Mathematics Grade 11 P2 National Exemplar November 2017 Memo

Downloaded from hlayiso.com testpapers.co.za NATIONAL SENIOR CERTIFICATE NASIONALE SENIOR SERTIFIKAAT GRADE/GRAAD 11 MATHEMATICS P2/WISKUNDE V2 NOVEMBER 2017 MARKING GUIDELINES/NASIENRIGLYNE MARKS/PUNTE: 150 These marking guidelines consist of 19 pages. Hierdie nasienriglyne bestaan uit 19 bladsye. Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Downloaded from hlayiso.com Mathematics P2/Wiskunde V2 2 DBE/November 2017 CAPS/KABV – Grade/Graad 11 – Marking guidelines/Nasienriglyne 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 'n kandidaat 'n vraag TWEE keer beantwoord het, sien slegs die EERSTE poging na. • As 'n kandidaat 'n 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 'n probleem op te los. QUESTION/VRAAG 1 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 1.1.1 min = 12  min + max Q1 = 17  median, Q1 and/en Q3 median / mediaan = 30 (2) Q3 = 38 max = 65 1.1.2 IQR = Q3 − Q1  answer/antw = 38 − 17 (1) = 21 1.1.3 Skewed to the right OR positively skewed  answer/antw Skeef na regs OF positief skeef (1) 5 8 10 17 20 29 32 48 50 50 63 y 107 1.2.1 439 + y Mean/Gemiddeld = 13 439 + y 439 + y 41 =  41 = 13 13 439 + y = 533  answer/antw y = 94 (2) 1.2.2 σ = 30,94  answer/antw (1) Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Downloaded from hlayiso.com Mathematics P2/Wiskunde V2 3 DBE/November 2017 CAPS/KABV – Grade/Graad 11 – Marking guidelines/Nasienriglyne 1.2.3 41 × 13 = 533 6 × 18 = 108 108  533 + 108 = 641  answer/antw 533 + 108 641 (3) = = 33,74 19 19 [10] QUESTION/VRAAG 2 2.1 CUMULATIVE AGE OF PERSON FREQUENCY/ SURVEYED/OUDERDOM FREQUENCY/FREKWENSIE KUMULATIEWE VAN PERSOON IN OPNAME FREKWENSIE 20 < x ≤ 30 7 7 30 < x ≤ 40 20 27 40 < x ≤ 50 25 52 50 < x ≤ 60 12 64 60 < x ≤ 70 8 72  20, 12 70 < x ≤ 80 4 76  8, 4  52 80 < x ≤ 90 4 80 76 (4) 2.2 n = 80 answ/antw (1) 2.3 40 < x ≤ 50  answ/antw (1) Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Downloaded from hlayiso.com Mathematics P2/Wiskunde V2 4 DBE/November 2017 CAPS/KABV – Grade/Graad 11 – Marking guidelines/Nasienriglyne 2.4 Ogive/Ogief 90 Grounding (20; 0) /Geanker by (20; 0) 80 upper limits/ boonste limiete 70  shape (smooth curve)/ 60 vorm Cumulative Frequency/Kumulatiewe Frekwensie (gladde kurwe) (3) 50 40 30 20 10 0 0 10 20 30 40 50 60 70 80 90 100 Age of person surveyed/Ouderdom van persoon in opname 2.5 80 – 58 = 22 Accept/aanvaar: 56 – 59 calls/oproepe 58 22 calls/oproepe × 100 = 27,5% 22 80  27,5% (3) [12] Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Downloaded from hlayiso.com Mathematics P2/Wiskunde V2 5 DBE/November 2017 CAPS/KABV – Grade/Graad 11 – Marking guidelines/Nasienriglyne QUESTION/VRAAG 3 y C B T 15 x O A(−2 ; −5) D 3.1 1  Standard BD y = − x+9 form/vorm 2 1 ∴ m BD = − 2  answ/antw ∴ m AC = 2 (2) 3.2 y − y1 = m( x − x1 ) y − (−5) = 2( x − (−2))  subst (–2 ; –5)  answ/antw y = 2x −1 (2) 3.3 1 1 2x − 1 = − x + 9  2x −1 = − x + 9 2 2 5 x = 10 2  x=4 x=4 y = 2(4) − 1  y=7 y=7 (3) T (4 ; 7 ) Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Downloaded from hlayiso.com Mathematics P2/Wiskunde V2 6 DBE/November 2017 CAPS/KABV – Grade/Graad 11 – Marking guidelines/Nasienriglyne 3.4.1 −2+ x 4= 2 8 = −2 + x  x = 10 x = 10 −5+ y 7= 2 14 = −5 + y  y = 19 y = 19 (2) C (10 ; 19 ) 3.4.2 AT = (4 − (− 2))2 + (7 − (− 5))2  subst. in distance/afstand = 180 form.  answer/antw =6 5 BT 2 + AT 2 = AB 2 (Pythagoras )  subst. in pyth BT = 15 − 180 2 ( ) 2  answer/antw =3 5 (4) 3.4.3 BC is the diameter/ middellyn [subt. right / ondersp. reg ∠ ] or/of S [conv.∠ in semi - circle/ omgk. ∠ in halfsirkel ] s s 15  answ/antw Radius = = 7,5 units/ eenh. 2 (2) [15] Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Downloaded from hlayiso.com Mathematics P2/Wiskunde V2 7 DBE/November 2017 CAPS/KABV – Grade/Graad 11 – Marking guidelines/Nasienriglyne QUESTION/VRAAG 4 B (4 ; 12) E O C A (–5 ; –3) 4.1 12 − (−3) 5  subst. in mAB = = gradient form. 4 − (−5) 3  answ/antw OR/OF (2) − 3 − 12 5 mAB = = −5−4 3 4.2 5 equation/verg. y − 12 = (x − 4) 3 5 0 − 12 = ( x − 4 ) y=0 3 16 x=− 5  16   answ/antw E − ; 0  5  (3) OR/OF 0 − 12 5 0+3 5 = = equating/verg. x−4 3 x+5 3 y=0 − 36 = 5 x − 20 9 = 5 x + 25 − 16 = 5 x − 16 = 5 x − 16 − 16 x= x=  answ/antw 5 5 (3)  16  E − ; 0  5  Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Downloaded from hlayiso.com Mathematics P2/Wiskunde V2 8 DBE/November 2017 CAPS/KABV – Grade/Graad 11 – Marking guidelines/Nasienriglyne 4.3 tan α = m AB 5  tan α = 5 3 tan α = 3  α = 59ο α = 59 ο (2) 4.4 76 ° + 59° =135° [ext ∠ of Δ]  135° BĈX =135° tan 135° = mBC  tan 135ο = mBC mBC = −1 = m//  answer/antw  subst (–3 ; –5) y − (− 3) = −1(x − (− 5))  answer/antw y = −x − 8 (5) [12] QUESTION/VRAAG 5 5.1 sin(90o − x). cos(180o + x) + tan x. cosx. sin ( x −180o )  cos x  − sin x sin x = cos x. (−cos x) + . cos x . (− sin x)  − cos x cos x sin x  = − cos 2 x − sin 2 x cos x = − (cos 2 x + sin 2 x) common = −1 factor/gemene fakt.  identity/identiteit (6) 5.2 o sin 315°. tan 210 .sin190° LHS =  − sin45° cos 100°.sin 120 o  tan30° = (− sin45°). (tan30o ).(− sin10°)  − sin10° (− sin 10°).(sin 60o )  – sin10° 1 1  sin 60° − .  subst. of special = 2 3 angles/inverv. van 3 sp hoeke 2 2 (6) =− 3 Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Downloaded from hlayiso.com Mathematics P2/Wiskunde V2 9 DBE/November 2017 CAPS/KABV – Grade/Graad 11 – Marking guidelines/Nasienriglyne y R β α O T x P 5.3.1 x2 + y2 = r 2 [Pythagoras] 2 2 2  subst in pyth ( x) + (3) = 5 x 2 = 16 x=4 x=4 R (4 ; 3) 3  answer/antw tan α = 4 (3) 5.3.2 sin β = sin (180 ο + α)  β = 180° + α  − sinα = −sinα −3  answer/antw = 5 (3) Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Downloaded from hlayiso.com Mathematics P2/Wiskunde V2 10 DBE/November 2017 CAPS/KABV – Grade/Graad 11 – Marking guidelines/Nasienriglyne 5.3.3 y −3 y =  10 5 10 y = −6 −3  ∴ x = −8 5  y = −6 P(− 8 ; − 6 )  x = −8 (4) 5.4. 2 sin θ − tan θ . cos θ LHS = cos θ − (1 − sin 2θ) sin θ sinθ  sin θ − . cos 2θ cos θ = cosθ  cos 2 θ cos θ − cos 2θ  common fact/ sin θ(1 − cosθ ) gemene fakt. = cos θ(1 − cosθ )  common fact/ = tan θ gemene fakt. = RHS (4) OR/OF sin θ − tan θ . cos 2 θ LHS = ( cos θ − 1 + 1 − cos 2 θ )  sin θ cos θ sinθ sin θ − . cos 2 θ  1 − cos 2 θ cos θ  common fact/ = cos θ − cos 2 θ gemene fakt. sin θ(1 − cosθ )  common fact/ = gemene fakt. cos θ(1 − cosθ ) = tan θ (4) = RHS [26] Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Downloaded from hlayiso.com Mathematics P2/Wiskunde V2 11 DBE/November 2017 CAPS/KABV – Grade/Graad 11 – Marking guidelines/Nasienriglyne QUESTION/VRAAG 6 6.1 sin( x − 30 ο ) = cos 2 x  sin(90ο − 2 x) sin( x − 30 ο ) = sin(90 ο − 2 x)  x − 30 = 90 − 2 x + 360 k ο ο ο x − 30 ο = 90 ο − 2 x + 360 ο k or x − 30 ο =180 ο − ( 90 ο − 2 x) + 360k ο ο ο ο  x = 40ο + 120ο k 3 x =120 + 360 k − x =120 + 360 k  ο ο x = 40 +120 k x = − 120 ο + 360 ο k , k ∈ Z x − 30ο =180ο − ( 90ο − 2 x) + 360k  x = − 120ο + 360ο k (5) OR/OF ( (  cos 90 ο − x − 30 ο )) cos(90 ο − (x − 30 ο )) = cos 2 x ( ) cos 120 ο − x = cos 2 x  120 ο − x = 2 x + 360 ο k ο ο ο ο 120 − x = 2 x + 360 k or 120 − x = −2 x + 360 k  x = 40ο + 120ο k − 3 x = −120 ο + 360 ο k x = −120° + 360 ο k  x = 40 ο + 120 ο k , k ∈ Z 120 ο − x = −2 x + 360 ο k NOTE/LET WEL:  x = 240 ο + 360 ο k x = −120° + k .360° is equivalent to/ekwivalent aan x = 240° + k .360° (5) 6.2.1 180°  answer/antw (1) 6.2.2 −1 ≤ y ≤ 1  values/waardes  notation/notasie OR/OF (2)  values/waardes y ∈ [−1; 1]  notation/notasie (2) Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Downloaded from hlayiso.com Mathematics P2/Wiskunde V2 12 DBE/November 2017 CAPS/KABV – Grade/Graad 11 – Marking guidelines/Nasienriglyne 6.2.3 y f x- intercept 1 f at/afsnit by 30o g  shape of/ vorm van f (180° ; 0,5) g x- x intercepts −90° −60° −30° 0 30° 60° 90° 120° 150° 180° at/afsnitte by 45o and/ en 135o (−90°;−0,7)  shape −1 of/vorm van g TP at/DP by (90 ο ;−1 ) (5) 6.2.4 x = − 80 ο ; x = 40 ο ; x = 160 ο one mark per answer/een punt per antw.(3) [16] QUESTION/VRAAG 7 Q θ 2x 3 P x R 7.1 x 2 = (2 x ) + (3) − 2(2 x )(3)cos θ  cos rule 2 2  subst 12 x cos θ = 3x 2 + 9 3x 2 + 9 cos θ =  simplify/vereenv 12 x (3) 3( x 2 + 3) cos θ = 12 x 2 x +3 cos θ = 4x Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Downloaded from hlayiso.com Mathematics P2/Wiskunde V2 13 DBE/November 2017 CAPS/KABV – Grade/Graad 11 – Marking guidelines/Nasienriglyne 7.2.1 cosθ = (2,4)2 + 3  subst  cos θ = 0,9125 4(2,4)  answer/antw. cosθ = 0,9125 (3) ο θ = 24,15 7.2.2 1 Area of/van ∆PQR = × PQ × QR × sin Qˆ 2 1 = × 4,8 × 3 × sin 24,15  subst 2 = 2,95 units/eenh 2  answer/antw. (2) 7.3 For/vir x > 0 , cosθ > 0 0° < θ < 90° x2 +3 x2 +3 0< <1  0 < <1 4x 4x x 2 + 3x < 4 x x2 − 4x + 3 < 0 (x −1)(x − 3) < 0 1< x< 3  1 < x < 3 (4) [12] Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Downloaded from hlayiso.com Mathematics P2/Wiskunde V2 14 DBE/November 2017 CAPS/KABV – Grade/Graad 11 – Marking guidelines/Nasienriglyne QUESTION 8/VRAAG 8 8.1 1  V = Ah Area of square/van 3 = (16 × 16 ) 1 vierk. 640 = × (16 × 16 ) × h 3 Subst in volume h = 7,5cm form (2) 8.2 slant height / skuinshoogte = 7 ,5 2 + 8 2 = 10 ,9658.... Subst in pyth answer/antw Subst in SA/ 1  BO form. Total surface / Totale buite area = (side × side ) + 4 b× ⊥ h  answer/antw 2  1  = (16 × 16 ) + 4 × 16 × 10,9658.... (4) 2  [6] = 606,91cm 2 QUESTION 9/VRAAG 9 O 1 2 R S T 9.1 Construction/Konstr: Draw/trek radii OR and/en OS Constr/Konstr In ΔOTR and/ en ΔOTS OR = OS (radii ) S (OT is common/gemeen) OT = OT (common side/ gemene sy ) S/R ( Tˆ = Tˆ = 90 ο ∠ s on straight line/ op 'n reguit lyn 1 2 ) ( ∆OTR ≡ ∆OTS 90 ο HS ) R S ∴ RT = TS (≡ ∆ ) s (5) Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Downloaded from hlayiso.com Mathematics P2/Wiskunde V2 15 DBE/November 2017 CAPS/KABV – Grade/Graad 11 – Marking guidelines/Nasienriglyne A 4 F 4 B x O D 9.2 line from centre ⊥ to chord/  S/R AF = FB = 4cm   8− x lyn van mdpt ⊥ aan koord  OD = OB = 8 − x (radii ) 2 2 2  (8 − x) 2 = x 2 + 4 2 OB = OF + FB ( Pythagoras ) ( 8 − x )2 = x 2 + 4 2 64 −16 x + x 2 = x 2 + 4 2  x =3 48 = 16 x x =3 length of/ lengte van radius = 8 − x Answer/antw (5) = 8−5 [10] = 3 units / eenh Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Downloaded from hlayiso.com Mathematics P2/Wiskunde V2 16 DBE/November 2017 CAPS/KABV – Grade/Graad 11 – Marking guidelines/Nasienriglyne QUESTION/VRAAG 10 V Q 2 1 2 1 M S 1 2 P R 10.1 Q̂ 2 = Rˆ 2 [ tangents from common point/ rk ln e van selfde punt ]  S R S Q̂ 2 + Rˆ 2 + 70 ο = 180 ο [sum ∠ Δ]  Rˆ 2 = 55ο 2 Rˆ 2 = 110 ο (4) Rˆ = 55 ο 2 10.2 Qˆ 2 + Qˆ 1 = 90 ο [tan/rkl ⊥ rad]  R  Qˆ1 = 35ο Qˆ = 35 ο 1 (2) OR/OF  R Rˆ1 + Rˆ 2 = 90 ο [tan/rkl ⊥ rad]  Qˆ1 = 35ο Rˆ = 35 ο 1 (2) Qˆ 1 = Rˆ1 == 35 ο [OR = OQ] 10.3 Ô1 + R̂ 1 + Q̂1 = 180 ο [sum ∠ Δ]  Ô1 = 110 ο Ô1 = 180 ο − 70 ο = 110 ο  Ô 2 = 110ο  R equal chords subtend = ∠ at the centre/  Ô 2 = 110 ο  (3)  gelyke koorde onrsp. = ∠ by mdpt  [9] Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Downloaded from hlayiso.com Mathematics P2/Wiskunde V2 17 DBE/November 2017 CAPS/KABV – Grade/Graad 11 – Marking guidelines/Nasienriglyne QUESTION/VRAAG 11 B 2 O 1 1 2 D A E 11.1 Bˆ1 = Bˆ 2 = x [BD bisect/halveer ∠ AB̂C ] S ABˆ C = 2 x Oˆ = 2 x [∠ at centre = 2 times ∠ at circumference/] 1 S  R [ midpts∠ = 2 × omtreks∠] R ∴ BC OD [corresponding ∠ are equal/ooreenk. ∠ is gelyk ] (4) OR/OF Bˆ1 = Bˆ 2 = x [BD bisect/halveer ∠ AB̂C ] S Dˆ = x [angle opp = sides/∠e to gelyke sye] S  R 1 R Dˆ 1 = Bˆ 2 = x (4) ∴ BC OD [alternate angles are equal/verw ∠e gelyk] OR/OF S Bˆ1 = Bˆ 2 = x [BD bisect/halveer ∠ AB̂C ] ABˆ C = 2 x S  R Oˆ = 2 x [angle at centre = 2 times angle at circumference ] 1 [ midpts∠ = 2 × omtreks∠] ˆ O2 = 180 − 2 x [∠ on a straight line/∠ op reguit lyn ] ο R Oˆ 2 + ABˆ C = 180 ο − 2 x + 2 x = 180 ο (4) ∴ BC OD [ co-int angles are suppl/ko-binne ∠ is suppl] Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Downloaded from hlayiso.com Mathematics P2/Wiskunde V2 18 DBE/November 2017 CAPS/KABV – Grade/Graad 11 – Marking guidelines/Nasienriglyne 11.2 ODˆ C = 90 ο [tan/rkl ⊥ rad ]  S/R  S R Cˆ = 90 ο [co − int / ko-binne ∠' s OD BC] (3) OR/OF  S/R  S R Dˆ 1 = x (3)  S/R Dˆ = 90 ο − x [tan/rkl ⊥ rad ] 2  S R ( ) Cˆ = 180 ο − 90 ο − x − x [int ∠' s of / van ∆ ] (3) = 90 ο [7] OR/OF EDˆ C = 90 ο [tan/rkl ⊥ rad ] Cˆ = 90 ο [corresp. / ooreenk . ∠' s OD BC] Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Downloaded from hlayiso.com Mathematics P2/Wiskunde V2 19 DBE/November 2017 CAPS/KABV – Grade/Graad 11 – Marking guidelines/Nasienriglyne QUESTION/VRAAG 12 A Q 1 2 x B F 1 2 6 5 T 4 7 P3 8 1 2 2 2 y 1 3 1 E D R C 12.1 P̂1 = x [tan − ch th/ rkl-kdst ] SR SR Ĉ 2 = x [alt / verw. ∠ s AD BC ] SR Ê = x [∠ in the same segment/dieselfde segment ] s  S/R (7) P̂5 = x [vert opp/ reg oorst ] 12.2 DCˆ B = x + y S R E Dˆ A = Dˆ = x + y [corresp / ooreenk. ∠ s = , AD 1 BC ] S R [ ∴ EP̂A = x + y ∠ s in the same segment/dieselfde segment ] (4) 12.3 Pˆ2 = y [ tan from a commom point/ rklyne v dieselfde pt ]  S/R DPˆ T = Pˆ + Pˆ + Pˆ 1 2 3 = x + y + ( x + y)  Ĉ = x + y = 2x + 2 y  Subst Ĉ = x + y  Answ/antw DPˆ T + Ĉ =180 ο [opp ∠ s of a cyclic quad/ teenoorst. ∠ s van kvhk] (4) 2 x + 2 y + x + y =180 ο [15] 3 x + 3 y =180 ο ∴ x + y = 60 ο TOTAL/TOTAAL: 150 Copyright reserved/Kopiereg voorbehou