THE
HEFFERNAN
GROUP
P.O. Box 1180
Surrey Hills North VIC 3127
Phone 03 9836 5021
info@theheffernangroup.com.au
www.theheffernangroup.com.au
MATHEMATICAL METHODS UNITS 1 & 2
TRIAL EXAMINATION 1
2020
Reading Time: 15 minutes
Writing time: 1hour
Instructions to students
This exam consists of 10 questions.
All questions should be answered in the spaces provided.
There is a total of 40 marks available.
If a question requires a numerical answer then an exact value must be given unless a decimal
approximation is specifically asked for.
Where more than one mark is allocated to a question working must be shown.
Students may not bring any notes or any calculators into this exam.
Diagrams in this exam are not to scale except where otherwise stated.
A formula sheet can be found on page 12 of this exam.
This paper has been prepared independently of the Victorian Curriculum and Assessment
Authority to provide additional exam preparation for students. The publication is in no way
connected with or endorsed by the Victorian Curriculum and Assessment Authority.
Ó THE HEFFERNAN GROUP 2020
These exam questions and solutions are licensed on a non transferable basis to the purchasing
school. They may be copied by the school which has purchased them. This license does not
permit distribution or copying of these exam questions and solutions by any other party.
2
Question 1 (3 marks)
Solve the following for x.
1
9
a.
34 x -1 =
b.
log 2 (2 x) - 3log 2 (4) + log 2 ( x 2 ) = 1, x > 0
1 mark
2 marks
Question 2 (2 marks)
Solve
é pö
3 tan(4 x) = 1 for x Î ê0, ÷ .
ë 2ø
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© THE HEFFERNAN GROUP 2020
Maths Methods 1 & 2 Trial Exam 1
3
Question 3 (3 marks)
Point A lies on the graph of y = sin(2x) at the point where x =
p
6
.
æxö
p
Point B lies on the graph of y = cosç ÷ at the point where x = .
è2ø
2
Find the midpoint of the line segment joining points A and B.
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© THE HEFFERNAN GROUP 2020
Maths Methods 1 & 2 Trial Exam 1
4
Question 4 (4 marks)
A spinner with three equal sections coloured red (R), white (W) and green (G) is shown
below.
red (R)
green (G)
white (W)
The spinner is spun and a fair coin is thrown simultaneously.
a.
i.
ii.
Draw a tree diagram below to represent the possible outcomes of this
experiment.
1 mark
List the outcomes that form the sample space for this experiment.
1 mark
Let R represent the event of the spinner landing on red.
Let T represent the event of a tail appearing.
b.
Find
i.
Pr( R Ç T ') .
1 mark
ii.
Pr( R ' È T ') .
1 mark
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© THE HEFFERNAN GROUP 2020
Maths Methods 1 & 2 Trial Exam 1
5
Question 5 (3 marks)
a.
Find the derivative of ( x -1)(3x + 2) with respect to x.
b.
Let g ( x) = x -
1 mark
3
, x > 0.
x2
Evaluate g '(2) . Express your answer in the form
positive integers.
a +b
where a, b and c are
c
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© THE HEFFERNAN GROUP 2020
Maths Methods 1 & 2 Trial Exam 1
2 marks
6
Question 6 (4 marks)
a.
Given that h ' ( x) =
1
- 2 x and h(1) = 0, find h( x).
x2
2 marks
4
b.
Evaluate
x3 - 2 x 2
dx .
x2
2
ò
2 marks
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© THE HEFFERNAN GROUP 2020
Maths Methods 1 & 2 Trial Exam 1
7
Question 7 (5 marks)
Let f : R ® R, f ( x) = ( x + 2)2 + 1 and g : (0, ¥) ® R, g ( x) = log e ( x).
The graphs of f and g are shown below.
y
y = f (x )
y = g ( x)
O
x
x =0
a.
State the range of f.
1 mark
b.
Find the distance between the turning point of the graph of f and the x-intercept of the
graph of g.
2 marks
c.
Sketch the graph of g -1, the inverse function of g, on the axes above, labelling any
axis intercept with its coordinates and any asymptote with its equation.
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© THE HEFFERNAN GROUP 2020
Maths Methods 1 & 2 Trial Exam 1
2 marks
8
Question 8 (6 marks)
Let f :[-2, ¥) ® R, f ( x) = x3 - x 2 - 5x - 3 .
a.
Show that x 3 - x 2 -5x - 3= (x - 3)(x +1) 2 .
1 mark
b.
Find the coordinates of the stationary points of x.
2 marks
c.
Sketch the graph of f on the axes below. Label the axis intercepts, stationary points
and endpoint with their coordinates.
3 marks
y
4
2
-4
-3
-2
-1
O
1
2
3
4
x
-2
-4
-6
-8
-10
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© THE HEFFERNAN GROUP 2020
Maths Methods 1 & 2 Trial Exam 1
9
Question 9 (5 marks)
A kitchen bench top is to have two rectangles cut out of it so that two sinks can be installed.
The length of the smaller sink is x cm and the width is y cm.
The length of the larger sink is (2 x + 5) cm and the width is 2y cm.
kitchen bench top
2x + 5
x
y
2y
sink cut-outs
The total length of cutting required through the bench top is 310 cm.
Find the length and width of the larger sink if the total area of the two sinks is to be a
maximum.
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© THE HEFFERNAN GROUP 2020
Maths Methods 1 & 2 Trial Exam 1
10
Question 10 (5 marks)
1
Let h : R \ {0} ® R, h( x) = 2 - .
x
The graph of h is shown below.
y
y = h( x )
y=2
x
O
x=0
a.
Find the instantaneous rate of change of h at the point where x =1.
b.
Find the average rate of change of h between x =
1
and x =1.
2
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© THE HEFFERNAN GROUP 2020
Maths Methods 1 & 2 Trial Exam 1
1 mark
2 marks
11
c.
The dilation T maps the graph of h(x) = 2 -
1
2
onto the graph of g(x) = 2 - , where
x
x
æ é x ù ö éa 0ù é x ù
T : R 2 ® R 2 , T çç ê ú ÷÷ = ê
ú ê ú and a and b are real constants.
è ë y û ø ë0 bû ë y û
Find the values of a and b.
2 marks
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© THE HEFFERNAN GROUP 2020
Maths Methods 1 & 2 Trial Exam 1
12
Mathematical Methods Units 1 & 2 formulas
Functions and graphs
distance formula
d = ( x2 - x1 ) 2 + ( y2 - y1 ) 2
midpoint formula
æ x + x y + y2 ö
midpoint = ç 1 2 , 1
2 ÷ø
è 2
Straight line graphs
general equation
y = mx + c
equation through point ( x1 , y1 )
y - y1 = m( x - x1 )
gradient
m=
y2 - y1
x2 - x1
Mensuration
circumference of a
circle
2p r
area of a circle
p r2
volume of a sphere
4
p r3
3
volume of a cylinder
p r 2h
Calculus
( )
d n
x = nx n -1
dx
1
ò x dx = n + 1 x
n
n +1
+ c, n ¹ -1
Probability
Pr( A) = 1 - Pr( A' )
Pr( A | B) =
Pr( A È B) = Pr( A) + Pr(B) - Pr( A Ç B)
Pr( A Ç B)
Pr(B)
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© THE HEFFERNAN GROUP 2020
Maths Methods 1 & 2 Trial Exam 1