1.1.1

?² − 4?−12=0
(? − 6)(? + 2)=0
? = 6 or ? = −2

✔ standard form
✔ ?=6 (CA applies)
✔ ?=−2 (CA applies)

(3)

1.1.2

3?²+2?−6=0
?=−(2) ± √(2)² − 4(3)(−6)
                2(3)
?=−2 ± √766
?=−1,79 or ?=1,12

✔ substitution

✔ x = −1,79 

✔ x = 1,12 

(3)

1.1.3 

 3?² − 1 = 27^-x
                 3
3?² − 1 = 3^−3? −1
∴ ?² − 1=−3? − 1
?² + 3? = 0
? (?+3) = 0
? = 0 or  ? = −3

OR

3?² − 1 = 27 − ?3
3?² − 1.3 = 27−?
3?² −1 + 1 = 3^−3?
∴ ?²=−3?
?² + 3? = 0
?(?+3) = 0
?=0 or ?=−3

✔ 3^−3?−1 
✔ equating exponents
✔factors/faktore 
✔ both x-values

✔?² −1 + 1 = 3^−3?
✔equating exponents 
✔ factors/faktore 
✔both x-values

(4)

✔ ? =−1
✔ ?=0

(2)

✔ manipulation of equation 
 ✔ standard form
✔ factors
✔ both x-values
✔ choosing x = 2 

(5) 
[17]

? − ? =3 ; ?? = 28
? − ? = 3…………………..(1)
?? = 28………………… (2)
From (1) ? = ? + 3
Substitute in (2)
?(?+3) = 28
?² + 3? − 28 = 0
(?−4)(?+7)  =0
?=4 or  ?=−7
?=7 or  ?=−4

OR

From  (1) ? = ?−3
Substitute in (2)
?(?−3) = 28
?² − 3? − 28 = 0
(?+4)(?−7) = 0
?=−4 or  ?=7
?=−7 or  ?=4

✔ ? = ? + 3
✔ substitute in (2)
✔ standard form
✔ factors
✔ y-value
✔ x-values

✔ ? = ? − 3✔ substitute in (2)
✔ standard form
✔factors
✔ x-values
✔ y-values

(6)

✔ standard form/standaardvorm
✔ factors/faktore 
✔ solution/oplossing  
−1 ≤ ? ≤4
✔ final answer/finale antwoord 
0 < ? ≤ 4

(4) 
[10]

✔ a = 1 
✔ b = −1 
✔ c = 1 
✔ ?_? = ?² − ? + 1
✔ 6321 

(5)

Row 80
6321 6323 6325 6327 ...
?_?=^?/₂[2(?) + (? − 1)?]
?₈₀ = ⁸⁰/₂ [2(6321)+(80−1)(2)] Row 80
?₈₀ = 512000

OR

Row 80 Term 80
?₈₀ = 6321 + (79×2)
?₈₀ = 6479
?_? = ?²[? + ?]
?₈₀ = 8⁰/₂[6321 + 6479] Row 80
?₈₀ = 512000

OR

2? = 2
? = 1
3? + ? = 4
3 + ? = 4
? = 1
?  + ? + ? = 1
1 + 1 + ? = 1
? = −1

?_? = ?² + ? − 1
?? = ?²+ ? − 1
?₈₀ = 80² + 80 −1
?₈₀ = 6479
?_?= ?/₂[?+?]
?₈₀=80/₂[6321+6479] Row 80
?₈₀=512000

✔ n= 80 
✔ d = 2 
✔ sub into correct formula   
✔ answer(4) 

✔ calculating term 80 of row  
✔ 6479 
✔ sub into correct formula 
✔ answer (4)

✔ ?_? = ?² + ? − 1
✔ ?₈₀ = 6479
✔ sub into formula
✔ answer 

(4) 
[9]

✔ substitution
✔ answer

(3)

? ;3? ;5? ;……………
? = 2?
?_?=^?/₂[2? + (? − 1)?]
?_?=^?/₂[2? + (? − 1)2?]
?_?=^?/₂(2? + 2?²−2?)
?_?=?³

OR

? = ?
? = 2?² − ?
?_? = ^?/₂ [? + ?]
??=^?/₂[?+2?² − ?]
?_?=?³

✔ first three terms
✔ ? = 2?
✔ substitution
✔ answer (4) 

✔ ? = ?
✔ ? = 2?² − ?
✔ substitution
✔ answer 

(4) 
[7]

? = ?+2
       ?
?3 = (?+2)²
           ?

✔ ratio
✔ answer

(2)

?∞=  ?  
     1−?
−8=       ?       
        1 − ^(?+2)/_?
−8=     ?²           
        ?−?−2
?²=16
?  = ±4

✔ both answers 

(4) 
[6]

✔ ?=¹⁵/₁₀₀ and ? =5
✔ sub into correct formula
✔ answer

(3)

✔ ?=^16.75/₁₂₀₀
✔ ?=−48
✔ sub into correct formula
✔ answer

(4)

?_? = ?[1 − (1+?)^−?]
                ?
?_? = 1436.29 [1 − (1+^16.75/₁₂₀₀)⁻¹⁸]
                          ^16.75/₁₂₀₀
?_? = ?22 721,97704
?_? = ?22 722

OR

Outstanding balance (OB)
OB = 50000(1+ ^16.75/₁₂₀₀)³⁰−[1436.29 [(1 + ^16.75/₁₂₀₀)³⁰−1]]
                                                                     ^16.75/₁₂₀₀
OB =  ? 22722,14
OB = ?22722

✔ ?=−18
✔ ?=^16.75/₁₂₀₀
✔ substitution
✔ answer
✔ rounding

(5)

✔ ?=30
✔ ?=^16.75/₁₂₀₀
✔ sub into both formulae
✔ answer
✔ rounding

(5)

✔ ? = 2? and ? = ?
✔ sub into correct formula
✔ using of logs
✔ answer

(4)
[16]

✔answer 

(1)

✔x > -2
✔x # 0

(2)

✔answer 

(1)

✔ sub of −4
✔ sub of point (2;5)
✔ ? = ±3
✔ answer with correct ? value

(4)

✔x = -2
✔y = -1

(2)

✔ sub of asymptotes
✔ sub of point (0;-3)
✔ ?=−4

(3)

✔? = ? + ?
✔? = −(? + 2) −1
✔? = −? − ?

(3)

✔swopping of x and y
✔answer

(2)

✔Shape 
✔y-intercept
✔any other correct point

(3)

g(x) = (½)^-x

OR

g(x) = 2x

✔✔Answer

✔✔Answer

(2)

✔✔x > 1

(2)
[25]

✔x = -3

(1)

✔ sub of turning point
(−3;5)
✔ sub of (0;4)
✔ simplification

(3)

✔ Δ=20
✔ irrational
✔ unequal

(3)

?(?) = 2?
?² + 6? + 4 = 2?
?² + 4? + 4 = 0
(?+2)² = 0
? = −2
?(−2)=−4

Point (−2;−4)

OR

?(?) = ?² + 6? + 4
?′(?) = 2? + 6 and ? = 2
2? + 6 = 2
2? = −4
? = −2
? = −4
Point (−2;−4)

✔ ?(?) = 2?
✔ equating equations
✔ ? = −2
✔ ? = −4

✔ derivative
?′(?) = 2? +6
✔ equating to gradient of g.
✔ x-value
✔ y-value

(4)
[11]

 ?(?) = 3?² − 1
?′(?) = lim   ?(?+ℎ)−?(?)
          ^ℎ→0          ℎ
=    lim  3(? + ℎ)² − 1 −(3?² − 1)
     ^ℎ→0                      h
=   lim  3(?² + 2xh + h²) − 1 - 3?² − 1
     ^ℎ→0                      h
=   lim  3?² + 6xh + 3h²− 1 - 3?² − 1
     ^ℎ→0                      h
=       lim     6?ℎ
        ^ℎ→0      h
= 6?

✔ formula
✔substitution of (?+ℎ)
✔ simplification
3?² + 6?ℎ + 3ℎ² − 1 − 3?² + 1
✔ =  lim     6?ℎ
        ^ℎ→0     h
✔ answer

(5)

✔?^½
✔10x
✔½?^−½

(3)

✔ 2
✔ −⁴/₃?⁻¹
✔ answer

(3)

✔ derivative
✔ 3?² ≥ 0

(2)
[13]

?(?) = ?³ − ?² − 8? + 12
(?−2)(?² + ? − 6) = 0
(?−2)(?−2)(?+3) = 0
?=2 or ?=2 or ? = −3
?(−3;0)

OR

                         ?+3               
?² − 4? + 4 √?³ − ?² − 8? + 12
?³ − 4?² +4?

     3?² − 12? + 12
    3?² − 12? + 12  
?(?) = (?² − 4? + 4)(? + 3)
?(−3;0)

✔(?−2)
✔(?² + ? − 6)
✔(?−2)(?−2)(?+3) 
✔coordinates of ? (−3;0)

✔?² − 4? + 4
✔✔? + 3
✔coordinates of ? (−3;0)

(4)

✔?′(?) 
✔?′(?)  = 0
✔correct ? value ? = ⁻⁴/₃
✔y = ⁵⁰⁰/₂₇ 

(5)

 ?′′(?) = 6? − 2
6? − 2 = 0
? =¹/₃

OR

?=⁻⁴/₃+2
        2
?=13

✔?′′(?) = 6? − 2
✔? =¹/₃
✔finding x value of  midpoint
✔? =¹/₃

(2)

✔? <⁻⁴/₃
✔? > 2

(2)

 ✔answer

(2)
[15]

✔?(0)=3 ?

(1)

✔?′(?)
✔?′(3)
✔=−¹⁵/₄ or -3.75

(3)

✔Decreasing

(1)

✔?′(? )= 0
✔factors
✔t-values
✔answer

(4)
[9]

12.1 

12.1.1 

P(A′) = 1 − P(A) 
= 1 − 0,35 
= 0,65

✔ P(A′) = 1 − P(A) 
✔ answer

(2)

12.1.2 

P(AandB) = 0 
P(AenB) = 0

✔ answer

(1)

12.1.3 

P(A or B) = 0,35 + 0,52 
= 0,87

✔  P(A or B) = P(A) + P(B)
✔ answer

(2)

12.2 

12.2.1 

6! = 720 

✔ 6! or/of 720 

 (1)

12.2.2

4! 
= 24

✔4! 
✔ 24 

(2)

12.2.3

2!.5! =  240   = 1
  6!       720      3

OR 

0,333

✔ 2! 
✔ 5! 
✔ 6! 
✔ answer

(4) 
[12]

TOTAL: 150