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MATHEMATICS PAPER 1 MEMORANDUM GRADE 12 - 2018 SEPTEMBER PREPARATORY EXAM PAPERS AND MEMOS
Share via Whatsapp Join our WhatsApp Group Join our Telegram GroupMATHEMATICS PAPER 1
NATIONAL SENIOR CERTIFICATE
GRADE 12
MEMORANDUM
SEPTEMBER 2018
NOTE:
- If a candidate answers a question TWICE, mark the FIRST attempt ONLY.
- Consistent accuracy applies in ALL aspects of the marking guideline.
- If a candidate crossed out an attempt of a question and did not redo the question, mark the crossed-out attempt.
- The mark for substitution is awarded for substitution into the correct formula.
QUESTION 1
| 1.1.1 | ½ ?2 − ? − 4 = 0 ?2 − 2? − 8 = 0 (? − 4)(? + 2) = 0 ? = 4 or/of ? = −2 | ✓✓ factors ✓?-values (3) |
| 1.1.2 | -3(x2 + 3x) + 7 = 0 −3?2 − 9? + 7 = 0 ? = −? ± √?2 − 4?? 2? ? = −(−9) ± √(−9)2 − 4(1)(7) 2(−3) ? = 0,64 or/of ? = −3,64 | ✓ standard form ✓ substitution ✓✓ ?-values (4) |
| 1.1.3 | 2?2 − 3? < 0 ?(2? − 3) < 0 0 < ? <3/2 | ✓ ?(2? − 3) ✓ critical values/kritiese waardes ✓ ✓ 0 < ? <3/2 (4) |
| 1.2 | ?−2?=3 …………………..(1) 4?2−3=−6?+5??……….(2) from (1) ?=2?+3 sub into (2) 2(2?+3)2−3=−6?+5?(2?+3) 4(4?2+12?+9)−3=−6?+10?2+15? 16?2+48?+36−3=10?2+9? 6?2+39?+33=0 2?2+13?+11=0 (?+1)(2?+11)=0 ?=−1 or ?=−11/2 ?=1 or ?=−8 | ✓ ?=2?+3 ✓ substitution ✓ standard form ✓ factors ✓ y-values s ✓ ?-values (6) |
| 1.3 | 2?2−(?−1)?+?−3=0 Δ =?2−4?? Δ =[−(?−1)]2−4(2)(?−3) Δ = ?2−2?+1−8?+24 Δ = ?2−10?+25 Δ = (?−5)2 (?−5)2≥0 ∴Roots are Real | ✓ Δ =[−(?−1)]2−4(2)(?−3) ✓ Δ = ?2−2?+1−8?+24 ✓ Δ = ?2−10?+25 ✓ Δ = (?−5)2 ✓ (?−5)2≥0 (5) |
| 1.4.1 | 32?= 3? 3−? 32?=3(1,5) 3−1,5 32?=3 ∴2?=1 ∴ ?=½ | ✓ sub. of ?=1,5 ✓ answer (2) |
| 1.4.2 | 32?= 3? 3−? 32(0)= 3? 3−? 1= 3? 3−? 3?=3−? 4?=3 ∴ ?=3/4 | ✓ sub. of ?=0 ✓ answer (2) [26] |
QUESTION 2
| 2.1 | 2?+2= ?−1 7?+1 2?+2 (2?+2)2=(7?+1)(?−1) 4?2+8?+4=7?2−6?−1 3?2−14?−5=0 (3?+1)(?−5)=0 ?=−1/3 or ?=5 | ✓ ?2 = ?3 ?1 ?2 ✓ standard form ✓ factors ✓ ?=−1/3 ✓ ?=5 (5) |
| 2.2 | 25 ;20 ;16;…. ?=25 ; ? =4/5 ?∞= ? 1−? ?∞= 25 1−4/5 ?∞=125? | ✓ ?=45 ✓ ?∞=?1−? ✓ substitution ✓?∞=125 ? (4) |
| 2.3 | ?=2 ?=29 ??=155 ??=?/2(?+?) 155=?/2(2+29) ?=10 29=2+(10−1)? 9?=27 ?=3 | ✓ sum formula of AS ✓sub. ? and ? ✓?=10 ✓ sub.n ?=10 ✓ ?=3 (5) |
[14]
QUESTION 3
| 3. | 2?=4 ?=2 24=2+?+? ............(1) 24=50+5?+? ........(2) 4?=−48 (2) – (1) ?=−12 24=2−12+? ?=34 ??=2?2−12?+34 See alternative answers | ✓ 2?=4 ✓ ?=2 ✓sub. into ?1 ✓sub. into ?5 ✓method of solving ✓ ?=−12 ✓ sub.of ? ✓ ?=34 (8) |
[8]
QUESTION 4
| 4.1 | 36=?2 ∴?=6 | ✓ sub of point ✓ ?=6 (2) |
| 4.2 | ?=6? | ✓ swop of ? and ?. ✓ ?=log6? (2) |
| 4.3 | 0<?≤1 | ✓✓ answer (2) |
| 4.4 | ?>2 | ✓✓ answer (2) |
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QUESTION 5
| 5.1 | ?(5;0) | ✓✓ answer (2) |
| 5.2 | ?=?(?−?)2+? ?=?(?−3/2)2+49/4 0=?(5−3/2)2+49/4 ?=−1 ?=−1(?−3/2)2+49/4 ?=−1(?2−3?+9/4)+49/4 ?=−?2+3?+10 OR ?=?(?−5)(?+2) Inspection 49/4=?(3/2−5)(3/2+2) ?=−1 ?=−1(?−5)(?+2) ?=−?2+3?+10 | ✓ sub of turning point ✓ sub of point B ✓ ?=−1 ✓?=−?2+3?+10 (4) ✓ sub of ?-intercepts ✓sub of turning point ✓ ?=−1 ✓?=−?2+3?+10 (4) |
| 5.3 | −?2+3?+10=−?+5 ?2−4?−5=0 (?−5)(?+1)=0 ?=5 or ?=−1 ?(−1;6) | ✓?(?)=?(?) ✓ standard form ✓ factors ✓?(−1;6) (4) |
| 5.4.1 | −1≤?≤5 | ✓✓ answer (2) |
| 5.4.2 | −?2+3?−2,25<0 −?2+3?+10<2,25+10 ∴ ?(?)<12,25 ?∈? ;?≠1,5 | ✓ ?(?)<12,25 ✓✓ ?∈? ;?≠1,5 accuracy (3) |
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QUESTION 6
| 6.1 |  | ✓ asymptote ✓ ?-intercept ✓ shape ✓ other point (4) |
| 6.2 | ?(?)=2?−1−1 ?′(?)=−2?−2 | ✓ ?′(?)=−2?−2−1 ✓ ?′(?)=−2?−2 (2) |
| 6.3 | ℎ(?)=−?−1 | ✓✓ answer (2) |
| 6.4 | ?′(?)= −2 and ℎ(?)=−?−1 | ✓ setting up of equation ✓ ?=√2 ✓ 2/√2−1 ✓sub of (√2; 2/√2−1) ✓ ?=2√2 (5) |
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QUESTION 7
| 7.1 | 2000(1+ 8 )12=2000(1+ ? )2 1200 200 √(1+ 8 )12=(1+ ? ) 1200 200 ?=8,13% | ✓ 8 and ? 1200 200 ✓ ?=12 and ?=2 ✓ ?=8,13% (3) |
| 7.2 | ?=?(1−?)? 4 500=9 500(1−7,7%)? ?= log 4500 9500 log(1−7,7%) ?≈9,325 It will take 10 years | ✓ correct formula ✓ sub. of A and P ✓ use of logs ✓?≈9,325 ✓ 10 years (5) |
| 7.3.1 | 75/100 ×170 500=?127 875 OR 25 ×170 500=42 625 100 Loan =170 500−42 625 Loan = R127 875 | ✓✓ answer (2) OR ✓ R 42 625 ✓ answer (2) |
| 7.3.2 | 127 875=?[1−(1+ 13,2 )−60] 1200 13.2 1200 ?=? 2 922,66 | ✓ 13.2 1200 ✓ ?=60 ✓ sub of i, n and 127 875 into correct formula ✓✓ answer (5) |
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QUESTION 8
| 8.1 | ?(?)=?−2?2 | ✓ formula ✓ substitution of (?+ℎ) ✓ simplification ✓ simplification to (−4?ℎ−2ℎ2+ℎ) ✓ common factor ✓ answer (6) |
| 8.2.1 | ?=1/9?−3+9? ?? =−1/3?−4+9 ?? Penalise 1 mark for incorrect notation. | ✓−1/3?−4 ✓ 9 (2) |
| 8.2.2 | ?= −1 +?3 | ✓ √?=?½ ✓ −12?−32 ✓ 34?−54 ✓ 3?2 (4) |
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QUESTION 9
| 9.1 | ℎ(?)= ?3−9?2+23?−15. ℎ′(?)=3?2−18?+23 ?=−?±√?2−4?? 2? ?=−(−18)±√(−18)2−4(3)(23) 2(3) ?=4,15 or ?=1,85 ?=1,85 at C | ✓ ℎ′(?)=3?2−18?+23 ✓ sub into formula ✓ both ? values ✓ ?=1,85 (4) |
| 9.2 | ℎ(?)= ?3−9?2+23?−15. ℎ(?)=(?−1)(?2−8?+15) ℎ(?)=(?−1)(?−3)(?−5) ∴?(5;0) | ✓(?−1)(?2−8?+15) ✓(?−1)(?−3)(?−5) ✓✓?(5;0) (4) |
| 9.3 | ℎ(?)= ?3−9?2+23?−15. | ✓ ℎ′′(?)=6?−18 ✓ 6?−18=0 ✓∴?=3 (3) |
| 9.4 | ℎ′(?)=3?2−18?+23 ℎ′(3)=3(3)2−18(3)+23 ℎ′(3)=−4 ?=−4?+? 0=−4(3)+? ?=12 ?=−4?+12 | ✓ ℎ′(3)=−4 ✓ sub of point D ✓?=−4?+12 (3) |
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QUESTION 10
| 10.1 | ?=?(50−½?)−(¼?2+35?+25) ?=50?−½?2−¼?2−35?−25 ?=−3/4?2+15?−25 | ✓ ?(50−½ ?) ✓ subtracting total cost (2) |
| 10.2 | ?? =−3/2?+15 ?? −3/2?+15=0 ?=10 | ✓ ?? =−3/2?+15 ?? ✓−3/2 ?+15=0 ✓ ?=10 (3) |
| 10.3 | ?=¼?2+35?+25 ? ?=¼?+35+25?−1 ?? =¼−25?−2 ?? ¼ −25?−2=0 25 = ¼ ?2 ?2=100 ?=10 ∴Minimum | ✓?= ¼?2+35?+25 ? ✓?=¼?+35+25?−1 ✓ ?? =¼ −25?−2 ?? ✓ ¼ −25?−2=0 ✓ ?=10 (5) |
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QUESTION 11
| 11.1.1 | ?(?)=1200 | ✓ answer(1) |
| 11.1.2 | ?(????)= 200 1600 ?(????)=1/8 | ✓ answer(1) |
| 11.1.3 | ?(?)×?(?)=3/4×1/8 = 3 32 3 = ? 32 1600 ?=150 | ✓ 3/4×1/8 ✓ 3 32 ✓ 3 = ? 32 1600 (3) |
| 11.1.4 | ?=1050 ?=50 ?=350 | ✓?=1050 ✓?=50 ✓?=350 (3) |
| 11.1.5 | ?(?/?)= 50 1600 ?(?/?)= 1 32 | ✓50 ✓ 1600 (2) |
| 11.2.1 | 9!=362880 | ✓✓ answer (2) |
| 11.2.2 | 4!×5!×6 =17 280 OR 6!×4!=17280 | ✓ 4!×5! ✓ ×6 ✓ 17280 (3) |
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TOTAL:150
ALTERNATIVE ANSWERS
| 1.1.1 | 12?2−?−4=0 (½?+1)(?−4)=0 ?=−2 or ?=4 OR 12?2−?−4=0 (½ ?−2)(?+2)=0 ?=4 or ?=−2 OR ?=−?±√?2−4?? 2? ?=−(−1)±√(−1)2−4(½)(−4) 2(½) ?=4 or ?=−2 | ✓✓ factors ✓ ?-values (3) ✓✓ factors ✓ ?-values (3) ✓✓sub into formula ✓ ?-values |
| 3.1 | 2?=4 ?=2 ?3=axis of symmetry ??=?(?+?)2+? ??=2(?+?)2+? ??=2(?−3)2+? 24=2(1−3)2+? ?=16 ??=2(?−3)2+16 ??=2(?2−6?+9)+16 ??=2?2−12?+34 | ✓ 2?=4 ✓ ?=2 ✓ ??=2(?+?)2+? ✓ ??=2(?−3)2+? ✓ 24=2(1−3)2+? ✓ ?=16 ✓ ??=2(?2−6?+9)+16 ✓ ??=2?2−12?+34 (8) |
| 5.4.2 | −?2+3?−9/4<0 ?2−3?+9/4>0 4?2−12?+9>0 (2?−3)(2?−3)>0 ∴?∈? ;?≠3/2 | ✓ factors ✓✓ answer (3) |
Published in Grade 12 September 2018 Preparatory Examinations