NOTE:

• If a candidate answered a question TWICE, mark the FIRST attempt ONLY.  
• Consistent accuracy(CA) 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. 

1.1.1 

x² - 7x = 0
x(x + 7) = 0
x = 0    or   x + 7 = 0
x = -7

✔ both factors 
✔ both x-values (2)

1.1.2 

Penalise 1 mark for  incorrect rounding off.
5 - 10x - 3x² = 0 

∴ x = 0,44   or   x = -3,77

✔ substitution
✔✔ x-values (3) 

1.1.3 

(2x - 1)(4 - x) ≥ 0
(2x - 1)(x - 4) ≤ 0
 

✔ critical values 
✔✔ answer (accuracy)   (3) 

1.1.4 

✔ squaring both sides 
✔ multiplication 
✔ standard form 
✔ factors 
✔ answers   (5) 

1.2 

y = -3x + 2.................(1) 
x² + y = xy + x ............(2) 
(1) in (2):
x² + (-3x + 2) = x(-3x + 2) + x
x² - 3x + 2 = -3x² + 2x + x 
4x² - 6x + 2 = 0
2x² - 3x + 1 = 0
(2x - 1)(x - 1) = 0
x = ½   or x = 1
y = ½   or y =  -1

✔ substitution
✔ standard form 
✔ factors 
✔ x-values 
✔ y-values  (5)

1.3.1 

✔ factors under root 
✔ simplification (this step must be seen) 
✔ answer  (3)

2.1.1 

12 ; 9 ; 6 ; . . . . .  
a = 12 and d = –3  

✔ d = –3  
✔ answer  (2) 

2.1.2 

T_n = -3(40) + 15
= -105

✔ substitution 
✔ answer (2) 

2.1.3 

✔ substitution
✔ answer    (2) 

2.2.1 

Quadratic Pattern : T₁ = 10 
1^st difference pattern :
-7; -5; -3;... 
∴ T₂ = 3 and T₃ = -2

✔ first differences 
✔ ✔ answers  (T₁ and T₂ )   (3)

2.2.2 

2a = 2   3a + b = -7   a + b  + c = 10
a = 1   3(1) + b = -7    (1) + (-10) + c = 10
b = -10   c = 19
∴T_n = n² - 10n +19

✔ value of a 
✔ value of b 
✔ value of c 
(✔✔✔can be awarded at formula)  (3)

2.2.3 

n² - 10n + 19 = 2019
n² - 10n - 2000 = 0
(n - 50)(n + 40) = 0
n  = 50 or  n = 40
∴T₅₀ = 2019

✔ equation 
✔ factors
✔ answer  (T₅₀)   (3) 

2.3.1 

S_n = 81 - 81(3)^-n
T₁ = S₁ = 81 - 81(3)^-n
= 54

✔ answer  (1)

2.3.2 

✔ T2 = 18
✔ r =  ¹/₃
✔ answer (3)

2.3.3 

Yes.
r = - 1 < ¹/₃ < 1 ;   r ≠ 0    

✔ YES
✔ reason   (2) 

2.3.4 

S_∞ =   a     or     S_∞ =81 - 81(3)^-∞
         1 - r
=     54       = 81
    1 - ¹/₃
= 81 - 81(¹/₃)^∞     = 81

✔ substitution
✔ answer  (2)

2.4 

6x + 18 + 12096 = 0
6x = -12114
x = -2019

✔ (2x + 3 + 2x + 6 + 2x + 9)  
✔ sum of GS  
✔ simplification  
✔ answer   (4) 

[27]

3.1 

(let y = 0) 
   2    +1  = 0
 x - 1
   2    = -1
x - 1
2 = -x + 1
x = -1
(-1 : 0)

Or

(let x= 0)
y =  2   + 1 
     0 -1
y = -2 + 1
y = -1
(0 ; -1)

✔ substitution (y = 0) 
✔ substitution (x = 0)  
✔ coordinates of A 
✔ coordinates of B   (4) 

3.2 

x =1 

✔ answer (1) 

3.3 

y ∈ R\{1}   OR   y ∈ R but y # 1

✔ answer  (1) 

3.4 

∴AB = 4 units

✔ coordinates of A and B  
✔ substitution 
✔ simplification  
✔ answer  (4) 

3.5 

h(x) = -      2       + 1
             (x + 5)

✔ – ve (reflection) 
✔ (x – 5) (shift )   (2) 

[12]

4.1 

Parabola
✔ x-intercepts
✔ Turning point
✔ Axis of symmetry 
✔ Shape 

Straight line 
✔ x-intercept 
✔ y-intercept  (6) 

4.2 

From sketch
−1 < x < 0      or    x∈−(-1 ; 0)
Algebraic solution 

-2x² - 2x > 0 
-2x(x + 1) > 0 
2x(x + 1)  <  0
Cv: 
-1 < x < 0

✔✔ answer  (2) 

4.3 

ST = -2x² - 2x - (3x - 3)
= -2x² - 5x + 3
ST'(x) = -4x - 5 = 0
-4x = 5

✔ ST = –2x² – 2x – (3x – 3)  
✔ ST’ = 0  
✔ x = -⁵/₄
✔answer   (4) 

[12]

5.1 

✔ substitution 
✔ answer  (2) 

5.2 

f-1(x) : x = 2^y
y = log₂x
g^-1(x) : x = 2y + 1
x - 1 = 2y
y = ½x - ½

✔ interchanging x and y  
✔ answer 
✔ interchanging x and y  
✔ answer   (4)

5.3 

f'(x).g(x) ≤ 0
f'(x) is always +ve
g(x) is -ve for x < -½
+ve for x > -½
∴x ≤ -½

✔ method  
✔ answer  (2)

[8]

6.1 

✔ substitution 
✔ answer (2) 

6.2 

✔ substitution 
✔ simplification
✔ answer (3) 

6.3.1 

∴effective rate = 6,70%

✔ formula
✔ substitution 
✔ answer  (3) 

6.3.2 

 
x= R997500,00

✔ 
✔ equation (= 2 000 000)  
✔ making x subject of the formula
✔ answer  (5) 

[13]

7.1 

f(x) = 5x² - 5x
f(x +h) = 5(x + h)² - 5(x + h)
= 5(x² + 12xh + h²) - 5x - 5h
= 5x² + 10xh +5h² - 5x - 5h

Answer only = 0 marks

✔ 5x² + 10xh +5h² - 5x - 5h
✔ substitution 
✔ simplification 
✔ common factor 
✔ answer (5)

7.2 

✔ 
 (3) 

7.3 

✔ f(-1)    and f(1)
✔ substitution 
✔ answer    (3)

[11] 

8.1.1 

f(x) = -(2x - 5)(x + 2)² = 0
-2x + 5 = 0   or x + 2 = 0
∴ x = ⁵/₂   or  x = -2
AB = 4,5 units

✔ x-intercepts
✔ answer (2)

8.1.2 

f(x) = -2x³ - 3x² + 12x + 20
f'(x) = -6x² - 6x + 12 = 0
x² + x - 2 = 0
(x + 2)(x - 1) = 0
x + 2 = 0  or x - 1 = 0
x = -2  or  x = 1
T(1;y)

✔ f'(x)
✔ factors 
✔ correct x-value for T (3) 

8.1.3 

m_p = -6(-3)² - 6(-3) + 12
= -24
y - y₁ = m(x - x1)
y - 11 = -24(x +3)
y =  -24x - 61

✔ gradient 
✔ substitution
✔ answer (3)

8.1.4 

T(1 ; y) 
y = -2(1)³ - 3(1)² + 12(1) + 20
 = 27  
∴ 0 < k < 27

✔ max. value
✔✔ answer (accuracy)   (3) 

8.2.1 

c'(x) -³/₂x² + 6x = 0
x(-³/₂x + 6) = 0
x =0   or x = 4
∴0< x < 4

✔ equating to 0 
✔ factors 
✔ answer (accuracy)   (3) 

8.2.2 

cn(x0 = -3 + 6 = 0      or   x = 0 + 4
                                                 2
x = 2                                 x = 2
∴ c(x) is concave up for x < 2
∴ c(x) is concave down for x > 2

✔ method
✔ x-value 
✔✔ conclusion/gevolgtrekking (4)

[18]

9.1 

h = 12 - 4x
          3

✔✔ answer (2)

9.2 

✔ substitution  
✔ answer 
✔ derivative
✔ f'(x) = 0
✔ answer  
✔ answer   (6) 

[8]

10.1.1 

P(S) = 1  

✔ 0,3 & 0,25  
✔ 0,35 & 0,25  
✔ 0,1   (3) 

10.1.2 

P(A or B) = P(A) + P(B) - P(A and B)
= 0,55 + 0,6 - 0,25
= 0,9

or (from sketch)
P(A or B) = 0,3 + 0,25 + 0,35
= 0,9

✔ method 
✔ answer (2)

10.1.3 

P(A and B') = 0,3

✔✔ answer  (2)

10.1.4 

No : P(A ∩ B) = 0,25 ≠  0

✔ answer  (1) 

10.1.5 

P(A ∪ B)' = 0,1 ≠  0

No :  or

P(A ∪ B) = 0,9 ≠  1

✔ answer  (1) 

10.2.1 

a = 20 
b = (40 - x)

✔ a = 20  
✔ b = (40 – x) (2) 

10.2.2 

79 - x + 20 + 19 - x + x + 11 + 16 + 40 - x = 173
-2x = 173 - 185
-2x = -12
x = 6

✔ equation
✔ answer  (2) 

10.2.3 

P (at least 2)
= 20 + 11 + 16 + 6 
              173
=  53  / 0,31 / 30,6%
   173

✔ adding correct values
 ✔ answer (2) 

[15] 

TOTAL: 150