Mathematics Grade 12 Paper 1 NSC June 2021 Memo

Downloaded from hlayiso.com SENIOR CERTIFICATE EXAMINATIONS/ SENIORSERTIFIKAAT-EKSAMEN NATIONAL SENIOR CERTIFICATE EXAMINATIONS/ NASIONALE SENIORSERTIFIKAAT-EKSAMEN MATHEMATICS P1/ WISKUNDE V1 MARKING GUIDELINES/NASIENRIGLYNE 2021 MARKS: 150 PUNTE: 150 These marking guidelines consist of 13 pages. Hierdie nasienriglyne bestaan uit 13 bladsye. Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Downloaded from hlayiso.com Mathematics P1/Wiskunde V1 2 DBE/2021 SC/SS/NSC/NSS – Marking Guidelines/Nasienriglyne NOTE:  If a candidate answers a question TWICE, only mark the FIRST attempt.  Consistent Accuracy applies in all aspects of the marking memorandum. LET WEL:  Indien 'n kandidaat 'n vraag TWEE keer beantwoord, merk slegs die EERSTE poging.  Volgehoue akkuraatheid is DEURGAANS op ALLE aspekte van die memorandum van toepassing. QUESTION/VRAAG 1 1.1.1 x 2  x  20  0 x  5x  4  0 factors x=5 x=5 or x = –4  x = –4 (3) 1.1.2 3x  2 x  6  0 2   2   22  43 6 substitution x 23 1  19 x simplification 3  x = –1,12 x = –1,12 or x = 1,79  x = 1,79 (4) 1.1.3 ( x  1) 2  9 standard form x 2  2x  8  0 ( x  4)( x  2)  0 Critical values: x = 4 or x=–2 critical values –2 4 x  2 or x  4  x  2 or x  4 (4) OR/OF OR/OF ( ;  2) or (4; )  ( ;  2) or (4; ) OR/OF OR/OF x  1  3 or x  1  3  x 1  3 x4 or x  2  x  1  3  x  4 or x  2 (4) Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Downloaded from hlayiso.com Mathematics P1/Wiskunde V1 3 DBE/2021 SC/SS/NSC/NSS – Marking Guidelines/Nasienriglyne 1.1.4 2 x6 2 x 2 x6  x2  isolating the surd 4( x  6)  ( x  2) 2  4 x  24  x 2  4 x  4 4 x  24  x 2  4 x  4 x 2  8x  20  0 ( x  10)( x  2)  0  x  10 x  10 or x  2  x  2 (4) 1.2 4 x  2  y . . . (1)  4x  2  y 4 x  y 2  8 . . . (2) substitution 2  y  y2  8 standard form y2  y  6  0 ( y  3)( y  2)  0 y = 3 or y = –2 y-values 1 x-values x or x = 1 4 (5) OR/OF OR/OF y   4 x  2 . . . (1)  y   4x  2 4 x  y 2  8 . . . (2) 4 x   4 x  2  8 2 substitution 4 x  16 x 2  16 x  4  8  0 16 x 2  12 x  4  0 4 x 2  3x  1  0 4x  1x  1  0 standard form 1 x or x = 1 x-values 4 y = 3 or y = –2 y-values (5) 1.3  2 x  3 y  23  3  6  2 3 3 2 x  3 y  218  36  218 or 3 6 2 2 x 18 and 3  3 y 6  x = 18 or y = 6 x = 18 and y=6  x  y  18  6 x  y  12  answer (A) (4) [24] Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Downloaded from hlayiso.com Mathematics P1/Wiskunde V1 4 DBE/2021 SC/SS/NSC/NSS – Marking Guidelines/Nasienriglyne QUESTION/VRAAG 2 2.1.1 72 ; 100 ; 120 ; 132 ; … 28 20 12 –8 –8  second differences = – 8 2a = – 8 a=–4 a=–4 3a + b = 28  b = 40 b = 40 a + b + c = 72 c = 36  c = 36 Tn   4n2  40n  36 (4) 2.1.2 T12  36  8n  36  8(12)  60  36 – 8n  4n 2  40n  36  60  – 60  standard form n 2  10n  24  0 (n  12)(n  2)  0 factors n  12  n  12 (5) 2.1.3 Tn  8n  40  0 /  8n  40 n  5 n5 Tn  4(5) 2  40(5)  36  136 136 (3) OR/OF OR/OF  b 40 substitution  5 2a 8 n  5 Tn  4(5)2  40(5)  36 136  136 (3) 2.1.4 Maximum value = 41 value (1) 2.2 2 sin 3x   11  15  2 sin 3x equating 4 sin 3x  4  4 sin 3x  4 sin 3x  1  sin 3x  1 3x  90 answer x  30 (4) [17] Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Downloaded from hlayiso.com Mathematics P1/Wiskunde V1 5 DBE/2021 SC/SS/NSC/NSS – Marking Guidelines/Nasienriglyne QUESTION/VRAAG 3 3.1.1 n 1 n 1 1 1 Tn  ar n 1  2000   2000  (1) 5 5 3.1.2 7 1 16 1 16  (1) T7  2000   125 5 125 3.1.3 n 1 16 1  2000   equating 15625 5 n 1 1 1   1953125  5  9 n 1 1 1 1  same base / use of log     OR n  1  log 1 5 5 5 1953125 n–1=9 n = 10  answer (3) 3.2 a a S   27   S   27  1 r 1 r a(1  r 3 ) a(1  r 3 ) S3   26  S3   26 1 r 1 r 27(1  r 3 )  26 substitution 26 1  r3  27 1 r3  27 1 1 r  r  3 3 (4) OR/OF OR/OF a S   27  1 r a  27(1  r )  a  27(1  r ) But a  ar  ar 2  26  a  ar  ar 2  26 a(1  r  r 2 )  26 substitution 27(1  r )(1  r  r 2 )  26 26 (1  r )(1  r  r 2 )  27 26 r 2  r 1 r3  r 2  r  27 26  r3 1  27 1 r3  27 1 r  1 r  (4) 3 3 [9] Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Downloaded from hlayiso.com Mathematics P1/Wiskunde V1 6 DBE/2021 SC/SS/NSC/NSS – Marking Guidelines/Nasienriglyne QUESTION/VRAAG 4 4.1 x  1  x  7  x  1  x  7 2 x  8  2 x  8 x  4  x  4  y  3  y  3 2  f ( x)  3 x4  p  4 and q  3 (4) OR/OF OR/OF p + q = 1 ……(1) p+q=1 –p + q = –7 q = p – 7 …….(2) q=p–7 subs. (2) into (1) p+p–7=1  substitution 2p = 8  simplification p=4 q = –3 (4) 4.2 2 y 3 x4 2 0 3 y=0 x4  2  3( x  4)  0  3x  14  0 14 14 x    x (2) 3 3 4.3  horizontal asymptote  vertical asymptote –4,67  y intercept –4 O  shape –3 –3,5 (4) [10] Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Downloaded from hlayiso.com Mathematics P1/Wiskunde V1 7 DBE/2021 SC/SS/NSC/NSS – Marking Guidelines/Nasienriglyne QUESTION/VRAAG 5 5.1  2 x 2  4 x  16  0 x 2  2x  8  0  factors ( x  4)( x  2)  0 x = 4 or x = – 2 x=–2 x=4  A(2 ; 0) and B(4 ; 0) (3) 5.2 f ( x)  2 x 2  4 x  16 b 4   1 1 2a  2(2) f (1)  2(1) 2  4(1)  16  18  C(1;18)  18 (2) OR/OF OR/OF f ( x)  2 x 2  4 x  16 f / ( x)  4 x  4  4x  4  0 x 1 1 f (1)  2(1) 2  4(1)  16  18  18  C(1;18) (2) 5.3 y  18  y  18 (1) OR/OF OR/OF y( ;18]  y( ;18] (1) 5.4 TP (1 ; 18) for f TP (2 ; 15) for h  TP for h at (2 ; 15)  p  1 q  3  p  1  q  3 (3) 5.5 y  2x  4 x  2y  4 swop x and y 1 1 y  x2  y  x2 2 2 (2) 1 5.6 g ( x)  0 or g ( x)  0 x = 4 or x = –2 (product 0 at x-intercepts) x=4  x = –2 (2) Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Downloaded from hlayiso.com Mathematics P1/Wiskunde V1 8 DBE/2021 SC/SS/NSC/NSS – Marking Guidelines/Nasienriglyne 5.7  2 x 2  4 x  16  k  2 x  4 equating  2 x 2  2 x  12  k  0  standard form b 2  4ac  0  b 2  4ac  0  substitution (2) 2  4(2)(12  k )  0 4  8(12  k )  0 100  8k  0 k  12,5  answer (5) OR/OF OR/OF g ( x)  2 /  g / ( x)  2 f / ( x)  4 x  4  2  f / ( x)  4 x  4 1 x 2 1 1 f    17,5  f    17,5 2 2 1 1 g   5  g   5 2 2 k  12,5  answer (5) [18] QUESTION/VRAAG 6 6.1.1 y  3x  swop x and y x  3y y  log 3 x  equation (2) x 4 6.1.2 h( x)  3  2 Transformation: 4 units left, 2 units down  x = 2 (A) P / (2 ; 9)  y = 9 (A) (2) x p 6.2 f ( x)  2  q q  16  q  16 16  2 p 3  16  substitute (3 ; 16) 2 p 3  32 2 p 3  2 5  2 p3  25 or p  3  log2 32  p35 p2  p2 (4) [8] Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Downloaded from hlayiso.com Mathematics P1/Wiskunde V1 9 DBE/2021 SC/SS/NSC/NSS – Marking Guidelines/Nasienriglyne QUESTION/VRAAG 7 7.1  i 16 13 080  10 0001   substitution into correct  4 16 formula  i 13080 n = 16 1     4 10000 i 16 13080  simplification 1  4 10000 i  0,0169227... 4 i  0,06769... i  6,77% answer (A) (4) 7.2.1 F  x 1  i   1 n  i  0,075  60  0,075 9 0001    1   12   12 F 0,075  substitution into correct 12 Formula F  R 652 743,95  answer (3) 7.2.2 60  9 000  R540 000  60  9 000  R540 000 A  P(1  i) n n  0,075  652743,95 1    190 214,14  540000  12   equation n  0, 075  730 214,14  652 743,95 1    12  1,1186....  1, 00625 n simplification n  log1,00625 1,1186  use of logs  n  18 months  18 months (6) Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Downloaded from hlayiso.com Mathematics P1/Wiskunde V1 10 DBE/2021 SC/SS/NSC/NSS – Marking Guidelines/Nasienriglyne OR/OF OR/OF Interest over 5 years = 652 743,95 – 9 000 × 60  60 × 9 000 = 112 743,95  interest on n years = 190 214,14 – 112 743, 95 = 77 470,19  answer n  0,075  equating 652 743,95 + 77 470,19 = 652743,95 1    12  1,1186....  1, 00625 simplification n n  log1,00625 1,1186  use of logs  n  18 months  18 months (6) [13] QUESTION/VRAAG 8 8.1 f ( x)  3 x 2 f ( x  h)  f ( x ) f / ( x)  lim h 0 h 3( x  h)2  3x 2  substitution f / ( x)  lim h 0 h 3x 2  6 xh  3h 2  3x 2 f / ( x)  lim  expansion h 0 h 6 xh  3h 2 = lim h 0 h  simplification h(6 x  3h) h(6 x  3h) = lim  lim h 0 h 0 h h = 6x 6x (5) 2 2 8.2.1 f ( x)  x  3  9 x 2  9x f / ( x)  2 x  18x 3 2x   18 x 3 (3) Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Downloaded from hlayiso.com Mathematics P1/Wiskunde V1 11 DBE/2021 SC/SS/NSC/NSS – Marking Guidelines/Nasienriglyne 8.2.2 g ( x)  ( x  3)( x  1) 1 1 x  2x 2 g ( x)  x  2 x  3 2 1   1 g / ( x)  1  x 2 1  x 2 (4) [12] QUESTION/VRAAG 9 9.1 f / ( x)  6 x 2  6 x  12  6 x 2  6 x  12 6 x 2  6 x  12  0 =0 x2  x  2  0 factors ( x  2)( x  1)  0 x  2 or x 1  x -values y = 20 or y=–7  y -values  A(–2 ; 20) and B(1 ; –7) (5) 9.2 f ( x)  12 x  6 //  12 x  6 12 x  6  0  f / / ( x)  0 12 x  6 1 x x   1 2 2 (3) OR/OF  2 1 1 OR/OF x  1 2 2 x   1 2 x   2 1  x   2 (3) 9.3 f (2)  24 /  f (2)/ Equation of the tangent: y  4  24( x  2)  24 y  24 x  44  equation (3) [11] Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Downloaded from hlayiso.com Mathematics P1/Wiskunde V1 12 DBE/2021 SC/SS/NSC/NSS – Marking Guidelines/Nasienriglyne QUESTION/VRAAG 10 y 10.1  x = –1 and x = 2  TP at x = –1 x 1     TP at x = 1  shape (4) 10.2.1 A B O 1 Area of segment = Area of big circle 4 1 =  (x  x 2 )2  1  (x  x2 )2 4 4 A B O Area triangle ABO counted 1 = Area   x  x 2 2   Area   1 2  x  x2 2  2 Area of shaded region  subtract areas 1 1  =  (x  x 2 )2 – x  x 2 4 2 2   2 = 4  x  x2 2   common factor (5)   2 2    x  2x  x 3 4   4  Copyright reserved/Kopiereg voorbehou Please turn over/Blaai om asseblief

Downloaded from hlayiso.com Mathematics P1/Wiskunde V1 13 DBE/2021 SC/SS/NSC/NSS – Marking Guidelines/Nasienriglyne 10.2.2 Area of shaded region    2 x 4  2 x 3  x 2  4 dA    2     4x  6x  2x   2 3 2    4x  6x  2x  3 2 dx  4   4  4x 3  6x 2  2x  0 x(2 x 2  3x  1)  0 x(2 x  1)( x  1)  0  factors 1 1 x  0 or x  or x  1  x  0; x  1; x  2 2 1  x (4) 2 [13] QUESTION/VRAAG 11 11.1 P(A) = 1 – P(not A) = 0,6  0,6 P(A and B) = P(A) × P(B)  P(A and B) = P(A) × P(B) = 0,6 × 0,3 9 = 50  answer (A)  0,18 (3) 11.2.1 15 15 a  0,1  (A) 150 150 (1) 11.2.2 m  1  0,7  0,3  0,3 (A) (1) 11.2.3 0,24  0,14  0,02  0,12  0,1  2b  0,7  addition 2b  0,08  simplification b  0,04  b  0,04 0,04  150  6 6 (4) 11.3.1 9 × 9 × 8 = 648 9 9 × 8 (2) 11.3.2 2 × 8 × 4 = 64 2 × 8 × 4 2 × 8 × 5 = 80 2 × 8 × 5 Total number = 64 + 80 = 144 144 (A) (4) [15] TOTAL/TOTAAL: 150 Copyright reserved/Kopiereg voorbehou