QUESTION 1
1.1.1
x² - 2x - 24 = 0
(x - 6)(x + 4) = 0
x = 6 or x = -4 (3)

1.1.2
2x² - 3x - 3 = 0
x = 3 ± √(-3)² - 4(2)(-3)
                2(2)
x = 3 ± √33
          4
x = 2,19 or x = -0,69 (3)

1.1.3
x² + 5x ≤ -4
x² + 5x + 4 ≤ 0
(x + 4)(x + 1) ≤ 0
Critical values: x = -4 or x = -1
-
4 ≤ x ≤ -1 OR x ∈ [–4 ; –1] (4)

1.1.4
√x + 28 = 2 - x
(√x + 28)² = (2 - x)²
x + 28 = 4 - 4x + x²
x² - 5x - 24 = 0
(x - 8)(x + 3) = 0
x ≠ 8 or x = -3

• squaring both sides
• standard form
• factors
• answers with selection (4)

1.2
2y = 3 + x
x = 2y - 3 … (1)
2xy + 7 = x² + 4y² … (2)
2y(2y - 3) + 7 = (2y - 3)² + 4y²
4y² - 6y + 7 = 4y² - 12y + 9 + 4y²
4y² - 6y + 2 = 0
2y² - 3y + 1 = 0
(2y  -1)(y - 1) = 0
y = ½ or y = 1
x = -2 or x = -1
OR
2y = 3 + x
y = ³/₂ + ^x/₂ …(1)
2xy + 7 = x² + 4y² … (2)
2x(³/₂ + ^x/₂) + 7 = x² + 4(³/₂ + ^x/₂)²
3x + x² + 7 = x² + 9 + 6x + x²
x² + 3x + 2 = 0
(x + 2)(x + 1) = 0
x = -2 or x = -1
y = ½ or y = 1
(6)

1.3
ⁿ/_m = ^p/_n
n² = mp
Δ = b² - 4ac
Δ = n² - 4mp , but n² = mp
Δ = n² - 4n² 
Δ = - 3n²
n² > 0
-3n² < 0
Δ < 0 ⇒ x is a non-real number
OR
Δ = mp - 4mp
Δ = -3mp
mp > 0
-3mp < 0
(4)
[24]

QUESTION 2

2.1
⁹⁰/_x - ⁸¹/₉₀
81x = 8100
x = 100 (2)
OR
x = 90 x ¹⁰/₉ 
x = 100 (2)

2.2
S_n = a(1 - rⁿ)
          1 - r
S_n = 100(1 - (0,9)ⁿ)
              1 - 0,9
S_n = 100(1 - (0,9)ⁿ)
                0.1
S_n = 1 000(1 - (0,9)ⁿ) (2)

2.3
S_∞ =  a  
       1 - r
S_∞ =  100  
        1 - ⁹/₁₀
S_∞ = 1000
OR
S_∞ = lim [1 000(1-(0,9)ⁿ)]
S_∞ = 1000
OR
substitution
answer (2)
OR
S_∞ = lim [1 000(1 - (0,9)ⁿ)]
       n→∞
S_∞ = 1000
[6]

QUESTION 3
3.1
–82 (1)

3.2 

2a = -2 ∴ a = -1
3a + b = 23 ∴ 3(–1) + b = 23 ∴ b = 26
a + b + c = -145 ∴ –1 + 26 + c = –145 ∴ c= -170
∴T_n = -n² + 26n - 170

OR
2a = -2 ∴ a = -1 
c = -145 + (-2) - 23 = -170
∴T_n = -n² + bn - 170
b = 26
∴T_n = -n² + 26n - 170
(3)

3.3
T_n = bn + c 
T_n = -2n + 25
- 2n + 25 = -121
- 2n =  -146
n = 73
Between T₇₃ and T₇₄
or
Tn = a + (n - 1)d
= 23 + (n - 1)(-2)
= 25 - 2n

OR
T_n+1 - T_n = -(n + 1)² + 26(n + 1) -170 - (-n² + 26n - 170)
-121 = -2n + 25
n = 73
Between T₇₃ and T₇₄

3.4
n = - b  = - 26  = 13
      2a      2(1)
T₁₃ = -1
∴ add 2

OR
T'n = -2n + 26 = 0
n = ₁₃
T₁₃ = -(13)² + 26(13) - 170 =-1
13
∴ add 2
[11]

QUESTION 4
4.1
a = 5 and d = 2
T₅₁ = 5 + (51 - 1)(2)
= 105
(3)

4.2
S_n = ⁿ/₂[2a + (n - 1)d ]
S₅₁ = ⁵¹/₂(5) + (51 - 1)2]
= 2 805
or
S₅₁ = ⁵¹/₂[5 + 105]
= 2 805 (2)

4.3

4.4
T₁ = -3     T_{4 999} = -2(4 999) - 1 = - 9 999

= (5 + 7 + 9 + … + 9 999 + 10 001 + 10 003) +(–3 – 5 – 7 – 9 – … – 9 999)
= 10 001 + 10 003 –3
= 20 001

OR
S_{4 999} = 4 999[2(-3) + (4 999 - 1)(-2)] = - 24 999 999
                  2
S_{5 000} = 5000((2)(5) + (5 000 - 1)(2)) = 25 020 000
                 2
 = 25 020 000 – 24 999 999
= 20 001 
(4)
[10]

QUESTION 5
5.1
x = 3
y = 2

5.2 
x ∈ R, x ≠ 3 
OR
x ∈ (-∞ ;3) ∪ (3;∞)
OR
x< 3 or x >3

5.3
0 =  -1   + 2 
     x - 3
-2x + 6 = -1
x = ⁷/₂
x-int (⁷/₂;0)

5.4
y-int (0;⁷/₃)

5.5

asymptotes
intercepts with the axes
shape
[10]

QUESTION 6
6.1
f (x) = log₄x
2 = log₄k
4² = k
∴k = 16
(2)

6.2
-1 = log₄x ∴ x = ¼
¼ ≤ x ≤ 16 or x ∈ [¼; 16]
(2)

6.3
f (x) = log₄x
y = log₄x
x = log₄y
y = 4^x
(2)

6.4
x < 0 
OR
x ∈ (-∞ ; 0)
[8]

QUESTION 7
7.1
B(–4 ; 0)
D(6 ; 0) 
(2)

7.2
f (x) = x² - 2x - 24
x_tp = - b OR 2x - 2 = 0 OR x = -4 + 6
         2a                                       2
x = -(-2)
      2(1)
x_tp = 1
y_tp = f(1)
= 1² - 2(1) - 24
= -25
C(1 ; –25)
(2)

7.3
y = -25
OR
y ∈ [-25 ;∞) 
(1)

7.4.1
m_AE = tan14, 04º = 0, 25 = ¼
(1)

7.4.2
m_tang = -4 
f ' (x) = 2x - 2 
2x - 2 = -4 
x_T = -1
y_T = -21

7.5
m_line = ¼
y + 9 =  ¼(x + 3) 
OR
-9 =¼(-3) + c

y + 9 =¼x + ¾
OR 
c = ³³/₄ =-8, 25
y = ¼x - ³³/₄ 
OR
y = 0, 25x = -8, 25

x² - 2x - 24 = ¼x - ³³/₄
4x² - 8x = 96 = x - 33
4x² - 9x - 63 = 0
(4x - 21)(x + 3) = 0
x = ²¹/₄ = 5,25 or x -3
(6)
[17]

QUESTION 8
8.1
A = P(1 - i)ⁿ
A = 980 000(1 - 0, 092)⁷
A = R498 685,82 (3)

8.2
A =  P(1 + i)ⁿ
116 253,50 = 75 000(1 + 0,068)⁴ⁿ
                                           4 
1,550 046 667 = (1, 017)⁴ⁿ
log(1,550 046 667) = 4n log(1, 017)
4n = log(1, 550 046 667) or 4n = log_1,017(1,550 046 667)
                log(1, 017) 
4n = 25,99 …
n = 6,50 years
(4)

8.3.1
F = x [(1 + i)ⁿ - 1]
                i
450 000 = 
x = R6 068,69
(3)

8.3.2

1. 
   
   P = R419 948,32
   OR
   Balance = A - F
   
   =R12 887 702,20 – R12 467 749,81
   =R419 952,39
   (3)
2. Total paid =11 058,85 x 21 x 12 = 2 786 830, 20 
   Loan Paid = 1 050 000 - 419 948,32 = 630 051, 68
   Interest paid = 2 786 830,20 – 630 051,68
   = R2 156 778,52
   OR
   Total paid =11 058,85x 21x12 = 2 786 830, 20
   Loan Paid = 1 050 000 - 419 952,39 = 630 047, 61
   Interest paid = 2 786 830,20 – 630 047,61
   = R2 156 782,59
   OR
   Interest paid
   = 11 058,85 × 21 × 12 – (1 050 000 – 419 948,32)
   = 2 786 830,20 – 630 051,68
   = R2 156 778,52
   [16]

QUESTION 9
9.1
f ' (x) = lim f (x + h) - f (x)
           h→0          h
f ' (x) = lim 2(x + h)² - 3(x + h) - (2x² - 3x)
           h→0                    h
f ' (x) = lim 2x² + 4xh + 2h² - 3x - 3h - 2x² + 3x
           h→0                        h
= lim 4xh + 2h² - 3h 
  h→0       h
= lim h(4x + 2h - 3)
   h→0      h
= lim(4x + 2h - 3)
   h→0
f ' (x) = 4x - 3

OR
f (x) = 2x² - 3x
f (x + h) = 2(x + h)² - 3(x + h)
f (x + h) = 2x² + 4xh + 2h² - 3x - 3h
f (x + h) - f (x) = 4xh + 2h² - 3h
f ' (x) = lim f (x + h) - f (x)
           h→0         h
= lim 4xh + 2h² - 3h 
  h→0        h
= lim h(4x + 2h - 3)
   h→0         h
= lim(4x + 2h - 3)
   h→0
∴ f ' (x) = 4x - 3

9.2.1
y = 4x⁵ - 6x⁴ + 3x
^dy/_dx = 20x⁴ - 24x³ + 3

9.2.2

(4)

[12]

QUESTION 10
10.1
h(x) = ax³ + bx²
h ' (x) = 3ax² + 2bx
h ' (4) = 0
48a + 8b = 0
6a + b = 0 ...(1)
h(4) = 32
64a + 16b = 32
4a + b = 2 ...(2)
(1) – (2): 6a + b = 0
4a + b = 2
2a = – 2
a = – 1
4(– 1) + b = 2
b = 6

10.2
h(x) = -x³ + 6x²
- x³ + 6x² = 0
x² (-x + 6) = 0
x = 0 or x = 6
∴A(6 ; 0)
(3)

10.3.1
0 < x < 4 or 0 ≤ x ≤ 4
OR
x (0 ; 4) or x [0 ; 4]
(2)

10.3.2
x > 2 
OR
x (2 ; )

10.4
f (x) = h(x - 1) = -(x - 1)³ + 6(x - 1)²
f (0) = 7
7 < k < 32 or k (7 ; 32) 
(3)
[15]

QUESTION 11
Time = ²⁰/_x
Cost = (water cost per hour × time) + (kms × R/km)
C(x) = 1,6 x (²⁰/_x) + 20(1,2 + ^x/₄₀₀₀)
C(x) = ³²/_x + 24 + ^x/₂₀₀
C ' (x) = - 32 +   1    = 0
               x²     200
x² = 6 400
x = 80 km/ h
[7]

QUESTION 12
12.1.1
No, because P(A and B) ≠ 0 (1)

12.1.2

1. P(A and B) = 0,3   P(only B) = 0,2
   P(A and B) = P(A) x P(B)
   0,3 = P(A) x 0,5
   P(A) = 0,6
   P(only A) = 0,3 (4)
2. 
   
   P(not A or not B) = 0, 2 + 0, 2 + 0,3 = 0, 7
   OR
   P(not A or not B) = 1-  P(A and B) = 1 - 0,3 = 0, 7
   OR
   P(A^/ or B^/) = P(A^/) + P(B^/) – P(A^/ and B^/)
   = 0,4 + 0,5 – 0,2 = 0,7

12.2.1
P(novel) = ³/₁₂ = ¼ (1)

12.2.2
12! = 479 001 600 (2)

12.2.3

(4)
P(start with poetry, end with drama AND all novels together)
= 5 x 3! x 8! x 4
          12!
=  1  
   99
[14]
TOTAL: 150