MATHS P1 GR10 MEMO NOV2020_Afr+Engl.pdf

NATIONAL SENIOR CERTIFICATE/ NASIONALE SENIOR SERTIFIKAAT GRADE/GRAAD 10 NOVEMBER 2020 MATHEMATICS P1/WISKUNDE V1 MARKING GUIDELINE/NASIENRIGLYN (EXEMPLAR/EKSEMPLAAR) MARKS/PUNTE: 100 This marking guideline consists of 10 pages. / Hierdie nasienriglyn bestaan uit 10 bladsye.

2 MATHEMATICS P1/WISKUNDE V1 (EC/NOVEMBER 2020) NOTE: • If a candidate answered a question TWICE, mark only the FIRST attempt. • If a candidate crossed out an answer and did not redo it, mark the crossed-out answer. • Consistent accuracy applies to ALL aspects of the marking guidelines. • Assuming values/answers in order to solve a problem is unacceptable. LET WEL: • As ʼn kandidaat ʼn vraag TWEE keer beantwoord het, sien slegs die EERSTE poging na. • As ʼn kandidaat ʼn antwoord deurgehaal en nie oorgedoen het nie, sien die deurgehaalde antwoord na. • Volgehoue akkuraatheid is op ALLE aspekte van die nasienriglyne van toepassing. • Dit is onaanvaarbaar om waardes/antwoorde te veronderstel om ʼn probleem op te los. QUESTION/VRAAG 1 1.1.1 4𝑦 2 − 16 ✓ answer/antwoord = 4(𝑦 2 − 4) = 4(𝑦 − 2)(𝑦 + 2) (1) OR/OF 4y2 – 16 ✓ answer/antwoord =(2y – 4)(2y + 4) =2(y – 2))2(y + 2) =4(y – 2)(y + 2) (1) 1.1.2 𝑥3 − 1 ✓ factorising/ 𝑥2 + 𝑥 + 1 faktoriseer (𝑥 − 1)(𝑥 2 + 𝑥 + 1) ✓ answer/antwoord = 𝑥2 + 𝑥 + 1 =𝑥−1 (2) 1.1.3 𝑥 − 1 + 𝑦 − 𝑥𝑦 ✓ common factor/ = (𝑥 − 1) + 𝑦(1 − 𝑥) gemene faktor = (𝑥 − 1) − 𝑦(𝑥 − 1) ✓ answer/antwoord = (𝑥 − 1)(1 − 𝑦) (2) 1.2.1 3 − 3𝑥 ✓ factorising 𝑥 2 − 3𝑥 + 2 numerator/ faktorisering 3(1 − 𝑥) teller = (𝑥 − 1)(𝑥 − 2) −3(𝑥 − 1) ✓ factorising = denominator/ (𝑥 − 1)(𝑥 − 2) faktorisering −3 = noemer 𝑥−2 ✓ answer/antwoord (3) Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

(EC/NOVEMBER 2020) MATHEMATICS P1/WISKUNDE V1 3 1.2.2 16−𝑥 . 12𝑥+1 ✓ separating bases/ 3𝑥 . 4−𝑥 opbreek van 12 and/en 16 4 −2𝑥 .4𝑥+1 .3𝑥+1 = ✓ addition of 3𝑥 .4−𝑥 exponents/optelling = 4−2𝑥+𝑥+1+𝑥 × 3𝑥+1−𝑥 van eksponente = 41 × 31 = 12 ✓ answer/antwoord (3) 1.3 𝑚 = 𝑥(𝑥 − 𝑦)2 ✓ expansion/uitbreiding = 𝑥(𝑥 2 − 2𝑥𝑦 + 𝑦 2 ) = 𝑥 3 − 2𝑥 2 𝑦 + 𝑥𝑦 2 ✓ substitution/vervanging = 3+4 =7 ✓ answer/antwoord (3) [14] Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

4 MATHEMATICS P1/WISKUNDE V1 (EC/NOVEMBER 2020) QUESTION/VRAAG 2 2.1.1 𝑥 3 = 9𝑥 ✓ factorisation/ 𝑥 3 − 9𝑥 = 0 faktorisering 𝑥(𝑥 2 − 9) = 0 ✓ factors/faktore 𝑥(𝑥 − 3)(𝑥 + 3) = 0 𝑥 = 0 or/of 𝑥 = 3 or/of 𝑥 = −3 ✓ answer/antwoord (3) 2.1.2 3 ✓ ÷ 𝑃𝑄 2 − 𝑃𝑞 2 𝑃 = 𝑥(𝑃𝑄 2 − 𝑃𝑞 2 ) 2 3 ✓ common factor/ ⇒ 𝑥(𝑃𝑄 2 − 𝑃𝑞 2 ) = 𝑃 2 P 3 𝑃 𝑥 = 𝑃𝑄2−𝑃𝑞2 gemene factor 2 𝑃 P = 𝑃(𝑄2−𝑞2) 2 ✓×3 𝑃 2 ∴ 𝑥= × 𝑃(𝑄 2 − 𝑞 2 ) 3 ✓ answer/antwoord 2 = 3(𝑄2−𝑞2) (4) 2.1.3 3 3𝑥 4 = 81 ✓ divide both 3 sides by 3 and 𝑥 4 = 27 4 3 both sides ( )3 / 𝑥 4 = 33 deel beide kante 4 3 3 4 deur 3 en beide (𝑥 ) = (33 )3 4 4 4 kante ( )3 x = 3 x = 81 ✓ answer/antwoord (2) 2.2.1 3(2 − 3𝑥) ≥ 15 ✓ simplify/ 6 − 9𝑥 ≥ 15 vereenvoudig −9𝑥 ≥ 9 ✓ (≤) 𝑥 ≤ −1 ✓ answer/antwoord (3) OR/OF 2 – 3x ≥ 5 – 3x ≥ 3 ✓ simplify/ x ≤ −1 vereenvoudig ✓ (≤) ✓ answer/antwoord (3) 2.2.2 ✓ answer/antwoord (1) Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

(EC/NOVEMBER 2020) MATHEMATICS P1/WISKUNDE V1 5 2.3 3𝑥 + 2𝑦 = 13 _______(1) ✓ substitution/ 3𝑥 = 2 − 𝑦 _______(2) vervanging 𝑦 = 2 − 3𝑥 _______(3) 3de vergelyking ✓ simplification/ Subs. (3) into (1) vereenvoudig 3𝑥 + 2(2 − 3𝑥) = 13 ✓ 𝑥-value/x-waarde 3𝑥 + 4 − 6𝑥 = 13 ✓ 𝑦-value/y-waarde 3𝑥 − 6𝑥 = 13 − 4 −3𝑥 = 9 𝑥 = −3 𝑦 = 2 − 3(−3) = 2+9 (4) 𝑦 = 11 OR/OF 3𝑥 + 2𝑦 = 13 _______(1) ✓ subtract (2) from (1)/ 3𝑥 + 𝑦 = 2 _______(2) Trek (2) af vanaf (1) (1) − (2): y =11 ✓ 𝑦-value/y-waarde Subs./Verv. 𝑦 = 11 into (2) 3𝑥 + 11 = 2 ✓ substitution/ 3𝑥 = −9 vervanging ∴ 𝑥 = −3 ✓ 𝑥-value/x-waarde (4) OR/OF ✓ multiply (2) x 2/ 3𝑥 + 2𝑦 = 13 _______(1) Maal (2) met 2 3𝑥 + 𝑦 = 2 _______(2) ✓ subtract (3) from (1)/ (2) x 2: 6𝑥 + 2𝑦 = 4 _______(3) Trek (3) af vanaf (1) (1)-(3) : 3𝑥 + 2𝑦 = 13 ✓ 𝑥-value/x-waarde 6𝑥 + 2𝑦 = 4 ✓ 𝑦-value/y-waarde −3𝑥 = 9 ∴ 𝑥 = −3 Subst. 𝑥 = −3 into (1)/Vervang 𝑥 = −3 in (1) 3(−3) + 2𝑦 = 13 −9 + 2𝑦 = 13 2𝑦 = 22 𝑦 = 11 (4) [17] Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

6 MATHEMATICS P1/WISKUNDE V1 (EC/NOVEMBER 2020) QUESTION/VRAAG 3 3.1 2(0) + 2 ; 3(0) + 4 ; 5(0) + 6 ; … ✓ fourth/vierde term 2;4 ;6 ;8 ✓ substitution/vervanging 𝑇𝑛 = 𝑚𝑛 + 𝑐 ✓ value of/waarde van c 8 = 2(4) + 𝑐 ⇒8+𝑐 =8 ✓ answer/antwoord ∴ 𝑐=0 ∴ 𝑇𝑛 = 2𝑛 (4) OR/OF 2 ; 4 ; 6 ; 8 ✓ fourth/vierde term d=2 T1 = 2(1) = 2 ✓ value of/waarde van d T2 = 2(2) = 4 T3 = 2(3) = 6 ✓ substitution/vervanging T4 = 2(4) = 8 ✓ answer/antwoord ∴ 𝑇n = 2n (4) 3.2 𝑇18 = 2(18) ✓ substitution/ = 36 vervanging ✓ answer/antwoord (2) 3.3 𝑇𝑛 = 2𝑛 ✓ 𝑇𝑛 = 108 108 = 2𝑛 ✓ answer/antwoord ⇒ 2𝑛 = 108 ∴ 𝑛 = 54 (2) 3.4 2𝑛 < 166 ✓ 2𝑛 < 166 𝑛 < 83 ✓ 𝑛 < 83 ✓ conclusion/afleiding 𝑇82 is the first term less than 166/ (3) T82 is die eerste term < 166 3.5 5; 10; 15; 20; 25; 30; … .. ✓ 16th even = 32nd number 32 × 5 in the pattern = 160 16de ewe = 32ste getal in die OR/OF patroon. ✓ 32 x 5 ✓ answer/antwoord (3) 10; 20; 30; ;…… The 16th even number / 16de ewe getal = 16 × 10 ✓ Even numbers/Ewe = 160 getalle ✓ 16 x 10 ✓ answer/antwoord (3) [14] Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

(EC/NOVEMBER 2020) MATHEMATICS P1/WISKUNDE V1 7 QUESTION/VRAAG 4 4.1.1 𝐷𝑒𝑝𝑜𝑠𝑖𝑡 30 ✓ deposit/deposito = × 9 899 𝑑𝑒𝑝𝑜𝑠𝑖𝑡𝑜 100 ✓ balance/balans = 𝑅2 969,70 𝐵𝑎𝑙𝑎𝑛𝑐𝑒 (2) : 𝑅9 899 − 𝑅2 969,70 = 𝑅6 929,30 𝑏𝑎𝑙𝑎𝑛𝑠 OR/OF 70 ✓ 100 × 9 899 70 ✓ answer/antwoord × 9 899 100 (2) = 𝑅6 929,30 4.1.2 𝐴 = 𝑃(1 + 𝑖𝑛) ✓ substitution/ 12 vervanging = 6 929,30 (1 + × 3) ✓ total payment/ 100 𝐴 = 𝑅9 423,85 totale paaiement 𝑅9 423,85 ✓ ÷ 36 𝑀𝑜𝑛𝑡ℎ𝑙𝑦 𝑝𝑎𝑦𝑚𝑒𝑛𝑡 = + 𝑅65,30 36 ✓ + R65,30 ✓ answer/antwoord 𝑀𝑎𝑎𝑛𝑑𝑒𝑙𝑖𝑘𝑠𝑒 𝑝𝑎𝑎𝑖𝑒𝑚𝑒𝑛𝑡 = 𝑅327,07 (5) 4.2.1 6800 Cost of machine/Koste van masjien ✓ 27,63 × 16,24 6 800 × 16,24 = £3 996,82 ✓ £3996,82 27,63 ✓ conclusion/ gevolgtrekking George will save money if he buys the machine in the USA / Machine is cheaper in the USA. / Machine is more expensive in England./ George sal geld bespaar as hy die masjien in Amerika koop. / Masjien is goedkoper in Amerika / Masjien is duurder in Engeland. (3) 4.2.2 £800 × 27,63 ✓ correct conversion / korrekte herleiding = 𝑅22 104,00 ✓ answer/antwoord (2) [12] Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

8 MATHEMATICS P1/WISKUNDE V1 (EC/NOVEMBER 2020) QUESTION/VRAAG 5 5.1.1 𝑎 ✓𝑞=1 𝑔(𝑥) = + 𝑞 𝑥 ✓ substitution/ 𝑎 2= +1 vervanging 4 ✓ answer/antwoord 𝑎 ⇒4+1=2 𝑎 =1 4 𝑎=4 4 𝑔(𝑥) = + 1 (3) 𝑥 5.1.2 ℎ(𝑥) = 𝑥 + 1 ✓ positive gradient/ positiewe gradiënt ✓ answer/antwoord (2) 5.2 ✓ asymptotes/ asimptote ✓ positive gradient of ℎ/positiewe gradiënt van h ✓ x-intercept of ℎ/ x-afsnitte van h ✓ points of intersection of g and h/snypunte van g en h (4) 5.3 4 ✓ equation of 𝑓 / 𝑓(𝑥) = − ( + 1) + 3 𝑥 vergelyking 4 van f = − −1+3 𝑥 ✓𝑥=0 4 ✓𝑦 = 2 𝑓(𝑥) = − + 2 𝑥 𝑥=0 𝑦=2 (3) 5.4.1 𝑥 = 2 𝑎𝑛𝑑/𝑒𝑛 − 2 ✓ 𝑥 = −2 ✓𝑥 =2 (2) 5.4.2 𝑥 ∈ [−2; 0) ✓✓ [−2; 0) (2) OR/OF −2 ≤ 𝑥 < 0 ✓✓ −2 ≤ 𝑥 < 0 (2) [16] Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

(EC/NOVEMBER 2020) MATHEMATICS P1/WISKUNDE V1 9 QUESTION/VRAAG 6 6.1 ✓ asymptote/ asimptote ✓✓ points of intersection/ snypunte ✓ shape of 𝑔/vorm van g ✓ shape of ℎ/vorm van h ✓ 𝑓 through origin / deur oorsprong (6) 6.2 1 (0,5; 0,75) ✓ 0,5 / 2 3 OR/OF ✓ 0,75 /4 1 3 accept/aanvaar ( ; ) 𝑥 ∈ (0,25; 0,5) / 2 4 1 1 (4 ; 2) 𝑦 ∈ (0,5; 0,8) / 1 4 (2 ; 5) (2) 6.3 𝑦 > −1 ✓ answer/antwoord (1) OR/OF 𝑦 ∈ (−1 ; ∞) ✓ answer/antwoord OR/OF (1) y ≠ −1 , 𝑦 ∈ ℝ ✓ answer/antwoord (1) 6.4 𝑥 ∈ (−∞; ∞) ✓✓ answer/ (2) OR/OF antwoord x ∈ ℝ ✓✓ answer/ (2) antwoord 6.5 𝑥 = −1 ✓✓ answer/ 𝑦=0 antwoord (2) 6.6 𝑥 ∈ (−∞; −2) ✓✓ answer/ (2) antwoord OR/OF ✓✓ answer/ 𝑥 < −2 antwoord (2) [15] Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

10 MATHEMATICS P1/WISKUNDE V1 (EC/NOVEMBER 2020) QUESTION/VRAAG 7 7.1.1 𝑃(𝑆) + 𝑃(𝑇) = 1 ✓ answer/antwoord (1) 7.1.2 𝑃(𝑇) = 𝑃(𝑆′) = 0,33 ✓ answer/antwoord (1) 7.2.1 ✓ 30 (intersection/ deursnee) ✓ 39 (H only/ alleenlik) ✓ 𝑥 − 30 (T only/alleenlik) ✓ 51 (outside/ buitekant) (4) 7.2.2 𝑥 − 30 + 30 + 39 + 51 = 180 ✓ equation/ 𝑥 + 90 = 180 vergelyking ∴ 𝑥 = 90 ✓ value of/waarde van x TB only: 90 − 30 TB alleenlik = 60 ✓ answer/antwoord (3) 7.2.3 (a) 𝑃(𝑇 𝑜𝑛𝑙𝑦) = 180 60 ✓ substitution of 60/vervanging P(T alleenlik) 1 met 60 (2) = 3 𝑜𝑟/𝑜𝑓 0,33 ✓ answer/antwoord 51 (b) 𝑃(no disease/𝑔𝑒𝑒𝑛 𝑠𝑖𝑒𝑘𝑡𝑒 ) = 180 ✓ answer/antwoord (1) [12] TOTAL/TOTAAL: 100 Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief