Mathematics Grade 12 P2 North West June 2024 Memo (Afr)

PROVINCIAL ASSESSMENT/ PROVINSIALE ASSESSERING GRADE 12/GRAAD 12 MATHEMATICS P2/WISKUNDE V2 JUNE/JUNIE 2024 MARKING GUIDELINES/NASIENRIGLYNE MARKS: 150/PUNTE: 150 These marking guidelines consist of 16 pages. Hierdie nasienriglyne bestaan uit 16 bladsye. Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Mathematics P2/Wiskunde V2 2 NW/June/Junie 2024 Grade 12/Graad 12 – Marking Guidelines/Nasienriglyne NOTE  If a candidate answers a question twice, only mark the FIRST attempt/ Indien `n kandidaat `n vraag twee keer antwoord, word slegs die EERSTE poging gemerk./  If a candidate has crossed out an attempt of a question and not redone the question, mark the crossed out version./Indien `n kandidaat `n poging doodgetrek het en NIE orgedoen het nie, word die doodgetrekte poging gemerk.  Consistent accuracy applies in ALL aspects of the marking memorandum. Stop marking at the second calculation error./Konsekwente akkuraatheid word dwarsdeur die memorandum toegepas. Met die tweede berekeningsfout word nasien gestop.  Assuming answers or values in order to solve a problem is NOT acceptable./ Die aanvaarding van antwoorde of waardes om `n problem op te los word NIE aanvaar nie./ GEOMETRY S A mark for a correct statement /`n Punt vir die korrekte bewering. (A statement mark is independent of a reason/ Hierdie punt is onafhanlik van die rede.) R A mark for a correct reason. /`n Punt vir die korrekte rede. (A reason mark may only be awarded if the statement is correct /Hierdie punt word net toegeken as die berwering korrek is.) S/R Award a mark if statement AND reason are both correct/ Ken die punt toe as die bewering SOWEL AS die rede korrek is. Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Mathematics P2/Wiskunde V2 3 NW/June/Junie 2024 Grade 12/Graad 12 – Marking Guidelines/Nasienriglyne QUESTION 1/VRAAG 1 y B (3; 6 ) C (9; 5 ) E θ α 2 N x 1 D A(– 2; – 6 ) 1.1 AC  (5  6) 2  (9  2) 2 substitution/vervanging AC  11 2  answer/antwoord (2) 1.2 56 m AC  92  substitution/vervanging m AC  1  m AC  1 (2) 1.3 y  y1  m( x  x1 ) C (9,5) or A (– 2; – 6) y 5 x9 y6x2  substituting point A or C /vervang punt A of C yx4 yx4  answer/antwoord (2) 1.4 Diagonals of a kite intersect at right angles/  answer/antwoord Hoeklyne van vlieër sny loodreg/ (1) So BE  AC. 1.5 m BD  1  m BD  1 B (3; 6)  substitution/vervanging y  6  1( x  3) y  x  9  answer/antwoord (3) Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Mathematics P2/Wiskunde V2 4 NW/June/Junie 2024 Grade 12/Graad 12 – Marking Guidelines/Nasienriglyne 1.6 At E, x  4  x  9  x  4  x  9 2 x  13 x  6,5  x  6,5 y  2,5  y  2,5 E (6,5; 2,5) (3) 1.7  x  10 D(10;– 1) [AC symmetry line of kite/AC lyn van  y  1 simmetrie van vlieër, so AB = BD]  motivation/motivering Answer only: 3/3 (3) 1.8 5 tan   m AD  12   22,62     22,62  N̂ 2    22,62   N̂ 2  22,62  tan   mCD  6   99,46     99,46  D̂1  76,84  [ext  of  /buite  van  ]  D̂1  76,84  AD̂C  103,16  [  s on a str line/reguit lyn/]  AD̂C  103,16  OR/OF AB = 13, BC = 37 and/en AC  11 2  correct lengths/korrekte lengtes ̂𝐶 AC 2 = AB 2 + BC 2 − 2AB × BCcosAB  substitution/vervanging (11 2 ) 2  13 2  ( 37 ) 2  2(13)( 37 ) cos AB̂C cosAB̂C  -0,227...  cosAB̂C  -0,227... AB̂C  103,16   AB̂C  103,16   answer/antwoord  AD̂C  103,16  OR/OF 7 2 BE  3,5 2  (3,5) 2   BE  7 2 2 2 3,5 2 cos AD̂E   cos AD̂E  3,5 2 13 13 AD̂E  67,62   AD̂E  67,62  3,5 2 cos CD̂E  37  CD̂E  35,54  CD̂E  35,54  answer/antwoord AD̂C  103,16  (5) [21] Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Mathematics P2/Wiskunde V2 5 NW/June/Junie 2024 Grade 12/Graad 12 – Marking Guidelines/Nasienriglyne QUESTION 2/VRAAG 2 2.1 ( x  2) 2  ( y  2) 2  25  equation of the circle/ vergelyking van sirkel (2  2) 2  (b  2) 2  25 substituting point T/ (b  2) 2  9 16  b 2  4b  4  25 vervang T or b  2  3 b 2  4b  5  0  simplification/ (b  5)(b  1)  0 vereenvoudiging b  5 or b  1 Factors /Faktore  answer /antwoord/ (4) 2.2.1 K(2; 2 – 5)  x2 K(2; – 3)  y  3 (2) 2.2.2 52 3 3 mMT    m MT   22 4 4 4 4  m PL   m PL  3 3 y  y1  m( x  x1 ) 4  substitution /vervanging/ y 5  ( x  2) 3 4 23 y  x  antwoord/answer 3 3 OR Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Mathematics P2/Wiskunde V2 6 NW/June/Junie 2024 Grade 12/Graad 12 – Marking Guidelines/Nasienriglyne 5  11 m PL   substitution/ vervanging  2  14 4 m PL  4  m PL  3 3 y  y1  m( x  x1 )  substituting P or T/ 4 y  5  ( x  2) vervang P of T 3 4 23  answer/ antwoord y  x 3 3 (4) 2.2.3 A(2; ) K(2; – 3) � P(– 14 ; – 11) H(2; – 11) 4 23 y x 3 3 4 23 y  ( 2)  3 3 31 31  yL  y 3 3 40 31 40  LK = LK = 3 3 3 3 Koordinate van P/Coordinates of P x2 y 3  6  4  x  14 2 2  y  11 x  14 y  5  PH = 18 PH –hoogte/Height = 2 – ( – 14) = 16  substitution/vervanging 1 40 Area of PKL =   16 2 3  answer/antwoord = 320 /106,67 vk eenhede/sq units (7) 3 2.3 The centres of the two circles lie on the same vertical line x  2 and the sum of the radii is 10/ Die middelpunte van die twee sirkels lê op  correct method/korrekte dieselfde vertikale lyn x  2 en die som van die metode radius = 10  sum of radii = 10/ n  2  10 2  n  10 som van ri = 10 of/or n  12 n  8  n  12 Answer only: full mark/  n  8 Slegs antwoord: vol punte (4) [21] Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Mathematics P2/Wiskunde V2 7 NW/June/Junie 2024 Grade 12/Graad 12 – Marking Guidelines/Nasienriglyne QUESTION 3/VRAAG 3 3.1 cos(90   x). sin( 540   x) tan( x  180  ). cos( x)   sin x  sin x. sin(180 0  x)  tan x  tan x. cos x  cos x  sin x.( sin x)   sin x  sin x sin x . cos x  cos x cos x  sin x  answer/antwoord (6) 3.2 cos(P  Q)  cosP.cosQ sinP.sinQ cos(2P)  cos(P  P)  cosP.cosP sinP.sinP   cosP.cosP sinP.sinP cos 2P  cos P.cos P  sin P.sin P  cos P  sin P 2 2  cos2 P  sin 2 P  cos2 P  (1  cos2 P)  1  cos P 2  cos2 P  1  cos2 P  2 cos P  1 2  2cos2 P  1 (4) 3.3.1 4 tan A  3,  A  90  ;360   r=5  correct sketch and quad/ korrekte skets en kwadrant 4 sin A    answer/antwoord 5 (3) 3.3.2 cos(A  30  )  cos A. cos30   sin A sin 30   expansion/uitbreiding 3 3    3  3   4  1  5  2           5  2   5  2   4  1      43 3   5  2   OR/OF  0,12 10  answer/antwoord (4) Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Mathematics P2/Wiskunde V2 8 NW/June/Junie 2024 Grade 12/Graad 12 – Marking Guidelines/Nasienriglyne 3.4.1 sin 78  cos12   cos12   p  answer/antwoord (2) 3.4.2 cos12   cos 2(6  )  cos 2(6  ) p  1  2 sin 2 6 0 p  1  2 sin 2 6 0  p  1  2 sin 2 6  1 p  sin 2 6 0 1 p 2   sin 2 6  2 1 p sin 6 0   2  answer/antwoord (4) [23] QUESTION 4/ VRAAG 4 4.1 sin 2 x  cos 2 x  1  tan x sin 2 x  cos 2 x  1 Both numerator and denominator/ Sowel teller as noemer sin 2 x  cos 2 x  1 LHS  sin 2 x  cos 2 x  1    2 sin x cos x  cos2 x  sin 2 x  1  2 sin x cos x   2 sin x cos x  cos2 x  sin 2 x  1  cos2 x  sin 2 x 2 sin x cos x  cos2 x  sin 2 x  1  2 sin x cos x  cos2 x  sin 2 x  1 2 sin x cos x  2 sin 2 x   2 sin x cos x  2 sin 2 x 2 sin x cos x  2 cos2 x  2 sin x cos x  2 cos2 x 2 sin x(cos x  sin x)  2 sin x(cos x  sin x)  2 cos x(sin x  cos x)  2 sin x(sin x  cos x) sin x sin x   cos x cos x  tan x (7) LHS = RHS Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Mathematics P2/Wiskunde V2 9 NW/June/Junie 2024 Grade 12/Graad 12 – Marking Guidelines/Nasienriglyne 4.2 sin 45   cos 45   1 tan 22,5   expansion/ uitbreiding sin 45   cos 45   1 2 2  1  2 2 2 2  substitution/vervanging  1 2 2 1  or 2  1  answer/antwoord 2 1 (3) 4.3 3 cos 2 x  1  5 cos x 3(2 cos2 x  1)  1  5 cos x  2 cos2 x  1 6 cos2 x  3  1  5 cos x 6 cos2 x  5 cos x  4  0  std form/std form (3 cos x  4)( 2 cos x  1)  0  factors/faktore 4 1 cos x  or cos x    both equations/beide 3 2 vergelykings No solution x  120   k.360  , k  Z  No solution/geen oplossing   x  240  k.360 , k  Z  Gen Solution 1/alg oplossing1  Gen Solution 2/ alg oplossing2/ OR/OF – 1 as/if k  Z is not included  1 x   cos1     k .360  nie aangedui  2 (7) x  120   k .360  , k  Z [17] Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Mathematics P2/Wiskunde V2 10 NW/June/Junie 2024 Grade 12/Graad 12 – Marking Guidelines/Nasienriglyne QUESTION 5/VRAAG 5 f ( x)  a sin x en/and g ( x)  sin x  b ; x  [360  ;360  ]. 5.1 a=2 a=2 b=1 b=1 (2) 5,2 Amplitude = 1  answer/antwoord (1)    5.3.1 x  270 of/or x  90  x  270  x  90  (2)       5.3.2  360  x  270 or  90  x  90 or   360  x  270 270   x  360    90   x  90  OF/ OR  270   x  360  x  (360  ;270  ) or x  (90  ;90  ) or OF/OR x  (270  ;360  )  x  (360  ;270  )  x  (90  ;90  )  x  (270  ;360  ) (3)       5.3.3  270  x  90 or 90  x  270   270  x  90  x   270  ;90    or x  90  ;270    90   x  270  OF/OR    x   270  ;90   x  90 ;270    (2) [10] Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Mathematics P2/Wiskunde V2 11 NW/June/Junie 2024 Grade 12/Graad 12 – Marking Guidelines/Nasienriglyne QUESTION 6/VRAAG 6 6.1 1  answer/antwoord A  p  r  sin2θ (1) 2 6.2 1 A   p  r  sin2θ 2 A r 1 A  r 1 2 psin2θ 2 psin2θ 2A r 2A  r psin2θ psin2θ (2) 6.3 h h tan    tan   r r h  r tan   h  r tan  2A sin  2A sin  h   h  p sin 2 cos p sin 2 cos 2A sin  h   2 sin  cos p  2 sin  cos cos A  answer/antwoord h (5) p cos2  6.4.1 A h pcos2 θ A  substitution/ vervanging 20  10cos 2 60   answer/antwoord A = 50 m 2 (2) 6.4.2 2A r  substitution/ vervanging p sin 2 2(50)  r = 11,55 m r 10 sin 2(60  ) r = 11,55 m  correct substitution t 2  10 2  (11,55) 2  2(10)(11,55) cos 2(60  ) korrekte vervanging t = 18,68 m  substitution/vervanging (4) [14] Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Mathematics P2/Wiskunde V2 12 NW/June/Junie 2024 Grade 12/Graad 12 – Marking Guidelines/Nasienriglyne QUESTION 7/VRAAG 7 R O 2 1 2 P 2 1 1 S 1300 Q 7.1.1 Ŝ1  P̂1 [  `e teenoor = sye/  s opp = sides]  S/R 2Ŝ1  130   180  [binne  `e van Δ /  s of a Δ]  S/R  answer/antwoord Ŝ1  25 (3) 7.1.2 R̂  Ŝ1  25 [omtrekshoeke/  s subt by same S R chord] (2) 7.1.3 Ô 2  260  [ mdpt  =2x omtreks  /angle at centre  S  R = 2  at circumf] Ô1 100  [omwenteling/  s around a point]  answer/antwoord (3) Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Mathematics P2/Wiskunde V2 13 NW/June/Junie 2024 Grade 12/Graad 12 – Marking Guidelines/Nasienriglyne 7.2 7.2.1 DC = 9x  answer/antwoord (1) 7.2.2 OM = OC – MC 9x  method/metode OM   2x 2 OM  2,5 x  answer/antwoord (2) 7.2.3 OB2  BM 2  OM 2  S/R BM  7 [line fron centre  to chord/radius  koord ] 2 2  9x   5x     7   2  substitution/vervanging  2   2 81x 2 25 x 2  49  4 4 14 x  49 2  x 2  3,5 x 2  3,5  x  1,87 x  1,87  radius = 8,42 9 eenhede/units radius = (1,87 )  8.42 units 2 (5) [16] Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Mathematics P2/Wiskunde V2 14 NW/June/Junie 2024 Grade 12/Graad 12 – Marking Guidelines/Nasienriglyne QUESTION 8/VRAAG 8 M h A k > B N > C 8.1 Const: Draw height k from A such that k  MB and  construction/konstruksie h from B such that h  AM. Konstr: Trek hoogte k vanaf A sodat k  MB en h vanaf B sodat h  AM. Proof/Bewys:: 1 1 MA  h  Area ΔMAB  MA  h area ΔMAB 2 2  area ΔANB 1 1 AN  h  Area ΔANB  AN  h 2 2 area ΔMAB AM area ΔMAB AM    area ΔANB AN area ΔANB AN 1 MB  k area ΔMAB 2 MB area ΔMAB MB     area ΔABC 1 BC area ΔABC BC BC  k 2 Maar/But area ΔANB = area ΔABC [same base and S R height/ selfde basis selfde hoogte of/or RP // TM] (7) MA MB  AN BC Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Mathematics P2/Wiskunde V2 15 NW/June/Junie 2024 Grade 12/Graad 12 – Marking Guidelines/Nasienriglyne 8.2.1 AC 36 4 AS 80 4 AC 4       TA 27 3 AR 60 3 TA 3 AC AS AS 4    TA AR AR 3 SC || RD [converse line || to one side of Δ/ R omgekeerd lyn || een sy van Δ OR Converse prop (3 theorem, SC || RD/ Omgekeerd eweredigheid stelling SC || RD OR lyn verdeel sy eweredig 8.2.2 RS = 80 – 60 = 20 cm  answer/antwoord (1) 8.2.3 3 RB  AR 2 3 RB  (60) 2  RB = 90 cm RB  90cm  SB = 70 cm SB = 70 cm CD 20  [line || to one side of Δ/lyn || een sy S R 30 70 van Δ OR/OF [prop theorem, SC || RD] OR/OF omgekeerd eweredigheid stelling SC || RD]  answer/antwoord CD = 8,57 cm (5) [16] Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Mathematics P2/Wiskunde V2 16 NW/June/Junie 2024 Grade 12/Graad 12 – Marking Guidelines/Nasienriglyne J QUESTION 9/VRAAG 9 1 2 M 3 T 1 N 1 2 2 3 R 1 1 2 2 K V L 9.1 L̂ 2  M̂ 2 [alt  s/verwiss  `e/ MN || KL]  S/R Ĵ 2  L̂ 2 [tan chord theorem/raaklyn-koord  S /R stelling] R JMRN is a cyclic quad/ is `n koordevierhoek (3) [converse of  s in same seg/ omgekeerde omtrekshoeke] 9.2 In ΔJNR and/en ΔLMK Ĵ 2  L̂ 2 [tan chord theorem/raaklyn-koord  S/R stelling]  S/R JN̂R  M̂1 [ext  of cyclic quad/buitehoek van kvh] R̂ 2  K̂ [  s of a Δ/binnehoeke van Δ]  3rd  of Δ/3de  van Δ ΔJNR ||| ΔLMK [AAA/HHH] or/of R (3) 9.3 JT  JN [Prop theorem/Eweredigheid stelling, TV NL , TN || VL OF/OR S R Line || to one side of Δ/Lyn || een sy van Δ] NL.JT JN   JN  NL.JT TV TV JN NR  [ΔJNR ||| ΔLMK] LM MK S R LM.NR JN  MK  JN  LM.NR NL.JT LM.NR MK   TV MK (6) [12] TOTAAL/TOTAL: 150 Copyright reserved

Mathematics P2/Wiskunde V2 17 NW/June/Junie 2024 Grade 12/Graad 12 – Marking Guidelines/Nasienriglyne Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief