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Victorian Certificate of Education
Year
SPECIALIST MATHEMATICS
Written examinations 1 and 2
FORMULA SHEET
Instructions
This formula sheet is provided for your reference.
Students are NOT permitted to bring mobile phones and/or any other unauthorised electronic
devices into the examination room.
© VICTORIAN CURRICULUM AND ASSESSMENT AUTHORITY 2016
Version 4 – April 2016
SPECMATH EXAM
2
Specialist Mathematics formulas
Mensuration
area of a trapezium
1 a+b h
(
)
2
curved surface area of a cylinder
2π rh
volume of a cylinder
π r2h
volume of a cone
1 2
π r h
3
volume of a pyramid
1 Ah
3
volume of a sphere
4 π r3
3
area of a triangle
1 bc sin (A )
2
sine rule
a
b
c
= =
sin ( A) sin ( B) sin (C )
cosine rule
c2 = a2 + b2 – 2ab cos (C )
Circular (trigonometric) functions
cos2 (x) + sin2 (x) = 1
1 + tan2 (x) = sec2 (x)
cot2 (x) + 1 = cosec2 (x)
sin (x + y) = sin (x) cos (y) + cos (x) sin (y)
sin (x – y) = sin (x) cos (y) – cos (x) sin (y)
cos (x + y) = cos (x) cos (y) – sin (x) sin (y)
cos (x – y) = cos (x) cos (y) + sin (x) sin (y)
tan ( x + y ) =
tan ( x) + tan ( y )
1 − tan ( x) tan ( y )
tan ( x − y ) =
tan ( x) − tan ( y )
1 + tan ( x) tan ( y )
cos (2x) = cos2 (x) – sin2 (x) = 2 cos2 (x) – 1 = 1 – 2 sin2 (x)
sin (2x) = 2 sin (x) cos (x)
tan (2 x) =
2 tan ( x)
1 − tan 2 ( x)
3
SPECMATH EXAM
Circular (trigonometric) functions – continued
Function
sin–1(arcsin)
cos–1(arccos)
tan–1(arctan)
Domain
[–1, 1]
[–1, 1]
R
 π π
− 2 , 2 


[0, �]
 π π
− , 
 2 2
Range
Algebra (complex numbers)
z = x + iy = r ( cos (θ ) + i sin (θ ) ) = r cis (θ )
z = x2 + y 2 = r
z1z2 = r1r2 cis (θ1 + θ2)
–π < Arg(z) ≤ π
z1 r1
= cis θ − θ
z2 r2 ( 1 2 )
zn = rn cis (nθ) (de Moivre’s theorem)
Probability and statistics
for random variables X and Y
E(aX + b) = aE(X) + b
E(aX + bY ) = aE(X ) + bE(Y )
var(aX + b) = a2var(X )
for independent random variables X and Y
var(aX + bY ) = a2var(X ) + b2var(Y )
approximate confidence interval for μ
s
s 

, x+z
x −z

n
n

mean
distribution of sample mean X
variance
( )
σ2
var ( X ) =
n
E X =µ
TURN OVER
SPECMATH EXAM
4
Calculus
d n
x = nx n − 1
dx
∫ x dx = n + 1 x
d ax
e = ae ax
dx
∫e
d
1
( log e ( x) ) =
x
dx
∫ x dx = log
d
( sin (ax) ) = a cos (ax)
dx
∫ sin (ax) dx = − a cos (ax) + c
d
( cos (ax) ) = −a sin (ax)
dx
∫ cos (ax) dx = a sin (ax) + c
d
( tan (ax) ) = a sec2 (ax)
dx
( )
( )
1
n
ax
dx =
n +1
+ c, n ≠ −1
1 ax
e +c
a
1
e
x +c
1
1
1
(
)
∫ sec (ax) dx = a tan (ax) + c
1
x
∫ a − x dx = sin  a  + c, a > 0
(
)
∫
d
1
sin −1 ( x) =
dx
1 − x2
d
−1
cos −1 ( x) =
dx
1 − x2
d
1
tan −1 ( x) =
dx
1 + x2
(
)
2
−1
2
∫
product rule
quotient rule
x
dx = cos −1   + c, a > 0
a
a −x
−1
2
2
x
dx = tan −1   + c
+x
a
1
(ax + b) n dx =
(ax + b) n + 1 + c, n ≠ −1
a (n + 1)
∫a
∫
2
a
2
2
(ax + b) −1 dx =
d
dv
du
( uv ) = u + v
dx
dx
dx
du
dv
−u
v
d u
dx
dx
 =
2
dx  v 
v
chain rule
dy dy du
=
dx du dx
Euler’s method
If
acceleration
a=
arc length
1
log e ax + b + c
a
dy
= f ( x), x0 = a and y0 = b, then xn + 1 = xn + h and yn + 1 = yn + h f (xn)
dx
∫
x2
x1
d 2x
dt
2
=
dv
dv d  1 
= v =  v2 
dt
dx dx  2 
2
1 + ( f ′( x) ) dx or
Vectors in two and three dimensions
∫
t2
t1
( x′(t ) )2 + ( y′(t ) )2 dt
Mechanics
r = x i + yj + zk

 

momentum
p = mv


r = x2 + y 2 + z 2 = r

d r dx
dy
dz
i
r = = i+
j+ k
dt dt dt dt equation of motion
R = ma


r 1 . r 2 = r1r2 cos (θ ) = x1 x2 + y1 y2 + z1 z2
 
END OF FORMULA SHEET