DURBAN GIRLS' COLLEGE
TRIALS EXAMINATION 2010
MATHEMATICS PAPER II
TIME: 3 HOURS
150 MARKS
EXAMINERS: Mrs M Dwyer
Mrs K Burne
MODERATOR: Mrs G Hawkey
PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY.
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This question paper consists of 10 pages plus a diagram sheet. All questions
must be answered in the answer book provided.
Answer Section A and Section B in separate booklets.
Answer all questions and show all working.
Number your answers exactly as the questions are numbered.
Do not redraw diagrams.
All answers must be given correct to one decimal place where necessary,
unless stated otherwise.
Approved calculators may be used, unless stated otherwise.
2
SECTION A
1. P(-4; a), Q(6; -3) and R(2;3) are 3 points in the Cartesian plane as shown
below, with R̂  90.
y
R(2; 3)

S
x
P(-4; a)
Q(6; -3)
a)
b)
c)
d)
e)
Determine the gradient of RQ
Calculate the size of 
Prove that a = -1
Determine the equation of RQ
Determine the co-ordinates of point S, the x intercept of RQ, and
show that it is also the midpoint of RQ
(2)
(2)
(3)
(3)
(4)
[14]
2.
y
M(-3; 2)
P(0; -2)
M(-3; 2) is the centre of the circle above. Determine the equation of
the tangent at point P(0; -2)
[4]
3. i) Give the co-ordinates of T(-2; 3) if it is transformed as follows:
a) reflected about x = 0
b) rotated through 180
c) translated according to the rule (x; y) → (x – 1; y + 2)
(2)
(2)
(2)
3
ii)
10
D
y
8
D
6
A
C C
B
4
A
A
B
2
B
D
C
-10
-5
5
-2
x
10
C
B
15
D
-4
A
-6
Describe each of these transformations shown in above:
a) ABCD  ABCD
b) ABCD  ABCD
c) ABCD  ABCD
(2)
(2)
(2)
y
iii)
y
B(- 6; 6)
Figure A: Small arrow
Figure C: Large arrow
B(- 3;3)
x
x
a) Write down the factor of enlargement converting B to B.
b) If the perimeter of A is 10, determine the perimeter of C
c) If the area of A is 15, determine the area of C
(1)
(1)
(1)
[15]
4
4. i)
The above is a box-and-whisker diagram showing the number of points scored
in 19 rugby matches.
a) Write down the 5 number summary.
b) Discuss whether the data is symmetrical or skewed.
(2)
(2)
ii)
Raw scores
x-
(x -
2
7
8
10
14
26

a)
b)
c)
d)

What is the value of n in the table above?
Calculate the mean.
Copy and complete the table.
Calculate the standard deviation.
(1)
(2)
(4)
(4)
[15]
5. i) Calculate the value of (sin A – cos A)2 – sin2 A if A = 20
(2)
ii) Solve for :
a) sin  =
3
2
(3)
b) sin2  + cos2  = 1
(2)
[7]
5
6. Simplify as far as possible:
a) (1 – sin2 x) tan2 x
b)
(3)
tan A. sin( 90  A )
sin(180  A )
(4)
[7]
7.
y
O
x
A
P(2; -2 3 )
Determine the value of:
a) OP
b) cos A
c) sin(180 - A)
(2)
(1)
(2)
[5]
8.
y
This graph shows f(x) = sin 2x
for x  [-90; 180].
A and B are the turning points
of the graph.
A
x
-90°
0°
90°
180°
D
C
B
Use the graph to answer the following questions:
a) Write down the period of f
b) Write down the co-ordinates of A and B
c) If D is the point (160; -0,6), and CD is parallel to the x-axis, write
down the co-ordinates of C
d) If the graph of f is shifted downwards by 1 unit, what will the equation
of this new graph be?
[8]
(1)
(4)
(2)
(1)
6
9.
12
11
1
10
2
9
3
8
The clock alongside shows 5
o’clock. If the hour hand (at 5)
is 4cm long and the minute
hand (at 12) is 5cm long,
calculate the distance between
the ends of the hands.
4
7
5
6
[4]
SECTION B
1.
a) A circle has centre M(3; -2). Show, using analytical methods,
whether it is possible for both points A(1; 2) and B(1,5; 1,5)
to lie on its circumference.
(5)
b)
N
M
Circle B
Circle A
Circles A and B touch each other externally. Circle A has centre M
and the equation x2 + y2 -2x -2y = 2. Circle B has centre N and
equation x2 + y2 -8x -10y = k, where k is a constant.
Determine the co-ordinates of M and N and hence calculate the value
of k.
(8)
7
c)
y
 A(2; 2 3 )
x
O
B
C
M (a;b)
Two circles with centre O(0; 0) and M(a; b) touch externally at B.
i) Determine the equation of circle O if it passes through A(2; 2 3 ).
(2)
ii) Circle centre M touches the y-axis at C(0; -8). Determine the
co-ordinates of M.
(6)
[21]
2.
Crop yield and rainfall
50
Yield in tons
40
30
20
10
0
0
50
100
150
200
Rainfall in mm
a) The scatter plot above shows the yield of maize and rainfall over a
period of time.
i)
ii)
What does this scatter plot suggest about the relationship between
crop yield and rainfall?
On the axes on the diagram sheet, draw in the line of best fit.
(2)
(2)
8
iii)
Estimate the yield when the rainfall is 200mm.
(2)
b) This cumulative frequency graph gives the age distribution of people
living in Village A
No.
of
people
Age in years
Use your graph to estimate
i) the number of people less than 25 years old
ii) the median age
iii) the inter-quartile range
(1)
(1)
(2)
[10]
9
3.
a) If sin 29 =
a , determine the value of the following in terms of a:
cos73cos15 + sin73sin15
b)
(3)
i) Prove that sin(45 + x).sin(45 - x) = ½ cos 2x
(5)
ii) Hence determine the value of sin75. sin15
(3)
c) Prove that
1  cos 2x  sin x
= tan x
sin 2x  cos x
(6)
d) Solve for x if cos(x - 30) = sin 3x where x  [-90; 90]
(6)
A
e)

In the triangle alongside, prove that
sin  = 2sin cos 


B
C
(4)
f)
T
AT is a vertical mast. TB and TC
are equal wire stays. TĈB  θ .
The angle of elevation of T from B
is . AB = p units.
A
p


B
C
i) Determine TB in terms of p and .
ii) Prove that BC =
(2)
2 p cos θ
cos α
(6)
[35]
10
4.
The BP logo alongside has been very visible
recently as a result of the disastrous oil spill in
the Gulf of Mexico.
In the diagram below, the logo has been drawn on a set of axes, with the
origin at its centre. A is the point(1,5; 3,8)
y
A(1,5; 3,8)
O
x
B
Determine the co-ordinates of B.
[5]
11
DIAGRAM SHEET
NAME: ___________________________________________
Hand this sheet in with your answer booklet for Section B.
SECTION B
2
Crop yield and rainfall
50
Yield in tons
40
30
20
10
0
0
50
100
Rainfall in mm
150
200