1
ST STITHIANS BOYS’ COLLEGE
GRADE 12 CORE MATHEMATICS
PAPER 3
AUGUST 2010
2 hours
100 marks
Name:
Teacher:
Instructions:
1. An approved, non-programmable calculator may be used unless stated
otherwise.
2. All answers are to be given correct to ONE decimal digit where necessary,
unless stated otherwise.
3. All working must be shown.
4. This examination paper consists of 13 pages.
5. All questions are to be answered on the question paper in the space provided.
6. There is a separate diagram sheet (on green paper) for you to use, but it is not
to be handed in.
7. There is a separate formula sheet.
Question 1
(4)
Question 2
(9)
Question 3
(14)
Question 4
(12)
Question 5
(6)
Question 6
(10)
Question 7
(8)
Question 8
(6)
Question 9
(10)
Question 10
(5)
Question 11
(9)
Question 12
(7)
Total:
100
2
Question 1:
a. Write down the recursive formula for the sequence
3 ; 3 3 ; 9 ; 9 3 ;
b. Given Tn  2(Tn 1 ) 2  1 and T1  5
1
, determine T2 and T3
4
(2)
(2)
Question 2:
A box of 40 pocket calculators is sent to a store. The owner of the store is not aware
that 5 of the pocket calculators are defective. Two pocket calculators are selected at
random from the box, the first one not being replaced before the second one is
selected.
a. What is the probability that the first one chosen is not defective?
(2)
3
b. What is the probability that of the two pocket calculators selected,
one calculator is defective and the other is not?
(4)
c. What is the probability that both the pocket calculators selected
are defective?
(3)
Question 3:
The table below indicates which distance in metres is required by a vehicle to get to a
standstill when it travels at a certain speed.
x
Speed (km/h)
20
40
60
80
100
120
140
y
Braking distance
required (metres)
6
16
30
48
70
80
110
a. Determine the equation of the regression line of best fit. Leave answers
correct to three decimal digits where necessary.
(4)
b. Determine the correlation coefficient.
(2)
4
c. Discuss the strength of the correlation between the speed and the distance
needed to come to a standstill.
(2)
d. Calculate the braking distance required for a speed of 80 km/h.
_
(2)
_
e. Does ( x ; y) lie on your line of best fit?. Validate your answer.
(4)
Question 4:
A survey of 80 students at a local library indicated the reading preference below:
44 read the National Geographic magazine
33 read the Getaway magazine
39 read the Leadership magazine
23 read both the National Geographic and the Leadership magazines
19 read both the Getaway and the Leadership magazines
9 read all three magazines
69 read at least one magazine
a. How many students did not read any of the magazines?
(1)
5
b. Draw a Venn diagram of the above information.
(5)
c. How many students read the National Geographic and the Getaway
magazines, but not the Leadership magazine?
(3)
d. What is the probability, correct to three decimal digits, that a student selected
at random will read at least two of the three magazines?
(3)
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Question 5:
Given P( A)  0,2
P( B)  0,5
P( A  B)  0,6
a. Are A and B mutually exclusive? Motivate your answer.
(3)
b. Are A and B independent? Motivate your answer.
(3)
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Question 6:
Sixty Grade 8 boys and girls were asked to run 50 metres and their times (in seconds)
were recorded.
Boys
Girls
7,4
7,6
8,1
8,2
7,6
7,8
8,3
8,3
7,8
7,9
8,3
8,4
7,9
8,0
8,4
8,4
8,0
8,0
8,4
8,4
8,0
8,1
8,5
8,5
8,1
8,1
8,5
8,5
8,1
8,2
8,5
8,5
8,2
8,2
8,6
8,6
8,3
8,3
8,8
8,8
8,3
8,3
8,9
8,9
8,5
8,5
8,9
9,0
8,6
8,6
9,0
9,3
8,8
8,8
9,5
9,7
8,8
9,1
9,7
9,7
a. Johnny says: ”All boys run faster than girls.”
Jill says: “Some boys run faster than girls.”
Using the above data, discuss the truth of both statements.
b. Consider the data for the girls.
The girls who ran a time of 9,7 s are removed from the sample.
How would this affect the following (if at all)?
- Mean
- Median
- Mode
Motivate your answers without doing Mathematical calculations.
(4)
(6)
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Question 7:
P
P , Q , R and S are points on the circle.
PR is a diameter and MRT is a tangent at
^
1 2
^
R . S 2  65 and R 3  35 .
Q
1
2
1
S
2
65 3
35
2 3
4
1
M
T
R
Determine, with reasons, the size of:
^
a. R 1
(1)
^
b. R 4
(2)
^
c. S 3
(2)
^
d. T
(1)
e. Hence prove that PR is a tangent to SRT .
(2)
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Question 8:
Two concentric circles centres at O have radii
of 5cm and 8,5cm respectively. QR  6cm and
OT  PS .
O
Determine the length of PS
(6)
P
1 2
Q
T
R
S
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Question 9:
In the figure, PQRS is a cyclic quadrilateral
and O is the centre of the circle. TR is a
tangent to the circle at R and TQP is a
Q
T
x2
P
1
2 3
1
^
straight line. RQ  RT and POR  y .
R
1 2
3
4
5
y
O
S
^
a. If TQR  x , name, with reasons,
THREE other angles that are each
equal to x .
(6)
b. If x  70 , prove, with reasons, that PTRO is NOT a cyclic quadrilateral. (4)
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Question 10:
Refer to the figure. Given ABC with
D on AB and E on AC such that BC // DE .
CE  2,5 , BD  x , AD  4 and AE  2x  1.
Determine, with reasons, the
value of x .
(5)
B
x
D
C
2,5
E
4
2x  1
A
12
P
Question 11:
Refer to the figure. Given PQR with
TS // RP and TE // QP .
1
E
2
S
2
1
Q
a. Prove, with reasons,
SQT /// PQR
b. If
1
2 3
R
T
(3)
PQ 5
 , find, with reasons, the value of :
SQ 2
i.
ST
PR
(2)
ii.
RE
RP
(4)
13
W
Question 12:
4
In the diagram, PQRS is a
parallelogram.
Side RS is extended to W.
WQ intersects PS at X. M is a point on
12
XQ such that MX  XW .
MT // XS , PQ  12cm and
WS  4cm .
Q
a. Calculate the length of TR.
b. Determine the value of
XM
XQ
X
P
S
M
T
R
(4)
(3)