAR JUDGE / BEOORDELAAR a = 0,0 b = 0,9 ŷ = 0,0 + 0,9x.. ŷ = 0,0 + 0,9(5) =,9, Yes OR they are consistent, because r = 0,9. (r = 0,89567 )/Ja OF hulle is konsekwent, want r = 0,9. (r = 0,89567 ) value a value b equation substitution answer statement r = 0,9 () () () []

Mathematics P/Wiskunde V 5 DBE/Feb. Mar./Feb. Mrt. 07 QUESTION/VRAAG y S R(0 ; 7) T(0 ; 4) O Q( ; 0) β M(5 ; ) x. m TQ 4 0 0 4 = answer. d x x ) ( y ) RQ RQ ( y (0 ) (7 0) 98 7.

5 Mathematics P/Wiskunde V 5 DBE/Feb. Mar./Feb. Mrt. 07 QUESTION/VRAAG y S R(0 ; 7) T(0 ; 4) O Q( ; 0) β M(5 ; ) x. m TQ = answer. d x x ) ( y ) RQ RQ ( y (0 ) (7 0) mfq mtq 8 4 k 4k 4 k 9 4 Equation of TQ: y x k + 4 k = 9 substitution/substitusie () answer in surd form () equating gradients/stel gradient gelyk 8 mfq k simplification/vereenvoudig answer gradient equation of TQ/vgl van TQ substitution of (k ; - 8) /substitusie van (k ; - 8) answer

6 Mathematics P/Wiskunde V 6 DBE/Feb. Mar./Feb. Mrt Using transformation/gebruik transformasie: S(7 ; ) x value/waarde y value/waarde Midpoint of TR = midpoint of SQ [diag m/hkle m] Midpoint of TR = (5 ; ) xs y 5 S 0 and xs 7 and y S S(7 ; ) x value/waarde of/van T y value/waarde of/van T x value/waarde of/van S y value/waarde of/van S Equation of TS: 7 y x 4 x Equation of RS: 4 y 7 ( x 0) 4 6 y x x 4 4 7x 49 x 7 y = S(7 ; ) x 6.5 TŜR TQˆ R [opp s of m/teenoorst e m] Tˆ QR 4 tan mtq α = 80 5, = 6,87 7 tan mrq 7 β = 45 T Qˆ R 6,87 45 = 8,87 T ŜR 8,87 equations of TS and RS/vgls van TS en RS equating / gelykstel x value/waarde y value/waarde α Tˆ QR tan m tan m answer TQ RQ (6)

7 Mathematics P/Wiskunde V 7 DBE/Feb. Mar./Feb. Mrt. 07 TQ = SR = 5 TR = RQ = TS = TQ RQ TR cosrqˆ T costŝr.tq.rq RQˆ T (5)( 98) 0,4... TŜR 8,87.6. MQ (5 ) ( 0).6. MQ MQ RQ 7 8 or , 9.QM. h area of ΔTQM [h same/dieselfde] area of TQR.QR. h QM QR 7 area of ΔTQM area of ΔTQM area of parm RQTS area of TQR 7 7 Answer only: full marks length of TQ OR SR length of TR length of RQ OR TS correct subst into cosine rule simplification answer (6) substitution/substitusie MQ 8 answer area of ΔTQM area of TQR 7 area parm RQTS area answer () TQR () area of ΔTQM area of TQR QM QR = 7 area of ΔTQM area of ΔTQM area of parm RQTS area of TQR = = 7 7 area of ΔTQM area of TQR 7 area parm RQTS area answer TQR ()

8 Mathematics P/Wiskunde V 8 DBE/Feb. Mar./Feb. Mrt. 07 QM. h area of ΔTQM area of parm RQTS RQ. h 7 7 QM. h RQ. h 7 answer () area of ΔTQM area of parm RQTS QT.QM.sin( ) area of ΔQTR QT.QM.sin( ) [.QT.QR.sin( )] 7 7 area parm RQTS area QT.QM.sin( ) [.QT.QR.sin( )] answer TQR () []

9 Mathematics P/Wiskunde V 9 DBE/Feb. Mar./Feb. Mrt. 07 QUESTION/VRAAG 4 y P S( ; 8) N T(0 ; 5) M O R x 4. line from centre to midpt of chord / lyn vanaf midpt na midpt van koord mst 0 m m [TS NP] ST NP m NP y = x c 8 = + c y y ( x x ) c = y 8 ( x ) y = x y x 4. P(0 ; ) [y-intercept of chord NP] radius is 6 units R(0 ; ) Equations of the tangents to the circle parallel to the x-axis/ Vgls van die raaklyne aan die sirkel aan die x-as: y and y 4.4 M( ; 0) [x intercept of/x-afsnit van NP] MT (0 ) (5 0) MT 46 =,08 answer () subst ( ; 8) and (0 ; 5) into gradient formula m ST m NP subst ( ; 8) into equation of a line equation (5) coordinates of P/ koördinate v P coordinates of R koördinate van R answers coordinates of M substitution answer

10 Mathematics P/Wiskunde V 0 DBE/Feb. Mar./Feb. Mrt MT = diameter/middellyn [conv in circle/omgek in sirkel] 46 radius = units Centre of circle/middelpunt v sirkel = Midpoint MT /Middelpunt MT 5 = ; Equation of circle through S, T and M: x 7 x 5 y 6, 04 5 y 46 4 radius of circle x value of M y value of M LHS of equation RHS of equation (5) [9] QUESTION/VRAAG 5 5. a b 5. f(x)= sin x 60 Period of f(x) = = 0 5. x [90 ; 5 ) {80 } 90 x < 5 or x = 80 Answer only: Full marks answer answer () 60 answer () 90 and 5 in interval form 80 as single value correct brackets () 90 and 5 in interval form 80 as single value correct inequalities () [7]

11 Mathematics P/Wiskunde V DBE/Feb. Mar./Feb. Mrt. 07 QUESTION/VRAAG sin (60 6 ) = sin 6 answer 6.. cos 7 cos( 6) 6. sin 6 tan R.T.P.: cos tan tan tan LHS tan sin cos cos sin cos cos cos = RHS Answer only: Full marks double angle/dubbelhoek answer writing as a single fraction/skryf as enkelbreuk quotient identity/ kwosiëntidentiteit () () denominator as a single fraction / Noemer as enkelbreuk square identity/vierkantidentiteit tan tan LHS tan sin cos cos sin cos cos cos cos sin cos cos = RHS writing as a single fraction/skryf as enkelbreuk quotient identity / kwosiëntidentiteit cos cos square identity/vierkantidentiteit quotient identity/

12 Mathematics P/Wiskunde V DBE/Feb. Mar./Feb. Mrt sin sin LHS cos cos sin cos cos cos sin sin cos cos sin cos RHS cos x 4 cos x or x 60 k. 60 x 0 k. 60 x 0 k. 70 x 40 k. 70 or or or or x 00 k. 60 or x 40 k. 60 x 600 k. 70 or x 480 k.70; k Z cos x 4 cos x or x 60 k. 60 or x 0 k. 60 x 0 k. 70 or x 40 k.70; k Z kwosiëntidentiteit writing as a single fraction/ skryf as enkelbreuk square identity/vierkantidentiteit simplification/vereenvoudiging cos x 60 and 00 0 and 40 4 write at least one general solution as x k. 60 write at least one general solution as x k. 70 ; kz (6) cos x 4 ±60 ±0 write at least one general solution as x k. 60 write at least one general solution as x k. 70 kz (6)

13 Mathematics P/Wiskunde V DBE/Feb. Mar./Feb. Mrt sin(a B) = cos[90 (A B)] cos[(90 A) ( B)] cos(90 A)cos( B) sin(90 A)sin( B) sin AcosB cos A( sinb) sin AcosB cos AsinB sin(a B) = cos[90 (A B)] cos[(90 B) A] cos(90 B)cosA sin(90 B)sinA sin BcosA cos BsinA sin AcosB cos AsinB 6.4. sin( x 64) cos( x 79) sin( x 9) cos( x 44) sin( x 64)cos( x 9) sin( x 9)[ cos( x 64)] sin( x 64)cos( x 9) cos( x 64)sin( x 9) sin[ x 64 ( x 9)] sin 45 co ratio/ko-verhouding writing as a difference of A & B/ skryf as verskil van A & B expansion/uitbreiding all reductions/alle reduksies co ratio/ko-verhouding writing as a difference of A & B/ skryf as verskil van A & B expansion/uitbreiding all reductions/alle reduksies cos( x 79) cos( x 9) cos( x 44) cos( x 64) compound formula identity/ saamgestelde identiteit sin 45 (6) []

14 Mathematics P/Wiskunde V 4 DBE/Feb. Mar./Feb. Mrt. 07 QUESTION/VRAAG 7 A C B 8, E D 7. CD sin 7 8,6 CD 8,6sin 7 CD,90 m substitution in correct trig ratio / substitusie in korrekte trig verh answer () 7. 0 cos 40 AE 0 AE cos 40 AE,05 m 7. AC CE AE CE.AE(cos AÊC) (8,6) (,05) (8,6)(,05)(cos 70) 67,49 AC,94 m substitution in correct trig ratio / substitusie in korrekte trig verh answer () correct use of cosine rule in ACE/ korrekte gebruik van reel in ACE correct subst into cosine rule AC answer [8]

15 Mathematics P/Wiskunde V 5 DBE/Feb. Mar./Feb. Mrt. 07 QUESTION/VRAAG 8 Q V P R 08 S 4 T 8. Qˆ 7 [opp s of cyclic quad/teenoorst e koordevh] S R 8. Rˆ Pˆ [s opp equal sides/e teenoor gelyke sye] 80-7 Rˆ [sum of s in /som v e in ] 54 S/R answer 8. Pˆ 4 [tan chord theorem/raakl-koordst] S R 8.4 Rˆ Pˆ Pˆ [ext of cyclic quad/buite van koordevh] = = 96 Rˆ [sum of/som vans/e in ] Rˆ 80 Rˆ Rˆ [s on str line/e op reguitlyn] = [sum of/som vans/e in ] = 96 R S S Rˆ 0 () () () () () [8]

16 Mathematics P/Wiskunde V 6 DBE/Feb. Mar./Feb. Mrt. 07 QUESTION/VRAAG 9 P Q W T S V ST SW = TQ WP = SV SW = VR WP = ST SV = TQ VR TV QR [prop theorem/eweredighst; TW QP] [prop theorem/eweredighst; VW RP] WS [both equal/beide gelyk ] PW [line divides sides of in prop/lyn verdeel sye van in dies verh] Tˆ Qˆ [corresp/ooreenkomst s/e; TV QR] S S answer S S R 9. VWS RPS RPS (any order) () 9.4 R WV PR SW SP 5 [ VWS RPS] R ratio answer () () () [0]

17 Mathematics P/Wiskunde V 7 DBE/Feb. Mar./Feb. Mrt. 07 QUESTION/VRAAG 0 0. P O A B Constr/ Konst : Draw line POand extend / Trek lyn Proof/ Bewys : OP OA Pˆ but Ô Similarly/ Netso, Ô Â Ô Pˆ Ô i.e. AÔB Pˆ Â (Pˆ APˆB Ô Pˆ Pˆ ) [radii] [ s opp/ teenoor sides/ sye] [ ext PO en verleng of ] construction S/R S/R S S (5)

18 Mathematics P/Wiskunde V 8 DBE/Feb. Mar./Feb. Mrt R y O P A S 4 Q T 0.. s in the same segment/ e in dieselfde sirkelsegment R 0.. Pˆ Ŝ y [s opp equal sides/e teenoor = sye] S R S R Ŝ Pˆ y [tan chord theorem/raakl-koordst] Pˆ Pˆ PQ bisects TPˆ S 0.. PÔQ Ŝ y [at centre = at circ/midpts= omtreks] S R 0..4 PˆA Pˆ Pˆ y [proved/bewys in..] T TPˆA P ÔQ [proved/bewys in..] PT = tangent [converse tan chord theorem/omgek raakl-koordst] R TPˆA P ÔQ () () ()

19 Mathematics P/Wiskunde V 9 DBE/Feb. Mar./Feb. Mrt OPˆQ OQˆ P 80 y [sum of/sum v s/e in ] O Qˆ P = 90 y [s opp equal sides/e to = sye; OP = OQ] In PAQ: Qˆ P Pˆ QÂP 80 O 90 - y y QÂP 80 [sum of/sum v s/e in ] QÂP 90 O ÂP 90 [s/e on straight line/op reguitlyn] O PˆT 90 [radius tangent/raaklyn] Pˆ 90 y Pˆ Ô OÂP 80 [sum of/sum v s/e in ] ( 90 y ) y OÂP 80 O ÂP = 90 S S R S S S R S S S (5) (5) POSQ is a kite/'n vlieër OQ PS O ÂP = 90 [diag of a kite/hoeklyne v vlieër] S R (5) In OAP and OAS OP = OS (radii) OA is common POA ˆ y Pˆ QOS ˆ OAP OAS (SAS) OAP ˆ OAS ˆ (s) OAP ˆ OAS ˆ 90 (s on str line) S S S R S (5) [9]

20 Mathematics P/Wiskunde V 0 DBE/Feb. Mar./Feb. Mrt. 07 QUESTION/VRAAG S L x P N T K R. Nˆ = 90 [ in semi-circle/halfsirkel] TPLN is a cyclic quad/ n koordevh [opp s of quad is suppl/ teenoore v vh is suppl] OR S R R () Nˆ = 90 [ in semi-circle/halfsirkel] TPLN is a cyclic quad [ext = int opp /buite = to binne ] S R R. Tˆ = P LˆN x [ext of cyclic quad/buite van koordevh] Kˆ 90 x [sum of/som v s/e in ] Nˆ Kˆ 90 x [tan chord theorem/raakl-koordst] Kˆ 90 x [sum of/som v s/e in ] Nˆ Kˆ 90 x [tan chord theorem/raakl-koordst] Nˆ x [tan chord theorem/raakl-koordst] Nˆ 90 [in semi circle/ halfsirkel] Nˆ 90 x [straight line/reguitlyn] R S R R S R R S S () () () ()

21 Mathematics P/Wiskunde V DBE/Feb. Mar./Feb. Mrt In KTP and KLN: PKˆ T LKˆ N [common/gemeen] K PˆT KNˆ L 90 [given/gegee] KTP KLN [] S S R () In KTP and KLN: PKˆ T LKˆ N [common/gemeen] K PˆT KNˆ L 90 [given/gegee] Tˆ PLˆ N x [proved in. OR sum of s in ] KTP KLN S S S ().. KT KP [ s] KL KN KT. KN = KP. KL But KL = KP [radii: PK = LP] KT. KN = KP. KP = KP = (KT TP ) [Theorem of Pythagoras] KT TP S/R S S S S (5) [4] TOTAL/TOTAAL: 50

NATIONAL/NASIONALE SENIOR CERTIFICATE/SERTIFIKAAT GRADE/GRAAD 12

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.