Mathematics Grade 12 Paper 2 NW June 2025 Memo (Afrikaans)

PROVINCIAL ASSESSMENT/ PROVINSIALE ASSESSERING GRADE 12/GRAAD 12 MATHEMATICS P2/WISKUNDE V2 JUNE/JUNIE 2025 MARKING GUIDELINES/NASIENRIGLYNE MARKS/PUNTE: 150 These marking guidelines consist of 18 pages. Hierdie nasienriglyne bestaan uit 18 bladsye. Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Mathematics P2/Wiskunde V2 2 NW/June/Junie 2025 Grade/Graad 12 Marking Guidelines/Nasienriglyne NOTE:  If a candidate answer a question TWICE, only mark the FIRST attempt.  If a candidate has crossed out an attempt of a question and not redone the question, mark the crossed-out version.  Consistent accuracy applies in ALL aspects of the Marking Guidelines. Stop marking at the second calculation error.  Assuming answer values in order to solve a problem is NOT acceptable. LET WEL:  As ‘n kandidaat ‘n vraag TWEE KEER beantwoord, sien slegs die EERSTE poging na.  As ‘n kandidaat ‘n antwoord van ‘n vraag doodtrek en nie oordoen nie, sien die doodgetrekte poging na.  Volgehoue akkuraatheid word in ALLE aspekte van die Nasienriglyne toegepas. Hou op nasien by die tweede berekeningsfout.  Aanvaar van antwoorde/waardes om ‘n probleem op te los, word NIE toegelaat nie. GEOMETRY S A mark for a correct statement/’n Punt vir die korrekte stelling (A statement mark is independent of a reason/’n stelling is onafhanklik 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/’n rede punt mag slegs toegeken word met korrekte stelling) S/R Award a mark if statement AND reason are both correct/ken ‘n punt toe as die stelling EN rede beide korrek is Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Mathematics P2/Wiskunde V2 3 NW/June/Junie 2025 Grade/Graad 12 Marking Guidelines/Nasienriglyne QUESTION/VRAAG 1 y B (3; 10) D C (13; 8) α θ x A (5; – 6) . E 1.1 10  8 m BC  3  13  substitution/vervanging 1 m BC    answer/antwoord 5 (2) 1.2 y  y1  m( x  x1 ) B (3, 10) or C (13; 8)) 1 1 y  10   ( x  3) y  8   ( x  13)  substitution/vervanging 5 or 5 1 53 1 53 y   x y   x  answer/antwoord 5 5 5 5 (2) 1.3 m AD  5 y  y1  m( x  x1 )  m AD  5 A (5; – 6) y  6  5( x  5)  substituting/vervang y  5 x  31 A or C  answer/antwoord (3) 1.4  x  x 2 y1  y 2  Midpoint/Middelpunt AC  1 ;   2 2  Midpoint/Middelpunt AC  13  5 ; 8  6   substitution/vervanging  2 2  Midpoint/Middelpunt AC (9; 1)  answer/antwoord (2) Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Mathematics P2/Wiskunde V2 4 NW/June/Junie 2025 Grade/Graad 12 Marking Guidelines/Nasienriglyne 1.5 1 m BC   5 11  8 11  10 m PC  or m BP   substitution/vervanging  2  13 23 1 1 m PC   m BP   gradient/gradiënt 5 5 m BC  m BP or mCP conclusion/gevolgtrekking The points are collinear/Die punte is kollinieêr (3) 1.6 E (15; – 8)  x  15  y  8 (2) 1.7 1 m BC   5 1 1 tan     tan    5 5   168,69     168 ,69  10  6 m AB  35 m AB  8  m AB  8 tan   8  tan   8   97,13    97 ,13  AB̂C  168 ,69   97 ,13  AB̂C  71,56   answer/antwoord (6) [20] Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Mathematics P2/Wiskunde V2 5 NW/June/Junie 2025 Grade/Graad 12 Marking Guidelines/Nasienriglyne QUESTION/VRAAG 2 y A .Q P . x R 2.1 ( x  15) 2  ( y  m) 2  64 x-int (15; 0)  substituting x-int/vervang x-afsnit (15  15) 2  (0  m) 2  64 m 2  64 simplification/vereenvoudiging m8  answer/antwoord OR/OF r 2  64  method/metode r 8 r=8 m  8  answer/antwoord OR/OF (m  3) 2  (15  3) 2  5  8  method/metode m  6m  9  144  169 2 m 2  6m  16  0  std form/standaardvorm (m  8)(m  2)  0  answer/antwoord m  8 or m  2 (3) 2.2 P (3; 3) P (3; 3) Q (15; 8)  Q (15; 8) (2) 2.3 PQ  (8  3) 2  (15  3) 2  substitution/vervanging PQ  13  answer/antwoord OR/OF PQ = 8 + 5  substitution/vervanging = 13  answer/antwoord Answer only/slegs antwoord : (2) full marks Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Mathematics P2/Wiskunde V2 6 NW/June/Junie 2025 Grade/Graad 12 Marking Guidelines/Nasienriglyne 2.4 ( x  3) 2  ( y  3) 2  25 ( x  3) 2  ( y  3) 2  25  x0 (0  3) 2  ( y  3) 2  25 9  ( y  3) 2  25 y 2  6 y  9  9  25  std form/vorm ( y  3) 2  16 y2  6y  7  0 y  3  4 ( y  7)( y  1)  0 y7 or y  1 A(0; 7)  answer/antwoord (3) 2.5 73 mAP  03 4 mAP   4 3  mAP   3 3  mtan  4 3  m tan  3 4 y  7  ( x  0) 4 3 substitution/vervanging y  x7 4 answer/antwoord (4) 2.6 P (3; 3) R(2; – 3) radius = 5  radius = 5 PR  (3  3) 2  (2  3) 2 = 37  PR = 6,08 = 6,08 radius <PR radius < PR conclusion/gevolgtrekking  P lies outside circle P (4) 2.7 0  x2 7  y2 3 3 2 2 x2  6 y 2  1 x = 6 R (6;– 1) y = – 1 3  substitution/vervanging y  xk 4 3  1  (6)  c 4 11 11 c  c 2 2 11 answer/antwoord   k  7 2 (6) [24] Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Mathematics P2/Wiskunde V2 7 NW/June/Junie 2025 Grade/Graad 12 Marking Guidelines/Nasienriglyne QUESTION/VRAAG 3 3.1 sin(90  x). tan(360  x) cos 240. tan 225 (cos x). tan x  cos x  cos(180  60  ). tan(180   45 )   tan x  sin x    cos 60  cos x   tan 45    cos x  sin x  cos 60 . tan 45  cos x sin x  1   and 1 1  1 2 2  2 sin x  answer/antwoord (7) 3.2.1 sin 223  sin(180  43 )      sin 43   sin 43   answer/antwoord  a (2) 3.2.2 sin 68,5  cos 21,5    cos 21,5  cos 43  cos 2(21,5 )    cos 2(21,5  ) cos 43  2 cos 2 21,5   1  2 cos 2 21,5   1 1  a 2  2 cos 2 21,5   1  1 a2 2 cos 2 21,5   1  a 2  1 1 a2 1 cos 2 21,5  2 1 a2 1  answer/antwoord cos 21,5    (5) 2 Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Mathematics P2/Wiskunde V2 8 NW/June/Junie 2025 Grade/Graad 12 Marking Guidelines/Nasienriglyne 3.3 y 5 3  correct sketch/korrekte A diagram –4 O x opp = 3 opp = 3/teenoorst = 3 sin( A  30  )  sinA.cos30   cosA.sin30   sinA.cos30   cosA.sin30   3   3   4  1    .       5   2   5  2   substitution/vervanging 43 3  10  answer/antwoord (5) [19] QUESTION/VRAAG 4 4.1.1 sin 2 x  tan x  tan x cos 2 x sin 2 x  tan x LHS  cos 2 x sin x  2 sin x cos x 2 sin x cos x   cos x sin x 2 cos 2 x  1  cos x 2 sin x cos 2 x  sin x  2 cos 2 x  1   (2 cos 2 x  1) cos x sin x(2 cos 2 x  1) 1 LCM/KGV   cos x 2 cos 2 x  1 common factor/gem faktor  tan x LHS = RHS (5) 4.1.2 sin 120  tan 60    tan 60   tan 60  cos120  = 3 answer/antwoord (2) Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Mathematics P2/Wiskunde V2 9 NW/June/Junie 2025 Grade/Graad 12 Marking Guidelines/Nasienriglyne 4.2 cos 24  . cos x  sin 24  . sin x  sin 2 x  cos( 24  x)  cos(24   x)  sin 2 x cos(24   x)  cos(90   2 x)  changing to cos/verander na 24   x  (90   2 x)  k .360  cos 24   x  90   2 x  k .360   first quad/1ste kwadrant x  66   k .360  ; k  Z or  Gen solution 1/alg opl 1 24   x  (90   2 x)  k .360  4th quad/4de kwadrant 24   x  90   2 x  k .360   3x  114   k .360   Gen solution 2/alg opl 1 x  38   k .120  ; k  Z – 1 if/as k  Z is not OR/OF included/nie deel van die cos 24  . cos x  sin 24  . sin x  sin 2 x antwoord is nie cos(24   x)  sin 2 x cos(24   x)  cos(90   2 x)  cos( 24  x)  90  2 x  (24  x)  k .360     changing to cos/verander na 90  2 x  24   x  k .360   cos  x  66   k .360  x  66   k .360  ; k  Z  first quad/1ste kwadrant or/OF 90   2 x  (24   x)  k .360   Gen solution 1/alg opl 1 90   2 x  24   x  k .360   3x  114   k .360  4th quad/4de kwadrant x  38   k .120  ; k  Z OR/OF cos 24  . cos x  sin 24  . sin x  sin 2 x  Gen solution 2/alg opl 2 cos(24   x)  sin 2 x – 1 if/as k  Z is not included/nie deel van die sin[90   (24   x)]  sin 2 x antwoord is nie sin( 66   x)  sin 2 x  cos( 24  x)  2 x  66   x  k .360  x  66   k.360  ; k  Z  changing to/verander na sin or  first quad/1ste kwadrant 2 x  180   (66   x)  k .360   Gen solution 1/alg opl 1 2 x  114   x  k .360  2th quad/4de kwadrant x  38  k .120 ; k  Z    Gen Solution 2/alg opl 2 OR/OF Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Mathematics P2/Wiskunde V2 10 NW/June/Junie 2025 Grade/Graad 12 Marking Guidelines/Nasienriglyne cos 24  . cos x  sin 24  . sin x  sin 2 x cos( 24   x) cos(24   x)  sin 2 x  changing to cos/verander na cos cos(24  x)]  cos(90  2 x)    first quad/1ste kwadrant 24   x  90   2 x  k .360 x  66   k .360 ; k  Z  Gen solution 1/alg opl 1 or 24   x  360   (90   2 x)  k .360  4th quad/4de kwadrant  3x  246   k .360   Gen Solution 2/alg opl 2 x  82   k .120  ; k  Z – 1 if/as k  Z is not included/nie deel van die antwoord is nie (6) [13] Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Mathematics P2/Wiskunde V2 11 NW/June/Junie 2025 Grade/Graad 12 Marking Guidelines/Nasienriglyne QUESTION/VRAAG 5 5.1 a=5 a=5 b=1 b=1 (2) 5.2 Amplitude = 4  answer/antwoord (1) 5.3 360   answer/antwoord (1) 5.4.1 x  90   answer/antwoord (1) 5.4.2 x  ( 90 ;90 ) or  90  x  90      answer/antwoord (2) 5.4.3 x  [180  ;180  ] or  180   x  180   answer/antwoord (2) 5.5 8 minimum when x  0  g ( x) 8 min  4 4 min  2  answer/antwoord (2) 5.6 k  5 or k 5  k  5 k  5 (2) [13] \ Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Mathematics P2/Wiskunde V2 12 NW/June/Junie 2025 Grade/Graad 12 Marking Guidelines/Nasienriglyne QUESTION/VRAAG 6 T h β G⅂ A 120o θ α B 6.1 Â 1    120   180   Â 1    120   180  Â 1  60     answer/antwoord Answer/antwoord only: full marks (2) 6.2 h h tan    tan   BG BG h h BG   BG  tan  tan  (2) 6.3 h h tan    tan   AG AG h h AG   AG  tan  tan  sin  sin( 60    ) sin  sin( 60    )    AG BG AG BG AG sin( 60    ) sin   BG h  sin( 60    ) tan substitution/vervanging sin   h tan tan sin( 60    ) sin   tan (4) Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Mathematics P2/Wiskunde V2 13 NW/June/Junie 2025 Grade/Graad 12 Marking Guidelines/Nasienriglyne 6.4 tan60  sin( 60   15  ) sin 15    substitution/vervanging tan tan60 sin( 60   15  )  tan   sin 15  tan   4,732...  tan   4,732 ...   78,07   answer/antwoord (3) [11] QUESTION/VRAAG 7 O 1 2 K 1 2 1 2 N L 640 1 2 M 7.1.1 Ô 1  M̂ 2  64  [alt  s /verw e ; KO//MN]  S/R N̂ 2  32  [  at centre = 2   at circumf]/ S R (3) midp   2  omtreks  7.1.2 L̂  148  [opp  s of a cyclic quad]/ S R teenoorst  van kvh (2) 7.1.3 K̂ 1  N̂ 2  32  [alt  s /verw e ; KO//MN] S R N̂ 1  32  [  s opp = sides/ℎ𝑜𝑒𝑘𝑒 𝑡. 𝑜. = 𝑠𝑦𝑒]  answer/antwoord OR M̂ 2  N̂ 1  N̂ 2  64  [  s opp = sides/ ℎ𝑜𝑒𝑘𝑒 𝑡. 𝑜. = OR 𝑠𝑦𝑒 ] S R N̂ 2  32  [proved/bewys in 7.1.1]  N̂ 1  32   answer/antwoord (3) Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Mathematics P2/Wiskunde V2 14 NW/June/Junie 2025 Grade/Graad 12 Marking Guidelines/Nasienriglyne 7.2 C F O . B P A Q 7.2. Draw diameter AOF and join FB/ Trek middellyn construction/konstruksie AOF en verbind met FB Proof:/Bewys S R FÂP  90  [rad  tan/rklyn] S/R AB̂F  90  [  sub by dia/  in a semi- S R circle/halfsirkel] But/maar CÂF  CB̂F [  s in same seg/ e in dieselfde sirkel segment]  CÂP  AB̂C (6) Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Mathematics P2/Wiskunde V2 15 NW/June/Junie 2025 Grade/Graad 12 Marking Guidelines/Nasienriglyne 7.3 Q P 1 2 2 1 W 2 1 1 2 1 T 2 3 R S 7.3.1  sub by dia/onderspan deur middellyn OR/OF R [  in semi-circle/  in halfsirkel] (1) 7.3.2 QP̂S  90  [rad  tan/rad  rklyn]  S/R S P̂2  90  x  R̂ 1  90  [  in semi-circle/  in halfsirkel] PR̂S  90  [  s on a str line/  op “n reguitlyn]  S/R e Ŝ  (90 - x)  90  180 [sum of  s of a Δ/som van     e S/R van Δ] Ŝ  x (4) 7.3.3 Ŝ  x [ proved in 7.3.2] Ŵ2  QP̂R  x [  s in same seg/  in dies seg] e S R  SRWT is a cyclic quad [ext  of a quad = int opp  R OR converse of ext  of cyclic quad] /  SWRT is ‘n kvh buite  van vierhoek  teenoorst binne van vierhoek  (3) OF omgekeerde buite  van kvh [22] Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Mathematics P2/Wiskunde V2 16 NW/June/Junie 2025 Grade/Graad 12 Marking Guidelines/Nasienriglyne QUESTION/VRAAG 8 A S P R T B C 8.1 AS AR  [line || to one side of Δ/lyn ||een sy v ∆ S R SP RB Prop theorem/ eweredigheid stelling RS || BP], AS 3 substitution/vervanging OR  SP 2  answer/antwoord AS 3   (4) SC 7 8.2 RT SP  S/R  [ line || to one side of Δ/lyn ||een sy v ∆ TC PC Prop theorem/ eweredigheid stelling RS || TP], substitution/vervanging RT 2  answer/antwoord  20 5 RT  8 units (3) 8.3 1 TC.PCsinT ĈP Area of TPC 2 1   TC.PC.sinTĈP Area of RSC 1 2 RC.SCsinR ĈS 2 1  RC.SC.sinR ĈS TC PC 2  . RC SC 5 5 5 5  .   . 7 7 7 7 25  answer/antwoord  49 (4) [11] Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Mathematics P2/Wiskunde V2 17 NW/June/Junie 2025 Grade/Graad 12 Marking Guidelines/Nasienriglyne QUESTION/VRAAG 9 A 2 >> C 1 1 2 2 2 3 D 4 1 1 >> T B P 9.1 In ΔPAT and ΔPCA P̂  P̂ [common/gemeenskaplik]  S/R Â 1  Ĉ1 [tan chord theorem/rklyn-koord stelling]  S /R AT̂P  PÂC [sum of  s of a Δ/som e v Δ]  ΔPAT ||| ΔPCA [AAA/    ]  R or/ 3rd angle/ of 3de  v  (3) 9.2 PA PT S R  [ ΔPAT ||| ΔPCA] PC PA PA 2  PC.PT S substitution/vervanging 6 2  ( x  5 )x x 2  5 x  36  0 (3) 9.3 x  5 x  36  0 2 ( x  9)( x  4)  0 factors/faktore x  9 or x  4 x-values/x-waardes  PT = 4 m answer/antwoord (3) Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Mathematics P2/Wiskunde V2 18 NW/ Junie 2025 Grade/Grade 12 – Marking Guidelines/Nasienriglyne 9.4 In ΔPDT and ΔPTA P̂  P̂ [common/gemeenskaplik] S T̂1  Ĉ 1 [corresp  s /ooreenk e ; BD// AC]  S/R Â 1  Ĉ 1 [tan chord theorem rklyn-koord stelling]  S/R  T̂1  Â 1 S D̂1  PT̂A [sum of  s of a Δ/som e v Δ]  ΔPDT ||| ΔPTA [    ]  S/R PD PT  [Δs ||| ] PT PA S PD 4  4 6  substitution/vervanging 8 PD = units or 2,67 units answer/antwoord 3 OR In ΔPDT and ΔPAC S P̂  P̂ [common/gemeenskaplik]  S/R T̂1  T̂4 [vert opp  s/regoorst e ] but T̂4  Ĉ1 [alt  s /verw e ; AC//TB]  S/R S  T̂1  Ĉ1  D̂1  PAC [sum of  s of a Δ/som v e v Δ]  ΔPDT ||| ΔPAC [    ]  S/R PD PT  [Δs ||| ] S PA PC PD 4   substitution/vervanging 6 9 8 answer/antwoord PD = units or 2,67 units 3 (8) [17] TOTAL/TOTAAL: 150 Copyright reserved/Kopiereg voorbehou