1
SECTION A
QUESTION 1
y
A
B
M

D(2;3)
C
O
X
ABCD is a rectangle with vertices on the circumference of circle centre M.
The equation of the circle is x2 + y2 − 10x − 8y + 31 = 0 and D is the point (2;3).
The equation of AD is y − 2x + 1 = 0.
a)
Prove that M is the point (5; 4)
(4)
b)
Write down the length of the diameter DB, leaving the
answer in surd form.
(1)
c)
Determine the co-ordinates of B.
(4)
d)
Determine the equation of BC.
(3)
e)
Determine the equation of the tangent to the circle at D.
(3)
[15]
2
QUESTION 2
A
2θ
2
1
Q 90°-θ
a
2
1
θ
B
R
The diagram represents an aerial view of a school’s grounds. R̂1   ,
Q̂  (90   ) and QR = a.
i) Express AR in terms of a and θ (in simplest form).
a
ii) Hence show that AB =
units.
2
 2  2 ,
(4)
(2)
[6]
QUESTION 3
The straight line joining A(2;3) and C(5;−3)
has an inclination θ. EC cuts the x-axis at D.
AĈE  36,8
a) Calculate θ correct to 1 decimal place.
(3)
b) Determine the size of D̂1
(2)
c) Determine the co-ordinates of D.
(4)
Y
A(2;3)
E
1
D
O
θ
B
X
36,8°
C(5;−3)
[9]
3
QUESTION 4
a) Solve for x if sin(2x + 30o) = 0,65 and x  [-180o ; 180o]
(6)
b) Determine the general solution if sin(  17o) = cos(20o + 2)
(7)
3
12
and cos B =
, A  [0°; 270°] and B  [−180°; 0°].
5
13
Calculate sin (A − B) without a calculator and with the use of a diagram. (7)
[20]
c) sin A = −
QUESTION 5
The cumulative frequency curve (ogive) below gives the final examination marks
(out of 100) of 50 learners.
Y
cumulative
frequency
50
40
30
20
10
X
O
30 40 50 60 70 80 90 100
Mark %
a) Use the information on the curve to draw a box and whisker diagram.
(5)
b) i) If the cumulative frequencies are 2; 4; 11; 24; 35; 46 and 50 complete the
following table. The table is to be filled in on the answer sheet.
4
Marks
30< x ≤40
40 <x ≤50
Cumulative
frequency
2
4
11
24
35
46
50
Frequency (f)
Midpoint (m)
2
35
(5)
ii) By filling in the blank column on the table, calculate the estimated
mean.
(2)
iii) What is the modal class?
(1)
iv) How many pupils achieved 50% or less for the examination?
(1)
v) What percentage of learners achieved above 80%?
(2)
[16]
QUESTION 6
a) Prove that
1  cos x  1
1 



1  cos x  sin x tan x 
2
b) For what values of x is the identity above undefined
if x Є[−180°;180°]?
(5)
(5)
[10]
P.T.O. ……SECTION B
5
SECTION B
QUESTION 7
The average daily maximum temperature in the Kruger National Park for the four
seasons is given below
Spring
30,2°C
Summer
32,3°C
Autumn
29,4°C
Winter
25,8°C
Give all answers correct to 1 decimal digit.
a)
Calculate the annual mean temperature.
(1)
b)
Calculate the variance
(3)
c)
If k (a constant value) is added to each of the four seasonal
averages, determine the effect of k on the annual mean
temperature and variance. Show your working.
QUESTION 8
P
(4)
[8]
Y
 Q(0 ; 2)
X
R(−8; −2)

Refer to the diagram.
The circle centre P touches the y-axis at the point Q(0;2) and
passes through R(−8;−2).
a) Find the equation of the circle.
(6)
b) The circle moves down into the third quadrant so that both the x- and y-axes
are tangents.
What is the equation of the new circle?
(2)
c) If the given circle is enlarged by a factor of 1,2 through
the origin, what is the radius of the new circle?
(1)
d) By what scale factor will the area of the new circle increase?
(1)
[10]
6
QUESTION 9
 PQR is an equilateral triangle with sides x units. QP, RQ and PR are produced
their own length to A, B and C respectively. The resulting  ABC is also an
equilateral triangle.
x
(a)
(b)
Find the area of  PQR in terms of x.
Find the length of AC in terms of x.
(c)
Find the ratio:
areaPQR
area ABC
(2)
(3)
(3)
[8]
7
QUESTION 10
Y
C
D
F
E
X
o
B (3 ; 1)
G
A (3 ; 2)
H
In the given sketch, A is the point ( 3 ;−2) and B the point ( 3 ; −1).
a)  OAB has been transformed by reflection and/or rotation, to create  OEF,
OCD, and ΔOGH
For each of the questions below, write down the co-ordinates of the required
vertex and describe the transformation of  OAB in words.
i) If C is the point (−2; 3 ); write down the co-ordinates of D.
ii) If F is the point (1; 3 ); write down the co-ordinates of E.
iii) If H is the point (−1; 3 ); write down the co-ordinates of G.
b)
(3)
(3)
(3)
Through what angle must ΔOEF rotate (anti-clockwise) to result in
ΔOGH?
(1)
[10]
QUESTION 11
a) i)
ii)
Express cos2A in terms of cos2A.
Hence without using a calculator show that
cos15° =
2 3
2
b) If tan40° = k, express
(5)
2 sin 20 cos 20
2  4 cos 2 20
in terms of k.
(4)
[9]
8
QUESTION 12
a) Prove that
cos3 = cos – 4sin2cos
b) Hence write down the maximum value of
11cos − 44sin2cos − 3
(4)
(3)
[7]
QUESTION 13
Circle (x − 1)2 + (y − 1)2 = 4 has centre A
and circle (x − 4)2 + (y − 5)2 = k + 41 has centre B.
Determine the value of k if the two circles touch each other externally.
[7]
QUESTION 14
4
V = 3 𝜋𝑟 3 ;
A sphere of radius r fits exactly
into a cone. The sphere touches
the sides and top of the cone
at P, R and Q respectively as
shown in figure 1.
AP=PC=CR=RB=AQ=x cm.
1
V = 3 𝜋𝑟 2 ℎ
Q
A
B
R
P
Fig 1
C
A cross-section of the sphere and cone is shown
in figure 2. O is the centre of the circular crosssection of the sphere.
12
a)
If the volume of the sphere is
,
3
show that r= 3
(3)
b)
If OC= 2 3 , calculate the value
of x.
(2)
c)
Hence calculate the volume of the cone in
terms of .
(3)
Q
A
x
Fig 2
B
O
r
R
P
x
C
[8]
9
QUESTION 15
ABCDE is a regular pentagon that has been superimposed on the flower.
N.B. The pentagon is NOT drawn to scale.
A(4 ; 5)
If the co-ordinates of A are (4;5), find the length of one side of the pentagon.
All calculations must be carried out correct to 1 decimal place.
[7]
10
ANSWER SHEET
NAME:
QUESTION 5
Y
cumulative
frequency
50
40
30
20
10
X
30 40 50 60 70 80 90 100
Marks
30< x ≤40
40 <x ≤50
Cumulative
frequency
2
4
11
24
35
46
50
Frequency (f)
2
Mark %
Midpoint (m)
35