Mathematics Grade 11 P2 National Exemplar November 2020 Memo

 NATIONAL SENIOR CERTIFICATE/ NASIONALE SENIOR SERTIFIKAAT GRADE/GRAAD 11 NOVEMBER 2020 MATHEMATICS P2/WISKUNDE V2 MARKING GUIDELINE/NASIENRIGLYN EXEMPLAR/EKSEMPLAAR MARKS/PUNTE: 150 This marking guideline consists of 14 pages./ Hierdie nasienriglyn bestaan uit 14 bladsye. 

2 MATHEMATICS P2/WISKUNDE V2 (EC/NOVEMBER 2020) QUESTION 1/VRAAG 1 1.1 0  ( 2) 2 9 substitution / vervanging m PS  36 3 9 answer / antwoord OR/OF 3y  2x 6 OR/OF 3y 2 x  6 2 2 y  x2 9y  x2 3 3 2 m PS  3 9 answer / antwoord (2) ସି଴ 1.2 40 4 9 9gradient / gradiënt mMR ଵ଴ିଷ 10  3 7 ෡ X ൌ ݉ୖ୑ 9tan RQ ˆ 4 tan RQX= 7 9answer / antwoord ˆ RQX=29,74q (4) 1.3 ˆ =-2  tan PQX ෡X ൌ െ ଶ 3 9tan PQ ଷ ˆ =146, 31q PQX ෡ X ൌ 146,31° 9 PQ T 146,31q  29, 74q 9answer / antwoord T 116,57q (3) 1.4 3 9݉ୖ୒ mRN 2 3 2 9product / produk mRN u mPS u 1 2 3 9െ1 RN A PS (3) 1.5 2 2 NR= 10  6  4  2 9substitution / vervanging NR= 52 9answer / antwoord QN = 13 9length QN / lengte QN 1 Area = u QN u NR 2 9choosing correct sides / 1 kies korrekte sye Area = u 13 u 52 9substitution / vervanging 2 Area = 13 units 2 / eenhede 2 9answer / antwoord (6)    Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

(EC/SEPTEMBER 2020)) MATHEMATICS P2/WISKUNDE V2 3  1.6 y0 4 9x = 0 x 3 7 9substitution / vervanging 7 y 12 9equation / vergelyking 12 9y-coordinate / y-koördinaat y  7 OR/OF OR/OF 4 y xc 9equation / vergelyking 7 Subst./vervang (3; 0) 4 0= 3 c 9substitution / vervanging 7 12 c  9value of c / waarde van c 7 12 y  9y-coordinate / y-koördinaat 7 (4) [22] Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

4 MATHEMATICS P2/WISKUNDE V2 (EC/NOVEMBER 2020) QUESTION 2/VRAAG 2  2.1 BC = 2 1  4  10  0 2 9substitution / vervanging BC = 25  100 9answer / antwoord BC = 125 5 5 (2) 2.2 6  0 mAB 9substitution / vervanging 2  6 3 mAB 9answer / antwoord 4 (2) 2.3 3 mCD mAB 9gradient / gradiënt 4 y mx  c 3 y xc 4 Sub C 1;10 9substitution / vervanging 3 10 1  c 4 43 c 4 3 43 y x 9answer / antwoord (3) 4 4   Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

(EC/SEPTEMBER 2020)) MATHEMATICS P2/WISKUNDE V2 5 2.4 M is the midpoint of both BD and AC / is die middelpunt van beide BC en AC 9 statement / stelling Midpoint of AC and BD /Middelpunt van AC en BD § 1  2 10  6 · M¨ ; ¸ 9 substitution / vervanging © 2 2 ¹ § 3 · 9answer / antwoord M ¨ ;2¸ © 2 ¹ (3) 2.5 x  4 3 y  0 ; 2 9 substitution / vervanging 2 2 2 x  4 3; y  0 4 9x- value/waarde 9y-value/waarde x 7; y 4 Answer only: Full marks/ Slegs antwoord: Volpunte (3) [13]  Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

6 MATHEMATICS P2/WISKUNDE V2 (EC/NOVEMBER 2020) QUESTION 3 / VRAAG 3  3.1 y B x O (– 12; – 5) 9diagram / diagram Therefore/d.w.s: r = 13 9value of r / waarde van r sin B  cos B 5 12  99substitution / vervanging 13 13 17 9answer / antwoord 13 (5) 3.2 sin 43q p 3.2.1 cos133q  cos(90q  43q)  sin 43q 9  sin 43q 9answer / antwoord p (2) 3.2.2 tan( 43q)   tan 43q 9  tan 43q p  99answer / antwoord 1  p2 (3) 3.3.1 sin(360q  x ) y tan( x  180q)  sin(90q  x )  sin x 9  sin x y tan x 9 cos x cos x 9 tan x  tan x y tan x  9  tan x 1 9 answer / antwoord (5) 3.3.2 sin 210q.cos150q.tan 25q tan 205q.cos 315q.sin135q  sin 30q.  cos 30.tan 25q 9  sin 30q 9  cos30q tan 25q.cos 45q.sin 45q 9 tan 25q 1 3 9 cos 45q .  2 2 9 sin 45q 1 1 . 2 2 9special angles / spesiale hoeke 3 9answer / antwoord 2 (7)  [22]   Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

(EC/SEPTEMBER 2020)) MATHEMATICS P2/WISKUNDE V2 7 QUESTION 4 / VRAAG 4 4.1 sin T  cos T .sin T tan T cos T  (1  sin 2 T ) sin T (1  cos T ) 9factorising / faktorisering LHS =  cos T  cos 2 T 9 cos2 T sin T (1  cos T ) = 9common factor / gemene faktor cos T (1  cos T ) 9answer / antwoord = tan T (4) 4.2 2sin x cos x  cos 2 x 0 cos x(2sin x  cos x) 0 9factors / faktore 9 cos x 0 cos x 0 or/of 2sin x cos x 1  9 tan x 1 2 cos x 0 or/of tan x 2 x 90q  360q.k or/of x 270q  360q.k 9 x 90q  360q.k  or/of x 26,57q  180q.k 9 x 270q  360q.k  9 x 26,57q  180q.k  (6) 4.3 2. sin D 1   1 sin D  2  1 sin D  1 4 sin D 4 D 14, 48q  D 14, 48q or/of D 165,52q D 165,52q (3) 4.4 § x y· 3 x y 9 45q  tan ¨ ¸ 1 and/en cos( x  y ) 2 © 2 ¹ 2 9 x  y 30q  x y 45q and/en x  y 30q  2 9setting up equations/ x  y 90q ...............(1) opstel van vergelykings x  y 30q ...............(2)  2 x 120q x 60q 9 x-value/waarde y 30q 9y-value/waarde (5) [18] Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

8 MATHEMATICS P2/WISKUNDE V2 (EC/NOVEMBER 2020) QUESTION 5 / VRAAG 5 5.1.1 0 d y d1 or > 0;1@ 90 91 (2) 5.1.2 Period = 720q 9answer (1) 5.2 9 shape / vorm 9 x-intercept / x-afsnit 9 y-intercept / y-afsnit 9 turning points / draaipunte (4) 5.3 180q d x d 150 or/of 30q d x d 180q 99 180q d x d 150  9 30q d x d 180q  (3) [10] Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

(EC/SEPTEMBER 2020)) MATHEMATICS P2/WISKUNDE V2 9 QUESTION 6 /VRAAG 6 6.1 PQ sin 60q PS 8 sin 60q 8 PS 9 sin 60q  PS 8 PS = 8 sin 60q 9 PS =  sin 60q 16 3 PS = 16 3 3 9 PS = 3 (3) 6.2 PQ In 'PQS: tan 60q QS 8 9 QS =  tan 60q 8 8 3 QS = m 8 3 tan 60q 3 9 QS = m 3 8 3 QR = m 8 3 3 9 QR = m 3 In 'RQS: RS2 QR 2  QS2  2.QR.QS.cos100q 9formula / 2 2 2 2 §8 3 · §8 3 · §8 3 · §8 3 · formule = ¨¨ 3 ¸¸  ¨¨ 3 ¸¸  2. ¨¨ 3 ¸¸ . ¨¨ 3 ¸¸ cos100q © ¹ © ¹ © ¹ © ¹ 9substitution / = 50,0756...... vervanging RS = 7,08 m 9simplification/ vereenvoudiging 9answer / antwoord (7) [10] Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

10 MATHEMATICS P2/WISKUNDE V2 (EC/NOVEMBER 2020) QUESTION 7 / VRAAG 7 B D C O A 7.1 ˆ =x Let/Laat A ˆ = x (angles opp = sides)/(hoeke teenoor BAD sye) 9S and/en R ˆ ADO = 2 x (angle at the centre)/(middelpuntshoek ) 99S and/en R Similarly, if you let / Net so, as jy: Cˆ = y; ˆ = 2y then/dan: CDO 9S ˆ = 2 x  2 y = 2( x  y ) ? ADC ˆ = 2 ABC 9S and conclusion en gevolgtrekking (5) 7.2.1 B̂3 10q (angles opp = sides; DB = DF) 9S 9R (hoeke teenoor sye; DB = DF) D̂ 2 20q (exterior angle of a 'BDF) 9S and/en R (buitehoek van 'BDF) (3) 7.2.2 ˆ = 90q (angles in a semi-circle) ABD 9S 9R (hoek in halwe sirkel ) Â = 70q (angles of a triangle) 9S 9R (hoeke van ' n driehoek ) (4) 7.2.3 Ô2 140q (angle at the centre) / (middelpuntshoek ) 9S 9R (2) 7.2.4 Ĉ1 110q (opposite angles of a c.q.)/(teenoorst. hoeke van k .v) 9S 9R OR  OF OR / OF ˆ O O ˆ O ˆ 220q (angles around a point/(omwenteling ) 1 4 3 9S and/en R Ĉ1 110q (angle at the centre)/(middelpuntshoek ) 9S and/en R (2)   Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

(EC/SEPTEMBER 2020)) MATHEMATICS P2/WISKUNDE V2 11 7.2.5 Ê = 70q (angles in the same segment)/(hoeke in dieselfde segment ) 9S 9R OR/OF OR/OF Ê = 70q (opposite angles of a c.q.)/(teenoorst . hoeke van k .v ) 9S 9R (2) 7.2.6 Ĉ2 70q (ext. ‘ of a c.q.)/ (buitehoek van k .v) 9S 9R OR/OF OR/OF Ĉ2  110q 180q (‘s on a straight line)/(hoeke op ' n reguitlyn) 9S 9R Ĉ2 70q (2) 7.2.7 Oˆ Oˆ 140q (vertically opp. ‘s)/(regoorstaande ‘e) 9S 9R 4 2 (2) [22]   Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

12 MATHEMATICS P2/WISKUNDE V2 (EC/NOVEMBER 2020) QUESTION 8 / VRAAG 8  P E A O C D F 8.1 ENG 9construction Draw diameter FP and join PA ˆ =x Let EFD 9S and R ˆ = 90q (tan A radius) OFD ˆ = 90q  x ? OFE 9S and R ˆ = 90q  x (angles in the same segment) ? PAE 9S and R ˆ = 90q (angles in a semi circle) PAF ˆ =x ? EAF 9conclusion ˆ =x ˆ =A ? EFD 8.1 AFR AFR Teken middellyn FP en verbind PA 9 konstruksie ˆ =x Laat EFD ˆ = 90q (raaklyn A radius ) OFD 9 S en R ˆ = 90q  x ? OFE ˆ = 90q  x (hoeke in dieselfde segment ) ? PAE 9 S en R 9 S en R ˆ = 90q (hoeke in ' n halwe sirkel ) PAF ˆ =x ? EAF 9 gevolgtrekking ˆ =x ˆ =A ? EFD (5)   Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

(EC/SEPTEMBER 2020)) MATHEMATICS P2/WISKUNDE V2 13 A E 1 2 3 B O 1 2 D 2 x 1 C F ෡ = ‫ ݔ‬ሺcorresponding angles; AB 8.2.1 A 9S 9R ‫ צ‬DCሻ/ሺ‫ ;݁݇݁݋݄ ݁݃݅ݐݏ݉݋݇݊݁݁ݎ݋݋‬AB || DCሻ ෡ଶ ൌ ‫ ݔ‬ሺtan-chordሻ / ሺ‫݊ݕ݈݇ܽܽݎ‬-݇‫݀ݎ݋݋‬ሻ E 9S 9R D෡ ଶ ൌ ‫ ݔ‬ሺangles opposite = sidesሻ / ሺ݄‫ ݎ݋݋݊݁݁ݐ ݁݇݁݋‬ൌ syeሻ 9S 9R ෡ Eଵ ൌ ‫ ݔ‬ሺalternate angles, AB 9S 9R ‫ צ‬DCሻ/ሺ‫ ;݁݇݁݋݄ ݈݁݀݊݁݁ݏݏ݅ݓݎ݁ݒ‬AB || DCሻ ෠Cଵାଶ ൌ E ෡ଵ ൌ ‫ ݔ‬ሺexterior angle of a c.q.ሻ/ሺܾ‫ ݊ܽݒ ݇݁݋݄݁ݐ݅ݑ‬Ԣ݊ ݇. ‫ݒ‬ሻ 9S 9R (10) ෡ 8.2.2 B = 180° െ ‫ ݔ‬ሺopposite angles of a c.q.ሻ 9S 9R ሺ‫ݐݏݎ݋݋݊݁݁ݐ‬. ݄‫ ݊ܽݒ ݁݇݁݋‬Ԣ݊ ݇. ‫ݒ‬ሻ ෡+෡ A B = ‫ ݔ‬൅ ሺ180° െ ‫ݔ‬ሻ ൌ 180° ‫ ׵‬AD ‫ צ‬BC ሺco-interior angles formed = ሻ ሺ݇‫݋‬-ܾ݅݊݊݁ ݄‫ = ݉ݎ݋ݒ݁݃ ݁݇݁݋‬180୭ ሻ 9R ‫ ׵‬ABCD is a parallelogram ሺopp. sides ‫צ‬ሻ ABCD is Ԣn parallelogram ሺ‫ݐݏݎ݋݋݊݁݁ݐ‬. ‫|| ݁ݕݏ‬ሻ 9R (4) [19]         Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

14 MATHEMATICS P2/WISKUNDE V2 (EC/NOVEMBER 2020)  QUESTION 9 / VRAAG 9  9.1.1 perpendicular to the chord / loodreg op die koord 9answer/antwoord (1) 9.1.2 interior opposite angle / teenoorstaande binnehoek 9answer/antwoord (1) 9.2 F 1 x O 1 2 3 1 2 L C E H 9.2.1 ෡ଶ ൌ E E ෡ଵ ൌ 90° ሺline from centreሻ 9S 9R ሺ݈‫ݐ݊ݑ݌݈݁݀݀݅݉ ݁݅݀ ݂ܽ݊ܽݒ ݊ݕ‬ሻ ෠ FCH ൌ 90° ሺangles in a semi-circleሻ 9S 9R ሺ౹‫ ݊' ݊݅ ݁݇݁݋‬౹݈ܽ‫݈݁݇ݎ݅ݏ ݁ݓ‬ሻ ෠ ෡ ‫ ׵‬FCH ൌ Eଶ ‫ ׵‬FC ‫ צ‬OE ሺcorresponding angles formed are ൌሻ 9R ሺ‫ ݁݃݅ݐݏ݉݋݇݊݁݁ݎ݋݋‬౹‫ ݏ݅ ݀ݎ݋ݓ ݉ݎ݋ݒ݁݃ ݐܽݓ ݁݇݁݋‬ൌሻ (5) 9.2.2 ˆ LFO = 90q (tan A radius) / (raaklyn A radius ) 9S and/en R Ê 2 90q (proven) / (reeds bewys ) 9S and/en R ? OFLE is a c.q. (converse exterior angle of a c.q.) 9R (omgekeerde buitehoek van k .v stelling ) (3) 9.2.3 Ĥ x (tan - chord) / (raaklyn - koord ) 9S 9R Ô1 2 x (angle at the centre) 9S (middelpuntshoek ) 9R (4) [14] TOTAL/TOTAAL: 150  Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief