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. Copyright reserved/kopiereg voorbehou Please turn over/blaai om asseblief

Mathematics/P/Wiskunde V DBE/ Feb. Mar./Feb. Mrt. 06 NSC/NSC Memorandum 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. Copyright reserved/kopiereg voorbehou Please turn over/blaai om asseblief

Mathematics/P/Wiskunde V 3 DBE/ Feb. Mar./Feb. Mrt. 06 NSC/NSC Memorandum QUESTION/VRAAG 3 6 0 30 40 50 60 70 75 80. The data is skewed to the left/die data is skeef na links. OR/OF The data is negatively skewed/die data is negatief skeef.. Range/Omvang = 80 0 = 60 answ/antw answ/antw max. min. answ/antw.3 5% of the learners failed/van die leerders het gedruip answ/antw () () ().4 445 + T 54 = 9 T = 4 4 4 0 8 36 4 6 69 75 75 80 0 4 6 75 80 (6) [] Copyright reserved/kopiereg voorbehou Please turn over/blaai om asseblief

Mathematics/P/Wiskunde V 4 DBE/ Feb. Mar./Feb. Mrt. 06 NSC/NSC Memorandum QUESTION/VRAAG.. (5) + 8(5) +...(85) 3080 Mean/Gemiddelde = = 60 60 = 5,33 messages per day/boodskappe per dag Cumulative Frequency/Kumulatiewe Frekwensie 70 65 60 55 50 45 40 35 30 5 0 5 0 5 OGIVE/OGIEF 3 080 3080 60 answ/antw (3) 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 (4) 0 0 0 0 30 40 50 60 70 80 90 00 Number of messages/getal boodskappe.3 Number of days/getal dae = 60 46 (see on graph above/sien op grafiek hierbo) = 4 days/dae OR/OF 46 (accept 45 49) answ/antw (accept 5) () Number of days/getal dae = + 3 + 8 = 4 days/dae add correct values/tel korrekte waardes by answ/antw () [9] Copyright reserved/kopiereg voorbehou Please turn over/blaai om asseblief

Mathematics/P/Wiskunde V 5 DBE/ Feb. Mar./Feb. Mrt. 06 NSC/NSC Memorandum QUESTION/VRAAG 3 y P( ; ) M N T x y = x + Q(0 ; ) x + 3y + 6 = 0 θ R 3. 3. m PQ ( ) = 0 = 3 QR: y = x 3 mqr = 3 mpq mqr = 3 3 = PQ QR PQˆ R = 90 subst ( ; ) & (0 ; ) answ/antw m QR = 3 m m = PQ QR () () Copyright reserved/kopiereg voorbehou Please turn over/blaai om asseblief

Mathematics/P/Wiskunde V 6 DBE/ Feb. Mar./Feb. Mrt. 06 NSC/NSC Memorandum 3.3 3.4 3 x = x + x = 4 3 x = 6 y = 4 R(6 ; 4) PR = PR = ( 6) 50 = 5 = ( 6) = 50 + ( ( 4)) Copyright reserved/kopiereg voorbehou PR = 50 = 5 + ( ( 4)) OR/OF 3.5 PR is a diameter/ n middellyn [chord subtends/kd onderspan 90 ] + 6 4 Centre of circle/midpt v sirkel: ; = 3 ; r = 50 OR 5 OR 3,54 7 3 50 5 x + y + = OR 4 3.6 m of/van radius = m of/van tangent/raaklyn = Equation of tangent/vgl van raaklyn: y y = ( x x ) y = x y = x 3.7 tan PNˆ T = mpr = PNˆ T = 35 tan PMˆ T = PQ 3 == m PMˆ T 7, 57 OR/OF OR,5 Pˆ = 63,43 [ext of /buite v ] θ = 6, 57 [sum of s in /som v e in ] OR/OF y = x + c = + c y = x equating/gelyk stel x-value/waarde y-value/waarde (3) subst into/in distance formula/ afstandsformule answ/antw in surd form/ wortelvorm () subst into/in distance formula/ afstandsformule answ/antw in surd form/ wortelvorm () S 3 ; Please turn over/blaai om asseblief r-value/waarde answ/antw m of tang/rkl (6) subst m & P( ; ) into/in eq of line/vgl v lyn answ/antw (3) tan PNˆ T = PNˆ T = 35 P Mˆ T = 7, 57 Pˆ = 63,43 answ/antw (5)

Mathematics/P/Wiskunde V 7 DBE/ Feb. Mar./Feb. Mrt. 06 NSC/NSC Memorandum Extrapolation of RQ to S/Verlenging van RQ na S: P S O M N T Q R tan PNˆ T = mpr = SNˆ R = 35 tan NŜR = m = RS 3 NŜR = 8, 43 θ = 80 (35 + 8,43 ) [sum of s in /som v e in ] = 6, 57 tan PNˆ T = SNˆ R = 35 tan NŜR = 3 N ŜR = 8, 43 answ/antw (5) PQ = PQ = + 3 0 = 0 PQ 0 sin θ = = = PR 50 θ = 6,57 QR = 6 QR = + 0 = 40 0 cos θ = = 50 θ = 6,57 5 5 OR/OF OR/OF OR/OF subst into/in distance formula/ afstandsformule distance/afst PQ correct trig ratio/ korrekte trig vh correct trig eq/ korrekte trig vgl answ/antw (5) subst into/in distance formula/ afstandsformule distance/afst PQ correct trig ratio/ korrekte trig vh correct trig eq/ korrekte trig vgl answ/antw (5) Copyright reserved/kopiereg voorbehou Please turn over/blaai om asseblief

Mathematics/P/Wiskunde V 8 DBE/ Feb. Mar./Feb. Mrt. 06 NSC/NSC Memorandum mrq m tanθ = + m. m RQ PR PR ( ) = 3 + ( )( ) 3 = θ = 6,57 correct formula/ korrekte formule m RQ = 3 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 = 45 O PˆM = θ = Pˆ [tan-chord theorem/raakl-kdst] tan PMˆ T = PQ 3 == m PMˆ T 7, 57 θ + 45 = 7, 57 [ext of /buite- v ] θ = 6,57 P ÔM = 45 R P Mˆ T = 7, 57 S θ = 6, 57 (5) [3] Copyright reserved/kopiereg voorbehou Please turn over/blaai om asseblief

Mathematics/P/Wiskunde V 9 DBE/ Feb. Mar./Feb. Mrt. 06 NSC/NSC Memorandum QUESTION/VRAAG 4 y V x + y = 0 O S x T R y = x + k P 4. OR TR [radius tangent/raakl] m m = 4. TR m = OR y = x x 5x OR + ( x) x + 4x x = 0 = 0 0 = 0 4 = 0 ( x + )( x ) = 0 x = y = () = 4 R( ; 4) S/R m of/van OR equation/vgl (3) subst eq of OR into circle eq/ subst vgl OR in sirkelvgl st. form/st. vorm x-value/waarde y-value/waarde (4) Copyright reserved/kopiereg voorbehou Please turn over/blaai om asseblief

Mathematics/P/Wiskunde V 0 DBE/ Feb. Mar./Feb. Mrt. 06 NSC/NSC Memorandum 4.3 Subst R( ; 4) into the equation of/in vgl van PRS: 4 = () + k k = 5 OT = 5 0 = x 5 x = 0 OS = 0 correct subst/ korrekte subst value of k y = 0 x-intercept/afsnit 4.4 Area/Oppervlakte = OS. OT = (0)(5) = 5 sq units/vk eenh V + 0 = x 4 and/en 0 = y V V( ; 4) T(0 ; 5)... from/van 4.3 VT = = = ( 0) 4 + 8 85 + (4 ( 5)) correct subst into area form/ subst korrek in opp-formule answ/antw (6) x-value/waardev y-value/waardev subst of points V and T into distance formula/ subst punte V en T in afst-form answ/antw (4) [7] Copyright reserved/kopiereg voorbehou Please turn over/blaai om asseblief

Mathematics/P/Wiskunde V DBE/ Feb. Mar./Feb. Mrt. 06 NSC/NSC Memorandum QUESTION/VRAAG 5 P( 7 ; 3) y θ θ O R x 6 S(a ; b) 5.. tanθ = 5.. sin ( θ) = sin θ OP OP = OP = 4 3 7 ( 7 ) = 6 sin ( θ ) = 3 4 + 3 answ/antw reduction/ reduksie OP = 4 answ/antw () (3) 5..3 a = cos θ 6 a = 6( sin 3 = 6 4 4 7 = 4 4 3 = 4 θ a = cos θ 6 a = 6( cos θ ) = = 4 3 = 4 7 4 4 4 ) 6 OR/OF OR/OF trig ratio/verh expansion/ uitbreiding 3 sin θ = 4 answ/antw trig ratio/verh expansion/ uitbreiding cosθ = answ/antw 7 4 (4) (4) Copyright reserved/kopiereg voorbehou Please turn over/blaai om asseblief

Mathematics/P/Wiskunde V DBE/ Feb. Mar./Feb. Mrt. 06 NSC/NSC Memorandum 5.. a = cos θ 6 a = 6(cos θ sin θ ) 7 3 = 6 4 4 = 6 6 3 = 4 4sin x.cos x (sin x.cos x) = sin x ( sin x) trig ratio/verh expansion/ uitbreiding cosθ = 3 sin θ = 4 answ/antw 7 4 & (4) = sin x cos x sin x cos x = tan x answ/antw 5.. 4sin5 cos5 = tan (5 ) sin 5 = tan30 = 3 = OR/OF 3 3 3 tan (5 ) answ/antw (3) () [3] Copyright reserved/kopiereg voorbehou Please turn over/blaai om asseblief

Mathematics/P/Wiskunde V 3 DBE/ Feb. Mar./Feb. Mrt. 06 NSC/NSC Memorandum QUESTION/VRAAG 6 6. sin (x + 60 ) + cos x = 0 sin x cos 60 + cos x sin 60 + cos x = 0 3 sin x + cos x + cos x = 0 3 sin x = cos x cos x sin x = 4cos x 3 cos x sin x = cos x( 4 sin x cos x( 4 = cos x cos x tan x = 4 3 6. tan x = 4 3 6.3. 3) 3) tan x = ( 4 + 3) ref = 80,0 x = 80, or/of 99,9 y f expansion/uitbreiding special angle values/ spesiale -waardes simpl/vereenv sin x = cos x( 4 80,0 99,90 80, (30 ; ) ( 60 ; 0) shape/vorm 3) (4) (3) g -0-60 60 0 80 40 0 x - - (3) 6.3. sin (x + 60 ) > cos x x ( 80,0 ; 99,90 ) OR/OF 80,0 < x < 99, 90 critical values/ kritiese waardes notation/notasie (3) [3] Copyright reserved/kopiereg voorbehou Please turn over/blaai om asseblief

Mathematics/P/Wiskunde V 4 DBE/ Feb. Mar./Feb. Mrt. 06 NSC/NSC Memorandum QUESTION/VRAAG 7 7.. Area of/oppervlakte van PQR = PQ.QR.sin Qˆ = x(0 4x)(sin 60 ) = 3 0x x = 5 3x 3x 7.. For maximum area/vir maksimum opp: (Area PQR) / = 0 5 3 3x = 0 OR/OF b xmax = a 5 3 = = ( 3) 5 or 3x = 5 3 5 x = max or or/of,5 or,5 subst into area rule/ subst in opp-reël subst & simpl/ subst en vereenv () (Area PQR) / = 0 5 3 answ/antw formula/e subst answ/antw 3x (3) (3) 7..3 OR/OF 5 3x 3x = 0 3x(5 x) = 0 x = 0 or 5 0 + 5 5 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 answ/antw subst into cosine rule/in cos-reël simpl/vereenv answ/antw (3) (3) Copyright reserved/kopiereg voorbehou Please turn over/blaai om asseblief

Mathematics/P/Wiskunde V 5 DBE/ Feb. Mar./Feb. Mrt. 06 NSC/NSC Memorandum 7. h In ABC: sin β = AB h AB = sin β In ABD: AB = BD and/en sin β sin(90 β ) = AD AB AB.sin β AD = sin(90 β ) OR/OF 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 = 4h AD = h OR/OF h + sin β h + sin β h + sin β A Dˆ B = 90 β [ s of/v = 80 ] h sin β h sin β h sin β.cos β ( sin + 4h 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] Copyright reserved/kopiereg voorbehou Please turn over/blaai om asseblief

Mathematics/P/Wiskunde V 6 DBE/ Feb. Mar./Feb. Mrt. 06 NSC/NSC Memorandum QUESTION/VRAAG 8 8. K 84 N 3 L T 8.. Kˆ = Mˆ = 40 [tan chord theorem/raakl-kdst] S R 8.. 8..3 Nˆ = Kˆ [ s in the same seg/ e in dies segm] Kˆ = 84 40 = 44 Nˆ = 44 40 S R Tˆ = Nˆ = 44 [alt/verw s/e; KT NM] S R 8..4 Lˆ = Kˆ + Tˆ [ext of /buite v ] = 40 + 44 = 84 M 8..5 In KLM: 44 + 84 + 40 + Lˆ = 80 [ s sum in / e som in ] () S (3) R () S () Lˆ = S () Copyright reserved/kopiereg voorbehou Please turn over/blaai om asseblief

Mathematics/P/Wiskunde V 7 DBE/ Feb. Mar./Feb. Mrt. 06 NSC/NSC Memorandum 8. A x + 40 E 08 C B D 8. Ĉ = 08 [opp s of m/tos e v m] x + 40 + 08 = 80 x = 3 [opp s of cyc quad/tos e v kdvh] x = 6 OR/OF Ĉ = 80 (x + 40 ) [opp s of cyc quad/tos e v kdvh] 80 (x + 40 ) = 08 [opp s of m/tos e v m] x = 3 x = 6 S R S R answ/antw S R S R answ/antw (5) (5) [5] Copyright reserved/kopiereg voorbehou Please turn over/blaai om asseblief

Mathematics/P/Wiskunde V 8 DBE/ Feb. Mar./Feb. Mrt. 06 NSC/NSC Memorandum QUESTION/VRAAG 9 A 3 B O F 3 D 3 3 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] 9.3 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 9.4 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 =] OR/OF 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 () (4) (5) (3) (3) [3] Copyright reserved/kopiereg voorbehou Please turn over/blaai om asseblief

Mathematics/P/Wiskunde V 9 DBE/ Feb. Mar./Feb. Mrt. 06 NSC/NSC Memorandum 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..3 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 3 S s/e reason/rede A Dˆ E = Qˆ A Dˆ E = Bˆ reason/rede S/R S () (3) () Copyright reserved/kopiereg voorbehou Please turn over/blaai om asseblief

Mathematics/P/Wiskunde V 0 DBE/ Feb. Mar./Feb. Mrt. 06 NSC/NSC Memorandum 0. 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 [,, ] OR/OF In ROP and/en RVS: Pˆ = VŜR [corresponding s/ ooreenkomstige 'e] Rˆ = Rˆ [common/gemeen] ROP RVS [,, ] answ/antw S R () S S & ; ; OR/OF 3 angles/hoeke (4) S R S S & ; ; OR/OF 3 angles/hoeke (4) Copyright reserved/kopiereg voorbehou Please turn over/blaai om asseblief

Mathematics/P/Wiskunde V DBE/ Feb. Mar./Feb. Mrt. 06 NSC/NSC Memorandum 0..3 In RVS and/en RST: V ŜR = STˆ R = 90 [ in semi-circle/ in halfsirkel] Rˆ is common/gemeen Vˆ = TŜR RVS RST [,, ] 0..4 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 & ; ; OR/OF 3 angles/hoeke RTS & STV S S S (with justification/met motivering) (3) RTS STV ratio/verh (6) [] TOTAL/TOTAAL: 50 Copyright reserved/kopiereg voorbehou