MATHEMATICAL STUDIES
NAME_______________________________ CLASS__________
Approved dictionaries, notes, calculators may be used.
INSTRUCTIONS TO CANDIDATES:
1. You will have 10 minutes to read the paper. In that time you may make notes on the
scribbling paper provided. You must not write in your script book nor use your
calculator until instructed to do so.
2. Answer all parts of Questions 1 to 14.
3. Correct answers (including appropriate working ) to all questions are required for full
marks
4. Diagrams, where given, are not necessarily drawn to scale.
5. Show all workings. (You are strongly advised not to use scribbling paper.
Work which you consider incorrect should be crossed out with a single line but should not be erased or rendered unreadable.). You may write on page 16, 19 and 24 if you
need more space, making sure to label each answer clearly.
6. Please use only black or blue pens for all work other than graphs and diagrams, which
may be done in pencil.
7. Complete the box on the top right-hand side of this page with information about the electronic
technology you are using in this examination.
8. Total marks is 150.
Question 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Marks 11 12 13 12 7 7 5 14 19 4 15 9 7 15
QUESTION 1
Find dx
dy if
(a) Find (Do not simplify your answer)
(b) 3
1)2ln (4x e y x
-=- (Do not simplify your answer)
(c) 2325x y xy =-
QUESTION 2
Consider the function )(x f y = where 186)(24--+-=x x x x f .
(a) F ind an expression for )(x f ' and classify (using an appropriate sign diagram)
all stationary points of the function.
(b) Find algebraically any non-stationary (non-horizontal) point(s) of inflection.
(c) For the interval 33≤≤-x , find all the values of x where
186)(24--+-=x x x x f attains its maximum and minimum values.
QUESTION 3
Let x
x f 31)(-
=. The graph of )(x f y =, for 0>x , and the chord through points )0,3(A and )2
1
,6(B are
shown below:
(a) (i) Find the slope of the chord AB .
(ii) Describe what the slope of the chord AB represents.
x y
(b) Find, from first principles , )6(f '.
(c) Check your answer to part (b) using direct differentiation.
(d) Give a geometric interpretation for the value of )6(f '.
(e) Find the equation of the normal to the curve at 6=x in the form E Dy Cx =+.
QUESTION 4
Below is the graph of )(x f y = where 161282)(++-=x x x f , for 40≤≤x .
(a) (i) Using two appropriate rectangles, obtain an upper estimate U A for
?4
)(dx x f .
(ii) On the graph above sketch the two rectangles you used for your upper
estimate U A . (1 mark)
(iii) Find the corresponding lower estimate L A for ?4
)(dx x
f .
x 1234y 48
y = f(x)
(b) The graph of )(x f y =, where 161282)(++-=x x x f , is now sketched for
b x ≤≤0, where b is an unknown positive constant. The areas of two regions of the graph are shown.
(i) Write an equation which shows the relationship between
?b
dx x f
)(, ?4
)(dx x f , ?12)(dx x f and ?b
dx x f )(
.
(ii) If
11)(=?b
dx x f , evaluate ?b
dx x f )(.
(iii) From the graph it appears that 1918<
scale of the graph being accurate.
x y
Mathematician Fibonacci wishes to analyse historical records of rabbit population in Australia. He discovers that in 1859 an early settler had a population of 24 rabbits sent out from England so that they could be released on his property and provide hunting sport. Fibonnacci finds the following data which estimates the population )(t
P of rabbits t years after the arrival of the initial 24 rabbits.
(a) (i) Consider the power, exponential and quadratic models. Explain clearly which
model you think is best for this data and why each of the other two are less
(ii) Write the equation of your chosen model.
(b) Fibonnaci discovers some new data for the 1900s which suggests that while the
population of rabbits continues to grow, the growth rate of the population of rabbits decreases. That is, since the food source is limited, the population tends to stabilise.
Name a type of model which can accomodate both the original data and this new information.
The graph of a polynomial function )(x f y = is shown below. The relevant points of the function have been indicated and labelled. The point A is a stationary (horizontal) point of inflection.
(a) On which interval(s) is the function (i) increasing? (ii) decreasing?
(b) On the axes above carefully sketch the derivative function )(x f y '= .
(3 marks)
(c) State the -x coordinate where the derivative function )(x f ' attains its least
x y
QUESTION 7 (5 marks)
(i)
Use graphical evidence to show that the exact value of
(2
212(2)dx π-+
=+?
(ii)
hence show that
(2 marks)
y
QUESTION 8 (10 marks)
Consider the following system of simultaneous equations with unknown variables x, y and z :
x +2y –3z =4
3x –y +5z =2 where a is a constant
4x +y +? ?a 2
–14???
z =a +2
(a) Show that the augmented matrix of this system can be reduced by row operations to the following form:
???????1
00
270
-3-14-16+a
2
????? 4
10-4+a ?
????
(4 marks)
(b) Using the results of (a) or otherwise solve the system of equations for
x+2y–3z=4
3x–y+5z=2
4x+y+2z=6
(c) For what value(s) of a does the original system not have a solution?
QUESTION 9
A boat is travelling in a straight line. Its position is given by
1405
400)(++-=t t s metres, where 0≥t is in seconds.
(a) (i) Find expressions for the velocity and acceleration of the boat. Clearly show
the units.
(ii) Find the initial position, velocity and acceleration of the boat hence
describe its motion.
(b) (i) Explain why the boat does not change direction for 0≥t .
(c) Explain what happens to the position s and velocity v of the boat as ∞→t .
(e) Given that 23
Cv dv
-=, find the value of the constant C
.
QUESTION 10
Suppose A 2 = A , where A is a 2x2 matrix.
(a) Under what condition is the following argument true? A 2 = A ? AA = A
? A -1AA = A -1A ? IA = A (b) If A = ??
? ?a a a a and A 2
= A , what value(s) can A be?
QUESTION 11
An industrial accident occurs in a gold refining plant. As a result a poisonous gas is released. The plant’s safety equipment detects the gas and measures C , the concentration of the gas in the air, in parts per million. The values for C , t hours after the accident occurs, are given below.
A scatter plot of this data is drawn.
Gas concentration in the air
01
2
3
4
5
6
7
8
9
5
10
15
20
25
30
t (hours)
C (p a r t s p e r m i l l i o n )
The safety engineer believes that a surge function will best model this data. a. Draw the approximate shape of such a model on the graph above.
2 marks
Susan uses modelling software to obtain ()C t , the surge function that best models the variation in C over time. Susan’s model is 0.237() 4.76t C t te -=. b. Determine, according to Susa n’s model, 0.237() 4.76t C t te -=,
i.
the length of time that the concentration C exceeds 1 part per million;
ii.the peak concentration of the gas and when it occurs.
Pedro develops a different surge function ()bt
=to model the variation in
C t Ate-
C over time. He notices that the points (1, 3.5) and (3, 7.1) lie on the graph of =.
()
y C t
c.Determine the values of A and b in Pedro’s model.
https://www.doczj.com/doc/fc4514197.html,pare the model, 0.196
=to Susan’s model
() 4.26t
C t t e-
0.237
() 4.76t
=. Which model fits the data best? Give one reason for
C t t e-
your answer.
QUESTION 12
A rectangle of length a units and breadth b units, where b a >, is divided into 3 sections as shown in the diagram below.
If x is the variable length that determines the size of the top left-hand section; a) Show that the shaded area is given by the function
)(2
1
2x bx ab A -+= unit 2
4 marks
b) Show that the area A is the maximum possible when 2b x =
3 marks
b
a
x
x