Download Paper 3

GRADE 12 MATHEMATICS PAPER 3
PRELIMINARY EXAMINATION SEPTEMBER 2011
PAGE 1 OF 12
SECTION A
Question 1 [10 marks]
a)
Find the first 4 terms of the sequence if, T 1 = 3 and Tn = ( – 1)n . 5 Tn – 1
(3)
b)
Write down a recursive formula for the following sequence: 12; 7; 2; – 3; …….
(2)
c)
T4 = 10 and T5 = 14
and
Tn = Tn-1 + Tn-2 (n > 2)
(i)
Find T6
(1)
(ii)
Find the first three terms.
(4)
GRADE 12 MATHEMATICS PAPER 3
PRELIMINARY EXAMINATION SEPTEMBER 2011
PAGE 2 OF 12
Question 2 [5 marks]
M
10
3
7
F
5
The Venn diagram shows how many students in a class of Grade 12s do Mathematics(M),
French(F) or neither.
a)
How many students are in the class?
(1)
b)
Find the probability that a student in the class does both subjects.
(1)
c)
Find
d)
Find the probability that a randomly selected student does not do French.
(1)
e)
Find the probability that a randomly selected student does only 1 of the subjects.
(1)
P(M  F).
(1)
GRADE 12 MATHEMATICS PAPER 3
PRELIMINARY EXAMINATION SEPTEMBER 2011
PAGE 3 OF 12
Question 3 [7 marks]
Jemima Puddle Duck has just obtained her driver’s license. She wants to buy a car and has
the following options from which to make her choice:
Model
Type
Colour
VW Polo or Toyota Yaris
Sedan or hatchback
White, black or red
a)
Draw a tree diagram that will show all possible choices available to Jemima.
b)
It is equally likely that Jemima may choose any model, any type of car or any colour
of car. She chooses her car at random.
(4)
(i)
What is the probability that she chooses a black, Polo hatchback?
(2)
(ii)
What is the probability that Jemima will chose a red car?
(1)
Question 4 [4 marks]
What is the probability that in 3 consecutive cricket matches, the captain will win the toss
a)
every time?
(2)
b)
at least once?
(2)
GRADE 12 MATHEMATICS PAPER 3
PRELIMINARY EXAMINATION SEPTEMBER 2011
PAGE 4 OF 12
Question 5 [12 marks]
Mrs Pepper wants to see if the prelim results for Mathematics Paper 3 can be used to predict
what the girls will get for their final Paper 3 examination. Unfortunately she only has the data
from a random sample of 10 students from 2010, so she knows that her results are going to be
unreliable (due to the small sample size).
These results are shown in the table below:
Prelim Paper 3
X
Final Paper 3
Y
29
49
60
65
72
79
84
88
90
100
40
45
55
64
60
66
92
82
93
75
a)
Draw a scatterplot of the data.
(3)
b)
Find the equation of the regression line which models the linear relationship between
x and y. You may use your calculator. Give all answers to 2 dp.
(4)
GRADE 12 MATHEMATICS PAPER 3
PRELIMINARY EXAMINATION SEPTEMBER 2011
PAGE 5 OF 12
c)
Draw the regression line for the data on the scatterplot.
(2)
d)
Use the regression line to estimate the final Paper 3 result for a student who got 75% for
Paper 3 in the preliminary examinations.
(1)
e)
Determine the correlation coefficient between the preliminary and the final examination
results, to 2 dp.
(1)
f)
Use the correlation coefficient to comment on the validity of using preliminary examination
results to predict final examination results.
(1)
Question 6 [11 marks]
Set A
Set B
45
35
40
30
Frequency
Frequency
35
30
25
20
15
25
20
15
10
10
5
5
0
0
5
15
25
35
45
55
65
75
Weight (in grams)
5
15
25
35
45
55
65
75
Weight (in grams)
The histograms show the distributions of weights of 100 insects from 2 locations.
Without doing any calculations, state
a)
Which set has the greater mean – give a reason.
(2)
b)
Which set has the smaller median?
(1)
GRADE 12 MATHEMATICS PAPER 3
c)
PRELIMINARY EXAMINATION SEPTEMBER 2011
PAGE 6 OF 12
Estimate the mean and (population) standard deviation of Set A, giving your
answers to 1 dp.
(4)
d)
Explain briefly why your answers to (c) are estimates.
(1)
e)
Explain how you can estimate the mean of Set B without doing any calculations.
(1)
(f)
If the standard deviation of Set B is actually 13, what would the value of the
standard deviation become if all the data values in Set B were transformed
(i)
by adding 5 (grams) to each value?
(1)
(ii)
(1)
by multiplying each data value by 3?
Question 7 [4 marks]
The test results of 31 students are shown in a box-whiskers diagram.
26
a)
b)
36
64
The results were read out in ascending order. If the 8th result was 26,
which student obtained a mark of 64?
(1)
How many students (to the nearest whole number) got a mark higher than 64?
(1)
GRADE 12 MATHEMATICS PAPER 3
c)
d)
PRELIMINARY EXAMINATION SEPTEMBER 2011
PAGE 7 OF 12
From the raw data, the mean is calculated to be 45 and the standard deviation is
calculated to be 19. How many students (to the nearest whole number) were more
than 1 sd away from the mean?
(1)
If the data were normally distributed, then about 99% of the data values would be within
3 sd’s if the mean. Assuming normality, estimate the maximum test score.
(1)
Question 8 [7 marks]
In a data set of 7 values
the median is 60
the range is 55
the data is bi-modal with modes of 25 and 70
the lowest value is 25
the mean is 52.
Calculate the 7 values in the data set and complete the table in ascending order.
GRADE 12 MATHEMATICS PAPER 3
PRELIMINARY EXAMINATION SEPTEMBER 2011
PAGE 8 OF 12
SECTION B - Diagrams in this section are not drawn to scale
Question 9 [5 marks]
In the diagram PT and QR are parallel. TP and TR are tangents to the circle PQRS.
PT̂R  RP̂Q  38 . NO REASONS REQUIRED FOR THIS QUESTION.
c
d
b
a
a)
What is the size of a ?
(1)
b)
What special name is given to ΔPTR?
(1)
c)
Hence, give the size of c, if TP̂R = c.
(1)
d)
Now find the size of b and d.
(2)
GRADE 12 MATHEMATICS PAPER 3
PRELIMINARY EXAMINATION SEPTEMBER 2011
PAGE 9 OF 12
Question 10 [8 marks]
In the diagram, O and T are centres of the circles PRST and POSV respectively. VTOR is a

straight line. PŜO  40 0 and PQR  50 0 .
VPQ is NOT a straight line
Q
50o
P
b c
a
V
T
d
O
R
40o
S
a)
b)
Determine the values of the following giving reasons
i)
a
ii)
(b+c)
ii)
d
Hence, prove that PQRO is a cyclic quadrilateral.
(6)
(2)
GRADE 12 MATHEMATICS PAPER 3
PRELIMINARY EXAMINATION SEPTEMBER 2011
PAGE 10 OF 12
Question 11 [12 marks]
Refer to the figure. QP and SR are two parallel chords of the circle with centre O.
PÔR  2x and PS and QR intersect at T. Give reasons for all working.
a)
Name 4 angles that each equal x.
(8)
b)
Hence show that T̂1  2 Q̂ .
(2)
c)
Prove that POTR is a cyclic quadrilateral.
(2)
GRADE 12 MATHEMATICS PAPER 3
PRELIMINARY EXAMINATION SEPTEMBER 2011
PAGE 11 OF 12
Question 12 [15 marks]
In the figure, BD is a diameter of circle centre O. MCT is a tangent to the circle at C.
∧
OM II DC and MC B = x . Give reasons for all answers.
x
Prove that
a)
BK = KC
(2)
b)
BOCM is a cyclic quadrilateral.
(4)
GRADE 12 MATHEMATICS PAPER 3
PRELIMINARY EXAMINATION SEPTEMBER 2011
PAGE 12 OF 12
c)
ΔBOK III ΔMCK
(4)
d)
Calculate BC if OK = 1 unit and MK = 7 units (leave answer in surd form)
(5)