1.1.1

(? − 2)(3? − 1) = 0
? − 2 = 0 or 3? − 1 = 0
? = 2 or ? = ¹/₃

✓✓ ?-values
(2)

1.1.2

2?² + 3? − 7 = 0                                Penalise 1 mark for incorrect rounding off
x = -b ± √b² - 4ac    
             2a
x = -(3) ± √(3)² - 4(2)(-7)
                  2(2)

x = -3 ± √65
            4
∴ x = 1, 27 or  x = -2,77

✓ substitution

✓✓ ?-values (3)

1.1.3

✓factors 

✓critical values with method

✓✓ answer (accuracy)

(4)

✓ simplifying LHS

✓ simplifying RHS 

✓ same base 

✓ equating exponents

✓ answer

(5)

✓ x subject of the formula  

✓ substitution 

✓ removing brackets

✓ standard form

✓ ?-values

✓ x-values

(6)

For equal roots  :
n² + 4mn = 0
n(n + 4m) = 0
∴ n = 0 or / of n = -4m

✓Δ  = 0

✓ both n-values

✓ substitution

✓ x = 0

✓ x = -2

(5)

2.1.1

15  ; 10  ; 7       ;     ?   ;     7
   −5   − 3   (? − 7)   (7 − ?)  - 1^st differences
          2      (? − 4) (−2? + 14) - 2^nd differences
x - 4 = 2  or   - 2x +14 = x - 4
x = 6  or   - 3x = -18
x= 6

✓ 2^nd differences 

✓ equating 

✓ answer 

(3)

2.1.2

2? = 2      3? + ? = −5     a + b + c = 1
? = 1        3(1) + ? = −5   1 - 8 + c = 15
                      ? = −8               c = 22
∴ ?? = ?² − 8? + 22

✓? = 1

✓ ? = −8

✓ ? = 22

✓ answer
(4)

2.1.3

T₅₀ = (50)²  - 8(50) + 22
= 2122

✓ substitution

✓ answer
(2)

2.2.1

a +14d = 74
a + 6d = 34
8d = 40
d = 5

✓ setting up 2 equations 

✓ method

✓ answer 

(3)

2.2.2

a + 6(5) = 34
a = 4

S₄₀ = 40 [2 (4) + (40 -1)(5)]
          2
= 4060

✓ value of a

✓ substitution into correct formula

✓ answer

(3)

2.2.3

✓✓ answer  (accuracy as one unit)
(2)

T_k = 3 ^k
       15
=  3 ^k  
   5 x 3

= 1  x 3^{k -1}
   5

✓ factors of 15 

✓ answer 

(2)

✓ a and r

✓ substitution into correct formula

✓ exponential equation  

✓ answer 

(4)

NO
r = 3 > 1
r not in the interval -1 < r < 1, r ≠ 0

✓ NO 

✓ reason

(2)

P = 9¹/₃ x 9¹/₉ x 9¹/₂₇ x ....to infinity
= 9¹/₃ + 9¹/₉ + 9¹/₂₇ ... to infinity

S_∞ =    a    
          1 - r

✓ adding exponents 

✓ sum to infinity

✓ answer 

✓ P = 9½  / √9

(4)

3.1.1

P (2; 4)
Q(0;1)

✓ coordinates of P 

✓ coordinates of Q

(2)

3.1.2

y = a(x - 2)² + 4
1 = a(0 - 2)² + 4
-3 = 4a
a = - 3
y = b^x
4 = b²
b = 2

✓ substitution

✓ answer

✓ substitution

✓ answer 

(4)

3.1.3

x ≥ 2 / x ≤ 2

✓✓ answer / antwoord

3.1.4

h(x) = 2 ^{f (x)} is a maximum when f (x) is a maximum
max valueof f (x) = 4 
max of h(x) = 2⁴ = 16 

✓ max. value of f(x) 

✓answer

(2)

3.2.1

? ≥ 1, ? ∈ ? OR ? ∈ (1; ∞)

✓ answer

(1)

3.2.2

p(x) = x² +1   /   r(x) = x² + 2x
p(x +1) - 2 = (x +1)² +1- 2
= x² + 2x +1+1- 2
= x² + 2x

Shift 1 unit to the left and 2units down 

OR

Turning Point of p(x) = (0 ; 1)
Turning Point of r(x) = (-1 ; -1)

Shift 1 unit to the left and 2 units down

✓ calculation

✓ 1 unit to the left

✓ 2 units down 

(3)

[14]

4.1

y =  -3   + 5
     0 +1
= 2

4.2

  -3    + 5 = 0
x +1

   -3    = -5
  x +1

-5x - 5 = -3
-5x = 2
x =- 2
      5

✓ simplification 

✓ answer 

(2)

4.3

✓ asymptotes

✓ x- and y-intercepts x

✓ shape

(3)

4.4

✓ substitution

✓ simplification

✓ reflection

(3)

[9]

5.1

f (x) = log₃ x
x = log₃ y
f ^-1(x) = 3^x

✓ interchanging x and y

✓ answer

(2)

5.2

f ^-1 is a reflection of f in the line y = x

✓ answer

(2)

5.3

y = log₃ x
-2 = log₃ k
∴ k = 3^-2

= 1 
   9

✓ substitution

✓ answer

(2)

5.4

0 < x < 1 
            9

OR 

log₃ x < -2 and x > 0
x < 1 and x > 0
      9
0 < x < 1 
            9

✓ ✓answer

(2)

5.5

x ≥ 1

✓✓ answer

(2)

[10]

6.1

effective rate =25,59% p.a

✓formula

✓ substitution

✓ answer

(3)

6.2

✓i and/en n

✓ substitution

✓ answer

(3)

6.3.1

✓ A and P

✓ substitution

✓ making n subject of the formula 

✓ answer (4)

6.3.2

✓ substitution

✓ answer 

(2)

6.3.3

Amount = R457 551,55 – R50 710.00
= R406 841,55

✓ answer

(1)

[13]

= 1 − 3(?² + 2?ℎ + ℎ²)
= 1 − 3?² − 6?ℎ − 3ℎ²

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

= lim   −6?ℎ − 3ℎ²
  ℎ→0         ℎ
 = lim    ℎ(−6? − 3ℎ)
   ℎ→0          ℎ
= lim(−6? − 3ℎ)
  ℎ→0 

= −6?  

Answer ONLY: 0 marks

✓ 1 − 3?² − 6?ℎ − 3ℎ²

✓ substitution

✓ common factor

✓ answer

(4)

✓ y = x + 2 + x ^-1

✓1 ✓ −?⁻²

(3)

y = 3x² - 2x +1
y' = 6x - 2

6x - 2 = 4
6x = 6
x = 1
y = 2 (1; 2)

✓ y ' = 6x - 2

✓ 6x - 2 = 4

✓ x-coordinate

✓ answer

(4)

8.1.1

f (x) = a(x + 2)(x - ²/₃ )(x - 3)
-16 = a(2 + 2)(2 -²/₃ )(2 - 3)
-16 = - 16 a
            3
a = 3

f (x) = 3(x + 2) ( x - ²/₃ )(x - 3)
= 3( x - ²/₃ )(x² - x - 6)
= 3(x³ - x² - 6x - ²/₃ x² + ²/₃ x + 4)
= 3x³ - 5x² -16x +12

✓ substitution of x-coordinates

✓ substitution of point

✓ value of a

✓ substitution

✓ removing brackets

(5)

8.1.2

?(?) = 3?³ − 5?² − 16? + 12

?^′(?) = 9?² − 10? − 16 = 0 (9? + 8)(? − 2) = 0
9? + 8 = 0 ??/?? ? − 2 = 0
? = − 8  ??  ? = 2
         9

? = 4900 (20,16)  
       243

B(− 8 ;  20,16)
      9

✓ ?^′(?) = 0

✓ factors

✓ x-values

✓ coordinates of P

(4)

8.1.3

✓x-intercepts

✓y-intercept

✓turning pts

✓ shape

(4)

f (x) = ax³ + bx² + 3x + 3  and g(x) =  f ⁿ(x) = 12x + 4

f '(x) = 3ax² + 2bx + 3
f ⁿ (x) = 6ax + 2b

6a = 12  and 2b = 4
a = 2   and  b = 2

✓ a = 2

✓ b = 2

(2)

12x + 4 = 0

x =- 1
       3
∴ x > - 1
          3

✓ x = - 1
           3

✓ answer

(2)

f ⁿ (x) < 0  for x < - 1     (concavedown )
                              3
f ⁿ(x) > 0   for  x > - 1     (concaveups )
                               3
x =- 1 is the x-coordinateof the point of inflection 3 where the concavity changes.
       3

✓ answer

✓ answer

(2)

9.1

EF = a - 2x
DE = tan 60⁰
BE

DE = x tan 60⁰
= √3x

Area = l x b
= √3x x (a - 2x)
= √3ax - 2√3x²

No mark for the answer

✓ EF = (a – 2x)

✓ tan ratio

✓ answer

✓ substitution

(4)

9.2

A(x) = √3ax - 2√3x²

A ' (x) = √3a - 4√3x = 0
x = -√3a
      -4√3

x = a 
      4

✓ derivative

✓ f ' (x) = 0

✓ answer

✓ substitution

✓ answer

(5)

10.1.1

P(F and S) =  67  / 0, 28
                     236

✓ answer

(1)

10.1.2

P(M ) ´ P(not S )
= 120 x 100
    236   236
= 0, 22

P(M ∩ not S ) =  51 
                          236
= 0, 22
∴ P(M ) x P(not S ) = P(M ∩ not S )
→ events are independent

✓ P(M) x P(not S)

✓ answer

✓ answer

✓ conclusion

(4)

10.2.1

✓ 1^st branch 

✓ 2^nd branch 

✓ outcomes

(3)

✓ setting up equation

✓ standard form

✓ answer

(3)