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NATIONAL SENIOR
CERTIFICATE/NASIONALE
SENIORSERTIFIKAAT
GRADE/GRAAD 12
NOVEMBER 2023
TECHNICAL MATHEMATICS P2/TEGNIESE WISKUNDE V2
MARKING GUIDELINE/NASIENRIGLYN
MARKS/PUNTE: 150
This marking guideline consists of 16 pages./
Hierdie nasienriglyn bestaan uit 16 bladsye.
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TECH MATH P2 GR11 MEMO NOV2023_Afr+English_hlayiso.com_.pdf
Technical Mathematics · Grade 11 · Eastern Cape November Exam · 2023 · English. Memorandum, 15 pages. Read online or download the PDF.
- Subject
- Technical Mathematics
- Grade
- Grade 11
- Language
- English
- Document type
- Memorandum
- Year
- 2023
- Exam period
- Eastern Cape November Exam
- Paper
- 2
- Pages
- 15
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- 682.3 KB
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2 TECHNICAL MATHEMATICS P2/TEGNIESE WISKUNDE V2 (EC/NOVEMBER 2023)
NOTE:
• Continuous accuracy (CA) applies only where indicated in this marking guideline.
• Assuming values/answers in order to solve a problem is unacceptable.
LET WEL:
• Volgehoue akkuraatheid (CA) is slegs van toepassing soos aangedui in hierdie nasienriglyn.
• Aanvaarding van waardes/antwoorde om ʼn probleem op te los, is onaanvaarbaar.
MARKING CODES / NASIENKODES
M Method / Metode
A Accuracy / Akkuraatheid
AO Answer only / Slegs antwoord
CA Consistent accuracy / Deurlopende akkuraatheid
F Formula / Formule
I Identity / Identiteit
R Rounding / Afronding
S Simplification / Vereenvoudiging
ST Statement / Bewering
RE Reason / Rede
ST RE Statement and correct reason / Bewering en korrekte rede
SF Substitution correctly in correct formula / Korrekte vervanging in die korrekte
formule
NPU No penalty for omitting units / Geen penalisering vir eenhede uitgelaat
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(EC/NOVEMBER 2023) TECHNICAL MATHEMATICS P2/TEGNIESE WISKUNDE V2 3
QUESTION/VRAAG 1
1.1 product / produk ✓ A
(1)
1.2 𝑦2 − 𝑦1 ✓F A
𝑚𝑃𝑄 =
𝑥2 − 𝑥1 ✓ SF A
6−2
𝑚𝑃𝑄 = 2−(−1)
4 ✓ Ans. / Antw. CA
𝑚𝑃𝑄 = 3 (3)
1.3 𝑚𝑃𝑄 × 𝑚𝑃𝑅 = −1 ✓M A
4 6−𝑘 ✓ SF A
× 2 − 6 = −1
3
4 6−𝑘
× −4 = −1 ✓ Simpl. / Vereenv. CA
3
6−𝑘
= −1
−3
6−𝑘 =3
−𝑘 = −3
𝑘=3 (3)
1.4 𝑀𝑄𝑅 (
𝑥2 + 𝑥1 𝑦2 + 𝑦1
; ) ✓F A
2 2
−1 + 6 2 + 3
✓ SF A
𝑀𝑄𝑅 ( ; )
2 2
5 5
𝑀𝑄𝑅 (2 ; 2) ✓ Ans. / Antw.CA
(3)
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4 TECHNICAL MATHEMATICS P2/TEGNIESE WISKUNDE V2 (EC/NOVEMBER 2023)
1.5 𝑀𝑄𝑅 = 𝑀𝑃𝑆
5 5 2 + 𝑥𝑆 6 + 𝑦 𝑆
( ; )=( ; )
2 2 2 2
5 2 + 𝑥𝑆 5 6+𝑦
∴2= 2 and/en ∴ 2 = 2 𝑆 ✓2=
5 2 + 𝑥𝑆
CA
2
5 = 2 + 𝑥𝑆 5 = 6 + 𝑦𝑆 5 6 + 𝑦𝑆
✓2= 2 CA
3 = 𝑥𝑆 −1 = 𝑦𝑆
∴ 𝑆(3 ; −1) ✓ value of/waarde van
x
CA
✓ value of/waarde van
y
CA
OR/OF
𝑃 → 𝑅 = 𝑃(2 ; 6) → 𝑅(6 ; 3)
∴ 𝑃 → 𝑅 = (𝑥 + 4 ; 𝑦 − 3) ✓M A
∴ 𝑄 → 𝑆 = 𝑄(−1 ; 2) → 𝑆(−1 + 4 ; 2 − 3) ✓S CA
∴ 𝑆(3 ; −1) ✓ value of/waarde van
x
CA
✓ value of/waarde van
y
CA
(4)
1.6 𝑚𝑃𝑅 = − 𝟒
𝟑
𝑚𝑃𝑅 = − 𝟒
𝟑 ✓ Gradient/gradient
𝟑 𝟑 CA
∴ 𝑦 = −𝟒𝑥 + 𝑐 ∴ 𝑦 − 𝑦1 = − 𝟒 (𝑥 − 𝑥1 )
𝟑 𝟑
(2 ; 6): 6 = − (2) + 𝑐 (2; 6): 𝑦 − 6 = − (𝑥 − 2)
𝟒 𝟒
𝟑 𝟑 𝟑 ✓ SF A
∴ 6 = −𝟐 + 𝑐 OR/OF ∴ 𝑦 − 6 = −𝟒𝑥 + 𝟐
10 𝟑 𝟑
∴ 3 =𝑐 ∴ 𝑦 = −𝟒𝑥 + 𝟐 + 6
𝟑 𝟏𝟓 𝟑 𝟏𝟓 ✓ c-value / -waarde
∴ 𝑦 = −𝟒𝑥 + 𝟐 ∴ 𝑦 = −𝟒𝑥 + 𝟐 CA
✓ Ans. /Antw. CA
(4)
1.7 𝑡𝑎𝑛𝜃 = 𝑚𝑃𝑅
3
𝑡𝑎𝑛𝜃 = −4 ✓ SF CA
3
∴ 𝑟𝑒𝑓/𝑣𝑒𝑟𝑤. ∠ = 𝑡𝑎𝑛−1 (4) ≈ 36,87° ✓ ref. ∠ / verw. ∠
∴ 𝜃 = 180° − 36,87° CA
∴ 𝜃 = 143,13° ✓ Ans. /Antw. CA
(3)
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(EC/NOVEMBER 2023) TECHNICAL MATHEMATICS P2/TEGNIESE WISKUNDE V2 5
1.8 RQ = √(𝑥2 − 𝑥1 )2 + (𝑦2 − 𝑦1 )2 ✓F A
✓ SF A
= √(6 + 1)2 + (3 − 2)2
✓ length/lengte PQ
= √50 CA
5 ✓ cos ratio/verh. CA
∴ cos Q =
√50 ✓ size of / grootte van
̂
∴ Q = 45° Q CA
OR/OF OR/OF
PR = √(𝑥2 − 𝑥1 )2 + (𝑦2 − 𝑦1 )2 ✓F A
✓ SF A
= √(6 − 2)2 + (3 − 6)2
✓ length/lengte PR
=5
CA
✓ tan ratio/verh. CA
5
∴ tan 𝑄 = ✓ size of / grootte van
5 Q CA
∴̂
Q = 45° (5)
[26]
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6 TECHNICAL MATHEMATICS P2/TEGNIESE WISKUNDE V2 (EC/NOVEMBER 2023)
QUESTION/VRAAG 2
2.1.1 tan(30,5˚ + 130,5˚) ≈ −0,34 ✓ SF A
✓ −0,34 A
AO: Full marks / Volpunte (2)
2.1.2 cosec(130,5° − 30,5°) ✓ SF A
1
= sin(100°) ✓ reciprocal ratio
/resiprook verh. A
≈ 1,02 ✓ 1,02 A
AO: Full marks / Volpunte (3)
2.2.1
54° (1)2 = 𝑘 2 + 𝑥 2 ✓ Pythagoras A
1 1 − 𝑘2 = 𝑥2
𝑘 ✓ Diagram A
±√1 − 𝑘 2 = 𝑥
36° ∴ 𝑥 = √1 − 𝑘 2
√1 − 𝑘 2 ✓𝑥 = √1 − 𝑘2 A
∴ cos36° = √1 − k2 ✓ √1 − 𝑘 2 CA
(4)
2.2.2 sin(216 °) = 𝑠𝑖𝑛(180° + 36°) ✓ Reduction/reduksie
sin(216 °) = −𝑠𝑖𝑛36° A
sin(216 °) = −𝑘
✓Ans. / Antw. CA
(2)
2.3 𝑡𝑎𝑛𝜃 = 2𝑠𝑖𝑛38,1°
𝑡𝑎𝑛𝜃 = 1,234 … ✓ 1,234 … A
𝑟𝑒𝑓/𝑣𝑒𝑟𝑤. ∠ ≈ 50,98° ✓ ref/verw. ∠ A
∴ 𝑄1: 𝜃 = 50,98°
✓ Quadrants/kwadrante
AND/EN A
✓ Both answers / beide
∴ 𝑄3: 𝜃 = 180° + 50,98° = 230,98° antwoorde CA
(4)
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(EC/NOVEMBER 2023) TECHNICAL MATHEMATICS P2/TEGNIESE WISKUNDE V2 7
QUESTION/VRAAG 3
3.1 1 ✓ 𝑐𝑜𝑠𝜃 A
cos(360° − 𝜃). . tan (360° + 𝜃)
cot(180° + 𝜃) 1
✓ 𝑐𝑜𝑡𝜃 A
cos(180° + 𝜃). 𝑡𝑎𝑛(180° − 𝜃)
✓ 𝑡𝑎𝑛𝜃 A
1
𝑐𝑜𝑠𝜃. 𝑐𝑜𝑡𝜃 . 𝑡𝑎𝑛𝜃 ✓ −𝑐𝑜𝑠𝜃 A
= ✓ −𝑡𝑎𝑛𝜃 A
−cos𝜃. −𝑡𝑎𝑛𝜃
1 ✓ 𝑡𝑎𝑛𝜃 CA
=
cotθ
= 𝑡𝑎𝑛𝜃
(6)
3.2 1 2 1+ 𝑠𝑖𝑛𝑥
(𝑡𝑎𝑛𝑥 + ) =
𝑐𝑜𝑠𝑥 1−𝑠𝑖𝑛𝑥
1 2
∴ 𝐿𝐻𝑆 = (𝑡𝑎𝑛𝑥 + ) 𝑠𝑖𝑛𝑥
𝑐𝑜𝑠𝑥
✓ 𝑐𝑜𝑠𝑥 A
𝑠𝑖𝑛𝑥 1 2
∴ 𝐿𝐻𝑆 = (𝑐𝑜𝑠𝑥 + 𝑐𝑜𝑠𝑥)
𝑠𝑖𝑛𝑥+1 2 (𝒔𝒊𝒏𝒙+𝟏)𝟐
∴ 𝐿𝐻𝑆 = ( 𝑐𝑜𝑠𝑥 ) ✓ CA
𝒄𝒐𝒔𝟐 𝒙
(𝑠𝑖𝑛𝑥 + 1)2
∴ 𝐿𝐻𝑆 = 𝑐𝑜𝑠2 𝑥 ✓ 1 − 𝑠𝑖𝑛2 𝑥 A
(𝑠𝑖𝑛𝑥 + 1)(𝑠𝑖𝑛𝑥 + 1)
∴ 𝐿𝐻𝑆 =
1−𝑠𝑖𝑛2 𝑥 ✓ LHS = RHS
(𝑠𝑖𝑛𝑥 + 1)(𝑠𝑖𝑛𝑥 + 1)
∴ 𝐿𝐻𝑆 = (1−𝑠𝑖𝑛𝑥)(1 + 𝑠𝑖𝑛𝑥)
1+𝑠𝑖𝑛𝑥
∴ 𝐿𝐻𝑆 = = 𝑅𝐻𝑆
1−𝑠𝑖𝑛𝑥 (4)
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8 TECHNICAL MATHEMATICS P2/TEGNIESE WISKUNDE V2 (EC/NOVEMBER 2023)
QUESTION/VRAAG 4
4.1 𝒇(𝒙):
✓ both x-int. /
albei x-afsnitte A
✓ y-int. / y-afsnit ✓
𝑔(𝑥) Shape / vorm A
𝒈(𝒙):
𝑓(𝑥) ✓ both x-int. /
albei x-afsnitte A
✓ y-int. / y-afsnit A
✓ Shape / vorm A
✓ Turning points /
draaipunte A
(7)
4.2 𝑦 ∈ [0 ; 2 ] ✓ Notation / notasie A
OR/OF ✓ start- and endpoints /
0≤𝑦≤2 begin- en eindpunte CA
(2)
4.3 360 ✓Ans. / Antw. A
(1)
4.4 90° ≤ 𝑥 ≤ 270° ✓ 90° ≤ CA
✓ ≤ 270° CA
(2)
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(EC/NOVEMBER 2023) TECHNICAL MATHEMATICS P2/TEGNIESE WISKUNDE V2 9
QUESTION/VRAAG 5
C
B
120°
1 600
m
40°
A 2 400 m D
5.1 BĈA = 180° − 120° − 40° ✓ ST ✓ RE
BĈA = 20° (int. ∠of ∆ / binne ∠ van ∆) (2)
5.2 AC AB
=
̂
sinABC sinBĈA ✓ SF CA
AC 1600
= ✓S CA
sin120° sin20°
1600 ✓ Rounded Ans. /
AC = × sin120°
sin20° Afgeronde Antw.
CA
AC = 4 051 m NPU
(3)
5.3 1
Area ∆ABC = 2 . a. b. sinC OR/OF
1 ✓F A
Area ∆ABC = . AB. AC. sinBA ̂C
2
1
Area ∆ABC = 2 (1600)(4051)sin40° ✓ SF CA
Area ∆ABC = 2 083 146,085 m2
✓Ans. / Antw. A
̂ D = 50° (compl. ∠’s / kompl. ∠’e)
CA
1
Area ∆ABC = 2 . a. b. sinC OR /OF
1
Area ∆ABC = . AC. AD. sinCA ̂D
2
1
Area ∆ABC = 2 (2400)(4051)sin50°
Area ∆ABC = 3 723 895,247 m2 ✓Ans. / Antw. A
∴ Total Area = 2 083 146,085 + 3 723 895,247 ✓M A
∴ Totale Opperv. = 5 807 041,33 m2 ✓ Ans. / Antw. A
NPU
(6)
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10 TECHNICAL MATHEMATICS P2/TEGNIESE WISKUNDE V2 (EC/NOVEMBER 2023)
QUESTION/VRAAG 6
6.1 Double or Twice/Dubbel of Twee keer ✓Ans. / Antw. (1)
6.2 A
48°
O
𝑥
1
2 𝑦 1
C 2
B
6.2.1 𝑥 = 48° × 2 = 96° ✓ ST ✓ RE
(∠ at centre = 2 × ∠ at circumf. / middelpts ∠ = 2 × omtreks ∠) (2)
6.2.2 OB = OC (radii)
Ĉ2 = ̂
B2 = 𝑦 (∠’s opp = sides / ∠’e teenoor = sye) ✓ ST ✓RE
180°−96°
∴𝑦= = 42° (int. ∠of ∆ / binne ∠ van ∆)
2 (2)
6.3 Equal / gelyk ✓Ans. / Antw. (1)
6.4
S
𝑧
𝑥 R
O
120° 1 𝑦
T Q
P
6.4.1 𝑥 = 120° (ext. ∠ of cq / buite ∠ van kvh) ✓ ST ✓ RE (2)
6.4.2 𝑦 = 90° (∠ in semi-circle / ∠ in semi-sirkel) ✓ ST ✓ RE (2)
6.4.3 𝑧 = 30° (ext. ∠ of ∆/ buite ∠ van ∆) ✓ ST ✓ RE (2)
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(EC/NOVEMBER 2023) TECHNICAL MATHEMATICS P2/TEGNIESE WISKUNDE V2 11
QUESTION/VRAAG 7
7.1 equal / gelyk ✓Ans. / Antw. (1)
7.2
7.2.1 𝐶̂4 = 25° (tan-chord / raaklyn koord) ✓ ST ✓ RE (2)
7.2.2 𝐶̂3 = 42° (∠’s in same seg. / ∠’e in dieselfde seg.) ✓ ST ✓ RE (2)
7.2.3 𝐵𝐶̂ 𝐷 = 90° (∠ in semi-circle / sirkel) ✓ ST ✓ RE
̂1 = 65° (int. ∠of ∆ / binne ∠ van ∆)
𝐷 ✓ ST ✓ RE (4)
7.3 outside / buite ✓Ans. / Antw. (1)
7.4 S P
K 1 4
23
2 3 M
4
O1
T 2
100° 1
3 10°
R
7.4.1 OQ = OR (radii)
𝑆̂3 = 10° (∠’s opp = sides / ∠’e teenoor = sye) ✓ ST ✓RE
∴ 𝑆̂2 = 40° ✓ ST
(∠ at centre = 2 × ∠ at circumf. / middelpts ∠ = 2 × omtreks ∠) ✓ RE (4)
7.4.2 𝑆̂4 + 𝑆̂3 = 90° ✓ ST
∴ 𝑆̂4 = 80° (tan ⊥radius) ✓ RE (2)
7.4.3 𝑅̂1 = 80° tan from same pt. / 𝑟𝑎𝑎𝑘𝑙𝑦𝑛 𝑢𝑖𝑡 𝑑𝑖𝑒𝑠𝑒𝑙𝑓𝑑𝑒 𝑝𝑡.) ✓ ST ✓ RE
𝑃̂ = 20° (int. ∠of ∆ / binne ∠ van ∆) ✓ ST ✓ RE
(4)
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12 TECHNICAL MATHEMATICS P2/TEGNIESE WISKUNDE V2 (EC/NOVEMBER 2023)
QUESTION/VRAAG 8
8.1 Perpendicular / loodreg ✓Ans. / Antw. (1)
8.2 B
E
8
D
𝑥
12
O
A
8.2.1 𝑂𝐸 = 𝑥 + 8 ✓ Ans. / Antw. (1)
8.2.2 𝑂𝐴2 = 𝑥 2 + (12)2 (Pyth) ✓ ST ✓RE
𝑂𝐴2 = 𝑥 2 + 144
𝑂𝐴 = ±√𝑥 2 + 144 ✓ Answer / Antw.CA
∴ 𝑂𝐴 = √𝑥 2 + 144 (3)
8.2.3 𝑥 + 8 = √𝑥 2 + 144 ✓M A
(𝑥 + 8)2 = 𝑥 2 + 144 ✓S A
𝑥 2 + 16𝑥 + 64 = 𝑥 2 + 144
16𝑥 = 80 ✓S CA
𝑥=5 ✓Ans. / Antw. CA
(4)
8.2.4 𝑟 = 13 ✓Ans. / Antw. (1)
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(EC/NOVEMBER 2023) TECHNICAL MATHEMATICS P2/TEGNIESE WISKUNDE V2 13
QUESTION/VRAAG 9
9.1 𝜔 = 2𝜋𝑛 ✓F A
𝜔 = 2𝜋(12) ✓ SF CA
𝜔 = 24𝜋 ✓Answer/Antw.CA
𝜔 ≈ 75,40 rad. / sec NPU
(3)
9.2 𝐷 = 80 mm ✓diameter/middellyn
𝑉 = 𝜋𝐷𝑛 A
𝑉 = 𝜋(80)(20) ✓F A
𝑉 = 1 600𝜋 ✓ SF CA
𝑉 ≈ 5 026,55 mm/min ✓Answer/Antw. CA
NPU
(4)
9.3 500 mm = 50 cm ✓conv. /herleid. A
4ℎ2 − 4𝑑ℎ + 𝑥 2 = 0 ✓F A
4ℎ2 − 4(56,6)ℎ + (50)2 = 0 ✓ SF CA
4ℎ2 − 226,4ℎ + 2500 = 0
÷ 4: ℎ2 − 56,6ℎ + 625 = 0
−𝑏 ± √𝑏 2 − 4𝑎𝑐
ℎ=
2𝑎 ✓ SF CA
−(−56,6) ± √(−56,6)2 − 4(1)(625)
ℎ=
2(1) ✓ Both answers/
∴ ℎ ≈ 41,56 or ℎ ≈ 15,04 beide antwoorde CA
∴ ℎ = 15,04 cm ✓ Answer/Antw. CA
NPU
(6)
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14 TECHNICAL MATHEMATICS P2/TEGNIESE WISKUNDE V2 (EC/NOVEMBER 2023)
9.4
2m
𝜃 2,5 m
9.4.1 𝑠 = 𝑟𝜃 ✓F A
2,5 = 2𝜃 ✓ SF CA
1,25 rad = 𝜃 ✓Answer/Antw.CA
(3)
9.4.2 Area =
r2 θ ✓F A
2 ✓ SF CA
(2)2 (1,25)
Area = 2
✓Ans./Antw. CA
Area = 2,5 cm2 NPU
(3)
OR/OF
rs
Area = 2 ✓F A
(2)(2,5) ✓ SF CA
Area = 2 ✓Ans./Antw. CA
Area = 2,5 cm2 NPU
(3)
9.4.3 𝐶 = 2𝜋𝑟 ✓F A
2,5 = 2𝜋𝑟 ✓ SF A
0,40 = 𝑟 ✓Answer/Antw.CA
(2)2 = ℎ2 + (0,40)2 ✓M A
(2)2 − (0,40)2 = ℎ2
3,84 = ℎ2
1,96 m = ℎ ✓Answer/Antw.CA
NPU
(5)
9.5 VolShell = Volouter − Volinner ✓M A
4 4
VolShell = 3 π(5,5)3 − 3 π(3,5)3 ✓ SF A
1331 343
VolShell = π− 6 π ✓Answer/Antw.CA
6
494 3
VolShell = 3 π cm
494 ✓M A
∴ Mass/𝐺𝑒𝑤𝑖𝑔 = 3 π × 30
∴ Mass/𝐺𝑒𝑤𝑖𝑔 = 4940π
∴ Mass/𝐺𝑒𝑤𝑖𝑔 = 15519,47 grams ✓Ans./Antw. CA
NPU
(5)
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(EC/NOVEMBER 2023) TECHNICAL MATHEMATICS P2/TEGNIESE WISKUNDE V2 15
QUESTION/VRAAG 10
12 cm 𝑥 cm 13 cm 11 cm 11,8 cm 18 cm
25 cm
𝑂 +𝑂
𝐴𝑟𝑒𝑎 = 𝑎 ( 1 2 𝑛 + 𝑂2 + 𝑂3 +. . . +𝑂𝑛−1 ) ✓F A
25 12+18
✓𝑎=5 A
329 = 5 ( 2 + 𝑥 + 13 + 11 + 11,8) ✓ SF CA
329 = 5(𝑥 + 50,8) ✓S CA
65,8 = 𝑥 + 50,8 ✓ 𝑥 = 15 CA
15 = 𝑥
OR / OF OR / OF
𝐴𝑟𝑒𝑎 = 𝑎(𝑚1 + 𝑚2 + 𝑚3 +. . . +𝑚𝑛−1 ) ✓F A
25 12 + 𝑥 𝑥 + 13 13 + 11 11 + 11,8 11,8 + 18 ✓𝑎=5 A
329 = 5 ( + + + + ) ✓ SF CA
2 2 2 2 2 25
12 + 𝑥 𝑥 + 13
𝑎= ✓S CA
329 = 5 ( + + 12 + 11,4 + 14,9) 5
2 2 𝑎=5 ✓ 𝑥 = 15 CA
25 + 2𝑥
329 = 5 ( + 38,3)
2
25 + 2𝑥
65,8 = + 38,3
2
25 + 2𝑥
27,5 = 2
55 = 25 + 2𝑥
30 = 2𝑥
15 = 𝑥 (5)
[5]
TOTAL/TOTAAL: 150
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