Mathematics Grade 11 P2 National Exemplar November 2020 Memo

NATIONAL SENIOR CERTIFICATE GRADE 11 NOVEMBER 2020 MATHEMATICS P2 EXEMPLAR MARKS: 150 TIME: 3 hours *Imat2* This question paper consists of 10 pages and an answer book of 20 pages.

2 MATHEMATICS P2 (EC/NOVEMBER 2020) INSTRUCTIONS AND INFORMATION Read the following instructions carefully before answering the questions. 1. This question paper consists of 9 questions. 2. Answer ALL the questions in the SPECIAL ANSWER BOOK provided. 3. Clearly show ALL calculations, diagrams, graphs, etc. which you have used in determining the answers. 4. Answers only will NOT necessarily be awarded full marks. 5. You may use an approved scientific calculator (non-programmable and non- graphical), unless stated otherwise. 6. If necessary, round off answers to TWO decimal places, unless stated otherwise. 7. Diagrams are NOT necessarily drawn to scale. 8. Write neatly and legibly. Copyright reserved Please turn over

(EC/NOVEMBER 2020) MATHEMATICS P2 3 QUESTION 1 In the diagram below, straight line PS is defined by 3 y  2x 6 and cuts the x-axis at Q(3; 0). MQR is a straight line which meets PR at R(10; 4). N(6; - 2) is a point on PS and RN is drawn. PQRˆ =T. 1.1 Determine the gradient of PS. (2) 1.2 Calculate the inclination angle of MR. (4) 1.3 Determine the value of T . (3) 1.4 Prove that RN A PS . (3) 1.5 Calculate the area of ' RQN . (6) 1.6 Calculate the y-intercept of MR. (4) [22] Copyright reserved Please turn over

4 MATHEMATICS P2 (EC/NOVEMBER 2020) QUESTION 2 ABCD is a parallelogram with A(-2; -6), B(4; 0), C(-1; 0) and D(x ; y) as shown below. 2.1 Calculate the length of BC. (2) 2.2 Determine the gradient of AB. (2) 2.3 Determine the equation of CD. (3) 2.4 Determine the coordinates of M, the midpoint of BD. (3) 2.5 Hence or otherwise, determine the values of x and y. (3) [13] Copyright reserved Please turn over

(EC/NOVEMBER 2020) MATHEMATICS P2 5 QUESTION 3 3.1 If 12 tan B  5 0 and 90q d B d 360q , determine the value of sin B  cos B with the aid of a sketch. (5) 3.2 If sin 43q p , determine the values of the following in terms of p, without a calculator. 3.2.1 cos133q (2) 3.2.2 tan(43q) (3) 3.3 Simplify each of the following fully, WITHOUT using a calculator: sin(90q  x) 3.3.1 y tan( x  180q) sin(360q  x) (5) tan 205q.cos 315q.sin135q 3.3.2 sin 210q.cos150q.tan 25q (7) [22] QUESTION 4 4.1 Prove that: sin T  cos T .sin T tan T cos T  (1  sin 2 T ) (4) 4.2 Determine the general solution of 2sin x cos x  cos2 x 0 (6) 4.3 Solve for D if : 2. sin D 1, where D  >0q;360q@ (3) 4.4 § x y· 3 If x and y are acute angles such that tan ¨ ¸ 1 and cos( x  y) , find the © 2 ¹ 2 values of x and y. (5) [18] Copyright reserved Please turn over

6 MATHEMATICS P2 (EC/NOVEMBER 2020) QUESTION 5 § x· Sketched below is a graph of f ( x) cos ¨ ¸ , where x  > 180q;180q@ . ©2¹ 5.1 For f ( x) , write down the: 5.1.1 Range (2) 5.1.2 Period (1) 5.2 Draw a graph of g ( x) sin( x  30q) for x  > 180q;180q@ on the axes provided in the ANSWER BOOK. Clearly show all intercepts with the axes, turning point, starting and ending points. (4) 5.3 For which values of x  > 180q;180q@ , is f ( x).g ( x) t 0 ? (3) [10] Copyright reserved Please turn over

(EC/NOVEMBER 2020) MATHEMATICS P2 7 QUESTION 6 A pole, 8 m tall, is held in a vertical position by two steel cables of equal length. In the sketch, PQ is the pole and PS and PR are the steel cables. The angle of elevation of the top of the pole from the anchor points R and S is 60q in both cases. P 8m Q 60q 60q R S 6.1 Determine the length of cable PS. Leave your answer in simplified surd form. (3) 6.2 ˆ =100q . Give your answer correct Determine the distance between R and S, if RQS to TWO decimal digits. (7) [10] Copyright reserved Please turn over

8 MATHEMATICS P2 (EC/NOVEMBER 2020) QUESTION 7 7.1 In the diagram below, D is the centre of circle ABC with radius BD produced to O. B D C O A ˆ = 2 u ABC Use the diagram to prove the theorem that states ADC ˆ . (5) 7.2 In the figure, O is the centre of circle ABCDE. DB = DF. AODF, BOE and BCF are ˆ = 10q . straight lines. CFD C F B 10q 3 1 2 1 2 3 2 1 D 2 1 3 O4 A E Calculate, giving reasons, the size of the following angles: 7.2.1 D̂ 2 (3) 7.2.2 Â (4) 7.2.3 Ô 2 (2) 7.2.4 Ĉ1 (2) 7.2.5 Ê (2) 7.2.6 Ĉ 2 (2) 7.2.7 Ô 3 (2) [22] Copyright reserved Please turn over

(EC/NOVEMBER 2020) MATHEMATICS P2 9 QUESTION 8 8.1 In the diagram below, O is the centre of circle AEF with CFD a tangent at F. E A O C D F ˆ. ˆ =A Use the diagram to prove the theorem which states that EFD (5) 8.2 In the diagram below, ADF is a tangent to the circle with points E, B, C and D on the circumference of the circle. AB || DC and EC = DC. A E 1 2 3 B O 1 2 D 2 x 1 C F 8.2.1 ˆ = x , name with reasons, FIVE other angles equal to x. If CDF (10) 8.2.2 Prove that ABCD is a parallelogram. (4) [19] Copyright reserved Please turn over

10 MATHEMATICS P2 (EC/NOVEMBER 2020) QUESTION 9 9.1 Complete the following theorem statements: 9.1.1 The line drawn from the centre of a circle to the midpoint of a chord is … (1) 9.1.2 The exterior angle of a cyclic quadrilateral is equal to the … (1) 9.2 In the diagram, FH is a diameter of the circle FCH with centre O. FC is a chord and LCH is a secant. LF is a tangent to the circle at F. E is a point on CH such that CE = HE. F 1 x O 1 2 3 1 2 L C E H 9.2.1 Prove that FC || OE. (5) 9.2.2 Prove that OFLE is a cyclic quadrilateral. (3) 9.2.3 If F̂1 x , express Ô1 with reasons in terms of x. (4) [14] TOTAL: 150 Copyright reserved Please turn over

LEARNER’S NAME: LEERDERNAAM: GRADE 11: GRAAD 11: NATIONAL/NASIONALE SENIOR CERTIFICATE/SERTIFIKAAT GRADE 11/GRAAD 11 NOVEMBER 2020 MATHEMATICS P2/WISKUNDE V2 SPECIAL ANSWER BOOK/SPESIALE ANTWOORDEBOEK (EXEMPLAR/EKSEMPLAAR) Marker/Merker Moderator’s Initials / Moderator se paraaf Question Mark Initial Marks S Marks D Marks P Marks NM Vraag Punt Parafeer Punte M Punte M Punte M Punte 1 2 3 4 5 6 7 8 9 TOTAL TOTAAL *Iwisab2* This special answer book consists of 20 pages./ Hierdie spesiale antwoordeboek bestaan uit 20 bladsye.

2 MATHEMATICS P2/WISKUNDE V2 (EC/NOVEMBER 2020) QUESTION 1/VRAAG 1 1.1 (2) 1.2 (4) 1.3 (3) Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

(EC/NOVEMBER 2020) MATHEMATICS P2/WISKUNDE V2 3 1.4 (3) 1.5 (6) 1.6 (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 (2) 2.2 (2) 2.3 (3) Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

(EC/NOVEMBER 2020) MATHEMATICS P2/WISKUNDE V2 5 2.4 (3) 2.5 (3) [13] Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

6 MATHEMATICS P2/WISKUNDE V2 (EC/NOVEMBER 2020) QUESTION 3/VRAAG 3 Sketch here/Teken hier: 3.1 (5) 3.2.1 (2) 3.2.2 (3) Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

(EC/NOVEMBER 2020) MATHEMATICS P2/WISKUNDE V2 7 3.3.1 (5) 3.3.2 (7) [22] Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

8 MATHEMATICS P2/WISKUNDE V2 (EC/NOVEMBER 2020) QUESTION 4/VRAAG 4 4.1 (4) 4.2 (6) Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

(EC/NOVEMBER 2020) MATHEMATICS P2/WISKUNDE V2 9 4.3 (3) 4.4 (5) [18] Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

10 MATHEMATICS P2/WISKUNDE V2 (EC/NOVEMBER 2020) QUESTION 5/VRAAG 5 5.1.1 (2) 5.1.2 (1) 5.2 (4) 5.3 (3) [10] Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

(EC/NOVEMBER 2020) MATHEMATICS P2/WISKUNDE V2 11 QUESTION 6/VRAAG 6 6.1 (3) 6.2 (7) [10] Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

12 MATHEMATICS P2/WISKUNDE V2 (EC/NOVEMBER 2020) QUESTION 7/VRAAG 7 B D C O A 7.1 (5) Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

(EC/NOVEMBER 2020) MATHEMATICS P2/WISKUNDE V2 13 7.2 C F B 10q 3 1 2 1 2 3 2 1 D 2 1 3 O4 A E 7.2.1 (3) 7.2.2 (4) 7.2.3 (2) 7.2.4 (2) Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

14 MATHEMATICS P2/WISKUNDE V2 (EC/NOVEMBER 2020) 7.2.5 (2) 7.2.6 (2) 7.2.7 (2) [22] Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

(EC/NOVEMBER 2020) MATHEMATICS P2/WISKUNDE V2 15 QUESTION 8/VRAAG 8 E A O C D F 8.1 (5) Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

16 MATHEMATICS P2/WISKUNDE V2 (EC/NOVEMBER 2020) A E 1 2 3 B O 1 2 D 2 x 1 C F 8.2.1 (10) 8.2.2 (4) [19] Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

(EC/NOVEMBER 2020) MATHEMATICS P2/WISKUNDE V2 17 QUESTION 9/VRAAG 9 9.1.1 (1) 9.1.2 (1) F 1 x O 1 2 3 1 2 L C E H 9.2.1 (5) Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

18 MATHEMATICS P2/WISKUNDE V2 (EC/NOVEMBER 2020) 9.2.2 (3) 9.2.3 (4) [14] TOTAL/TOTAAL: 150 Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

(EC/NOVEMBER 2020) MATHEMATICS P2/WISKUNDE V2 19 Additional Space/Addisionele ruimte Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

20 MATHEMATICS P2/WISKUNDE V2 (EC/NOVEMBER 2020) Additional Space/Addisionele ruimte Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

 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 = ¨¨ ¸¸  ¨¨ ¸¸  2. ¨¨ ¸¸ . ¨¨ ¸¸ cos100q © 3 ¹ © 3 ¹ © 3 ¹ © 3 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 ABD = 90q (angles in a semi-circle) 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 ෡ Eଶ ൌ ‫ ݔ‬ሺtan-chordሻ / ሺ‫݊ݕ݈݇ܽܽݎ‬-݇‫݀ݎ݋݋‬ሻ 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) ෡ = 180° െ ‫ ݔ‬ሺopposite angles of a c.q.ሻ 8.2.2 B 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 ෡ Eଶ ൌ 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