MARKING CODES

A

Accuracy

AO

Answer only

CA

Consistent accuracy

M

Method

R

Rounding

NPR

No penalty for rounding

NPU

No penalty for units omitted

S

Simplification

F

Correct formula

SF

Substitution in correct formula

QUESTION 1

1.1

1.1.1

x² - 8x - 33 = 0
( x -11)( x + 3) = 0  or   
∴ x = 11 or  x = -3
            

(8)2

2(1)

 4(1)  33

? Factors
SF        A
? x = 11
? x = -3        CA

(3)

1.1.2

x² - 7x = 10 (-3x -1)
x² - 7x = -30x -10
x² + 23x+10 = 0

x = - 0, 44 or      x = - 22, 56

?S       A

? SF    CA

?both values of x   CA

1.1.3

-2x² + 9x+ 5 < 0
(-2x - 1)( x - 5) < 0
C.V / K .W : - ½ and  5
Solution g : x < - ½ or x > 5

SF        A
?Critical Values CA
? x< - ½        CA
? x > 5 CA

1.2

P = 2(l + w)
90 = 2 (l + x)
45 - x = l
41² = x² + (45 - x)²
1681 = x² + 2025 - 90x + x²
0 = 2x² - 90x + 344
0 = x² - 45x + 172

x = 40, 78 or 4, 22
∴ width  is 4, 22 cm

OR

x = 40, 78 or 4, 22
∴ width is 4, 22 cm

?M (Pyth.)       CA
?S       CA
?S       CA
?SF     CA
?width          CA

OR 

?length in terms of x      A
?M (Pyth.)       CA
?S       CA
?S       CA
?SF     CA
?width / wydte            CA

1.3

x = y+ 3           and            y - x² = -2x - 3
substitute x into y - x² = -2x - 3
y - ( y+ 3)²  =  -2(y+ 3) - 3
y - y² - 6 y - 9 = -2y - 6 - 3
- y² - 3y = 0
(- y )( y+ 3) = 0
y = 0 or  y = -3
x = 0 or x = 3

?S   CA

? Factors  SF   CA

? Both y-values   CA

? Both x-values   CA

OR
x = y+ 3           and          y - x² = -2x - 3
x = 3 = y         and           y = x² - 2x - 3
equate y:           x - 3 = x² - 2x - 3
0 = x² - 3x
0  = x ( x - 3)
x = 0    or   x =3

y = 0    or  y = -3

OR

(5)

? equating Y

? S

CA

? Factors

CA

? x-values

CA

? y-values

CA

1.4

1.4.1

K = 8 + 32 + 1 = 41

? value of K

A

(1)

1.4.2

41 = 1 0 1 0 0 1₂

? 1 0 1 0 0 1₂
From 1.4.1

CA

(1)

[23]

QUESTION 2

2.1

Δ = b² - 4ac < 0
4 + 8m < 0
m < -½

?Discriminant  < 0    A
?S       CA

(2)

2.2

Δ = b² - 4ac
= (-b)²  - 4 (a)[-½]
= 0 + 4 = 4
Roots are real, rational and unequal

? SF    A
? S      CA
?Real and unequal   CA
?rational roots CA

(4)

[6]

QUESTION 3

3.1

3.1.1

? Exponential form /

Eksponensiële vorm

? S

A CA

3.1.2

OR

?Log property 

?Exponential form

?S

?Log property

OR

?Log property

?S

?Log property

?Log property

A

CA

CA

CA

A

CA

CA

CA

3.2

-2 (log 25 - log 4) = 4
   log 2 - log 5
RHS = -2 (log 5² - log 2² )
               log 2 - log 5
= -2 × 2(log 5 - log 2)
        -(log 5 - log 2)
=  4 = LHS

?Exponential form

?Log property

?Factors

A

CA

CA

(3)

3.3

3.3.1

?M

?Exponential property

?Exponential property

A

CA

CA

3.3.2

log₃ (x - 3) - log₃ 5 = 1
log₃ ( x - 3) = log₃3
           5
( x - 3) = 3
    5
x - 3 = 15
x = 18

OR

log₃ (x - 3) - log₃ 5 = 1
log₃ (x - 3) = log₃ 3 + log₃ 5
log₃ ( x - 3) = log₃ (3× 5)
x - 3 = 15
x = 18

?Log property

?S

OR

?Log property

?Log property

?S

CA

CA

A

CA

CA

3.4

x - 3(5i + 2) = 4 - 3i + yi
x - yi = 3(5i + 2) + 4 - 3i
= 15i + 6 + 4 - 3i
= 10 + 12i
x = 10
y =  -12

OR

x - 3(5i + 2) = 4 - 3i + yi
x - 15i - 6 = 4 + i (-3 + y)
x - 6 - 15i = 4 + i (-3 + y)
x - 6 = 4   and   -15 = -3 + y
x = 10   and   -12 = y

?x-value

?y-value

OR

?S

?x-value

?y-value

CA

CA

A

CA

CA

3.5

V = 110, 4 + 46,1i

θ =  22, 7º or θ = 180º + 22, 7º
= 202, 7º
z = 119, 64 cis (22, 7º) or   z = 119, 64 cis 202, 7º

?Substitution         A
?S       CA
?Ratio     CA
? θ    CA
?Polar form        CA

(5)

[23]

QUESTION 4

4.1

4.1.1

and           h(x) = ^a/_x   or   xy = a
3 = ^a/₁   3×1=a
∴3 = a

? SF

?value of  r

?value of a

4.1.2

0 ≤ y ≤ √10

? 0 and  √10   CA from 4.1.2
?Correct notation   A

(2)

4.1.3

B(-√10;0)

? Coordinates of  B   CA from 4.1.1

(1)

4.1.4

y= 0

x= 0

? x= 0  A 

? y= 0  A

(2)

4.2

4.2.1

k(x) = 2( x - 2)² - 2
k(x) = 2( x - 2)² - 2
k(0) = 2(0 - 2)² - 2
= 6
y- int. = 6

OR 

k(x) = 2( x - 2)² - 2
= 2(x² - 4x+ 4) - 2
= 2x² - 8x+ 6
y - intercept  6

?y-int   CA

OR

?S       A
?y-int. CA

(2)

4.2.2

TP (2; -2)

?x-coordinate     ?y-coordinate

(2)

4.2.3

k(x) = 2 ( x - 2)² - 2
= 2(x²  - 4x + 4) - 2
= 2x² -  8x+ 6
x - int . ; k(x) = 0
0 = x² - 4x + 3
= ( x - 1)( x - 3)
x= 1 or    x= 3

?Equate to 0          CA

?Factors         CA

?Both x-values            CA

(4)

4.2.4

x ∈ R

? x ∈ R       A

(1)

4.2.5

f:
?Shape   A
?y-intercept     CA

k:
?Shape   A
?both x-intercepts            CA
?y- intercept     CA
?Turning point        CA

(6)

4.2.6

y ≥ -2

? y ≥ -2   CA

(1)

[24]

QUESTION 5

5.1

A = P(1+ ni)
= 35(1 + 9 × 0,15)
= R82, 25

?SF

?S

A

CA

(2)

5.2

A = P(1+i )ⁿ
= 13 565 (1 + 0, 065)⁸
= 22 450

?F

?SF

?S

A

CA

CA

(3)

5.3

? SF  A

?S  CA

? Sum  CA

? SF  A

?S  CA

?Difference   CA

?SF  CA

?S  CA

? SF  A

?S  CA

? SF  A

? S  CA

?SF  A

?S  CA

?S  CA

?S  CA

[13]

QUESTION 6

6.1

?F  A

?SF  CA

?S  CA

?S  CA

? f '(x) = -2    CA

6.2

6.2.1

?3a   A

?a^{- 4}    CA

(2)

6.2.2

?S    A

? ^x/₂    CA

? 3x³   CA

6.2.3

S = ½ ft ²
ds = ft
dt
= πt

? πt   CA

6.3.3

f (x) = 3x²
f'(x) = 6x
Av.grad. = f (8)- f (2)
                     8 - 2
= 192 - 12
       8 - 2
6x = 30
x = 5

?6x     A

?S       CA

?Equating derivative and av. gradient   CA

? x= 5             CA

(4)

[16]

QUESTION 7

7.1

g(x)   = x³ - 12x - 16
g(-2) = (-2)³  -12 (-2) -16
= 0
∴ (x - 2) is a factor of (x)

?substitution by -2        A

?S       CA

(2)

7.2

g(x) = x³ - 12x - 16
0 = ( x + 2)(x² - 2x - 8)
0 = ( x + 2)(x + 2)(x - 4)
x = -2 or x = 4

?Equating to 0 A
?Quadratic factor   A
?Factors of quadratic factor CA
?values of x   CA

(4)

7.3

(0; -16)

? y-intercept         A

(1)

7.4

f(x) = x³ -12x -16
f'(x) = 3x² - 12
0 = 3x² - 12
0 = x² - 4
0 = ( x - 2)( x + 2)
x = 2 or  x = - 2
g (2) = (2)³  - 12(2) - 16 = -32
g (-2) = (-2)³  - (-2) -16 = 0
TP(-2; 0) and (2; -32)

?Derivative           A
?0       CA
?Factors       CA
?Both x values   CA
? (-2; 0)     CA
? (2; -32)   CA

7.5

?Shape    A

?y-intercept    CA

?x-intercepts    CA

?Both turning points   CA

(4)

7.6

h(x) = (x - 2)³  - 12(x - 2) -16

?h(x)   A

(1)

7.7

-2 > x or  x < 2

√  -2 > x   CA

? x < 2   CA

(2)

[20]

QUESTION 8

8.1

8.1.1

q = 820 - p

? q = 820 - p

A

(1)

8.1.2

Z= pq
= p (820 - p)
= 820 p - p²

?Substitution CA
?S       CA

(2)

8.1.3

Z = 820 p - p²
dZ = 820 - 2 p
dp
0 = 820 - 2 p
∴ p= 410

?S       CA

8.2

8.2.1

R ( x) = -50x² + 3200x -1860
R (15) = -50 (15)²  + 3200 (15) -1860
= R34 890

?Substitution  A
?S       CA

(2)

8.2.2

R (x) = -50x² + 3200x -1860
R' (x) = -100x + 3200
0 = -100x + 3200
100x= 3200
x = 32
R (x) = -50x² + 3200x - 1860
R (32) = -50 (32)²  + 3200 (32) -1860
= R49340

= artisan's maximum earnings

?S       CA
?Substitution CA
?S       CA

[11]

QUESTION 9

9.1

9.1.1

? 2x^½     A
? 4x^½    CA
? πx       A
? c         A

9.1.2

?S           A
? x³       CA
?x²        CA
  2
? -2x - c     CA

(4)

9.2

?A definite integral formula   A
? 2x²        CA
? -x³         CA
     3
?Substitution in A by 3     CA
?Substitution in A by 1,5  CA
?S       CA

[14]

TOTAL:

150