Technical Mathematics P1 Nov 2025 MG Afr & Eng

` NATIONAL SENIOR CERTIFICATE NASIONALE SENIOR SERTIFIKAAT GRADE 12/GRAAD 12 TECHNICAL MATHEMATICS P1/TEGNIESE WISKUNDE V1 NOVEMBER 2025 MARKING GUIDELINES/NASIENRIGLYNE MARKS/PUNTE: 150 MARKING CODES/NASIENKODES A Accuracy/Akkuraatheid AO Answer only/Slegs antwoord CA Consistent accuracy/Volgehoue akkuraatheid F Formula/Formule M Method/Metode R Rounding/Afronding NPR No penalty for rounding/Geen penalisering vir afronding nie NPU No penalty for units omitted/Geen penalisering vir eenhede weggelaat nie S Simplification/Vereenvoudiging SF Substitution in correct formula/Vervanging in korrekte formule These marking guidelines consist of 18 pages. Hierdie nasienriglyne bestaan uit 18 bladsye. Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Technical Mathematics/P1/Tegniese Wiskunde/V1 2 DBE/November 2025 NSC/NSS – Marking Guidelines/Nasienriglyne NOTE: • If a candidate answers a question TWICE, mark only the FIRST attempt. • Consistent accuracy (CA) applies in all aspects of the marking guidelines where indicated. LET WEL: • Indien 'n kandidaat 'n vraag TWEE keer beantwoord, sien slegs die EERSTE poging na. • Volgehoue akkuraatheid (CA) is deurgaans op alle aspekte van die nasienriglyne van toepassing waar aangedui. QUESTION/VRAAG 1 1.1.1  4 2x  x −  = 0  9 ✓0 A 4 4 x = 0 or / of ✓ 9 9 A (2) 1.1.2 6 + ( 2 x − 5)( x + 2 ) = 0 ✓ product/produk A 6 + 2 x 2 − x − 10 = 0 2 x2 − x − 4 = 0 ✓ std form/vorm CA −b  b 2 − 4ac x= 2a − ( −1)  ( −1) − 4(2) ( − 4 ) 2 = ✓ SF CA 2(2)  x  1, 69 or / of x  − 1,19 ✓ both each x – values/ beide x-waardes CA (4) 1.1.3 ( 3 − x )( x + 2 )  0 Critical values/Kritieke waardes: 3 and/en −2 ✓ both critical values/ kritieke waardes A  − 2  x  3 OR/OF x ( −2;3) OR/OF ✓ correct notation/ korrekte notasie x  − 2 and/en x  3 A OR/OF A AO: Full marks/Volpunte (2) Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Technical Mathematics/P1/Tegniese Wiskunde/V1 3 DBE/November 2025 NSC/NSS – Marking Guidelines/Nasienriglyne 1.2.1 y = x −1 ✓ subject/onderwerp A (1) 1.2.2 y − x + 1 = 0 and x2 + x y = 3 x 2 + x ( x −1) = 3 ✓ subst./vervang CA x2 + x2 − x − 3 = 0 2 x2 − x − 3 = 0 ✓ std form/vorm CA ( 2 x − 3)( x + 1) = 0 ✓ Factors/Faktore/SF CA OR/OF ( −1) − 4(2)(−3) 2 −( −1)  x= 2(2)  x= 3 or/of x = − 1 ✓ both x-values/ 2 beide x-waardes CA 3 1  y = −1 = OR/OF y = − 1 − 1 = − 2 2 2 ✓ both y-values/ beide y-waardes OR/OF OR/OF CA x = y +1 ✓ subst./vervang ( y + 1) + y ( y + 1) = 3 2 A 2 y2 + 3y − 2 = 0 ✓ std form/vorm CA ( 2 y − 1)( y + 2 ) = 0 ✓ factors/ faktore/SF OR/OF CA ( 3 ) − 4(2)( −2) 2 − (3)  y= 2(2) 1  y= or/of y = −2 ✓ both y-values/ CA 2 beide y-waardes 1 3  x= +1= or/of x = − 2 + 1 = −1 ✓ both x-values/ CA 2 2 beide x-waardes (5) Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Technical Mathematics/P1/Tegniese Wiskunde/V1 4 DBE/November 2025 NSC/NSS – Marking Guidelines/Nasienriglyne 1.3.1 BP = 2  N T BP N= 2π T A ✓ dividing/deling (1) 1.3.2 BP N= 2π T 117 366,54 = ✓ SF CA 2π ( 560,44 )  33, 33 r / s ✓S CA OR/OF OR/OF BP = 2  N T ✓ SF A 117 366,54 = 2  N ( 560,44 ) N  33, 33 r / s ✓S CA (2) 1.4 81 = 10100012 ✓ 10100012 A (1) 1.5 81  110112 ✓ 27 CA = 81  27 =3 ✓3 CA AO: Full marks/ Volpunte (2) [20] Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Technical Mathematics/P1/Tegniese Wiskunde/V1 5 DBE/November 2025 NSC/NSS – Marking Guidelines/Nasienriglyne QUESTION/VRAAG 2 2.1.1  = b2 − 4ac ✓ discriminant formula/ diskriminant formule A (1) 2.1.2  = b 2 − 4 ac = ( −2 ) − 4 (1)( 2 ) 2 ✓ SF A = −4 ✓ discriminant/diskriminant CA (2) 2.1.3 Roots are non-real OR imaginery/Wortels is nie-reële OF imaginêr ✓ non-real/nie-reële A (1) 2.2 x2 + 2 x − 4 = m x2 + 2 x − 4 − m = 0 ✓ standard form/ standaardvorm CA  = b2 − 4ac ( 2 ) − 4 ( 1)( −4 − m ) = 0 2 ✓ SF CA 20 + 4 m = 0 ✓ =0 A m = −5 ✓ value of/waarde van m CA (4) [8] Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Technical Mathematics/P1/Tegniese Wiskunde/V1 6 DBE/November 2025 NSC/NSS – Marking Guidelines/Nasienriglyne QUESTION/VRAAG 3 3.1.1 3 27 p12 ✓3 A = 3 p4 ✓ p4 A (2) 3.1.2 3  2x 2x+2 − 2x 3  2x 3  2x ✓ product/ A = OR/ OF = 2 x  22 − 2 x 4 . 2x − 2x produk 3  2x 3 . 2x = = ✓ common factor/gemene 2 x ( 4 − 1) 3 . 2x faktor/subtract like terms/ =1 verskil tussen gelyksoortige terme CA ✓S CA (3) 3.2.1 1 ✓S A a2 (1) 3.2.2 1 2 log a a 2 = 2 1 log a a ✓ log law/prop./ 2 log-wet/eienskap A =1 ✓1 CA (2) 3.3.1 log 27 ✓ exp.prop./eksp.eienskap A = log 33 = 3 log 3 = 3q ✓S CA (2) 3.3.2 log 60 = log ( 2  3 10 ) = log 2 + log 3 + log10 ✓ log prop. A ✓ p+q = p + q+ 1 CA ✓1 CA (3) Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Technical Mathematics/P1/Tegniese Wiskunde/V1 7 DBE/November 2025 NSC/NSS – Marking Guidelines/Nasienriglyne 3.4 log 3 x + log 3( x + 2 ) = 1 log 3 x ( x + 2 ) = 1 ✓ log prop./ log-wet A log 3( x + 2 x ) = log 3 3 OR/OF x + 2 x = 3 2 2 1 ✓ log property/ log. eienskap CA x2 + 2 x − 3 = 0 ✓ standard form/ CA ( x + 3)( x −1) = 0 standaardvorm x = − 3 or / of x = 1 x =1 ✓ x −3 CA ✓ x =1 CA (5) 3.5.1 V = r cis  V = 2 cis 120 ✓ sub./ vervang A (1) 3.5.2 V = 2 ( cos120 + i sin120 ) ✓ real CA V = −1 + 3 i ✓ imaginary CA (2) 3.6 a + 7 bi = −21i 2 + 21i a + 7 b i = −21( −1) + 21i ✓ i 2 = −1 A a + 7 b i = 21 + 21 i ✓ a = 21 CA  a = 21 and/en  b = 3 A ✓ b=3 (3) [24] Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Technical Mathematics/P1/Tegniese Wiskunde/V1 8 DBE/November 2025 NSC/NSS – Marking Guidelines/Nasienriglyne QUESTION 4/VRAAG 4 4.1.1 x=0 ✓ x=0 A y=3 ✓ y=3 A (2) 4.1.2 x  R;x  0 ✓ xR ; x0 A OR/OF x  ( −  ; 0)  ( 0 ;  ) (1) 4.1.3 y -int:/ y -afsnit: ✓ y -intercept/ y=3 y afsnit A x -int:/ x -afsnit: 0 = 3x + 3 ✓ x -intercept/  x = −1 A x -afsnit (2) 4.1.4 3 f ( x) = +3 x 3 ✓ f ( x) = 0 A 0= +3 x  x = −1 ✓ x = −1 CA (2) 4.1.5 For/Vir f : ✓ shape of f / vorm van f A ✓ both asymptotes/ beide assimptote CA ✓ intercept of f / CA afsnit van f For/Vir g : ✓ shape of g / vorm van g A ✓ both intercepts of g / beide afsnitte van g CA (5) Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Technical Mathematics/P1/Tegniese Wiskunde/V1 9 DBE/November 2025 NSC/NSS – Marking Guidelines/Nasienriglyne 4.1.6 −1  x  0 ✓ critical values/ kritieke waardes CA OR/OF ✓ notation/notasie A x  −1 and / en x  0 OR/OF x[ − 1; 0) (2) 4.2.1 x = −1 ✓ x = −1 A (1) 4.2.2 B ( 2; 0 ) ✓2 CA ✓0 A (2) 4.2.3 f ( x) = − ( x + 1) + q 2 ✓ subst. the value of p / A vervanging van p Subst: A ( − 4;0 ) : OR/ OF Subst: B ( 2;0 ) : 0 = − ( −4 + 1) + q 2  ✓ substitute A or B/ CA 0 = − ( 2 + 1) + q 2  vervang A of B  0 = −9 + q ✓ value of/ CA waarde van q  q=9 OR/ OF OR/OF ✓ subst. the x-intercepts A f ( x) = − ( x + 4 )( x − 2 ) ✓ subst. the value of p / = − x2 − 2x + 8 vervanging van p f (−1) = − (−1)2 − 2(−1) + 8 OR/OF ✓ value of/ CA 4( −1)(8) − ( −2) 2 waarde van q q= CA 4 ( −1) q = 9 (3) Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Technical Mathematics/P1/Tegniese Wiskunde/V1 10 DBE/November 2025 NSC/NSS – Marking Guidelines/Nasienriglyne 4.2.4 y  9 OR/ OF y  ( −  ; 9] OR/OF ✓ correct interval CA −  y  9 (1) 4.2.5 y=6 ✓ y=6 A (1) 4.2.6 g ( x) = a x + 6 Subst / Vervang C ( −1;9 ) : ✓ substitute C / −1 9= a +6 vervang C CA 1 3= a a= 1 ✓ value of / 3 waarde van a CA x 1 g ( x ) =   + 6 OR / OF g ( x ) = 3− x + 6 3 (2) [24] Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Technical Mathematics/P1/Tegniese Wiskunde/V1 11 DBE/November 2025 NSC/NSS – Marking Guidelines/Nasienriglyne QUESTION/VRAAG 5 5.1.1 A = P (1 − i n) F A (1) 5.1.2 A = P (1 − i n) = 4 990 1 − ( 0, 0589  7 )  ✓ SF A =  R 2 932, 62 ✓S CA (2) 5.2 A = P (1 + i ) n ✓F A = R 32 000 (1 + 0, 0715 ) 4 ✓ SF A  R 42 181,18 ✓S CA (3) 5.3.1 1 ✓ litres/liter A 5 000  = 2 500 litres/liter 2 (1) 5.3.2 A = P (1 − i ) n ✓F A 2 500 = 5 000 (1 − i ) 35 ✓SF A 2 500 = (1 − i ) 35 5 000 2 500 35 = 1− i 5 000 2 500 ✓ make i the subject/ i = 1 − 35 maak i die onderwerp CA 5 000 i = 0, 0196... OR/ OF r  1, 96...% ✓ decimal value of i OR r/ desimale waarde van i OF r CA NPR (4) 5.3.3 A = P (1 − i ) n ✓ n = 60 A = 5 000 (1 − 0, 0196... ) 60 ✓ SF CA  1 524,64 litres/liter ✓S CA  Yes, more than 1 500 litres will be left in the tank/ ✓ conclusion/gevolgtrekking CA Ja, meer as 1 500 liter sal in die tenk oor (4) wees. [15] Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Technical Mathematics/P1/Tegniese Wiskunde/V1 12 DBE/November 2025 NSC/NSS – Marking Guidelines/Nasienriglyne QUESTION/VRAAG 6 6.1 1 f (x) = 4 + x 3 f ( x + h)− f ( x) f / ( x ) = lim ✓ definition/definisie A h→0 h 1  1  4 + ( x + h) −  4 + x  3  3  ✓ SF A = lim h→0 h 1 1 1 4+ x + h − 4− x = lim 3 3 3 h→0 ✓S CA h 1 h = lim 3 ✓S CA h→0 h 1 = lim   h→0 3 1 1 ✓  f ( x) = / 3 CA 3 (5) 6.2.1 y = −3 x6−4 ✓S A y = − 3 x2 (1) 6.2.2 dy = −6 x ✓ −6 x dx CA (1) 6.3.1 D x  5 x 8 − 11 = 40x 7 ✓ 40 x 7 A ✓ 0 (Implied/Aanvaar) A (2) 6.3.2 d  10  −  d x x  = d dx ( − 10 x −1 ) ✓ 10 x −1 A = 10x −2 ✓ 10 x −2 CA (2) Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Technical Mathematics/P1/Tegniese Wiskunde/V1 13 DBE/November 2025 NSC/NSS – Marking Guidelines/Nasienriglyne 6.3.3 4 x 4 −5 f ( x)= − + x 3 5 − 5 − 4x ✓ x 4 =− + x 4 A 3 4 A 4 5 − 9 ✓− f /( x ) = − − x 4 3 3 4 5 − 9 ✓ − x 4 CA 4 (3) 6.4.1 y = 100 + 15 x  m = 15 ✓ m = 15 A (1) 6.4.2 y = 100 + 15 x 25 = 100 + 15 x ✓ subst./vervang y = 25 A  x = −5 ✓ x = −5 CA h / ( x ) = 3 a x 2 +12 x = 15 ✓ derivative/afgeleide A h / ( −5 ) = 3 a ( − 5 ) + 12 ( − 5 ) = 15 2 ✓ h / ( −5 ) = 15 CA 75 a = 75 a = 1 ✓ value of a /waarde van a CA (5) [20] Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Technical Mathematics/P1/Tegniese Wiskunde/V1 14 DBE/November 2025 NSC/NSS – Marking Guidelines/Nasienriglyne QUESTION/VRAAG 7 7.1 y = 20 ✓ y = 20 A (1) 7.2 g (5) = 0 ✓ equating to 0/vervanging en A gelykstel aan 0 a (5)3 − 2(5)2 − 19(5) + 20 = 0 ✓ Substitution/vervanging A 125 a = 125 a =1 (2) 7.3 x − 2 x −19 x + 20 = 0 3 2 ✓M A ( x − 5)( x2 + 3x − 4) = 0 ( x − 5 ) ( x − 1) ( x + 4 ) = 0  x = 5 or / of x = 1 or / of x = − 4 ✓✓✓ x-intercepts/afsnitte CA OR/OF OR/OF ( x + 4)( x 2 − 6 x + 5) = 0 ✓M A ( x + 4) ( x − 1) ( x − 5 ) = 0  x = − 4 or / of x = 1 or / of x = 5 ✓✓✓ x-intercepts/afsnitte CA OR/OF OR/OF ( x − 1)( x 2 − x − 20) = 0 A ✓M ( x − 1) ( x − 5 ) ( x + 4 ) = 0 CA  x = 1 or / of x = 5 or / of x = − 4 ✓✓✓ x-intercepts/afsnitte AO Full marks/Volpunte (4) 7.4 g ( x) = 3x − 4 x − 19 = 0 / 2 ✓ derivative/afgeleide A ✓ equating derivative to 0/ A stel afgeleide gelyk aan 0 − ( − 4)  ( − 4 ) − 4 ( 3)( −19 ) 2 x= ✓ factors/formula/ CA 2 ( 3) faktore/formule x = 3, 27 or / of x = − 1, 94 ✓ both values of/ CA beide waardes van x g ( 3, 27 ) = ( 3, 27 ) − 2 ( 3, 27 ) − 19 ( 3, 27 ) + 20 3 2  − 28,55 g ( −1,94 ) = ( −1,94 ) − 2 ( −1,94 ) − 19 ( −1,94 ) + 20 3 2 ✓ both values of/beide  42, 03 waardes van y CA ( 3, 27 ; − 28,55) or / of ( −1,94 ; 42, 03) (5) Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Technical Mathematics/P1/Tegniese Wiskunde/V1 15 DBE/November 2025 NSC/NSS – Marking Guidelines/Nasienriglyne 7.5 ✓ y-intercept/afsnit CA ✓ all x-intercepts/ alle x-afsnitte CA ✓ both turning CA points/beide draaipunte ✓ shape/vorm CA (4) [16] Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Technical Mathematics/P1/Tegniese Wiskunde/V1 16 DBE/November 2025 NSC/NSS – Marking Guidelines/Nasienriglyne QUESTION/VRAAG 8 8.1 100 ✓ 100 A (1) 8.2 F (5) = 20(5) − ( 5 ) 2 ✓ substitution/vervanging A ✓ F = 75 CA = 75 (2) 8.3 Amount made after 5  t  p / Bedrag na 5  t  p : = 25  R2,50 ✓ 25 CA ✓S CA = R62,50 (2) 8.4 F ( t ) = 20t − t 2 ✓ derivative/afgeleide A F / (t ) = 20 − 2t For/Vir minimum: F / (t ) = 0 ✓ equating derivative to 0/ stel afgeleide gelyk aan 0 A 20 − 2t = 0 20 t= 2 t = 10 min ✓ value of/waarde van t CA  p = 10 min OR/OF OR/OF ✓F A −b t= ✓ SF A 2a −(20) = ✓ answer/ antwoord CA 2(−1) = 10 min  p = 10 min OR/OF OR/OF A 20 t − t 2 = 100 ✓ equating to 100/ stel gelyk aan 100 CA t 2 − 20t + 100 = 0 ✓ factors/ faktore/SF ( t − 10 ) = 0 OR/OF 2 CA − ( − 20 )  ( − 20 ) − 4 (1)(100 ) 2 t= 2 (1) t = 10  p = 10 min ✓ answer/ antwoord (3) [8] Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Technical Mathematics/P1/Tegniese Wiskunde/V1 17 DBE/November 2025 NSC/NSS – Marking Guidelines/Nasienriglyne QUESTION/VRAAG 9 9.1.1  x dx 3 x4 x 4 ✓ A = +C 4 4 A ✓ C (2) 9.1.2  3x 1    2 + x ( x − 2 )  dx OR/OF  2 + x ( x − 2 ) dx 3x −2 2  1 2 ✓S =   23 x + − 2  dx A  x x  1 ✓ − 2 x −2 CA =  23 x + − 2 x −2 dx OR/OF  23 x + x −1 − 2 x −2 dx x 23 x ✓ A 23 x  2 x −1  3ln 2 = + ln x −  + C ✓ + ln x 3 ln 2  −1  CA 23 x ✓ + 2x −1 + C = + ln x + 2 x −1 + C CA 3 ln 2 (5) Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Technical Mathematics/P1/Tegniese Wiskunde/V1 18 DBE/November 2025 NSC/NSS – Marking Guidelines/Nasienriglyne 9.2 f ( x) = −2 x2 + 8x = 0 A: Shaded area from/Skakeerde area van x = 0 to x = 1 1 A A =  ( − 2 x 2 + 8 x ) dx ✓ area notation/ 0 area notasie A 1  2  2 ✓ x3 + 4 x 2 A =  − x3 + 4 x 2   3 0 3  2   2  ✓ SF CA =  − (1)3 + 4 (1) 2  −  − (0)3 + 4 (0) 2   3   3  10 10 = units 2 / eenhede 2 ✓ units 2 / eenhede 2 CA 3 3 B: Unshaded area from/ Onskakeerde area van x = 1 to x = 4 4 A B =  ( − 2 x 2 + 8 x ) dx 1 4  2  =  − x3 + 4 x 2   3 1 CA ✓ SF  2   2  =  − (4)3 + 4 (4) 2  −  − (1)3 + 4 (1) 2   3   3  64 10 = − 3 3 ✓ 18 units 2 / eenhede2 CA = 18 units / eenhede2 2 OR/OF OR/OF A + B: Shaded area/ Skakeerde area van + Unshaded area Onskakeerde area van x = 0 to x = 4 4 A A+B =  ( − 2 x 2 + 8 x ) dx 0 4  2  =  − x3 + 4 x 2   3 0  2  = − ( 4) + 4 ( 4 ) − 0 3 2 ✓ SF CA  3  64 = 3 64 10  AB = − = 18 units 2 / eenhede 2 CA 3 3 ✓ 18 units 2 / eenhede2 Ratio/Verhouding:  10  CA ✓ ratio/verhouding A  3  = = 0,1851 B 18 ✓ conclusion/ The ratio is less than 0, 2 /Die verhouding is minder as 0, 2 . gevolgtrekking CA (8) [15] TOTAL/TOTAAL: 150 Copyright reserved/Kopiereg voorbehou