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MATHEMATICS SPECIALIST
Unit 1 and Unit 2
Formula Sheet
(For use with Year 11 examinations and response tasks)
2015/75004v9
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This document is valid for teaching and examining from 1 July 2015.
MATHEMATICS SPECIALIST
UNIT 1 AND UNIT 2
1
FORMULA SHEET
Measurement
Circle:
C = 2πr = π D, where C is the circumference,
r is the radius and D is the diameter
A = πr 2, where A is the area
Triangle:
1
A = bh, where b is the base and h is the perpendicular height
2
Parallelogram:
A = bh
Trapezium:
A=
Prism:
V = Ah, where V is the volume and A is the area of the base
Pyramid:
V = 1 Ah
3
Cylinder:
S = 2 πrh + 2 πr 2, where S is the total surface area
V = πr 2h
Cone:
S = πrs + πr 2, where s is the slant height
1
V = 3 πr 2h
Sphere:
S = 4πr 2
4
V = 3 πr 3
1
(a + b)h, where a and b are the lengths of the parallel sides
2
MATHEMATICS SPECIALIST
UNIT 1 AND UNIT 2
2
FORMULA SHEET
Combinatorics
Combinations
Number of arrangements: (of n different objects in an ordered list)
n(n 1)(n  2)  .....  3  2 1  n!
Number of combinations: (of r objects taken from a set of n distinct objects)
æ n ö
ç 0 ÷ =1
è
ø
n  n 
 
;
r  nr
n
n!
;
 
 r  r ! n  r !
Number of permutations: (of r objects taken from a set of n distinct objects)
nP  n n 1 n  2 ... n  r  1  n!
   
  n  r !
r
n!
r1 !r2 !r3 ! ...
Number of permutations with some identical objects:
Inclusion – exclusion principle:
| A  B || A |  | B|-|A  B |
| A B C  A  B  C – A B – AC – B C  A B C |
Vectors in the Plane
Representing vectors
Magnitude of a vector:
a = ( a1 ,a2 ) = a12 + a22
Algebra of vectors
Unit vector:
Scalar product:
aˆ 
a
a
𝐚⦁𝐛 = |𝐚||𝐛|𝒄𝒐𝒔𝜃
or
̂ )𝐛
̂ = |𝐚|𝑐𝑜𝑠𝜃 𝐛
̂
Vector projection (of a on b): 𝐩 = (𝐚⦁𝐛
𝐚⦁𝐛 = 𝑎1 𝑏1 + 𝑎2 𝑏2
MATHEMATICS SPECIALIST
UNIT 1 AND UNIT 2
3
FORMULA SHEET
Trigonometry
Basic trigonometric functions
sin     sin 
cos    cos
 
sin     cos 
 2
 
cos     sin 
 2
tan     tan 
Cosine and sine rules
For any triangle ABC with corresponding length of sides a,b,c
Cosine rule:
c 2  a 2  b 2  2ab cos C
Sine rule:
a
b
c
=
=
sin A sin B sinC
Area of  :
1
A  ab sin C
2
1
 s ( s  a)( s  b)( s  c) where s  (a  b  c)
2
Circular measure and radian measure
In a circle of radius
r for an arc subtending angle  (radians) at the centre
Length of arc:
Area of sector:
 r
A  12 r 2
Length of chord:
l  2r sin 12 
Area of segment:
A  12 r 2 (  sin  )
Compound angles
Angle sum and difference identites:
sin  A  B   sin A cos B  cos Asin B
cos  A  B   cos A cos B sin Asin B
tan  A  B  
Double angle identities:
tan A  tan B
1 tan A tan B
sin 2 A  2sin A cos A
cos2 A  cos2 A  sin 2 A  2cos2 A 1  1  2sin 2 A
tan 2 A =
2 tan A
1  tan 2 A
MATHEMATICS SPECIALIST
UNIT 1 AND UNIT 2
4
FORMULA SHEET
Reciprocal trigonometric functions
sec  
1
, cos   0
cos 
cosec 
1
, sin  0
sin
cot  
1
, tan   0
tan 
Trigonometric identities
Pythagorean identities: sin 2   cos2   1
1  tan 2   sec2 
cot 2   1  cosec2
1
cos A cos B  cos  A  B   cos  A  B  
2
1
sin A sin B  cos  A  B   cos  A  B  
2
1
sin A cos B  sin  A  B   sin  A  B  
2
1
cos A sin B  sin  A  B   sin  A  B  
2
Product identities:
Auxiliary angle formulae:

𝑎 sin 𝑥 ± 𝑏 cos 𝑥 = 𝑅 sin(𝑥 ± 𝛼) for 0< < , where R 2 = a 2  b 2 , tan  
2
Triple angle identities: sin  3 A 
 3sin A  4sin 3 A
cos (3𝐴) = 4 cos3 𝐴 − 3 cos 𝐴
tan  3 A
=
3tan A  tan 3 A
1  3tan 2 A
b
a
MATHEMATICS SPECIALIST
UNIT 1 AND UNIT 2
5
FORMULA SHEET
Matrices
Matrix arithmetic
Identity matrix:
If A is invertible, AA
1
= I where I is the identity matrix
1
Inverse matrix:
a b 
1  d b 
 c d   ad  bc  c a 




Determinant:
If A = ê
é a b ù
ú then det A  ad  bc
c
d
ë
û
Transformation Matrices
Dilation:
é a 0 ù
ê
ú
ë 0 b û
Rotation:
cos 
 sin 

Reflection:
cos 2
 sin 2

 sin  
where  is an anti-clockwise rotation about the origin
cos  
sin 2 
where the reflection is in the line y  x tan
 cos 2 
Real and Complex numbers
Number Sets
Natural Numbers:
Integer Numbers:
ℕ ∶= {1,2,3, … . . }
ℤ ∶= {… − 2, −1,0,1,2, … }
Rational Numbers:
ℚ ∶= {𝑞: 𝑞 = 𝑏 , where a and b are integers, 𝑏 ≠ 0}
Irrational Numbers:
Numbers that cannot be expressed as the ratio of two integers
Real Numbers:
Complex Numbers:
The set of all rational and irrational numbers (ℝ)
ℂ ∶= {𝑧 ∶ 𝑧 = 𝑎𝑖 + 𝑏, where 𝑎, 𝑏 ∈ ℝ , 𝑖 2 = −1 }
𝑎
Complex Numbers
For 𝑧 = 𝑎𝑖 + 𝑏, where 𝑎, 𝑏 ∈ ℝ , 𝑖 2 = −1
Modulus:
mod z = z = a + ib = a 2 + b2
Product:
z1z2 = z1 z2
Conjugate:
z  a  ib , zz = z , z1 + z2 = z1 + z2 ,
2
z1z2 = z1z2
MATHEMATICS SPECIALIST
UNIT 1 AND UNIT 2
6
FORMULA SHEET
Other useful results
Binomial expansion:
Binomial coefficients:
Index laws:
For a, b >0 and m,n real,
a mb m  (ab) m
a
m
1
 m
a
a m a n  a mn
(a m ) n  a mn
am
 a mn
n
a
a0  1
m
For a > 0, m an integer and n a positive integer, a n  n a m 
Arithmetic sequences
For initial term a and common difference d:
 a
n
m
Tn  a   n  1 d , n  1
Tn1  Tn  d , where T1  a
Sn 
Geometric sequences
For initial term a and common difference r:
n
 2a   n  1 d 
2
Tn+1 = rTn , where T1 = a
Tn  ar n1 ,n  1
Sn 
a 1  r n 
1 r
a
S 
,
1 r
r 1
Lines and Linear relationships
(
)
(
For points P x1 ,y1 and Q x2 ,y2
)
Mid-point of P and Q :
Gradient of the line through P and Q :
Equation of the line through P with slope m:
æx +x y +y ö
M =ç 1 2 , 1 2÷
2 ø
è 2
y y
m 2 1
x2  x1
y  y1  m  x  x1 
Parallel lines:
m1 = m2
Perpendicular lines:
m1m2  1
General equation of a line:
ax + by + c = 0 or y = mx + c
MATHEMATICS SPECIALIST
UNIT 1 AND UNIT 2
7
FORMULA SHEET
Quadratic relationships
For the general quadratic equation ax2  bx  c  0, a  0
b  
b2 


c

Completing the square: ax  bx  c  a  x 



2a  
4a 

2
2
Discriminant:
  b2  4ac
Quadratic formula:
x
b  b 2  4ac
2a
Graphs and Relations
Equation of a circle:
 x  a
2
  y  b  r2
2
( )
where, a,b is the centre and r is the radius
Note: Any additional formulas identified by the examination writers as necessary will be included in
the body of the particular question.