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Technical Mathematics P1 Nov 2022 MG Afr Eng_hlayiso.com_.pdf

Subject: Technical MathematicsGrade 12202219 pages

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Downloaded from hlayiso.com NATIONAL SENIOR CERTIFICATE NASIONALE SENIOR SERTIFIKAAT GRADE 12/GRAAD 12 TECHNICAL MATHEMATICS P1/TEGNIESE WISKUNDE V1 NOVEMBER 2022 MARKING GUIDELINES/NASIENRIGLYNE MARKS/PUNTE: 150 MARKING CODES/NASIENKODES A Accuracy/Akkuraatheid CA Consistent accuracy/Volgehoue akkuraatheid 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 PR Penalty for rounding/Penalisasie vir afronding 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 Technical Mathematics/P1/Tegniese Wiskunde/V1 2 DBE/November 2022 NSC/NSS – Marking Guidelines/Nasienriglyne NOTE:  If a candidate answers a question TWICE, only mark the FIRST attempt.  Consistent accuracy marking to be applied where indicated.  # Shows questions where Tolerance Range will be applied: Q 3.3 ; Q 5.3 ; Q 6.2.3 ; Q 9.1.3 LET WEL:  Indien 'n kandidaat 'n vraag TWEE keer beantwoord, sien slegs die EERSTE poging na.  Volgehoue akkuraatheidnasien moet toegepas word soos aangedui.  # Toon vrae waar Toleransie Wydte (Verdraagsaamheids omvang) toegepas word: V 3.3 ; V 5.3 ; V 6.2.3 ; V 9.1.3 QUESTION/VRAAG 1 1.1.1 x 7  x   0  x 0 A x 0 or/of x  7 x  7 A (2) 1.1.2 4 x 2  5x  4  0  b  b 2  4ac x 2a   5   52  4(4)(4)   SF A 2(4) OR/OF  5  52  4( 4)(4)  positive x- value / CA  positiewe x- waarde 2( 4)  negative x- value / negatiewe x- waarde CA  x  1, 80 or / of x   0 , 55 NPR AO full marks /volpunte (3) 1.1.3 2x  8  0 2 2 ( x  4)  0 2 2 ( x  2) ( x  2)  0 OR/OF  factors/faktore A Formula / Formule ( x  2) ( x  2)  0 Critical values/kritiese waardes: 2 and/en  2 Both critical values/ CA  x  2 or/of x  2 beide kritiese waardes OR/OF A x  ( ;  2 ) or/of x  (2 ; )  notation/notasie OR/OF AO full marks /volpunte (3) Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Downloaded from hlayiso.com Technical Mathematics/P1/Tegniese Wiskunde/V1 3 DBE/November 2022 NSC/NSS – Marking Guidelines/Nasienriglyne 1.2 y  5x  2 and/en y  x 2  4 x  8 x 2  4 x  8  5x  2  equating/gelykstel A x2  x  6  0  standard form/ standaardvorm CA ( x  3) ( x  2)  0 OR/OF  factors or formula/ faktore of formule CA  x3 or/of x  2  both x-values/ beide x- waardes CA OR/OF y  53  2 or/of OR / OF y  3  43  8 or/of 2 y  5 2  2 y   2  4 2  8 2  y  13 or/of y   12 both y-values/ beide y-waardes CA OR/OF OR/OF y2 x 5  y  2  y  2 2 y  4 8  5   5   substitution/ vervanging A y2  4y  4 4y  8 y  8 25 5 25 y  y 2  4 y  4  20 y  40  200 y 2  y  156  0  standard vorm/ standaardvorm CA ( y  13) ( y  12)  0 OR/OF  factors or formula/ faktore of formule CA  y  13 or / of y  12 both y-value/ 13  2  12  2 beide y-waardes CA x or / of x  5 5 x  3 and / en x   2 both x values/ beide x-waardes CA (5) Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Downloaded from hlayiso.com Technical Mathematics/P1/Tegniese Wiskunde/V1 4 DBE/November 2022 NSC/NSS – Marking Guidelines/Nasienriglyne  d2 L 1.3.1 SV  4 4 SV   d 2  L M A 4 SV  L OR/OF  L subject/L-onderwerp CA d2 AO full marks /volpunte (2) 4 SV 1.3.2 L d 2 4  1020 , 5   SF CA   102  13 cm  value of L/ CA waarde v L OR/OF OR/OF SV L   d2 4 1020,5  SF  CA  10  2 4  13 cm value of L/ CA waarde v L OR/OF OR/OF  d2 L SV  4  102  L 1020,5  4  SF A L  13 cm value of L/ CA waarde v L PR in this question only/ slegs in hierdie vraag (2) Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Downloaded from hlayiso.com Technical Mathematics/P1/Tegniese Wiskunde/V1 5 DBE/November 2022 NSC/NSS – Marking Guidelines/Nasienriglyne 1.4.1 25 24 23 22 21 20 P 1 0 1 0 =10  10 A (1) 1.4.2 1010 2  10000 2  10100000 2  A AO full marks /volpunte OR/OF 25 24 23 22 21 20  16 Q 1 0 0 0 0 =16 A  101000002 P  Q  10  16  160 = 10100000 2 CA 27 26 25 24 23 22 21 20 1 0 1 0 0 0 0 0 (2) [20] QUESTION/VRAAG 2 2.1.1 x 2  2x  6  0   b 2  4ac   2   4 16  2  SF A   20 S CA AO full marks /volpunte (2) 2.1.2 Non-real/Imaginary/Nie-reël/imaginêr  non-real/imaginary/ CA nie-reël/imaginêr (1) 2.2 x 2xk 0 2   2  4  1k   SF A 2 22  4  1k   0    0 A 4  4k  0 k 1 value(s) of k/ CA waarde(s) van k (3) [6] Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Downloaded from hlayiso.com Technical Mathematics/P1/Tegniese Wiskunde/V1 6 DBE/November 2022 NSC/NSS – Marking Guidelines/Nasienriglyne QUESTION/VRAAG 3 8 x3 y 2 3.1.1 16 xy 4 x2 1 2 A =  x 2 or/of x 2y2 2  2 y 2 or/of y2 A (2) 48  12 3.1.2 27 4 32 3 22 3  2 3  OR/OF  OR/OF simplified surd forms/ A 3 3 3 3 vereenvoudigde 1 wortelvorme 6 3 3 (4  2) 3 (4  2) 2 S CA    1 3 3 3 3 2 3. 3 2 S CA OR/OF OR/OF 48  12 27 4 3 2 3 simplified surd forms/   vereenvoudigde 3 3 3 3 wortelvorme A 6  3 S CA 2 S CA AO one mark/slegs een punt (3) 3.2.1 log 25  log 5 2 OR/OF 2 log 5 OR/OF log 5  log 5  log(exp) property/- A log(eksp,)eienskap  2m S CA AO full marks /volpunte (2) 3.2.2 log 2  10  M  log   OR/OF  log 2  log 5  log 5 A 5 log property/ eienskap  log 10  log 5 A S 1  m CA AO full marks /volpunte (3) Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Downloaded from hlayiso.com Technical Mathematics/P1/Tegniese Wiskunde/V1 7 DBE/November 2022 NSC/NSS – Marking Guidelines/Nasienriglyne 3.3 log 2 ( x  3)  3   log 2 ( x  4 ) log 2 ( x  3)  log 2 ( x  4 )  3 log 2 ( x  3) ( x  4 )  3  log property/-eienskap A ( x  3)( x  4 )  23  8  exponential form/ CA eksponentvorm x  x 12  8 2 x 2  x  20  0  standard form/ CA ( x  4) ( x  5 )  0 OR/OF standaardvorm  (1)   12  4(1)(20)  factors or formula/ CA x faktore of formule 2(1)  correct value of/ CA  x  5 or/of x   4 korrek waarde van x OR/OF OR/OF log 2 ( x  3)  3   log 2 ( x  4 )  log property/-eienskap A log 2 ( x  3)  3 log 2 2  log 2 ( x  4 ) 1 log 2 ( x  3)  log 2 8  log 2 ( x  4 ) 1 ( x  3) 1 log 2  log 2  log property/-eienskap CA 8 x  4 x3 1  8 x4 x  3x  4  1  8 x 2  x 12  8  standard form/ x 2  x  20  0 standaardvorm CA ( x  4) ( x  5 )  0 OR/OF  (1)   12  4(1)(20)  factors or formula/ CA x faktore of formule 2(1)  x  5 or/of x   4  correct value of/ CA korrek waarde van x OR/OF OR/OF Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Downloaded from hlayiso.com Technical Mathematics/P1/Tegniese Wiskunde/V1 8 DBE/November 2022 NSC/NSS – Marking Guidelines/Nasienriglyne log 2 ( x  3)  3  log 2 ( x  4 ) log 2 ( x  3)  log 2 8  log 2 ( x  4 )  log property/-eienskap A 8 log 2 ( x  3)  log 2  log property/-eienskap  x  4 CA x3 8  1 x4 x  3x  4  8 x 2  x 12  8 x 2  x  20  0  standard form/ CA ( x  4) ( x  5 )  0 OR/OF standaardvorm  factors or formula/ CA faktore of formule  correct value of/  x  5 or/of x   4 CA korrek waarde van x (5) 3.4.1 z   1  3i conjugate/toegevoegde A (1) 3.4.2 z 2  2 cis 135°  2 cos 135° + i 2 sin 135°  expansion/uitbreiding A  1  i  1  i A AO full marks /volpunte (2) 3.4.3 z1  z2   1  3 i   1 i  OR/OF substitution/vervanging CA  2i S CA AO full marks /volpunte (2) 3.5 x  yi  (1  i)  4  5i x  yi 1  i  4  5i S A x  y i  5  4i S CA  x  5 and/ en y  4  value of x/waarde van x CA  value of y/waarde van y OR/OF OR/OF CA x  yi  (1  i)  4  5i S x  yi 1  i  4  5i A S CA ( x  1)  ( y  1) i  4  5i x 1  4 and / en y 1  5  value of x/waarde van x  value of y/waarde van y CA CA (4) [24] Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Downloaded from hlayiso.com Technical Mathematics/P1/Tegniese Wiskunde/V1 9 DBE/November 2022 NSC/NSS – Marking Guidelines/Nasienriglyne QUESTION/VRAAG 4 4.1.1(a) y   9 OR/OF y  [9 ;  ) OR/OF  9  y    y   9 / y  [9 ;  ) / A  9 y  (1) 4.1.1(b) Q (4 ;  5)  x4 A  y  5 A (2) 4.1.2(a) f ( x )  x2  4 x  5 x  int s. / afsn. : x2  4 x  5  0  y0 A ( x  1) ( x  5)  0 OR/OF factors/form/   4   42  41 5 faktore/formule A x 21  both values of x/ x   1 or / of x 5 beide waardes van x CA AO full marks /volpunte (3) 4.1.2(b) The length of/lengte van AB  6 units/eenhede 6 CA (1) 4.1.3 g ( x )  mx  c through/deur A  1; 0 ; Q (4;5) y y 50  SF m 1 2  CA x1  x2 4  (1)  1  m  1m  1 CA Must be negative/ g ( x )   1x  c OR/OF y  y1  1( x  x1 ) Moet negatief wees subst./ verv. : (1; 0) 0   1(1)  c y  0  1( x  1) c  1 OR/OF subst./ verv. : (4 ;  5)  5   1(4)  c OR/OF y   5  1( x  4)  c  1 CA c  1 (3) Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Downloaded from hlayiso.com Technical Mathematics/P1/Tegniese Wiskunde/V1 10 DBE/November 2022 NSC/NSS – Marking Guidelines/Nasienriglyne 4.1.4 x  5 ; x  1  x5 CA  x  1 CA OR/OF OR/OF x  ( ; 5) ; x   1  ( ; 5) CA  x  1 CA OR/OF OR/OF x  ( ;  1) or / of (1 ; 5)  ( ;  1) CA  (1 ; 5) CA (2) 4.2.1  13  x  13 OR/OF x    13 ; 13   critical values/ A kritiese waardes  notation/notasie A OR/OF  3 , 61  x  3 , 61 OR/OF x   3 , 61 ; 3 , 61  OR/OF x  3 , 61 and / en x  3 , 61 OR/OF x   13 and / en x  13 (2) 4.2.2(a) x  0 ; y 1  x0 A  y 1 A (2) 3 4.2.2(b) 0 1  y0 A x 3 1  x x  3 value of x/waarde van x A AO full marks /volpunte (2) Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Downloaded from hlayiso.com Technical Mathematics/P1/Tegniese Wiskunde/V1 11 DBE/November 2022 NSC/NSS – Marking Guidelines/Nasienriglyne 4.2.3 k:  shape/vorm A  x-int./afsn. CA  CA asymptote/asimptote h:  shape/vorm A  both/beide x- CA int./afsn. (5) 4.3  shape/vorm A  (0;0) A  A asymptote/asimptoot (3) [26] Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Downloaded from hlayiso.com Technical Mathematics/P1/Tegniese Wiskunde/V1 12 DBE/November 2022 NSC/NSS – Marking Guidelines/Nasienriglyne QUESTION/VRAAG 5 5.1 A  P 1  i  F A n A  R 8 000 1  13%  SF 3 S CA  R 11 543 ,18 (3) NPR 5.2.1 10 621  10 621 A (1) 5.2.2 g  g A (1) n F 5.2.3 A  P 1  i  A  SF A 3459  10 6211  0,128  n 3459 S CA   0,872  n 10 621  3 459  log   10 621  3459 CA n OR/OF n  log 0,872 log( 0,872) 10 621  log form/-vorm CA  n  8,1905 years/jaar  value of/waarde van n (5) NPR 5.3 Value of investment end of 24 months/ # n  6%  2  12 Waarde van belegging einde van 24 maande: A  P 1  i   R20000 1    12   R 22 543,19552  SF A Invested/Belê R5 000 R22 543, 19552  R 5 000  R27 543,19552 Value of investment end of 5 years/ Waarde van belegging einde van 5 jaar: M + R5 000 A  5%  3  2 A  P1  i   27543,19552 1  n   2   R31 941,66  R31 941 , 66  R 35 0 00  SF CA He will NOT have enough money/Hy sal NIE  R 31 941, 66 CA genoeg geld hȇ NIE. \ OR/OF conclusion/gevolgtrekking CA OR/OF Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Downloaded from hlayiso.com Technical Mathematics/P1/Tegniese Wiskunde/V1 13 DBE/November 2022 NSC/NSS – Marking Guidelines/Nasienriglyne  6%  212  5%  2  3  6%  2 12 A  R20 000 1   1    1    12  A  12   2   5%  2  3  5%  2  3  R5 000 1   1   A   2   2   M + R5 000 A  R31 941, 66  R31 941, 66 CA  R 31 941 , 66  R 35 000 He will NOT have enough money/Hy sal NIE conclusion/gevolgtrekking CA genoeg geld hȇ NIE. OR/OF OR/OF R35 000 P   6%  212  5%  23  6%  212 1   1    1   A  12   2   12  5000  5%  23   1    6%  212  2  A 1    12   M  R5 000 A  R 22 339, 68  R 22 339, 68 CA  R 20 000  R 22 339, 68 He will NOT have enough money/Hy sal NIE conclusion/gevolgtrekking genoeg geld hȇ NIE. CA NPR (5) [15] Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Downloaded from hlayiso.com Technical Mathematics/P1/Tegniese Wiskunde/V1 14 DBE/November 2022 NSC/NSS – Marking Guidelines/Nasienriglyne QUESTION/VRAAG 6 6.1 f ( x )  5  8x f ( x  h)  f ( x) f / ( x )  lim  definition/definisie A h0 h 5  8( x  h )  ( 5  8 x)  SF A  lim Penalty of one mark for h0 h incorrect notation 5  8 x  8h  5  8 x S CA  lim Penaliseer een punt h0 h indien notasie foutief is.  8h  lim S CA h0 h AO only one mark/slegs  lim (  8) h0 een punt f / ( x)   8 –8 CA (5) 6.2.1 f ( x )  3 x5   x  15 x 4 A f / ( x )  15 x 4     A (2) 6.2.2  y  x 2 4 x  2 x 1  y  4 x3  2 x  S A dy  12x 2 CA 12 x 2  2  2 CA dx (3) 6.2.3  2  4 # D x  5 x 4  2  8t 4 x   x5 A  5x  2 A   x 2  4  5  2 2 4   Dx x  x 5  8t x 4 5 1  5   x CA   5 1 4 4   x 3 CA  x 5  x 3  8t 4 4 5 5 5  8t 4 CA (5) 6.3.1 g /  x   12 x  3  derivative of/afgeleide van g A 12  p   3   21  equat.deriv/afgeleide = – 21 A 12 p   24 p  2  p 2 CA (3) 6.3.2 y  6  2  3  2 2  y coordinate/y-koördinaat CA  y  18 y m xc OR/OF y  y1  m x  x1  18   21(2)  c y  18   21x  2  SF CA c   24 y  18   21 42  y   21 x  24  eqn. of tangent/vergelyk. van CA raaklyn (3) [21] Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Downloaded from hlayiso.com Technical Mathematics/P1/Tegniese Wiskunde/V1 15 DBE/November 2022 NSC/NSS – Marking Guidelines/Nasienriglyne QUESTION/VRAAG 7 7.1 OA = 1 unit/eenheid length/lengte A (1) 7.2 0  1  3 1  9 1  k 3 2  subst./verv. A 1; 0 A k  5 (1) 7.3 x - intercepts/afsnitte ; y  0 ( x  1) ( x  1 ) ( x  5 )  0  ( x  1) A  ( x  1) ( x  5 ) A  x- coordinates of B/  B   5 ; 0 koördinate van B CA  y-coordinates of B/ koördinate van B A OR/OF OR/OF ( x  1)( x 2  4 x  5)  0  linear factor/ A OR/OF  quadratic factor/ ( x  5)( x 2  2 x  1)  0 A ( x  1) ( x  1 ) ( x  5 )  0  x- coordinates of B/ CA  x  1 or/of x   5 koördinate van B  y-coordinates of B/ A  B   5 ; 0 koördinate van B AO full marks /volpunte (4) 7.4 f ( x)  3 x  6 x  9  0 / 2 derivative/afgeleide A equating derivative to 0/ A x 2  2x  3  0 stel afgeleide gelyk aan 0 factors/formula/faktore/ CA ( x  3) ( x  1)  0 OR/OF formule  x   3 or/of x  1 x value of D/ CA x-waarde van D f  3    3  3  3  9  3  5 3 2  32  y value of D/ CA  D  3 ; 32 waarde van D AO two marks/twee punte (5) Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Downloaded from hlayiso.com Technical Mathematics/P1/Tegniese Wiskunde/V1 16 DBE/November 2022 NSC/NSS – Marking Guidelines/Nasienriglyne 7.5 3 x 1 OR/OF x  3 ; 1  critical values/ kritiese waardes CA OR/OF  notation/notasie A x   3 and/en x 1 (2) 7.6  1 ;  2  x coordinate/ A -koördinaat y coordinate/ A -koördinaat (2) [15] Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Downloaded from hlayiso.com Technical Mathematics/P1/Tegniese Wiskunde/V1 17 DBE/November 2022 NSC/NSS – Marking Guidelines/Nasienriglyne QUESTION/VRAAG 8 8.1 37,5  C  37,5  C A NPU (1) 8.2 T / (t )  7  t  derivative/afgeleide A T / (4)  7  4  Subst./verv. t = 4 A  3 oC/ s  3 oC / s CA NPU (3) 8.3 7  t  0 OR/OF  7   7t 0 / t  A  1 2   2  t  7s  t  7s CA T ( 7 )  37,5  7 ( 7 )  0,5( 7 ) 2  62 o C  62 o C CA NPU (3) 8.4 7t 0 t 7 t  7 CA 7  t  10 OR/OF t  (7 ;10]  restricting to 10/ beperking tot 10 A OR/OF t  7 and / en t  10 AO full marks /volpunte (2) [9] Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Downloaded from hlayiso.com Technical Mathematics/P1/Tegniese Wiskunde/V1 18 DBE/November 2022 NSC/NSS – Marking Guidelines/Nasienriglyne QUESTION/VRAAG 9  3 ln x  3x 9.1.1 1 A dx  C A = 3 ln x  C (2)  4  2  dx 9.1.2 x 2 x 2 x  4x A  4x   C OR/OF 4x  C ln 2 1 ln 2 x x 1 1 value of 2nd integral/ A     OR/OF  4 x    4x    2 2 waarde van die tweede  C OR/OF C ln 2 1 integraal ln 2 If C is omitted : NO penalty/ Indien C weggelaat is : GEEN penalisering (2) 9.1.3 8x 4  x 2 #  2 x dx  1     4 x3  x  dx  S A  2   x4 CA 1 2 1  x4  x C  x 2 CA 4 4 No simplification: 0 marks/ Geen vereenvoudiging: 0 punte No penalty if C omitted/Geen penalisering indien C uitgelaat (3) 4   x 2  2 x  8 dx Area notation using 4 9.2 A   h( x) dx  integrals/Oppervlakte- 2 2 A 4 notasie met gebruik van  x3  integrale    x2  8x   3  2 x3    x 2  8x A 3  (4)3   ( 2)3     (4) 2  8 (4)    ( 2) 2  8 ( 2) CA  3   3   SF 28  area CA   9, 33 3 /oppervlakte  28     % calculation /  36      100 %  25,93 % OR/OF 20%  36     7,2 3 berekening CA 36  5 25,93 % > 20 % 7,2  9,33  The learner's statement is NOT correct/ conclusion/ Die leerder se bewering is NIE korrek nie. Afleiding. CA Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Downloaded from hlayiso.com Technical Mathematics/P1/Tegniese Wiskunde/V1 19 DBE/November 2022 NSC/NSS – Marking Guidelines/Nasienriglyne OR/OF OR/OF 2   x 2  2 x  8 dx 2 A  36   h( x) dx  36   Area notation using 2 2 intergrals/ Oppervlakte - A notasie met gebruik van 2 integrale  x3   36     x  8 x  2 x3 A    x 2  8x  3  2 3  (2) 3   (  2) 3   SF CA  36    (2) 2  8 (2)    (  2) 2  8 ( 2)  3   3  28   9, 33  area/oppervlakte CA 3  28     36   % calculation /     100 %  25,93 % OR/OF 20%  36     7,2 3 CA berekening 36  5 25,93 % > 20 % 7,2  9,33  The learner's statement is NOT correct/ CA Die leerder se bewering is NIE korrek nie. conclusion/ afleiding (7) [14] TOTAL/TOTAAL: 150 Copyright reserved/Kopiereg voorbehou