Alternative Education
Equivalency Scheme (AEES)
YEAR 12
ADVANCED
MATHEMATICS
PART C
Practice Test
Version 01
PRACTICE TEST INSTRUCTIONS
1.
There are three Practice Test components – A, B and C (and 3 test components in the final
test).
2.
There are ELEVEN extended response questions in this Practice test component (and 11
questions in each final test component).
3.
You must show your calculations in your answer.
4.
Answer all the questions in the PRACTICE TEST BOOKLET.
5.
Give yourself 30 minutes to complete this Practice test component (as there are 30
minutes for each final test component). Allow 1.5 hours in total to complete all three
Practice Test components.
6.
We suggest setting up a test environment and timing yourself so it’s similar to the final test.
7.
Do not waste time if you do not know the answer. Move on to the next question and if you
have time at the end of the test, go back to the questions you skipped.
8.
Work as quickly and accurately as you can.
9.
We strongly recommend using a silent, battery-operated, non-programmable scientific
calculator. To obtain an accurate result, no other resource or device should be used for
this Practice test.
10.
Formulae are provided at the start of each Practice test component (and in each final test
component booklet).
11.
Diagrams are not necessarily drawn to scale.
12.
After you finish the test, use the solutions provided at the back of each Practice Test
component to check your answers.
13.
The score you achieve on the Practice Test is an indication only of how you may perform
on the final test.
14.
The Practice Test score cannot be used in place of sitting the final test.
15.
The topics listed on the Answer Key can give you an indication of the areas you can
improve.
Page 2 of 16
Alternative Education Equivalency (AEE) Assessments
Year 12 Advanced Mathematics Part C – Version 01
FORMULAE SHEET
The following formulae may be used in your calculations:
Quadratic Equations
If ax 2  bx  c  0 then x 
b  (b2  4ac)
2a
Series
where a is the first term, L is the last, d is the common difference and r is the common ratio
n
n
a  (a  d)  (a  2d)  ...  (a  (n  1)d)  (2a  (n  1)d)  (a  L)
2
2
n
a(1  r )
a  ar  ar 2  ...  ar n1 
, r 1
1 r
Arithmetic
Geometric
Space & Measurement
In any triangle ABC,
a
b
c


sin A sin B sin C
Area 
1
ab sin C
2
a 2  b2  c2  2 bc cos A
cos A 
Trapezium:
b2  c 2  a 2
2 bc
Area =
1
(a  b)  height, where a and b are the lengths of the parallel sides
2
Prism:
Volume = Area of base  height
Cylinder:
Total surface area = 2 r h  2 r 2
Pyramid:
Volume =
Cone:
Total surface area =  r s   r 2 , s is the slant height
Sphere:
Total surface area = 4  r 2
Volume =  r 2  h
1
 area of base  height
3
Volume =
Volume =
1
 r2  h
3
4
 r3
3
Volume of solids of revolution about the axes:   y 2 dx and   x 2 dy
Rate: If y  ky, then y  Aekx
Page 3 of 16
Alternative Education Equivalency (AEE) Assessments
Year 10 English - Version 01
Temperature conversion formula
Degrees Celsius to degrees Fahrenheit: °F = (°C x 1.8) + 32
Theorem of Pythagoras: In any right-angled triangle c 2  a 2  b2
Index Laws
For a , b  0 and m ,n real,
am an  am  n
am 
a m bm  (ab)m
am
1
a
m
a
n
(a m )n  a m n
 am  n
a0  1
m
an 
For m an integer and n a positive integer
n
am 
 a
n
m
Calculus
Function notation
y
y
Leibniz Notation
y
y
du
dv
vu
dx
dx
Product rule
f ( x ) g ( x)
f ( x) g ( x)  f ( x) g ( x)
uv
Quotient rule
f ( x)
g ( x)
f ( x) g ( x)  f ( x) g ( x)
( g ( x))2
u
v
Chain rule
f ( g ( x))
f ( g ( x)) g ( x)
y  f (u) and u  g ( x)
Fundamental Theorem of Calculus:
du
dv
vu
dx
dx
2
v
dy du

du dx
d x
b
 f (t ) dt  f ( x) and a f ( x) dx  f (b)  f (a)
dx a
Standard Derivatives
If y  f(x)  x n , then y ' 
dy
 f '(x)  nx n1
dx
dy
dy
1
 f '(x)  e x If y  f(x)  loge x then y ' 
 f '(x) 
dx
dx
x
dy
If y  f(x)  sin(ax), then y ' 
 f '(x)  a cos(ax)
dx
dy
If y  f (x)  cos(ax), then y ' 
 f '(x)  a sin(ax)
dx
If y  f(x)  e x , then
Standard Integrals
1 n1
x , n  1 , and x  0 if n  0
n 1
1
1
ax
= ln x, x  0
= e ax , a  0
 e dx
 x dx
a
1
1
 cos ax dx = a sin ax , a  0  sin ax dx =  a cos ax , a  0
x
Page 4 of 16
n
dx
=
Alternative Education Equivalency (AEE) Assessments
Year 10 English - Version 01
Probability Laws
P( A)  P( A)  1
P( A  B)  P( A)  P( B)  P( A  B)
P( A  B)  P( A) P( B / A)  P( B) P( A / B)
Pr( A / B) 
Pr( A  B)
Pr(B)
Trigonometry:
In any right-angled triangle:
sin θ =
opposite
hypotenuse
cos θ =
adjacent
hypotenuse
Hypotenuse
Opposite side
θ
Adjacent side
tan θ =
opposite
adjacent
Growth decay and interest formulae:

Simple growth or decay: A  P(1  ni )

Compound growth or decay: A  P(1  i )n
Where:
A = amount at the end of n years
P  principal
n  number of years
r% = interest rate per year, i 

r
100
Compound interest, where the interest is compounded t times per year:
i
A  P(1  ) nt
t
Where:
t  number of interest periods per year

Future value of an annuity: F 
OR F 
x[(1  i )n  1]
Contributions at end of each period
i
x[(1  i )n  1]  (1  i )
Contributions at beginning of each period
i
Where:
F = future value of annuity
i = interest rate per compounding period, as a decimal fraction
n = number of compounding periods
Page 5 of 16
Alternative Education Equivalency (AEE) Assessments
Year 10 English - Version 01
Question 1 - algebra
(1 marks)
Fully simplify the expression (3x)2 + (x - 1)(x + 3)
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______________________________________________________________________________
______________________________________________________________________________
Question 2 – Real Functions
For the basic following functions: f(x) =
2x  1
and g(x) =
1 x
(2 marks)
x  1 find the composite function, g( f (x)) in
simplest terms:
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______________________________________________________________________________
______________________________________________________________________________
Question 3 – applications of geometrical properties
(2 marks)
The ∆ABC is reflected in the line y = x. Show the line y = x and the image of ∆ABC (with the coordinates of
the vertices labelled) on the graph below.
Page 6 of 16
Alternative Education Equivalency (AEE) Assessments
Year 10 English - Version 01
Question 4 - logarithmic and exponential functions
(4 marks)
Sketch the graph of 𝑦 = 𝑙𝑜𝑔𝑒 (𝑥 − 3) + 2. State the domain and range, the equation of the asymptote and
the coordinates of any x and y intercepts.
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______________________________________________________________________________
______________________________________________________________________________
Question 5 – Linear Functions
(3 marks)
Find the equation of the line that passes through the point (4,6) and is perpendicular to the line
y = 2x + 4
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______________________________________________________________________________
______________________________________________________________________________
Page 7 of 16
Alternative Education Equivalency (AEE) Assessments
Year 10 English - Version 01
Question 6 - Probability
(2 marks)
A 6-sided dice is thrown and unfortunately it is later found to be weighted unfairly so that Pr(6) =
1
. What
3
is the exact probability of throwing at least one ‘6’ in three throws with this die?
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______________________________________________________________________________
______________________________________________________________________________
Question 7 - plane geometry – geometrical properties
(4 marks)
On the diagram below, BC and AC are tangents to the circle
B
o
210
O
A
(a)
Find BCO
C
(2 marks)
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______________________________________________________________________________
______________________________________________________________________________
(b)
Prove that ∆OBC is congruent to ∆OAC
(2 marks)
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______________________________________________________________________________
______________________________________________________________________________
Page 8 of 16
Alternative Education Equivalency (AEE) Assessments
Year 10 English - Version 01
Question 8 - coordinate methods in geometry
(3 marks)
Using coordinate geometry methods or otherwise, find the equation of the locus of all points P(x,y),
equidistant from A(p,3) and B(-p,3).
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
Question 9 - the quadratic polynomial and the parabola
(7 marks)
A parabola is represented by the function f(x) = x2 + 2x - 8
a)
Find the coordinates of the x- intercepts
(2 marks)
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
b)
State the coordinates of the turning point of the parabola
(2 marks)
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
c)
Find the coordinate(s) of the point(s) of intersection of this parabola and the line y = 2x – 4 (3 marks)
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
Page 9 of 16
Alternative Education Equivalency (AEE) Assessments
Year 10 English - Version 01
Question 10 – Geometrical Applications of Differentiation
Y
(2 marks)
(3,y)
•
1
5
X
If Y represents g’(x), sketch a possible graph of the primitive function, g(x), on the axes below. Clearly show
the x-coordinates of any intercept values on the X-axis and of any maxima, minima or points of inflection.
Y
X
Question 11 - tangent to a curve and derivative of a function
(4 marks)
x4
For the hyperbola y = x  3 , find an expression for the equation of the tangent to the curve at x = 2.
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______________________________________________________________________________
______________________________________________________________________________
END OF TEST
Page 10 of 16
Alternative Education Equivalency (AEE) Assessments
Year 10 English - Version 01
SOLUTIONS
Solution 1

9x2 + x2 + 2x – 3 = 10x2 + 2x – 3
1 mark
Solution 2
 2x  1 
 2x  1 
g( f (x))  g 
 1
 
 1 x 
 1 x


x 2
1 x


Solution 3
1 mark correct triangle with coordinates
1 mark correct line y = x
Page 11 of 16
Alternative Education Equivalency (AEE) Assessments
Year 10 English - Version 01
Solution 4

Graph shape

domain 𝑥 > 3 (both correct)
range 𝑅
equation of asymptote 𝑥 = 3
x intercept (3.14 , 0)


(no y intercept – no need to state)
Solution 5
m = -1/2
1
y–6=- x+4
2
1
y = - x + 10
2




or 2y + x = 20
Page 12 of 16
Alternative Education Equivalency (AEE) Assessments
Year 10 English - Version 01
Solution 6
Pr(  one ‘6’) = 1 – Pr(No ‘6’s) = 1 – (
= 1-
8
19
=
27 27
2 3
)
3


Solution 7
(a) <BOA = 360o - 210o = 150o
By symmetry, <BOC = ½ <BOC = 75o , <OBC = 90O (tangent)
<BCO = 180 – 75 – 90 = 15o

(b) OB = OA = radius (S), OC common (S), BC = AC symmetry (S)
(or SAS or SAS)
Must give reason SSS etc



Solution 8
( x  p) 2  ( y  3) 2 =
( x  p) 2  ( y  3) 2
-2xp = 2xp
0 = 4xp
 x = 0 is equation of the locus
Solution 9




a) f(x) = (x + 4)(x – 2)
so x-intercepts at x = -4 and 2

b) By symmetry of x-intercepts the TP will be at x = -1
so y = (-1+4)(-1-2) = -5

or (1,-9)
c) x2 + 2x - 8 = 2x – 4
 x2 = 4, x = 2 or -2
 (2, 0) and (-2, -8)
Page 13 of 16




Alternative Education Equivalency (AEE) Assessments
Year 10 English - Version 01
Solution 10
MIN
POI
1
X
5
Solution 11
dy 1( x  3)  1( x  4)
dx =
( x  3) 2
dy  1 6  7
at x = 2, dx =
1 = 1
and y = -6



Equation of tangent is y + 6 = -7(x -2)
y = -7x +8
Page 14 of 16

Alternative Education Equivalency (AEE) Assessments
Year 10 English - Version 01
Notes
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Page 15 of 16
Alternative Education Equivalency (AEE) Assessments
Year 10 English - Version 01
© 2014 Vocational Education and Training Assessment Services
Level 5, 478 Albert Street, East Melbourne Victoria 3002.
All rights reserved. No part of this test may be reproduced
without written permission from VETASSESS.
18-8-2014