NATIONAL SENIOR CERTIFICATE GRADE/GRAAD 12
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1 NATIONAL SENIOR CERTIFICATE GRADE/GRAAD MATHEMATICS P/WISKUNDE V FEBRUARY/MARCH/FEBRUARIE/MAART 06 MEMORANDUM MARKS: 50 PUNTE: 50 This memorandum consists of pages./ Hierdie memorandum bestaan uit bladsye.
2 Mathematics/P/Wiskunde V DBE/ Feb. Mar./Feb. Mrt. 06 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 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 DBE/ Feb. Mar./Feb. Mrt. 06 QUESTION/VRAAG The data is skewed to the left/die data is skeef na links. The data is negatively skewed/die data is negatief skeef.. Range/Omvang max. min.. 5% of the learners failed/van die leerders het gedruip () () (). 5 + T 5 9 T (6) []
4 Mathematics/P/Wiskunde V DBE/ Feb. Mar./Feb. Mrt. 06 QUESTION/VRAAG.. (5) + 8(5) +...(85) 080 Mean/Gemiddelde , messages per day/boodskappe per dag Cumulative Frequency/Kumulatiewe Frekwensie OGIVE/OGIEF () grounding at (0 ; 0) plotting at upper limits plotting. cumulative f smooth shape of curve geanker by (0 ; 0) stip by boonste limiete plot kumulatiewe f gladde vorm van kurwe () Number of messages/getal boodskappe. Number of days/getal dae 60 6 (see on graph above/sien op grafiek hierbo) days/dae 6 (accept 5 9) (accept 5) () Number of days/getal dae days/dae add correct values/tel korrekte waardes by () [9]
5 Mathematics/P/Wiskunde V 5 DBE/ Feb. Mar./Feb. Mrt. 06 QUESTION/VRAAG y P( ; ) M N T x y x + Q(0 ; ) x + y θ R.. m PQ ( ) 0 QR: y x mqr mpq mqr PQ QR PQˆ R 90 subst ( ; ) & (0 ; ) m QR m m PQ QR () ()
6 Mathematics/P/Wiskunde V 6 DBE/ Feb. Mar./Feb. Mrt x x + x x 6 y R(6 ; ) PR ( 6) PR 50 5 ( 6) 50 + ( ( )) PR ( ( )).5 PR is a diameter/ n middellyn [chord subtends/kd onderspan 90 ] + 6 Centre of circle/midpt v sirkel: ; ; r 50 5 OR OR, x + y + OR OR,5.6 m of/van radius m of/van tangent/raaklyn Equation of tangent/vgl van raaklyn: y y ( ) x x y x + c y x + c y x y x m.7 tan PNˆ T mpr PNˆ T 5 tan PMˆ T PQ PMˆ T 7, 57 Pˆ 6, [ext of /buite v ] θ 6, 57 [sum of s in /som v e in ] equating/gelyk stel x-value/waarde y-value/waarde () subst into/in distance formula/ afstandsformule in surd form/ wortelvorm () subst into/in distance formula/ afstandsformule in surd form/ wortelvorm () S ; r-value/waarde m of tang/rkl (6) subst m & P( ; ) into/in eq of line/vgl v lyn () tan PNˆ T PNˆ T 5 P Mˆ T 7, 57 Pˆ 6, (5)
7 Mathematics/P/Wiskunde V 7 DBE/ Feb. Mar./Feb. Mrt. 06 Extrapolation of RQ to S/Verlenging van RQ na S: P S O M N T Q R tan PNˆ T mpr SNˆ R 5 tan NŜR m RS NŜR 8, θ 80 (5 + 8, ) [sum of s in /som v e in ] 6, 57 tan PNˆ T SNˆ R 5 tan NŜR N ŜR 8, (5) PQ + 0 PQ 0 PQ 0 sin θ PR 50 θ 6,57 QR 6 QR 0 0 cos θ 50 θ 6, subst into/in distance formula/ afstandsformule distance/afst PQ correct trig ratio/ korrekte trig vh correct trig eq/ korrekte trig vgl (5) subst into/in distance formula/ afstandsformule distance/afst PQ correct trig ratio/ korrekte trig vh correct trig eq/ korrekte trig vgl (5)
8 Mathematics/P/Wiskunde V 8 DBE/ Feb. Mar./Feb. Mrt. 06 mrq m tanθ + m. m RQ PR PR ( ) + ( )( ) θ 6,57 correct formula/ korrekte formule m RQ correct subst/ subst korrek tan θ θ 6, 57 (5) P θ O M N T Q R tangent OP goes through the origin/raakl OP gaan deur oorsprong P ÔM 5 O PˆM θ Pˆ [tan-chord theorem/raakl-kdst] tan PMˆ T m PQ PMˆ T 7, 57 θ + 5 7, 57 [ext of /buite- v ] θ 6,57 P ÔM 5 R P Mˆ T 7, 57 S θ 6, 57 (5) []
9 Mathematics/P/Wiskunde V 9 DBE/ Feb. Mar./Feb. Mrt. 06 QUESTION/VRAAG y V x + y 0 O S x T R y x + k P. OR TR [radius tangent/raakl] m TR mor m OR y x. x + ( x) 0 x + x 0 5x 0 0 x 0 ( x + )( x ) 0 x y () R( ; ) S/R m of/van OR equation/vgl () subst eq of OR into circle eq/ subst vgl OR in sirkelvgl st. form/st. vorm x-value/waarde y-value/waarde ()
10 Mathematics/P/Wiskunde V 0 DBE/ Feb. Mar./Feb. Mrt. 06. Subst R( ; ) into the equation of/in vgl van PRS: () + k k 5 OT 5 0 x 5 x 0 OS 0 correct subst/ korrekte subst value of k y 0 x-intercept/afsnit. Area/Oppervlakte OS. OT (0)(5) 5 sq units/vk eenh V + 0 x and/en 0 y V V( ; ) T(0 ; 5)... from/van. VT ( 0) + ( ( 5)) correct subst into area form/ subst korrek in opp-formule (6) x-value/waardev y-value/waardev subst of points V and T into distance formula/ subst punte V en T in afst-form () [7]
11 Mathematics/P/Wiskunde V DBE/ Feb. Mar./Feb. Mrt. 06 QUESTION/VRAAG 5 P( 7 ; ) y θ θ O R x 6 S(a ; b) 5.. tanθ sin ( θ) sin θ OP ( 7 ) + OP 6 OP sin ( θ ) reduction/ reduksie OP () () 5.. a cos θ 6 a 6( sin 6 7 θ a cos θ 6 a 6( cos θ ) 7 ) 6 trig ratio/verh expansion/ uitbreiding sin θ trig ratio/verh expansion/ uitbreiding 7 cosθ () ()
12 Mathematics/P/Wiskunde V DBE/ Feb. Mar./Feb. Mrt a cos θ 6 a 6(cos θ sin θ ) sin x.cos x (sin x.cos x) sin x ( sin x) trig ratio/verh expansion/ uitbreiding cosθ sin θ 7 & () sin x cos x sin x cos x tan x 5.. sin5 cos5 tan (5 ) sin 5 tan0 tan (5 ) () () []
13 Mathematics/P/Wiskunde V DBE/ Feb. Mar./Feb. Mrt. 06 QUESTION/VRAAG 6 6. sin (x + 60 ) + cos x 0 sin x cos 60 + cos x sin 60 + cos x 0 sin x + cos x + cos x 0 sin sin x x cos x cos x sin x cos x( ) sin x cos x( ) cos x cos x tan x 6. tan x tan x ( + ) ref 80,0 x 80, or/of 99,9 6.. cos x cos x y f expansion/uitbreiding special angle values/ spesiale -waardes simpl/vereenv sin x cos x( 80,0 99,90 80, (0 ; ) ( 60 ; 0) shape/vorm ) () () g x - - () 6.. sin (x + 60 ) > cos x x ( 80,0 ; 99,90 ) 80,0 < x < 99, 90 critical values/ kritiese waardes notation/notasie () []
14 Mathematics/P/Wiskunde V DBE/ Feb. Mar./Feb. Mrt. 06 QUESTION/VRAAG Area of/oppervlakte van PQR PQ.QR.sin Qˆ x(0 x)(sin 60 ) 0x x 5 x x 7.. For maximum area/vir maksimum opp: (Area PQR) / 0 5 x 0 x 5 5 x max or or/of,5 b xmax a 5 5 or or,5 ( ) subst into area rule/ subst in opp-reël subst & simpl/ subst en vereenv () (Area PQR) / 0 5 formula/e subst x () () x x 0 x(5 x) 0 x 0 or x max or/of,5 RP QP 0 8,5 RP 9,0 + QR +,5.QP.QR.cosQ (0)(,5) cos 60 x-intercepts/ x-afsnitte subst subst into cosine rule/in cos-reël simpl/vereenv () ()
15 Mathematics/P/Wiskunde V 5 DBE/ Feb. Mar./Feb. Mrt h In ABC: sin β AB h AB sin β In ABD: AB BD and/en A Dˆ B 90 β [ s of/v 80 ] sin β sin(90 β ) AD AB AB.sin β AD sin(90 β ) h sin β.cos β sin β cos β h h In ABC: sin β AB h AB sin β In ABD: AB BD AD AB + AB AB.AB.cos β h sin β h sin β h sin β h AD h h + sin β h + sin β h + sin β h.cos β sin β h ( sin sin β h + h sin β β ) Split isosceles triangle ABQ into two congruent triangles AEB and DEB. Then ABC BAE (AB AC, A Bˆ E BÂC β, h) AE ED BC h AD h AB ito h and/en β A Dˆ B 90 β correct subst into cosine rule/subst korrek in cos-reël AD as subject/ onderwerp expansion/uitbrei sin (90 β) cos β answer ito h (7) AB ito h and/en β correct subst into cosine rule/subst korrek in cos-reël expansion/uitbrei multiplication/ vermenigv simpl/vereenv answer ito h (7) (7) [5]
16 Mathematics/P/Wiskunde V 6 DBE/ Feb. Mar./Feb. Mrt. 06 QUESTION/VRAAG 8 8. K 8 N L T 8.. Kˆ Mˆ 0 [tan chord theorem/raakl-kdst] S R Nˆ Kˆ [ s in the same seg/ e in dies segm] Kˆ 8 0 Nˆ 0 S R Tˆ Nˆ [alt/verw s/e; KT NM] S R 8.. Lˆ Kˆ + Tˆ [ext of /buite v ] M () S () R () S () 8..5 In KLM: Lˆ 80 [ s sum in / e som in ] Lˆ S ()
17 Mathematics/P/Wiskunde V 7 DBE/ Feb. Mar./Feb. Mrt A x + 0 E 08 C B D 8. Ĉ 08 [opp s of m/tos e v m] x x [opp s of cyc quad/tos e v kdvh] x 6 Ĉ 80 (x + 0 ) [opp s of cyc quad/tos e v kdvh] 80 (x + 0 ) 08 [opp s of m/tos e v m] x x 6 S R S R S R S R (5) (5) [5]
18 Mathematics/P/Wiskunde V 8 DBE/ Feb. Mar./Feb. Mrt. 06 QUESTION/VRAAG 9 A B O F D C G E 9. ABCD is a m [diags of quad bisect each other/ hoekl v vh halveer mekaar] 9. ED FE DB AF [Prop Th/Eweredigh st; DF BA] ED GE DB CG [Prop Th/Eweredigh st; DG BC] 9. FE GE AF CG [proved/bewys] AC FG [line divides two sides of in prop/ lyn verdeel sye van eweredig] Ĉ Fˆ [alt/verw s/e; AC FG] Â Ĉ [alt/verw s/e; AB CD] Â Fˆ 9. Â Â [diags of rhombus/hoekl v ruit] Â Fˆ [ Â Fˆ ] ACGF cyc quad/kdvh [ s in the same seg / e in dies segm ] Ĉ Â [ s opp equal sides of rhombus/ e to gelyke sye v ruit] Â Ĝ [alt/verw- s/e; AC FG] Ĉ Ĝ ACGF is a cyc quad/kdvh [ s in the same seg / e in dies segm ] R S R S R S S R S S S S R S S R () () (5) () () []
19 Mathematics/P/Wiskunde V 9 DBE/ Feb. Mar./Feb. Mrt. 06 QUESTION/VRAAG 0 0. A P D E Q R B C 0.. In ADE and/en PQR: AD PQ [construction/konstr] Â Pˆ [given/gegee] AE PR [construction/konstr] ADE PQR [S S] 0.. A Dˆ E Qˆ [ s corres/ooreenk s/e ] But Bˆ Qˆ [given/gegee] A Dˆ E Bˆ DE BC [corres/ooreenk s/e ] 0.. AB AC AD AE [Prop Th/Eweredigh st; DE BC] But/Maar AD PQ and/en AE PR [construction/konstr] AB AC PQ PR all/al S s/e reason/rede A Dˆ E Qˆ A Dˆ E Bˆ reason/rede S/R S () () ()
20 Mathematics/P/Wiskunde V 0 DBE/ Feb. Mar./Feb. Mrt R V O T P S 0.. line from centre to midpt of chord/lyn van midpt na midpt van koord 0.. OP VS [Midpt Theorem/Midpt-stelling] In ROP and/en RVS: Rˆ Rˆ [common/gemeen] Ô Vˆ [corresp/ooreenk s/e; OP VS] ROP RVS [,, ] In ROP and/en RVS: Pˆ VŜR [corresponding s/ ooreenkomstige 'e] Rˆ Rˆ [common/gemeen] ROP RVS [,, ] S R () S S & ; ; angles/hoeke () S R S S & ; ; angles/hoeke ()
21 Mathematics/P/Wiskunde V DBE/ Feb. Mar./Feb. Mrt In RVS and/en RST: V ŜR STˆ R 90 [ in semi-circle/ in halfsirkel] Rˆ is common/gemeen Vˆ TŜR RVS RST [,, ] 0.. In RTS and/en STV: R Tˆ S VTˆ S 90 [ s on straight line/ e op rt lyn] Rˆ 90 T ŜR T ŜV T ŜR Vˆ RTS STV [,, ] RT TS ST VT ST VT.TR S R S & ; ; angles/hoeke () RTS & STV S S S (with justification/met motivering) RTS STV ratio/verh (6) [] TOTAL/TOTAAL: 50
NATIONAL SENIOR CERTIFICATE GRADE/GRAAD MATHEMATICS P/WISKUNDE V FEBRUARY/MARCH/FEBRUARIE/MAART 06 MEMORANDUM MARKS: 50 PUNTE: 50 This memorandum consists of pages./ Hierdie memorandum bestaan uit bladsye.
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