ST STITHIANS BOYS COLLEGE
MATHEMATICS DEPARTMENT
Grade 12
July 2008
MATHEMATICS: Paper 1
(LO 1 AND LO 2)
Time: 3 hours
150 marks
PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY
1.
This question paper consists of 10 pages, including a Formula Sheet.
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 a specific question prohibits the use of a calculator.
6.
Round off your answers to one decimal digit where necessary.
7.
It is in your best interest to work neatly and legibly.
8.
All the necessary working details must be clearly shown.
_____________________________________________________________________
1
QUESTION 1
The first four terms of a sequence are:
1;
(a)
(b)
10;
25;
46; .......;
Write down the next two terms.
Determine the formula Tn  an 2  bn  c for the sequence.
(2)
(5)
[7 marks]
QUESTION 2
x
Calculate the value of x if:
 (10  2n)  242
[6 marks]
n 1
QUESTION 3
1
27
The second term of a geometric series is 10 , the fifth term is 1250 .
th
(a)
Find the n - term.
(b)
Find the sum to infinity of the series.
(5)
(2)
[7 marks]
QUESTION 4
(a)
Convert an effective interest rate of 12,3 % p.a. to a nominal annual interest
rate compounded daily (correct to ONE decimal digit).
(4)
(b)
Farmer Brown purchased a tractor for R441 000. The tractor depreciates at
the rate of 20 % p.a. on a reducing balance. The cost of a new tractor
increases by 16% compound interest p.a.
(i)
Find the value of the tractor in 6 years time.
(2)
(ii)
If the old tractor is traded in, calculate how much extra Farmer Brown
will have to pay in, to buy a new tractor in 6 years time.
(3)
(iii)
Farmer Brown decides to deposit a fixed amount into a sinking
fund at the end of each month for the purchase of a new tractor in six
years time. The first payment will be made immediately and the last
payment will be made at the end of the 6 year period. Assuming that he
needs R958 845,28, calculate the monthly payment if interest is earned
at the rate of 9,2 % p.a. compounded monthly.
(5)
[14 marks]
2
QUESTION 5
On the set of axes provided on the diagram sheet:
(d)
Sketch the graph of y  f ( x)  3x .
Label at least two defining points clearly.
Determine the inverse of f and write it in the form f 1 ( x)  .........
Sketch the graph of f 1 on the same set of axes as f. Label at least two
defining points clearly.
Is f 1 a function? Give a reason for your answer.
(e)
If g(x) = f (x - 4) – 3
(a)
(b)
(c)
(i) Describe the transformation of f to g.
(2)
(1)
(2)
(2)
(1)
(ii) Sketch the graph of g. Show any new asymptotes and label at least two
defining points clearly.
(3)
[11 marks]
QUESTION 6
Sketched are the graphs of f ( x)  x 2  4 x  12
and g ( x)  2 x  12 .
y
x
QB is a line perpendicular to the x-axis, which lies between A and P.
(a)
Find the maximum length of QB.
(4)
(b)
Find the x co-ordinate of the point at which the tangent to f(x) will be
parallel to g(x)
(3)
[7 marks]
3
QUESTION 7
(a)
Sketch the parabola y = f(x) which has the following properties:
(i)
(ii)
f (0) = 6
(iii)
f (1)  0
(iv) f ( x)  0
when x  -1
f ( x) < 0
when x  1
(3)
Determine the equation of f(x).
(3)
[6 marks]
(v)
(b)
f (-3) = 0
QUESTION 8
The graphs of f(x) = cos x + p and g(x) = cos qx are sketched for
90  x  90 .
y
(a)
Write down the value of p if the graph of f touches the x-axis.
(1)
(b)
Write down the amplitude of f.
(1)
(c)
Write down the value of q if the period of g is half the period of f.
(1)
(d)
Write down the minimum value of f.
(1)
(e)
Use the graphs to solve:
g ( x) . f ( x)  0
(2)
[6 marks]
4
QUESTION 9
(a)
(b)
Evaluate, without the aid of a calculator:
(i)
3 x 1  3 x  2
3 5 x.3 4
(4)
(ii)
log 6  log 2
log 9(2log 5  log 4)
(4)
Solve for x in the following:
(i)
( x  1)( x  5)  12
(3)
(ii)
4
 0
x
(1)
(iii)
45000  155000 (1  0,18) x
(3)
(iv)
log5 ( x  2)  log5 ( x  1)  log5 4
(4)
[18 marks]
QUESTION 10
Given: f ( x) 
x 2
x4
(a)
For which values of x is f(x) undefined?
(b)
The following table gives values of f(x) as x approaches 4.
(1)
x
3,9
3,99
3,999
3,9999 4,0001
4,001
f(x) 0,251582 0,250156 0,250016
0,249998 0,249984
?
(c)
Give, correct to 6 decimal places, the missing value of f (3,9999) .
(1)
Hence, or otherwise find the value of lim f ( x)
(1)
x4
[3 marks]
QUESTION 11
1
x
Find the derivative, g ( x) from first principles.
Given: g ( x) 
(a)
(b)
(5)
Hence find the equation of the tangent to g(x) at the point where x = 3.
(4)
[9 marks]
5
Question 12
Determine
(a)
(b)
dy
in each of the following:
dx
1
2 x2
xy  y  x 2  1
y  3 x4 
(3)
(3)
[6 marks]
QUESTION 13
The graph shows the curve of : f ( x)   x3  4 x 2  11x  30
The graph has an x-intercept of (5; 0) and has a y-intercept of (0; -30).
(a)
Calculate the co-ordinates of A
(4)
(b)
(c)
Calculate the co-ordinates of B
(4)
4
John says that x  is the x co-ordinate at which this function has a point of
3
inflection. Show all working to validate this statement.
(5)
(d)
Write down the values of x for which f ( x)  0
(e)
If k < - 36, how many real roots will the equation
 x3  4 x 2  11x  30  k have?
(2)
(1)
[16 marks]
6
QUESTION 14
It is estimated that the circulation of the magazine “ Surfer Girl” can be represented
by the equation:
M (t )  50t 3  300t  4000 , where t is the
number of years of publication.
(a)
How many magazines were sold at the time
of the initial publication?
(b)
Derive an equation for the rate at which the
circulation will change in t years.
(c)
(1)
(1)
At what rate will the circulation be changing in 3 years
time?
(d)
(e)
(1)
Determine the average rate of change of the circulation during the
second year of publication, that is between t = 1 and t = 2.
(3)
When will the magazine have the least sales?
(2)
[8 marks]
QUESTION 15
A surfboard company makes two types of surfboards, namely airbrushed and clear.
It takes eight hours to make an airbrushed board and four hours to make a clear board.
The company can only work for a maximum of 72 hours per week.
At least 10 boards must be made weekly. No more than 12 clear boards can be
produced weekly.
The profit on one airbrushed board is R250,00 and the profit on a clear board is R100.
(a)
If two of the constraints are: 8 x  4 y  72 and y  12, write down the
third constraint in terms of x and y. Let x be the number of airbrushed boards
and y the number of clear boards.
(1)
(b)
Represent the inequalities which model this problem on the axes provided on
the detachable Diagram Sheet. Clearly indicate the feasible region.
(4)
(c)
How many of each type of board should be made weekly for maximum profit?
(2)
(d)
What is this profit?
(2)
[9 marks]
7
QUESTION 16
The curves with equations y 
properties:
(i)
(ii)
(iii)
4
 2 and y  ax 2  bx  c have the following
x
there is a common point where x = 2
there is a common tangent where x = 2
both curves pass through the point (1;6).
Find the values of a, b and c.
[7 marks]
QUESTION 17
Refer to the figure that shows how the Spierpinski Triangle is formed. The original
shaded triangle is an equilateral triangle.
In Step 1, the midpoints of the sides of the shaded area are joined and the resulting
triangle is coloured grey.
In Step 2, the midpoints of the sides of the shaded triangles are joined and the three
triangles so formed are coloured grey.
This process is continued. All the triangles formed are similar to each other and the
original.
Step 0
Step 1
Step 2
(a)
Write down how many white and how many shaded triangles there will be in
Step 3.
(1)
(b)
Find an expression for the number of white triangles in Step n.
(c)
Verify for the first 3 steps, that an expression for the number of shaded
3n  1
triangles is given by the formula: Tn 
.
2
(d)
(1)
(2)
At what step are there a total of 9841 triangles, both white and shaded?
(3)
[7 marks]
8
QUESTION 18
If
1 
1
1
1
1
 2  2  2  ........  x ,
2
2
3
4
5
Find in terms of x the value of :
1
1
1
1
 2  2  2  ........
[3 marks]
2
2
4
6
8
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1 
9