MATHEMATICS: PAPER I JULY 2008
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ST STITHIANS GIRLS COLLEGE
MATHEMATICS: PAPER 1
GRADE 12
DATE: 17 July 2008
TIME: 3 hours
MARKS: 150
NAME:
TEACHER:
PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY
1.
This question paper consists of 8 pages with a Formula Sheet.
Please check that your paper is complete.
2.
Read the questions carefully.
3.
Answer all the questions.
4.
Number your answers exactly as the questions are numbered.
5.
You may use an approved non-programmable and non-graphical calculator, unless otherwise stated.
6.
Round off your answers to one decimal digit where necessary, unless otherwise stated.
7.
All the necessary working details must be clearly shown.
8.
It is in your own interest to write legibly and to present your work neatly.
MATHEMATICS: PAPER I JULY 2008
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SECTION A
QUESTION 1
The first five terms of a sequence are 1 ; 5 ; 5 ; 9 ; 13
The rule for this sequence is: Next term = sum of the previous two terms – 1
a) Work out the next two terms in the sequence.
(2)
b) Two consecutive terms are 357 and 577. Find the value of the term before these.
(2)
4 marks
QUESTION 2
Solve for x:
a) 2log x  log1  log100
(3)
b) x  log 4 8  log9 3
(4)
7 marks
QUESTION 3
You have R12 500 to invest and your financial advisor suggests two options:
1. “Saver’s Special”: 12% p.a. interest, compounded quarterly
2. “Real Returns”: 11% p.a. interest, compounded monthly
a)
Calculate the effective interest rate for each investment.
b)
How much money would you accumulate if you invested your money
in the Saver’s Special account for 10 years?
c)
(4)
(3)
How much longer would you have to invest your money in the Real
Returns account, to accumulate the same amount as investing in the
Saver’s Special for 10 years? Give your answer to the nearest month.
(5)
12 marks
MATHEMATICS: PAPER I JULY 2008
Page 3 of 8
QUESTION 4
Differentiate f ( x)  x 2 by first principles.
5 marks
QUESTION 5
a) If f ( x)   3 x 2  x  , find f ( x)
(4)
1
b) Find Dx [3 x 2  ]
x
(3)
2
c) If y 
d) Find
2 x 2  3x  2
dy
, find
x2
dx
(3)
d  4 x3  3x 2 


dx  2 x 
(4)
14 marks
QUESTION 6
In the figure, sketch graphs of f, a hyperbola and g, an exponential function, are shown.
Point P is the reflected image of Q (4 ; -2) about the line y  x .
y
10
P
5
f
-20
y
g
-10
x
10
20
x
-5
a) Find the co-ordinates of P.
(2)
-10
b) Find the equations of f and g.
(5)
c) If h is the reflected image of f about the y-axis, write down the equation of h.
(2)
d) Will the graph of g 1 pass through Q. Motivate your answer.
(3)
12 marks
MATHEMATICS: PAPER I JULY 2008
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QUESTION 7
Lee is designing a small bridge and she decides that it will be supported by a parabolic arch,
1
constructed on the horizontal plane PQ. The equation y   x 2 is used as a mathematical model to
2
design the arch (refer to the diagram).
2
yy
1
xx
-4
-3
-2
-1
1
2
3
4
-1
-2
-3
-4
-5
P
Q
-6
Lee investigates different transformations of her original model:
a) The bridge needs to be built 2 metres to the left of the original plan.
(2)
b) The bridge needs to be widened.
(2)
c) The bridge needs to be heightened, but its shape has to be preserved.
(2)
Write down the number(s) of the equation(s) from the list below, which could model the position
of the bridge for each of the situations a), b) and c).
1
1. y   x 2  2
2
4. y 
1 2
x
2
7. y   x 2
2. y  
1
2
 x  2
2
1
5. y   x 2
4
1
3. y   x 2  2
2
 1

6. y    x  2 
 2

2
8. y  2 x 2
6 marks
MATHEMATICS: PAPER I JULY 2008
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QUESTION 8
The cross section of a hilly region can be drawn as the graph of y  x3  8x 2  16 x  8 for 0  x  5 ,
where x is measured in kilometers and y is the height above sea level in metres.
a) Draw the cross section for 0  x  5 . Show all calculations.
(6)
b) The peak is called Triblik and the base of the valley is Vim Tarn. Mark these two features
on your cross section and calculate the height of Triblik above Vim Tarn.
(3)
c) Determine the gradient of the hill at the point P where x  1.
(2)
d) Hence, determine the equation of the tangent to the hill at P.
(4)
15 marks
MATHEMATICS: PAPER I JULY 2008
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SECTION B
QUESTION 9
In order to find Tn in a second order linear difference sequence, the following formula (which appears
on your formula sheet) can be used:
Tn  T1   n  1 f 
 n  1 n  2 
2
s
where f is the first term of the first difference and s is the second difference.
The following sequence is given: 2 ; 12 ; 20 ; 26 ;.......
a)
Use the formula to show that for this sequence , Tn  n 2  13n  10 .
(5)
b)
Hence, find which term(s) have a value of 10 .
(4)
c)
Which term(s) have a value of at least 26?
(5)
d)
Show by completing the square, that no term in this sequence has a value greater than 33. (5)
19 marks
QUESTION 10
Felix takes out a loan of R350 000 in order to start his business. He repays R150 000 two years later.
Four years after taking out the loan, he expands his business and borrows a further R560 000.
He pays off his total debt by means of a payment of Rx three years later. The interest rate is 18% p.a.
compounded monthly. Calculate x.
6 marks
MATHEMATICS: PAPER I JULY 2008
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QUESTION 11
If log 3  p and log 5  q , express the following in terms of p and q:
a) log 2
b) log 5 45
(3)
(5)
8 marks
QUESTION 12
A function given by y  ax 2 
b
has a minimum value of 12 at x  2 .
x
a) Find a and b.
(7)
b) Give the equation of the vertical asymptote of the function.
(1)
8 marks
QUESTION 13
The following characteristics apply to f  x   ax 2  bx  c:
f  1  1, f   1  0, f  0   4, f   x   0 if x  1 and f   x   0 if x  1.
a)
Use the information to sketch f  x  .
(4)
b)
Determine the values of a, b and c.
(4)
c)
For which value(s) of k will f  k   0 ?
(2)
d)
For which value(s) of a will f  x   a have at most one solution?
(2)
e)
Use your graph to solve for x: f  x  . f   x   0 .
(2)
14 marks
MATHEMATICS: PAPER I JULY 2008
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QUESTION 14
During an experiment, the volume of water in a dam is found to be
V  t   60  8t  3t 2 , where V  t  represents the volume of water in
the dam, in kilolitres, t days after the experiment has begun.
There is a suspected leak in the dam wall.
a)
Calculate the rate of change of volume of water after 3 days.
(3)
b)
When did the volume of water start decreasing?
(3)
c)
When will the dam be empty?
(4)
10 marks
QUESTION 15
A spherical drop of mercury splashes and breaks into equal spheres with radii one-quarter that
of the original sphere.
4 3
r
3
Surface area of a sphere = 4 r 2
Volume of a sphere =
a) What is the ratio of the volume of one of the new spheres to that of the original sphere?
(3)
b) How many new spheres are there?
(1)
c) What is the ratio of the surface area of one of the new spheres to that of the original?
(3)
d) What is the ratio of the total surface area of all the new spheres to the surface area of the
original sphere?
(3)
10 marks