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THOMAS MORE COLLEGE
Matric Trials.
ADVANCED
PROGRAMME
MATHEMATICS
Time : 3 Hours.
August 2008.
Marks : 270.
INSTRUCTIONS AND INFORMATION
Read the following instructions carefully
1.
This question paper consists of 8 pages in two sections. In addition, a
formula sheet is provided for use by all candidates.
Please check that your paper is complete.
SECTION A:
Calculus & Algebra (200 marks)
SECTION B:
Statistics (70 marks)
2.
Answer all the questions.
3.
Answer each section on a separate answer sheet.
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.
Give answers correct to 2 decimal places.
7.
All the necessary corking details must be clearly shown.
8.
It is in your own interest to write legibly and to present your work neatly.
1
SECTION A :
Algebra and Calculus [200 marks]
QUESTION 1
Prove by induction that 9n + 3 is divisible by 4, n  
[9]
QUESTION 2
Use the Newton-Rhapson method to find a positive solution for the equation
 
1 + x = 2tanx in the interval  0;  . Use 0,8 as a starting value and give the answer
 2
correct to 3 decimal places.
[13]
QUESTION 3
3.1
3.2
Simplify the following:
3.1.1
In e 2
(1)
3.1.2
In(8x) 12 + In4x 2 – In(16x) 12
(4)
Solve for x 
(answers to 2 decimals where necessary):
1
86
3.2.1
In(x + 1) =
(3)
3.2.2
e – 2x =
3.2.3
The streptococci bacteria population N at time t (in months) is given by
4
12
(3)
N = N0e2t
where N0 is the initial population
How long would it take for a population to double?
Does the initial population have any effect on this time? Explain.
2
(4)
3.2.4
Jack planted a mysterious bean just outside his kitchen window.
It immediately sprouted 2,56 cm above the ground.
Jack kept a careful log of the plant’s growth. He measured the
height of the plant each day at 8:00am and recorded this data:
Day
Height (cm)
3.2.4.1
3.2.4.2
3.2.4.3
3.2.4.4
0
1
2
3
4
2,56
6,4
16
40
100
Define variable and write an exponential equation for this pattern.
If the pattern were to continue, what would the heights be on the
fifth and sixth days?
(8)
Jack's younger brother measured the plant at 8:00pm on the
evening of the third day and found it to be about 63,25 cm tall.
How would you have found this result mathematically?
(3)
Find the height of the sprout (to the nearest centimetre) at
12:00 noon on the sixth day.
(3)
On what day and at what time (to the nearest hour) will the plant
reach a height of 1 km?
(3)
[32]
QUESTION 4
4.1
Solve for x :
| x – 4 | = 2x
(5)
4.2
Solve for x :
6
| x 1|
(5)
4.3
 3
k
x
and f (x) = 2 | x – 1 | – 3. A is one of the points
of intersection of the two graphs.
Determine:
The sketch shows the graphs of g ( x ) 
4.3.1
the value(s) of k
(3)
4.3.2
the co-ordinates of P
(1)
4.3.3
the co-ordinates of A
(5)
4.3.4
the x – intercepts of f
(3)
[22]
3
QUESTION 5
Prove the following identity:
1


2 sec 2 A = sec 2 A 1 
 +1
2
cos ec A 

[8]
QUESTION 6

x 2  1 x  1
Given f ( x )  

 1 x
x 1
6.1
Prove that the graph of f is continuous at x = 1
(6)
6.2
Does the function f have an inverse function? Explain
(3)
6.3
Is the function differentiable at x = 1? Explain
(4)
[13]
QUESTION 7
Let f be a function that is continuous on the interval [0; 4). The function f can be
differentiated twice except at x = 2. The function f and its derivatives have the properties
indicated in the table below. DNE indicates that the derivate “Does Not Exist”.
x
0
0<x<1
1
1<x<2
2
2<x<3
3
3<x<4
f(x)
–1
Negative
0
Positive
2
Positive
0
Negative
f '(x)
4
Positive
0
Positive
DNE
Negative
–3
Negative
f ''(x)
–2
Negative
0
Positive
DNE
Negative
0
Positive
7.1
For 0 < × < 4, find all the values of x at which f has a relative maximum or
minimum. With justification, state whether f has a relative maximum or minimum
at these values.
(3)
7.2
On the axes provided on the diagram sheet, sketch the graph of a function that
has all the characteristics of f.
(9)
[12]
4
QUESTION 8
8.1
dy
sin 2 x
Solve
if y =
dx
x2
(6)
8.2
d 
x x 2  1

dx 
(6)
8.3
Consider the curve given by x 2 + 4y 2 = 7 + 3xy
dy
dx

3y  2x
8y  3x
8.3.1
Show that
(8)
8.3.2
Show that when x = 3 there is a point P at which the tangent to the
curve at P is horizontal. Find the y-value of P
(5)
[25]
QUESTION 9
Petrol stations only run out of petrol when there is a major price increase. Generally they
manage their petrol levels in the tanks to ensure that they have sufficient stock. The petrol is
stored in an underground cylindrical tank and the amount left in the tank is checked using a
dipstick. Our problem is to calibrate the dipstick.
It is easy to measure the height, say h, of the petrol left in the tank using a dipstick. However
the volume is proportional to the cross-sectional area not just the height.
9.1
Determine cos  in terms of h
(4)
9.2
Show that the cross sectional area of petrol in the tank is determined by
A = 4 – 2sin(2)
(6)
9.3
What fraction of the tank is full when  =

?
4
(4)
[14]
5
QUESTION 10
Evaluate the following integrals:
10.1
 cos3  sin 2 d
10.2
 sin
10.3
x
2
(6)
x  sec 4 x dx
(10)
x  4 dx
(8)
[24]
QUESTION 11
The graph below is of the function y =
x
Use Riemann sums to determine the area below the curve between [ 2 ; 6 ]
[10]
QUESTION 12
Let f and g be the function given by f (x) = 2x (1 – x)
and g (x) = 3 (x – 1) x for 0  x  1.
The graphs of f and g are shown in the figure alongside
12.1
Find the area of the shaded region enclosed
by the graphs of f and g.
(10)
12.2
Find the volume of the solid generated when
the shaded region is revolved about the
line y = 2.
(Hint: Use a transformation on the functions to
make this rotation simpler.)
(8)
6
[18]
SECTION B :
Statistics [70 marks]
QUESTION 1
x2
10
P (X = x) =
x = 1, 3, k, 9
(Note that k is not necessarily  [ 3 ; 9 ]
Find the value of k.
[10]
QUESTION 2
A discrete random variable X has the following probability distribution:
x
0
1
2
3
4
P(X = x)
P
2p
2p2
4p2
p – p2
2.1
Find the value of p and plot the probability distribution.
2.2
Find P(X < 3)
(13)
(4)
[17]
QUESTION 3
A field mushroom has its spores deposited around it by the wind and these germinate
when conditions are favourable. If the random variable X is the distance (in metres)
from the "parent" that the mushroom germinates, the probability density function is
given by:
f (x) =
{
k (1 – 0, 04x2) 0  x  5
0
elsewhere
3.1
Determine the value of k.
(10)
3.2
Determine the probability that the mushroom germinates more than 2 metres
away from its "parent".
(8)
[18]
7
QUESTION 4
The probability density function for the lifespan of a certain insect species is given by
f (x) =
{
3 2 3
x 
16
4
0
0  x  m
elsewhere
Find m, the maximum lifespan of these insects.
[10]
QUESTION 5
The digits 2; 3; 4; 4; 4; 5 are given.
5.1
How many 6-digit numbers can be formed from the given digits?
(6)
5.2
What is the probability that a random number generated using these digits is
larger than 400 000?
(9)
[15]
8