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
1 a+b h
area of a trapezium
2
( )
curved surface area of a cylinder 2rh
volume of a cylinder r2h
1 2
volume of a cone r h
3
volume of a pyramid 1 Ah
3
4 r3
volume of a sphere
3
1 bc sin (A )
area of a triangle
2
a b c
sine rule = =
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) + tan ( y ) tan ( x) tan ( y )
tan ( x + y ) = tan ( x y ) =
1 tan ( x) tan ( y ) 1 + tan ( x) tan ( y )
cos(2x) = cos2(x) sin2(x) = 2 cos2(x) 1 = 1 2 sin2(x)
2 tan ( x)
sin(2x) = 2 sin(x) cos(x) tan (2 x) =
1 tan 2 ( x)
3 SPECMATH EXAM
Circular (trigonometric) functions continued
Function sin1(arcsin) cos1(arccos) tan1(arctan)
Domain [1, 1] [1, 1] R
Range 2 , 2 [0, ] ,
2 2
Algebra (complex numbers)
z = x + iy = r ( cos ( ) + i sin ( ) ) = r cis ( )
z = x2 + y 2 = r < Arg(z)
z1 r1
z2 r2 ( 1 2 )
z1z2 = r1r2 cis(1 + 2) = cis
zn = rn cis(n) (de Moivres theorem)
Probability and statistics
E(aX + b) = aE(X) + b
for random variables X and Y 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)
s s
approximate confidence interval for x z , x+z
n n
mean ( )
E X =
distribution of sample mean X 2
variance var ( X ) =
n
TURN OVER
SPECMATH EXAM 4
Calculus
d n
dx
( )
x = nx n 1
x dx = n + 1 x
n 1 n +1
+ c, n 1
d ax
dx
( )
e = ae ax
e
ax
dx =
1 ax
a
e +c
d 1 1
dx
( log e ( x) ) =
x x dx = log e x +c
d 1
dx
( sin (ax) ) = a cos (ax) sin (ax) dx = a cos (ax) + c
d 1
dx
( cos (ax) ) = a sin (ax) cos (ax) dx = a sin (ax) + c
d 1
( tan (ax) ) = a sec2 (ax) sec (ax) dx = a tan (ax) + c
2
dx
d
dx
( )
sin 1 ( x) =
1
1 x2
1
2
x
a x dx = sin a + c, a > 0
2
1
d
dx
(
cos 1 ( x) =) 1
1 x2
1 x
dx = cos 1 + c, a > 0
2 a
2
a x
d a x
dx
(
tan 1 ( x) =) 1
1 + x2 a +x2
dx = tan 1 + c
2
a
1
(ax + b) n dx =
a (n + 1)
(ax + b) n + 1 + c, n 1
1
(ax + b) 1 dx =
a
log e ax + b + c
d dv du
product rule ( uv ) = u + v
dx dx dx
du dv
v u
quotient rule d u dx dx
= 2
dx v v
dy dy du
chain rule =
dx du dx
dy
Eulers method If = f ( x), x0 = a and y0 = b, then xn + 1 = xn + handyn + 1 = yn + hf(xn)
dx
d 2x dv dv d 1
acceleration a= 2
= = v = v2
dt dt dx dx 2
x2 t2
2
arc length 1 + ( f ( x) ) dx or ( x(t ) )2 + ( y(t ) )2 dt
x1 t1
Vectors in two and three dimensions Mechanics
r = x i + yj + zk momentum p = mv
r = x2 + y 2 + z 2 = r equation of motion R = ma
i d r dx dy dz
r = = i+ j+ k
dt dt dt dt
r 1 . r 2 = r1r2 cos ( ) = x1 x2 + y1 y2 + z1 z2
END OF FORMULA SHEET
