Mathematics Grade 12 Paper 1 NW June 2025 Memo (Afrikaans)

PROVINCIAL ASSESSMENT/ PROVINSIALE ASSESSERING GRADE/GRAAD 12 MATHEMATICS P1/WISKUNDE V1 JUNE/JUNIE 2025 MARKING GUIDELINES/NASIENRIGLYNE MARKS/PUNTE: 150 This marking guidelines consists of 12 pages. Hierdie nasienriglyne bestaan uit 12 bladsye. Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Mathematics P1/Wiskunde V1 2 NW/June/Junie 2025 Grade/Graad 12 – Marking Guidelines/Nasienriglyne NOTE:  If a candidate answers 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 answers/values in order to solve a problem is NOT acceptable. LET WEL:  Indien ‘n kandidaat ‘n vraag TWEE KEER beantwoord, sien slegs die EERSTE poging na.  Indien ‘n kandidaat ‘n antwoord van ‘n vraag doodtrek en nie oordoen nie, merk die doodgetrekte poging.  Volgehoue akkuraatheid isdeurgans op ALLE aspekte van die nasienriglyne van toepassing. Hou op nasien by die tweede berekeningsfout.  Aanvaar van antwoorde/waardes om ‘n probleem op te los, word NIE toegelaat nie. QUESTION/VRAAG 1 1.1.1 𝑥 2 − 𝑥 − 12 = 0 (𝑥 + 3)(𝑥 − 4) = 0 factors/faktore 𝑥 = −3 or/of 𝑥 = 4 𝑥 = −3 𝑥 = 4 (3) 1.1.2 5𝑥 2 + 2𝑥 = 9 5𝑥 2 + 2𝑥 − 9 = 0 std form/vorm −2 ± √22 − 4(5)(−9) 𝑥= subst 2(5) 𝑥 = 1,16 = 1,16 or/of 𝑥 = −1,56 𝑥 = −1,56 (4) 2 1.1.3 18 − 3𝑥 − 𝑥 ≥ 0 (6 + 𝑥)(3 − 𝑥) ≥ 0 factors/faktore  critical values/ −6 ≤ 𝑥 ≤ 3 kritiese waardes answer/antw OR/OF OR/OF 18 − 3𝑥 − 𝑥 2 ≥ 0 𝑥 2 + 3𝑥 − 18 ≤ 0 (𝑥 + 6)(𝑥 − 3) ≤ 0 factors/faktore  critical values/ −6 ≤ 𝑥 ≤ 3 kritiese waardes answer/antw (4) Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Mathematics P1/Wiskunde V1 3 NW/June/Junie 2025 Grade/Graad 12 – Marking Guidelines/Nasienriglyne 1.1.4 6 − √𝑥 + 4 = 𝑥 + 4 −√𝑥 + 4 = 𝑥 − 2 isolate/isoleer √ 𝑥 + 4 = 𝑥 2 − 4𝑥 + 4 square/kwadreer 𝑥 2 − 5𝑥 = 0 std form/vorm 𝑥(𝑥 − 5) = 0 factors/faktore 𝑥 = 0 or/of 𝑥 ≠ 5 𝑥 = 0 or/of 𝑥 = 5 𝑥 ≠ 5 (6) 1.2 2𝑦 − 𝑥 = 3 . . . . . . . . . ① 𝑦 2 + 3𝑥 − 2𝑥𝑦 = 0 . . . . ② 2𝑦 − 3 = 𝑥 . . . . . . . . . ③ equation/vergelyking ∴ 𝑦 2 + 3(2𝑦 − 3) − 2𝑦(2𝑦 − 3) = 0 subst 𝑦 2 + 6𝑦 − 9 − 4𝑦 2 + 6𝑦 = 0 −3𝑦 2 + 12𝑦 − 9 = 0 𝑦 2 − 4𝑦 + 3 = 0 std form/vorm (𝑦 − 1)(𝑦 − 3) = 0 factors/faktore y-values/waardes 𝑦=1 or/of 𝑦 = 3 𝑥 = 2(1) − 3 or/of 𝑥 = 2(3) − 3 x-values/waardes = −1 =3 (6) 1.3 1 3 1 (3 − ) (9 + + ) √5 √5 5 1 3 = 33 − ( ) cubes/derdemagte √5 1 1 √5 rationalise/rasionaliseer = 27 − ∙ × 5 √5 √5 1 𝑎 = 27 = 27 − √5 25 𝑏 = 5 ∴ 𝑎 = 27 and/en 𝑏 = 5 (4) [27] QUESTION/VRAAG 2 2.1 2; 7; 16; . . . . . . . ; 781; 862 5 9 13 ∴ 𝑇4 = 29 29 (1) 2.2 2𝑎 = 4; 𝑎 = 2  2𝑎 = 4; 𝑎 = 2 3(2) + 𝑏 = 5; 𝑏 = −1  𝑏 = −1 2 − 1 + 𝑐 = 2; 𝑐 = 1 𝑐 =1 𝑇𝑛 = 2𝑛2 − 𝑛 + 1 (4) 2.3 𝑇𝑛 = 2𝑛2 − 𝑛 + 1 862 = 2𝑛2 − 𝑛 + 1 subst 0 = 2𝑛2 − 𝑛 − 861 factorise/faktoriseer 0 = (𝑛 − 21)(2𝑛 + 41) 41 𝑛 = 21 or/of 𝑛 = − 2 term 21 ∴ 𝑇21 = 862 N/A (3) [8] Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Mathematics P1/Wiskunde V1 4 NW/June/Junie 2025 Grade/Graad 12 – Marking Guidelines/Nasienriglyne QUESTION/VRAAG 3 3.1.1 2 + 6 + 18 + 54 + . . . 𝑇𝑛 = 2.3𝑛−1 2.3𝑛−1 (1) 𝑚 3.1.2 2 ∑ ∙ 3𝑛 = 59 048 3 𝑛=1 ∴ 2 + 6 + 18 + 54 + . . . 𝑎; 𝑟 𝑆𝑚 = 59 048; 𝑎 = 2; 𝑟 = 3 𝑎(𝑟 𝑚 − 1) = 59 048 𝑟−1 2(3𝑚 − 1) subst = 59 048 3−1 3𝑚 = 59 049 3𝑚 = 310 exp law/eksp wet 𝑚 = 10 answer/antwoord (4) 3.2.1 1 𝑎 = 3; 𝑟 = 2 3 3 3 + + + . . . 2 4 3 4 3 ∴ 𝑇3 = (1) 4 3.2.2 𝑎 𝑆∞ = 1−𝑟 3 subst = 1 1−2 6 =6 (2) 3.3 1; 2; 3; 4; . . . . ; 8 000 8 000 𝑆8000 = (1 + 8 000) 2 = 32 004 000 answer/antwoord Powers of 6/Magte van 6: 6; 36; 216; 1 296; 7 776 sequence/ry 6(65 − 1) subst 𝑆5 = 6−1 = 9 330 ∴ 𝑆 = 32 004 000 − 9 330 = 31 994 670 answer/antwoord OR/OF (4) Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Mathematics P1/Wiskunde V1 5 NW/June/Junie 2025 Grade/Graad 12 – Marking Guidelines/Nasienriglyne 1; 2; 3; 4; . . . . ; 8 000 OR/OF 8 000 𝑆8000 = (1 + 8 000) 2 = 32 004 000 Powers of 6/Magte van 6: 6; 36; 216; 1 296; 7 776 answer/antwoord 𝑆5 = 6 + 36 + 216 + 1296 + 7776 = 9 330 series/ry ∴ 𝑆 = 32 004 000 − 9 330 add terms to 9 330/ = 31 994 670 tel terme op na 9 330 answer/antwoord [12] QUESTION/VRAAG 4 4.1.1 𝑇17 = 3 × 17 − 14 3 x 17 = 37  − 14 (2) 4.1.2 White tiles/wit teëls 5; 7; 9; . . . ∴ 𝑇𝑛 = 5 + (𝑛 − 1)2 = 2𝑛 + 3 2𝑛 3 (2) 4.1.3 Black tiles/Swart teëls: 1 + 2 + 3 + . . . + 25 𝑛 𝑛 ∴ 𝑆𝑛 = [𝑎 + 𝑙] = [2𝑎 + (𝑛 − 1)𝑑] 2 2 25 25 subst 𝑆25 = (1 + 25) = [2(1) + 24(1)] 2 2 = 325 = 325 325 White tiles/Wit teëls: 5 + 7 + 9 + . . . + 53 𝑛 𝑛 ∴ 𝑆𝑛 = [𝑎 + 𝑙] = [2𝑎 + (𝑛 − 1)𝑑] 2 2 25 25 subst 𝑆25 = (5 + 53) = [2(5) + 24(2)] 2 2 = 725 = 725 725 ∴Tiles all together/Teëls altesaam = 1 050 1 050 (5) Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Mathematics P1/Wiskunde V1 6 NW/June/Junie 2025 Grade/Graad 12 – Marking Guidelines/Nasienriglyne OR/OF 6 + 9 + 12 + . . . . 25 terms/terme series/reeks 𝑎 = 6; 𝑑 = 3 a d 25 𝑆25 = [2(6) + (25 − 1)(3)] subst 2 = 1 050 1 050 4.2 1 2 19 + 18 + 17 + . . . 5 5 4 𝑎; 𝑑 𝑑 = − ; 𝑎 = 19 5 4 subst 𝑇𝑛 = 19 + (𝑛 − 1) (− ) 5 4 4 = 19 − 𝑛 + 5 5 99 4 = − 𝑛 5 5 99 4 99 4 − 𝑛<0 5 − 5𝑛 < 0 5 5 4𝑛 99 − <− 5 5 𝑛 > 24,75  𝑛 > 24,75 ∴First negative term is/Eerste negatiewe term is T25 answer/antwoord (5) [14] Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Mathematics P1/Wiskunde V1 7 NW/June/Junie 2025 Grade/Graad 12 – Marking Guidelines/Nasienriglyne QUESTION/VRAAG 5 Given/Gegee: 𝑓(𝑥) = 𝑥 2 + 2𝑥 − 8 and/en 𝑔(𝑥) = 𝑚𝑥 + 𝑐 5.1 𝑝 = (−6)2 + 2(−6) − 8 = 16 16 (1) 5.2 𝑥 −intercepts/afsnitte: 𝑦 = 0 𝑥 2 + 2𝑥 − 8 = 0 𝑓(𝑥) = 0 (𝑥 + 4)(𝑥 − 2) = 0 factorise/faktoriseer 𝑥 = −4 or/of 𝑥 = 2 ∴ 𝐴(−4; 0) 𝐴(−4; 0) ∴ 𝐵(2; 0) 𝐵(2; 0) (4) 5.3 Turning point/draaipunt: 𝑏 2 𝑥=− =− = −1 𝑥 −value/waarde 2𝑎 2(1) 𝑦 = (−1)2 + 2(−1) − 8 = −9 𝑦 −value/waarde ∴ 𝐷(−1; −9) (2) 5.4 𝑐 = −8 𝑐 = −8 𝑚 = −2 𝑚 = −2 (2) 5.5 𝑚 = 𝑓 ′ (𝑥) = 2𝑥 + 2 𝑓 ′ (𝑥) = 2𝑥 + 2 𝑚 = 𝑓 ′ (0) subst = 2(0) + 2 =2 Subst 𝑚 = 2 and/en 𝐶(0; −8) into/in 𝑦 = 𝑚𝑥 + 𝑐 ∴ 𝑦 = 2𝑥 − 8 equation/vergelyking (3) 5.6 (−∞; 9] values/waardes notation/notasie OR/OF OR/OF 𝑦 ≤ 9; 𝑦 ∈ ℝ answer/antwoord (2) 5.7 𝑓(𝑥) − 𝑔(𝑥) > 0 𝑓(𝑥) > 𝑔(𝑥) ∴ 𝑥 < −4 or/of 𝑥 > 0  𝑥 < −4 𝑥>0 (2) 2 5.8 𝑥 + 2𝑥 + 𝑘 = 0 𝑥 2 + 2𝑥 − 8 = −𝑘 − 8 ∴ −𝑘 − 8 < −9 ∴ 𝑘>1 answer/antwoord (2) [18] Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Mathematics P1/Wiskunde V1 8 NW/June/Junie 2025 Grade/Graad 12 – Marking Guidelines/Nasienriglyne QUESTION/VRAAG 6 6 Given/Gegee: ℎ(𝑥) = 𝑥−2 − 1 6.1 𝑥 = 2 𝑥 = 2 𝑦 = −1 𝑦 = −1 (2) 6.2 𝑥 −intercept/afsnit: 𝑦 = 0 6 ℎ(𝑥) = 0 0= −1 𝑥−2 6 1= 𝑥−2 𝑥−2=6 𝑥=8 ∴ (8; 0)  (8; 0) 𝑦 −intercept/afsnit: 𝑥 = 0 6 𝑦= −1 −2 = −4 ∴ (0; −4)  (0; −4) (3) 6.3 shape/vorm asymptotes/asimptote x- and y- intercept/afsnit (3) 6.4 ℎ(𝑥) < 0; 𝑥 ∈ [0; ∞) ∴ 0 ≤ 𝑥 < 2 or/of 𝑥 > 8 0 ≤ 𝑥 < 2 𝑥 > 8 (2) [10] Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Mathematics P1/Wiskunde V1 9 NW/June/Junie 2025 Grade/Graad 12 – Marking Guidelines/Nasienriglyne QUESTION/VRAAG 7 Given/Gegee: 𝑝(𝑥) = 𝑏 𝑥 and/en 𝑝−1 (𝑥) S(27; -3) lies on 𝑝−1 /lê op 𝑝−1 K(-3; 27) lies on 𝑝/lê op 𝑝 7.1 𝑝(𝑥) = 𝑏 𝑥 27 = 𝑏 −3 subst (27; -3) 1 = 27 𝑏3 1 𝑏3 = 27 1 𝑏 = 3 1 ∴𝑏= (2) 3 𝑥 7.2 1 𝑓(𝑥) = 𝑦 = ( ) 3 1 𝑦 𝑓 −1 (𝑥): 𝑥 = ( ) swapping/omruil 3 𝑦 = log 1 𝑥 equation/vergelyking 3 Answer only: 2/2 (2) 7.3 𝑄(0; 1) 𝑄(0; 1) 𝑅(1; 0) 𝑅(1; 0) (2) 7.4.1 0 < 𝑝(𝑥) ≤ 1 ∴𝑥≥0 𝑥 ≥ 0 (1) 7.4.2 log 1 𝑥 < −3 3 𝑥 > 27 ∴ 𝑥 > 27 (1) 7.5 𝑇(27; 1) T(27; 1) ∴ 𝑇𝑄 = 27; 𝑇𝑆 = 4 TQ; TS 1 ∴ Area ∆𝑄𝑇𝑆 = 2 ∙ 𝑇𝑄 ∙ 𝑇𝑆 1 = × 27 × 4 subst 2 = 54 units2/eenh2 answer/antwoord (4) [12] Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Mathematics P1/Wiskunde V1 10 NW/June/Junie 2025 Grade/Graad 12 – Marking Guidelines/Nasienriglyne QUESTION/VRAAG 8 8.1 𝑓(𝑥) = 3𝑥 2 − 2 𝑓(𝑥 + ℎ) = 3(𝑥 + ℎ)2 − 2 = 3𝑥 2 + 6𝑥ℎ + 3ℎ2 − 2 𝑓(𝑥 + ℎ) value/waarde 𝑓(𝑥 + ℎ) − 𝑓(𝑥) 𝑓 ′ (𝑥) = lim ℎ→0 ℎ 3𝑥 2 + 6𝑥ℎ + 3ℎ2 − 2 − 3𝑥 2 + 2 subst = lim ℎ→0 ℎ simplification/vereenv ℎ(6𝑥 + 3ℎ) = lim ℎ→0 ℎ = lim (6𝑥 + 3ℎ) factors/faktore ℎ→0 = 6𝑥 answer/antwoord (5) 8.2.1 1 𝑓(𝑥) = 3𝑥 4 − 𝑥 2 + 5 2 12𝑥 3 𝑓 ′ (𝑥) = 12𝑥 3 − 𝑥 −𝑥 (2) 8.2.2 √𝑥 1 𝑦= − 2 9𝑥 3 1 1 1 1 1 1 = 𝑥 2 − 𝑥 −3 2 𝑥 2 9 𝑥 −3 2 9 1 𝑑𝑦 1 − 1 1 −4 1 1 ∴ = 𝑥 2 + 𝑥 4 𝑥 − 2 + 3 𝑥 −4 𝑑𝑥 4 3 (4) 8.2.3 2𝑥 2 − 3𝑥 − 5 𝑓(𝑥) = 𝑥+1 (2𝑥 − 5)(𝑥 + 1) = factors/faktore (𝑥 + 1) 2𝑥 − 5 = 2𝑥 − 5 𝑓 ′ (𝑥) = 2 ′ (𝑥) ∴𝑓 =2 (3) 1 8.3 𝑦 = −𝑥 2 + 4𝑥 and/en 𝑦 = 2𝑥 − 4 𝑦 ′ = −2𝑥 + 4 𝑚⊥ = −2 𝑚 = −2𝑥 + 4 ∴ 𝑚⊥ = −2 eqauting/Vergelyk ∴ −2 = −2𝑥 + 4 𝑥 = 3 ∴ 𝑥=3 𝑦 = −(3)2 + 4(3) = 3 (3; 3) ∴ (3; 3) and/en 𝑚 = −2 ∴ 3 = −2(3) + 𝑐 subst ∴𝑐=9 answer/antwoord ∴ 𝑦 = −2𝑥 + 9 (7) [21] Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Mathematics P1/Wiskunde V1 11 NW/June/Junie 2025 Grade/Graad 12 – Marking Guidelines/Nasienriglyne QUESTION/VRAAG 9 Given/Gegee: 𝑓(𝑥) = −2𝑥 3 + 5𝑥 2 + 4𝑥 − 3 9.1 𝑓(𝑥) = −2𝑥 3 + 5𝑥 2 + 4𝑥 − 3 0 = −(2𝑥 3 − 5𝑥 2 − 4𝑥 + 3) 2𝑥 − 5𝑥 2 − 4𝑥 + 3 = 0 3 (𝑥 + 1)(2𝑥 2 + 𝑘𝑥 + 3) = 0 method to factorise/ 2 2 2 faktoriseer metode 2𝑥 + 𝑘𝑥 = −5𝑥 𝑘𝑥 2 = −7𝑥 2 (𝑥 + 1)(2𝑥 2 − 7𝑥 + 3) = 0 (𝑥 + 1)(𝑥 − 3)(2𝑥 − 1) = 0 1 𝑥 = −1 or/of 𝑥 = 3 or/of 𝑥 = 2 each 𝑥 −value/ elke 𝑥 −waarde (4) 9.2 𝑓(𝑥) = −2𝑥 3 + 5𝑥 2 + 4𝑥 − 3 𝑓 ′ (𝑥) = −6𝑥 2 + 10𝑥 + 4 = 0 𝑓 ′ (𝑥) = 0 3𝑥 2 − 5𝑥 − 2 = 0 (𝑥 − 2)(3𝑥 + 1) = 0 factors/faktore 𝑥 = 2 or/of 𝑥 = − 3 1 𝑥 −values/waardes 𝑓(2) = −2(2)3 + 5(2)2 + 4(2) − 3 =9 1 1 3 1 2 1 𝑓 (− ) = −2 (− ) + 5 (− ) + 4 (− ) − 3 3 3 3 3 100 1 100 =−  𝐵 (− 3 ; − 27 ) 27 1 100 ∴ 𝐵 (− 3 ; − 27 ) and/en 𝐸(2; 9)  𝐸(2; 9) (5) 9.3 𝑓(𝑥) = −2𝑥 3 + 5𝑥 2 + 4𝑥 − 3 𝑓 ′ (𝑥) = −6𝑥 2 + 10𝑥 + 4 𝑓"(𝑥) < 0 𝑓"(𝑥) < 0 ∴ −12𝑥 + 10 < 0 𝑓"(𝑥) = −12𝑥 + 10 −12𝑥 < −10 5 ∴ 𝑥> 5 6 𝑥 >6 (3) Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Mathematics/P1 12 NW/June 2025 Grade/Graad 12– Marking Guidelines/Nasienriglyne 9.4 𝑥. 𝑓 ′ (𝑥) > 0 1 0 < 𝑥 < 2 0 < 𝑥 < 2 or/of 𝑥 < − 3 1 𝑥 < −3 (3) 9.5 2𝑥 3 − 5𝑥 2 − 4𝑥 + 3 = 𝑘 ∴ −𝑓(𝑥) = 𝑘 𝑓(𝑥) = −𝑘 𝑓(𝑥) = −𝑘 100 ∴− < −𝑘 < 9 27 100 notation/notasie ∴ −9 < 𝑘 < values/waardes (3) 27 [18] QUESTION/VRAAG 10 10.1 Q is the maximum t.pt/maksimum dr.pt Q max/maks P is the minimum t.pt/minimum dr.pt P minimum (2) 10.2 𝑥>𝑎 ∴ 𝑓(𝑥) is increasing/stygend increase/stygend (2) 10.3 𝑓 ′ (𝑥) = 3𝑝𝑥 2 + 2𝑞𝑥 + 𝑟 𝑓′(𝑥) ∴ (𝑜; 𝑟) coordinate/koőrdinaat (2) 10.4 If 𝑥 = 𝑏 ⇒ 𝑓 ′ (𝑥) = 0 𝑓 ′ (𝑥) = 0 ∴turning point/draaipunt Q Q t.pt/dr.pt (2) 10.5 𝑎+𝑏 ( ; 0) 2 answer (2) [10] TOTAL/TOTAAL: 150 Copyright reserved/Kopiereg voorbehou