GRADE/GRADE 12 SEPTEMBER 2016 MATHEMATICS P2 / WISKUNDE V2 MEMORANDUM
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1 NATIONAL SENIOR CERTIFICATE GRADE/GRADE SEPTEMBER 06 MATHEMATICS P / WISKUNDE V MEMORANDUM MARKS / PUNTE: 50 This memorandum consists of 6 pages. Hierdie memorandum bestaan uit 6 bladsye.
2 MATHEMATICS P (EC/SEPTEMBER 06) QUESTION / VRAAG Day/Dag Weight / Gewig correct points 5-7 correct points plotting all points 4 punte korrek 5 7 punte korrek alle punte afgesteek. a 4,84 b 5,83 y 4,84 5,83x.3 x ; y 8;5;75,5.4 y-int 4,84 4,84 5,83x 80 A B equation / vergelyking 8 ;5;75,5 and/en y-int / y-afsnit 4,84 regression line / regressielyn () substitution / vervanging 5,83x 44,84 x 7,69 answer / antwoord On the morning of the 8 th day the bar of soap will be less than 80 grams. Op die oggend van die 8 ste dag sal steen seep minder as 80 gram wees. ().5 r 0, 998 answer / antwoord ().6 Very strong negative correlation. Baie sterk negatiewe answer / antwoord korrelasie. () [] Kopiereg voorbehou Blaai om asseblief
3 (EC/SEPTEMBER 06) WISKUNDE V 3 QUESTION / VRAAG learners / leerders answer / antwoord (). 3 pages / bladsye answer / antwoord ().3 x 8, 9 answer / antwoord.4 3, answer / antwoord.5 8,9 3,; 8,9 3, 5,07;4,3 8 learners are outside one standard deviation 8 leerders is buite een standaardafwyking interval / interval 8 learners / leerders 8,57% () () ,0% [8] Copyright reserved Please turn over
4 4 MATHEMATICS P (EC/SEPTEMBER 06) QUESTION 3 / VRAAG mpr m SP PT SQ y 4 x y x 3.3 Equation of PR / Vergelyking van PR y x 4 y x 3 x x 3 4x 4 x 6 5x 0 x y subst. P and R into correct formula verv. van P en R in korrekte formule m PR m SQ subst. m and Q into correct formula verv. van m en Q in korrekte formule y x substituting m and P into equation of a str. line / vervanging van m en P in die vergelyking van reguitlyn equation of PR / vergelyking van PR equating PR and SQ / gelykstel van PR en SQ x-value / x-waarde y-value / y-waarde () OR/OF substituting m and R into equation of a Kopiereg voorbehou Blaai om asseblief
5 (EC/SEPTEMBER 06) WISKUNDE V 5 y 5 x 4 y0 x 4 y x6 y x3 x x 3 4x 4 x 6 5x 0 x y 3.4 x y4 x 5 and/ en y 8 S ( 5 ; 8) 3.5 SQ PT Area PQS unit / eenhede str. line / vervanging van m en R in die vergelyking van reguitlyn equation of PR / vergelyking van PR equating PR and SQ / gelykstel van PR en SQ x-value / x-waarde y-value / y-waarde substituting into correct formula vervanging in korrekte formule x-value / x-waarde y-value / y-waarde subt. into correct form / verv. in korrekte formule SQ = 6 5 PT = 5 Subt into correct form. / verv. in korrekte formule 5 units / eenhede (5) OR/OF area of PQS = ST PT unit / eenhede SQ = 6 5 ST = 3 5 PT = 5 subst into form / verv. in formule 5 units / eenhede (5) [8] Copyright reserved Please turn over
6 6 MATHEMATICS P (EC/SEPTEMBER 06) QUESTION 4 / VRAAG 4 4. x 8x 6 y x 4 y 3 M 4 ; y 3 y y 3 tan/ rkl 6y y Q (0; ) 3 mradius 4 0 m y x 0 y x [tangent radius] [raaklyn completing square kwadraatsvoltooiing x 4 y 3 0 x-coordinate / x-koördinaat y-coordinate / y-koördinaat (4) subst. x = 0 into circle equation verv. x = 0 in sirkel vergelyking y 3 4 y m rad m / m raaklyn = tan subst. mtan/ rkl and Q into correct form. / verv. mtan/ rkl en Q in korrekte formule. equation / vergelyking 4.4 y 6 answer / antwoord () (4) Kopiereg voorbehou Blaai om asseblief
7 (EC/SEPTEMBER 06) WISKUNDE V x 5 x 5 U ; 6 6 mau mad 0 0 mau mda AU DA Â 90º DQˆ U 90º [tangent radius] / [raaklyn radius] 6 = x + 5 x s QUAD is a cyclic quad.[opp. add up to 80º] QUAD is ʼn koordevierhoek [ teenoorst. e se som is 80º] m AU m AD m m AU DA Â 90 D Q ˆU 90 R () (6) [0] Copyright reserved Please turn over
8 8 MATHEMATICS P (EC/SEPTEMBER 06) QUESTION 5 / VRAAG sin cos.9 5. sin 5 sin 5 answer t cos 9 cos 38 cos9 cos9 sin 74 sin 74 4 OR/OF cos 38 cos 38 t cos(80 x).sin(80 x).sin 74 sin( x 360).sin 37.sin 53.sin( x 90) ( cos x).sin x.sin 74 sin x.sin 37.cos 37.( cos x) cos.9 cos 38 cos.9 cos 9 simplification / vereenvoudiging answer / antwoord (4) cos x sin x sin x cos37 cos x sin 74 answer / antwoord cos(80 x).sin(80 x).sin sin( x360 ).sin 37.sin 53.sin( x90 ) ( cos x).sin x.sin sin x.sin 37.cos 37.( cos x) 0 sin sin 37.cos37 4sin 37.cos sin 37.cos cos x sin x sin x cos37 cos x sin 37 0 answer / antwoord (7) Kopiereg voorbehou Blaai om asseblief
9 (EC/SEPTEMBER 06) WISKUNDE V cos x 0 cos x 0 cos x x 0 k.360 x k x 0 k.80; k x k; 5.3. L.H.S/LK sinx cosx sin x cos x sin x sin x sin x sin x sin x sin x sin x k (4) sin x removing brackets verwyder hakies sin x [] Copyright reserved Please turn over
10 0 MATHEMATICS P (EC/SEPTEMBER 06) QUESTION 6 / VRAAG sin x 60 sin 90 x x x 360. k or/ of x x 360. k or/ x k of x k 0 0 x 0 0. k ; k Z x 30 ; 0 ; OR/OF cos x cos 30 x x 30 x 360. k or 3x k or x x 30; 0; g: f: x 30 x k 360. k ; k co-ratio / kofunksie both gen. solns beide algemene oplossings co-ratio / kofunksie both gen. solns beide algemene oplossings (5) x-intercept. x-afsnit y-intercept. y-afsnit shape / vorm x-intercept. x-afsnit y-intercept. y-afsnit shape / vorm (6) answer / antwoord () 6.4 h x cos x 90 sin x substitution / vervanging sin x () [4] Kopiereg voorbehou Blaai om asseblief
11 (EC/SEPTEMBER 06) WISKUNDE V QUESTION 7 / VRAAG NP ON NP 77,86 PQ OP cos(0) using Pyth theorem correctly korrekte gebruik van stelling van Pythagoras answer / antwoord subst. into cosine rule verv. in cosinus formule () answer / antwoord PQ 78, ,86 77, cos Nˆ Nˆ 60.0 substitution / vervanging (77,86)(77,86) Nˆ () () [6] Copyright reserved Please turn over
12 MATHEMATICS P (EC/SEPTEMBER 06) QUESTION 8 / VRAAG 8 8. Bisects the chord / Halveer die koord answer / antwoord () Qˆ 90 [ in semi circle.] / [ in halwe sirkel] s Lˆ Qˆ 90 [corresp., QN LO] e [ooreenk., QN LO] QL = LP [line from centre perp. to chord] [lynstuk vanaf middelpt is loodreg op koord] 8.. 8ML 8..3 OP OL LP 6ML 9ML 49 7ML 49 ML 7 S R S/R R (4) MF () using Pyth correctly 4ML 3ML 7 korrekte gebruik van Pythagoras simplification / vereenvoudiging ML [9] Kopiereg voorbehou Blaai om asseblief
13 (EC/SEPTEMBER 06) WISKUNDE V 3 QUESTION 9 / VRAAG 9 9. Dˆ 3 30 [ s opp equal sides] /[ e teenoor gelyke sye] S R Ô 0 [sum of s of a.] / [som v/d e S/R van ʼn ] 9. 0 Â 60 [ at centre = at circumf.] / S R [middelpunts = omtrekshoek] () 9.3 Ĉ 0 [opp. s of cyclic quad.] / [teenoorst. e van k.v] S R () 9.4 A Dˆ B 70 [tan chord theorem.] / [raaklyn koord stelling] S R () [9] Copyright reserved Please turn over
14 4 MATHEMATICS P (EC/SEPTEMBER 06) QUESTION 0 / VRAAG 0 0. constr. konstr. Û 90 [tan radius.] / [raaklyn radius] Û Ẑ 90 [ in semi circle] / [ in halwe sirkel] Ŝ 80 (90 Û ) 90 [sum of s of a ] [som van die e van ʼn ] Ŝ Û Ŝ Ŷ [ s in same segment] / [ e in dieselfde segment] Û Ŷ S/R S/R S/R S/R (5) Kopiereg voorbehou Blaai om asseblief
15 (EC/SEPTEMBER 06) WISKUNDE V Â x [ s opp. = sides] / [ e teenoor gelyke sye] S/R S/R S/R Ĉ Â x [tan chord theo.] / [raaklyn koord stelling] Â Ĉ [tan chord theo.] / [raaklyn koord stelling] Ĉ Ŝ [alt. s, CB TS] / [verw. e, CB TS] S/R Ĉ Tˆ [corresp s, CB TS] / [ooreenk. e, CB TS] S/R (5) 0.. Ŝ Tˆ x [proven in 0..] / [bewys in 0..] S CS = CT [sides opp. = s ] / [sye teenoor gelyke hoeke] R () 0..3 AR AT S/R [line to one side of a ] BR CT [lynstuk aan een sy van ʼn ] CS = CT CS = CT [proved in 0..] / [bewys in 0..] 3 AT 4 AT 6cm substitution vervanging AT (4) [6] Copyright reserved Please turn over
16 6 MATHEMATICS P (EC/SEPTEMBER 06) QUESTION / VRAAG F R B A C Z D P. Â Ĉ [tan chord theo.] / [raaklyn koord stelling] S R Ẑ [tan chord theo] / [raaklyn koord stelling] S/R Ĉ Ẑ [both = x] / [beide = x] BC RZ [corresp. s =] / [ooreenk. e =] R (4). Ẑ Pˆ [ s in same segment] / [ e in dieselfde segment] S/R Ĉ [corresp. s ; BC RZ] / [ooreenk. e S ; BC RZ] BC is a tangent to circle ACP [conv. of tan chord theorem] BC is ʼn raaklyn aan sirkel ACP [ omgekeerde van raaklyn koord stelling] R.3 Bˆ Dˆ [ext. of a cyclic quad.] S R [buite van koordevierhoek] Rˆ Bˆ [corresp. s, BC RZ] [ooreenk. e, BC RZ] S/R S/R.4 Rˆ Dˆ Pˆ Ẑ s [ in same segment] [ e in dieselfde segment] Â Ĉ 4 [3 rd ] / [3 de ] RZA DPC [equiangular or ] [gelykhoekig of ] ZA RA PC DC [similar s ] / [gelykvormige e ] ZADC RA....() PC AR AZ AB AC [line to one side of a ] [lynstuk aan een sy van ʼn ] AZ AB AR...() AC ZADC AZ AB PC AC DC AC PC AB N R S/R ZADC RA PC S/R AZ AB RA AC simplification (5) (5) [7] TOTAL/TOTAAL: 50 Kopiereg voorbehou Blaai om asseblief
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