Page 1 of 22

SECTION A

QUESTION 1

In the diagram below:

DC CB

A is the centre of the circle.

E is the midpoint of AB.

The equation of line BA is: 3 7y x

DF is a tangent to the circle at F.

(a) Find the co-ordinates of E. (2)

4 2 5 1;

2 2

(3;2)

E

E

(b) Determine the equation of the circle, centre A, passing through point E.

Give the equation on the form 2 2 2( ) ( )x a y b r (3)

2 2 2

2

2

2 2

(4 3) (5 2)

1 9

10

( 4) ( 5) 10

r

r

r

x y

B(2;-1)

E

A(4;5)

F

D

C(10;3)

y

x

Page 2 of 22

(c) Find the gradient of line BC. (2)

3 1 1

10 2 2BCm

(d) Hence, or otherwise, determine the equation of line DC. (3)

2

2

(10;3)

3 2(10)

23

2 23

DCm

y x c

sub

c

c

y x

(e) Show, by calculation, that the co-ordinates of D are (6 ;11) . (3)

3 7 2 23

5 30

6

3(6) 7 11

D(6;11)

x x

x

x

y

(f) Find the size of ˆEBC . (5)

3

tan 3

71,6

1

2

1tan

2

26,6

ˆ 71,6 26,6 45

EB

BC

m

m

EBC

Page 3 of 22

(g) Find the length of DF. (6)

2 2 2

2

2 2 2

2

ˆ 90 (tan )

10

(6 4) (11 5)

40

( )

40 10

30

DFA chord

FA

AD

AD

DF AD FA pythag

DF

DF

[24]

Page 4 of 22

QUESTION 2

(a) Simplify:

2

2

(sin cos ) 1

sin 1

x x

x

(4)

2 2

2

2

sin 2sin .cos cos 1

cos

2sin .cos

cos

2sin

cos

2 tan

x x x

x

x x

x

x

x

x

(b) Solve for x in the interval [ 180 ;180 ] :

cos( 30 ) sin3x x . (7)

cos( 30 ) cos(90 3 )

30 (90 3 ) 360

30 90 3

4 120

30 90

30 (90 3 )

2 60

30 180

150 ; 60 ; 30 ;120

x x

x x k

x x

x

x k

or

x x

x

x k

x

[11]

Page 5 of 22

G

1

1

1

1

21

E

D

C

B

A

QUESTION 3

(a) Refer to the figure:

Below are three statements that refer to the above figure. What additional

information would be required, if any, to make each individual statement

true?

(1) 1 1 1ˆ ˆ ˆA E D

D must be on the circumference

(2) 1 1ˆB C

GB = GC or G is the centre of the circle

(3) 1 2ˆ ˆ 90C C

BE is the diameter

(3)

Page 6 of 22

O

S

R

Q

P

(b) Use the figure below to prove that ˆ ˆ 180S Q .

O is the centre of the circle. (6)

Given: Cyclic quad PQRS, O centre of the circle

RTP: ˆ ˆ 180S Q

Const.: Join PO and OR

Proof:

ˆ

ˆ 2 (2 )

ˆ 360 2 ( ' int)

ˆ 180 ( )

ˆ ˆ 180

Let S x

POR x at circum

reflexPOR x s around a po

Q x half at centre

S Q

[9]

Page 7 of 22

32

1

21

2

1

D

FE

C

B

A

QUESTION 4

In the figure:

ABCD is a cyclic quadrilateral.

AB = AF

(a) Prove that DE = EF. (3)

1

3

ˆ

ˆ ( )

ˆ ( )

Let F x

B x AB AF

D x ext of cyclic quad

ED EF

(b) If it is further given that ED bisects ˆCDF , prove that FB bisects ˆABC .

(3)

2

2

1 2

ˆ ( sec )

ˆ ( )

ˆ ˆ

ˆsec

D x bi ted

B x in same seg

B x B

FB bi ts ABC

[6]

Page 8 of 22

BDA

O

C

QUESTION 5

In the figure:

O is the centre of the circle.

AC BC

OD CB

Prove that:

(a) / / /ODB ACB (3)

ˆˆ1) 90 ( )

ˆ ˆ2) ( )

ˆ ˆ3) (3 )

/ / / ( )

In ODB and ACB

OBD C given

B B common

DOB CAB rd of triangle

ODB ACB AAA

(b) 22 .AD OA BC (4)

2

(/ / / ' )

( )

( )

2

2 .

OD DB OBs

AC CB AB

AD DB OD AB

OB OA radii

AD OA

CB AD

AD OA BC

[7]

Page 9 of 22

QUESTION 6

A consumer testing company studied three different brands of washing machines

to see how much water was used during each wash. Each washing machine was

tested 15 times. The box and whisker plots below show the results of this study.

Washing machine A

Washing machine B

Washing machine C

Number of litres used by washing machine.

(a) Which brand of machine (A, B or C) used up most water on average? (1)

C

(b) Which brand of machine (A, B or C) is the most predictable? (1)

B

(c) The results of Washing Machine C are shown below:

81 85 85 88 89 90 92 101 104 105 106 106 108 112 112

(1) Determine the standard deviation of the litres of water used. (2)

10,4

(2) Based on this data, how many litres of water would be used in 67%

of the washing loads? (2)

[97,6 10,4 ; 97,6 10,4]

[87,2 ; 108]

[6]

Page 10 of 22

5 10 15 20 25 30

1

2

3

4

5

6

7

8

9

10

QUESTION 7

The distance (𝑥) in kilometres that the staff at a certain school in Durban travel to

work each day is summarised in the histogram below:

(a) Is the data positively or negatively skewed? (1)

Skewed to the right, positively skewed

(b) Which of the intervals is the modal interval? (1)

Mode 5 - 10

(c) Use the histogram to complete the given table. (3)

Intervals Frequency Cumulative Frequency

0 ≤ 𝑥 < 5 7 7

5 ≤ 𝑥 < 10 9 16

10 ≤ 𝑥

< 15

8 24

15 ≤ 𝑥

< 20

3 27

Kilometres

Fre

qu

ency

Page 11 of 22

20 ≤ 𝑥

< 25

2 29

25 ≤ 𝑥

< 30

1 30

(d) Use your frequency table to draw an ogive below. Label your axes. (3)

(e) Determine the following using your ogive:

(1) the interquartile range (2)

14,1- 5,6 = 8,5

(2) the percentage of these staff members that stayed between 4km and

14km from the school? (2)

Page 12 of 22

22 4

100 60%30

[12]

75 marks

Page 13 of 22

3

3

2

1

2

1

32

1

F

E

D

C

BA

SECTION B

QUESTION 8

(a) In the figure:

ABC is a tangent to the circle at B.

1 3ˆ ˆD D

Prove that:

(1) DCBF is a cyclic quadrilateral. (4)

1 2 3

2 1 3 1

ˆ ˆ (tan )

ˆ ˆ ˆ( )

( int )

B D AC chord EB

D D D

DCBF cyclic ext opp

(2) FC is a tangent to the circle FED. (4)

3

3 2

2

ˆ ˆ (tan )

ˆ ˆ ( )

ˆ ˆ

tan (tan )

E B AC chord BD

B F in same seg

F E

FC to circle FED chord converse

Page 14 of 22

xx

y

30°

CA

D

B

(b) In the figure:

ABCD is a cyclic quadrilateral.

AD = DC = x

AC = y

ˆ 30B

Show that 2 3y x (5)

2 2 2

2 2 2

2 2

ˆ 150 ( )

2 . cos150

2 3

(2 3)

2 3

D opp of cyclic quad

y x x x x

y x x

y x

y x

Page 15 of 22

Q

O

P

K

L

M

N

(c) In the figure (not drawn to scale):

OM // PL

KN : KO = 8 : 3

LM = 3KL

Calculate:

(1) Area of KMO

Area of NMO

(3)

1.

321 5

.2

KO hKO

ONON h

(2) LQ

QN (3)

( / / )

3 .3 9

4 4

( / / )

994

5 20

PO MLPL OM

OK MK

p kPO p

k

LQ POPL OQ

QN ON

pLQ

QN p

[19]

Page 16 of 22

QUESTION 9

The sketch below shows the graphs of 𝑓(𝑥) = − sin 𝑥 and 𝑔(𝑥) = cos𝑥

2 for

𝑥 ∈ [−180°; 180°]. The coordinates of 𝐴, the point of intersection of the two

graphs, are (𝑡;√3

2).

(a) Determine the value of 𝑡. (2)

3sin

2

60

t

t

(b) Use the sketch to solve the following equation: cos𝑥

2= −2 sin

𝑥

2cos

𝑥

2

for 𝑥 ∈ [−180°; 180°]. (3)

cos 2sin .cos sin

2 2 2

180 ; 60 ;180

x x xx

x

(c) If 𝑓 is translated 10° to the left and 2 units up, what will the new equation

of 𝑓 be? (2)

sin( 10 ) 2y x

[7]

(𝑡;√3

2)

𝑓

𝑔

𝐴

Page 17 of 22

QUESTION 10

(a) Simplify: sin170 .cos10

sin(20 ).cos cos(20 ).sinx x x x

(4)

sin10 .cos10

sin(20 )

1sin 20

2

sin 20

1

2

x x

(b) Prove the following identity:

cos cos2 1 1

sin sin 2 tan

x x

x x x

(5)

2

2

cos (2cos 1) 1

sin 2sin cos

cos 2cos

sin (1 2cos )

cos (1 2cos )

sin (1 2cos )

cos

sin

1

tan

x xLHS

x x x

x x

x x

x x

x x

x

x

x

RHS

Page 18 of 22

(c) Determine the general solution for :

27sin 2cos 5 0 (7)

2

2

2

7sin 2(1 sin ) 5 0

7sin 2 2sin 5 0

2sin 7sin 3 0

(2sin 1)(sin 3) 0

1sin sin 3

2

180 30 360

210 360

360 30 360

330 360

or

k

k

or

k

k

(d) If 31

sinsin

x ax and

31cos

cosx b

x ,

Prove that tan .b

xa

(5)

2 3 2 3

2 2

3 3

2

2 33

2 3 2

3

3

3 2

33

3

1 sin sin 1 cos cos

1 sin 1 cossin cos

1 sin

sin costan

1 coscos sin

tantan

tan

tan

x a x x b x

x xx x

a b

xx x bax

xx a x

b

bx

a x

bx

a

bx

a

[21]

Page 19 of 22

A

BO

Q(4;3)

P

y

x

QUESTION 11

The two circles in the diagram represent two interlocking gears, which touch at

point Q (4 ; 3).

The circles have the following equations:

x2 + y

2 – 25 = 0 and x

2 – 12x + y

2 – 9y + 50 = 0

(a) Show that the co-ordinates of P are (6; 4½). (3)

2 2

2 2

2

2

1 112 36 9 4 50 36 4

2 2

1 25( 6) 4

2 4

16;4

2

x x y y

x y

P

(b) Determine the equation of common tangent AB. (4)

3 4

4 3

4(4;3)

3

43 .4

3

25 4 25

3 3 3

OB ABm m

y x c sub

c

c y x

Page 20 of 22

(c) If the larger gear makes one full revolution, how many times will the

smaller gear turn completely? (4)

52 2 5

2

5 10

small bigC C

C C

twice

(d) Find the area of AOB . (3)

2

25 250; ;0

3 4

1 25 25 625

2 4 3 24

A B

Area units

(e) Another tangent to circle O, drawn from A touches the circle at C.

Determine the length of CQ. (2)

(4;3) ( 4;3)

8

Q C

CQ units

[16]

Page 21 of 22

QUESTION 12

The diagram shows a pyramid

shaped ‘cone’.

Each face is an isosceles triangle

with base angles of 640. The base

is a square of side 6cm.

EG is the slant height of the

pyramid.

EF is the perpendicular height of

the pyramid.

𝑉𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑎 𝑝𝑦𝑟𝑎𝑚𝑖𝑑 = 1

3× 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑏𝑎𝑠𝑒 × 𝑝𝑒𝑟𝑝𝑒𝑛𝑑𝑖𝑐𝑢𝑙𝑎𝑟 ℎ𝑒𝑖𝑔ℎ𝑡

(a) Determine the length of edge AE. (3)

3 ( )

3cos64

6,8

BG GA cm EG BA in isos

AE

AE cm

(b) Calculate the height EF. (4)

2 2 2

2 2 2

tan 643

6,2

( )

6,2 3

5,4

EG

EG cm

EF EG FG pythag

EF

EF cm

6 cm

64F

G

D

E

C

B

A

Page 22 of 22

(c) Determine the volume of the pyramid. (2)

3

1.(6 6).

3

136 5,4

3

64,8

V EF

V

V cm

(d) The pyramid is to be wrapped in a single layer of gold foil, with no

overlaps.

Calculate the total area of foil that would be needed. (3)

2

14 6 6 6

2

14 6 6,2 6 6

2

110,4

TSA EG

TSA

TSA cm

[12]

75 marks

Total: 150 marks