Page 1 of 15 Alternative Education Equivalency Assessments (AEEA) Year 12 Advanced Mathematics Sample Questions Year 12 Advanced Mathematics Sample Questions Alternative Education Equivalency Assessments (AEEA) ADVANCED MATHEMATICS The following examples show the types of items in the test, but do not necessarily indicate the full range of items or test difficulty. For the Advanced Mathematics test, you may use a silent, battery-operated, non-programmable scientific calculator (not CAS/graphics calculator) and a ruler. See the Solutions pages for answers to these sample questions. Formulae The following formulae may be used in your calculations: Quadratic Equations If ax 2 bx c 0 then x b (b 2 4ac) 2a Series where a is the first term, L is the last, d is the common difference and r is the common ratio Arithmetic a (a d ) (a 2d ) ... (a (n 1) d ) n 2 (2a (n 1) d ) n 2 (a L) Geometric a ar ar 2 ... ar n1 a(1 r n ) 1 r , r 1 Space & Measurement In any triangle ABC, 1 sin 2 Area ab C Trapezium: Area = height, where a and b are the lengths of the parallel sides Prism: Volume = Area of base height Cylinder: Total surface area = Volume = Pyramid: Volume = area of base height sin sin sin a b c A B C 2 2 2 2 cos a b c bc A 2 2 2 cos 2 b c a A bc 1 ( ) 2 a b 2 2 2 rh r 2 r h 1 3
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Page 1 of 15 Alternative Education Equivalency Assessments (AEEA)
Year 12 Advanced Mathematics Sample Questions
Year 12 Advanced Mathematics
Sample Questions Alternative Education Equivalency Assessments (AEEA)
ADVANCED MATHEMATICS
The following examples show the types of items in the test, but do not necessarily indicate the full
range of items or test difficulty. For the Advanced Mathematics test, you may use a silent,
battery-operated, non-programmable scientific calculator (not CAS/graphics calculator) and a
ruler. See the Solutions pages for answers to these sample questions.
Formulae
The following formulae may be used in your calculations:
Quadratic Equations
If ax2 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
Arithmetic
a (ad) (a2d) ... (a (n1)d)n
2(2a (n1)d)
n
2(a L)
Geometric
aar ar2 ...arn1 a(1 rn)
1 r, r 1
Space & Measurement
In any triangle ABC,
1sin
2Area ab C
Trapezium: Area = height, where a and b are the lengths of the parallel sides
Prism: Volume = Area of base height
Cylinder: Total surface area = Volume =
Pyramid: Volume = area of base height
sin sin sin
a b c
A B C
2 2 2 2 cos a b c bc A
2 2 2
cos2
b c aA
bc
1( )
2a b
22 2r h r 2r h
1
3
Page 2 of 15 Alternative Education Equivalency Assessments (AEEA)
Year 12 Advanced Mathematics Sample Questions
Space & Measurement (cont’d)
Cone: Total surface area = , s is the slant height Volume =
Sphere: Total surface area = Volume =
Volume of solids of revolution about the axes: and
Rate: If y’ = ky, then y = Ae kx
Temperature conversion formula
Degrees Celsius to degrees Fahrenheit: ( 1.8) 32F C
Theorem of Pythagoras
In any right-angled triangle: 2 2 2c a b
Index laws
For and m ,n real,
For m an integer and n a positive integer
Calculus
Function notation Leibniz Notation
Product
rule
Quotient
rule
Chain rule
Fundamental Theorem of Calculus: and
Standard Derivatives
If y f (x) xn , then y '
dy
dx f '(x) nxn1
If y f (x) ex , then
dy
dx f '(x) ex
If y f (x) log
ex then y '
dy
dx f '(x)
1
x
If y f (x) sin(ax), then y '
dy
dx f '(x) a cos(ax)
If y f (x) cos(ax), then y '
dy
dx f '(x) a sin(ax)
2r s r 21
3r h
24 r 34
3r
2y dx2x dy
, 0a b
m n m na a a ( )m m ma b ab ( )m n mna a
1m
ma
a
m
m n
n
aa
a
0 1a
m
mn nmna a a
y y y y
( ) ( )f x g x ( ) ( ) ( ) ( )f x g x f x g x uvdu dv
v udx dx
( )
( )
f x
g x 2
( ) ( ) ( ) ( )
( ( ))
f x g x f x g x
g x
u
v 2
du dvv u
dx dx
v
( ( ))f g x ( ( )) ( )f g x g x ( ) ( )andy f u u g x dy du
du dx
( ) ( )x
a
df t dt f x
dx ( ) ( ) ( )
b
af x dx f b f a
Page 3 of 15 Alternative Education Equivalency Assessments (AEEA)
Year 12 Advanced Mathematics Sample Questions
Standard Integrals
= , , and if
= = ,
= , = ,
Probability laws
Trigonometry
In any right-angled triangle:
sin θ = 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
cos θ = 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
tan θ = 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
Growth, decay and interest formulae
Simple growth or decay: (1 )A P ni
Compound growth or decay: (1 )nA P i
Where:
A = amount at the end of n years
P = principal
n = number of years
r % = interest rate per year, i = r
100
dxxn
1
1
1
nxn
1n 0x 0n
dxx
1,ln x 0x dxeax
axe
a
10a
axcos dx axa
sin1
0a axsin dx axa
cos1
0a
( ) ( ) 1P A P A
( ) ( ) ( ) ( )P A B P A P B P A B
( ) ( ) ( / ) ( ) ( / )P A B P A P B A P B P A B
)Pr(
)Pr()/Pr(
B
BABA
Opposite side
Adjacent side
Hypotenuse
Page 4 of 15 Alternative Education Equivalency Assessments (AEEA)
Year 12 Advanced Mathematics Sample Questions
Growth, decay and interest formulae (cont’d)
Compound interest, where the interest is compounded t times per year:
nt
t
iPA )1(
Where:
t number of interest periods per year
Future value of an annuity: [(1 ) 1]nx i
Fi
contributions at end of each period
OR [(1 ) 1] (1 )nx i i
Fi
contributions at beginning of each period
Where:
F = future value of annuity
i = interest rate per compounding period, as a decimal fraction
n = number of compounding periods
Page 5 of 15 Alternative Education Equivalency Assessments (AEEA)
Year 12 Advanced Mathematics Sample Questions
Real functions
Example 1 For the basic following functions: f(x) = x
x
1
12 and h(x) = 1 – 2x find the
composite function , ( ( ))f h x in simplest terms: (2 marks)