A-Math Formula List - Additional Math (4047)
*Formulas highlighted in yellow are found in the formula list of the exam paper.
Quadratic Equation
b2 - 4ac > 0
Real and Distinct Roots
/ Unequal Roots
b2 - 4ac = 0
Real and Equal
Roots/Repeat Roots/
Coincident Roots.
b2 - 4ac < 0
Imaginary roots
Also known as Complex
Roots.
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A-Math Formula List - Additional Math (4047)
Roots of Quadratic Equation
The Quadratic Equation has solutions 𝜶 and 𝜷
Sum of Roots is
𝑏
Product of Roots is 𝛼𝛽
𝛼 + 𝛽 = −𝑎
=
𝑐
𝑎
The Equation is x2 – (Sum of Roots) x + (Product of Roots) = 0
𝛼 − 𝛽 = ±√(𝛼 − 𝛽)2
𝛼 2 + 𝛽2 = (𝛼 + 𝛽)2 − 2𝛼𝛽
𝛼 4 + 𝛽4 = (𝛼 2 + 𝛽2 )2 − 2(𝛼𝛽)2
Indices
Same Base Number
Same Power
xa  xb  xa b
am bm   ab m
x a a b
x
b
x
m
am  a 
 
m
b
b
b
 x a   xab
 
 
b
a


NOTE:  x   xa  xb
 
Base Number-Same → Power Add
or Subtract
Power Same→ Base Number
Multiply or Divide.
Other Laws of Indices
x a 
1
xa
1
b
x b  x1
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A-Math Formula List - Additional Math (4047)
xa
a

b
x y 
yb
1
1
1
x b

1 b 1
x
b
x
1
 xa
x a
x
 y
 
a
b
x b  xa
a
1
1
x b

a b a
x
xb

a
a
 y
 
x

Surds
a  b  ab
a a 
 
a
2
a b
a
b
m a  n b  m  n ab
a
m a  n a  (m  n) a
m a  n a  (m  n) a
Rationalizing Denominator

 

n a
n a
1
n a



2
n a n a
n2 a
n2  a


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A-Math Formula List - Additional Math (4047)

 

n a
n a
1
n a



2
n a n a
n2 a
n2  a


1
n a


n a
n a
 n a    n a 
2
2
na
 n   a 
1
n a


n a
n a
 n a    n a 
2
2
na
 n   a 
Partial Factions
Linear Factor
mx  n
A
B


(ax b)(cx d ) (ax b) (cx  d )
Repeat Factors
mx  n
(ax b)(cx d )2

A
B
C


(ax b) (cx  d ) (cx  d )2
Quadratic Factors
mx  n
A
Bx C

(ax b)(cx 2 d ) (ax b) (cx 2  d )

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A-Math Formula List - Additional Math (4047)
Note: If the highest coefficient of the NUMERATOR is the SAME or LARGER that
the DENOMINATOR. Do LONG DIVISION before partial fractions.
Logarithm
ln1 = 0
log b 1 = 0
by = x
y = log b x
log b b = 1
ln e x = x
log b bx = x
ln e xa = xa
Note: log b m + log b n = log b (m × n)
ln x = log e x
log e e=1
thus ln e =1
log b m + log b n ≠ log b m × log b n
log b m – log b n = log b (
Remember:
m
)
n
log b m a = a × log b m
log v u 
log a u
log a v
log v u 
log u u
1

log u v log u v
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A-Math Formula List - Additional Math (4047)
Binomial Theorem
n
n!
 
 r  r !(n  r )!
n
n n
   Cr
r
n
 n  n(n 1)  n  n(n 1)(n  2)
123
Remember:    1,    n,   
,  
0
1
2
1

2
 
 
 
 3
Since:
n
n
n
n
(a  bx) n  a n    a n 1 (bx)1    a n  2 (bx) 2    a n 3 (bx)3 ....   a n r (bx) r  (bx) n
1
 2
 3
r
Therefore:
n
(n)(n  1) n  2
(n)(n  1)(n  2) n 3
(a  bx) n  a n  a n 1 (bx)1 
a (bx) 2 
a (bx)3 ....  (bx) n
1
1 2
1 2  3
Alternate formula if a=1
1  kx 
n
n
n
n
 n  n ( n 1)
n
0
1
2
n 1
n
  1n  kx    1n 1  kx    1n  2  kx   .... 
 kx    1nn  kx 
1
0
1
 2
 n  1
 n
OR
n
n
 n 
n
1
2
n 1
n
1

kx

1

kx

kx

....


     

  kx    kx 
1
 2
 n  1
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A-Math Formula List - Additional Math (4047)
Trigonometry
cos ec 
1
sin 
sin 2   cos 2   1
sec 
1
cos
1  cot 2   cos ec 2
cot  
1 cos

tan  sin 
1  tan 2   sec 2 
Compound Angle Formula
sin  A  B   sin A cos B  cos A sin B
cos  A  B   cos A cos B sin A sin B
tan  A  B  
tan A  tan B
1 tan A tan B
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A-Math Formula List - Additional Math (4047)
Double Angle Formula
sin 2 A  2sin Acos A
cos 2 A  2 cos 2 A  1 / cos 2 A  cos 2 A  sin 2 A / cos 2 A  1  2sin 2 A
tan 2 A 
2 tan A
1  tan 2 A
Half Angle Formula
sin
A
1  cos A

2
2
cos
A
1  cos A

2
2
tan
A
1  cos A 1  cos A


2
1  cos A
sin A
R-Formula
aCos  bSin  RCos(  )
aSin  bCos  RSin(   )
Where
R  a 2  b2
tan  
b
a
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A-Math Formula List - Additional Math (4047)
Coordinate Geometry
Gradient(m) =
y2  y1
x2  x1
General Equation Y  y1  m( X  x1 )
where ( x1 , y1 ) is a point on the
graph.
Alternatively you may use y=mx+c
Mid-point of a line
(
x1  x2 y1  y2
,
)
2
2
Distance between two points
( x2  x1 )2  ( y2  y1 ) 2
If two lines have the same gradient
If two line have perpendicular
gradient
m1  m2
i.e. 90° to each other.
m1  
1
m2
or
m1  m2  1
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A-Math Formula List - Additional Math (4047)
Perpendicular Bisector
1. The two lines AB and PQ must
intersect at 90°
m1 ( AB )  
1
m2 ( PQ )
2. One Line (AB) will cut the mid-point of the other line (PQ)
x2  x1 y2  y1 
,

2 
 2
Mid-point PQ = 
Area of Plane Figure (Polygon Figure)
Vertices A(x1,y1), B(x2,y2),C (x3,y3)
1 𝑥1
|
2 𝑦1
𝑥2 𝑥3
𝑦2 𝑦3
𝑥4
𝑦4
𝑥1
𝑦1 |
1
|(𝑥 𝑦 + 𝑥2 𝑦3 + 𝑥3 𝑦4 + 𝑥4 𝑦1 ) − (𝑥2 𝑦1 + 𝑥3 𝑦2 + 𝑥4 𝑦3 + 𝑥1 𝑦4 )|
2 1 2
*It does NOT matter if you calculate in a clockwise or anti-clockwise direction.
The is a |modulus| in the formula
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A-Math Formula List - Additional Math (4047)
Circles
Radius of a circle
r  ( x  a)2  ( y  b)2
Equation of a Circle
(𝑥 − 𝑎)2 + (𝑦 − 𝑏 )2 = 𝑟 2 (Standard Form)
or
𝑥 2 + 𝑦 2 + 2𝑔𝑥 + 2𝑓𝑦 + 𝑐 = 0 (General Form)
where 𝑎 = −𝑔, 𝑏 = −𝑓
and 𝑟 = √𝑔2 + 𝑓 2 − 𝑐
Proofs in Plane Geometry
Midpoint Theorem
Tangent –Chord Theorem
(Alternate Segment Theorem)
If X and Y are midpoints,
then YZ // WX
Angle W = Angle X
Angle Y = Angle Z
YZ = ½ WX
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A-Math Formula List - Additional Math (4047)
Differentiation
dy
(ax n )  anx n 1
dx
Where’ a’ is a constant
Sum/Difference of Function
dy
(a)  0
dx
d
du dv
(u  v) 

dx
dx dx
and ‘n’ is an integer.
Chain Rule
dy dy du


dx du dx
Quotient Rule
du
dv
u
d u
dx
dx
 
2
dx  v 
v
v
Product Rule
d
dv
du
(uv)  u  v
dx
dx
dx
d
(sin x)  cos x
dx
d
(tan x)  sec 2 x
dx
d
(cos x)   sin x
dx
Chain Rule is use to differentiate the functions below.
d
 a sin  bx  c    a  cos(bx  c)  b
dx
d
 a cos  bx  c    a   sin(bx  c)  b
dx 
d
 a tan  bx  c    a  sec 2 (bx  c)  b
dx
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A-Math Formula List - Additional Math (4047)
d
 a sin n  bx  c    a  n  sin n 1 (bx  c)  cos(bx  c)  b
dx
d
 a cos n  bx  c    a  n  cos n 1 (bx  c)   sin(bx  c)  b
dx
d
 ata n n  bx  c    a  n  tan n 1 (bx  c)  sec 2 (bx  c )  b
dx
Exponential/Natural Logarithm Function
d ax b
( e )  a e ax  b
dx
d x
(e )  e x
dx
‘a’ is a constant
d
1
(ln x)  (where x>0)
dx
x
d
a
ln(ax  b) 
dx
ax  b
(where ax +b>0)
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A-Math Formula List - Additional Math (4047)
Integration
ax n 1
 ax dx  n  1  c
  ax  b 
n
n
 a dx  ax  c
where n≠-1
 ax  b 
dx 
n 1
(n  1)(a)
c
n≠1,a≠0; a & b are constants
Product of constant and a function
Sum and Difference of function
 af ( x)dx  a  f ( x)dx
  f ( x)   g ( x)dx    f ( x)dx    g ( x)dx
 a cos bx dx 
where f(x) is a function
a sin bx
c
b
1
 x dx  ln x  c
 a sin bx dx  
where x>0
a cos bx
c
b
∫
2
 a sec bx dx 
a tan bx
c
b
1
ln(𝑎𝑥 + 𝑏)
𝑑𝑥 =
+𝐶
𝑎𝑥 + 𝑏
𝑎
1
ln(𝑎𝑥 𝑛 + 𝑏)
∫ 𝑛
𝑑𝑥 =
+𝐶
𝑎𝑥 + 𝑏
𝑎 × 𝑛 × 𝑥 𝑛−1
x
x
 e dx  e  c
∫𝑒
𝑎𝑥+𝑏
𝑒 𝑎𝑥+𝑏
𝑑𝑥 =
+𝐶
𝑎
𝑛
∫𝑒
𝑎𝑥 𝑛 +𝑏
𝑒 𝑎𝑥 +𝑏
𝑑𝑥 =
+𝐶
𝑎 × 𝑛 × 𝑥 𝑛−1
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A-Math Formula List - Additional Math (4047)
Integration of Area
Area between
Curve and X-axis
Area between
Curve and X-axis
𝑏
𝑏
∫ 𝑓 (𝑥 )𝑑𝑥
− ∫ 𝑓(𝑥 )𝑑𝑥
𝑎
𝑎
Area between
Curve and Y-axis
Area between
Curve and Y-axis
𝑏
𝑏
∫ 𝑓(𝑦)𝑑𝑦
− ∫ 𝑓(𝑦)𝑑𝑦
𝑎
𝑎
Area between the Curves f(x) (u-shape) and g(x)
(n-shape).
𝑏
𝑏
∫ 𝑔(𝑥)𝑑𝑥 − ∫ 𝑓 (𝑥)𝑑𝑥
𝑎
𝑎
Area between the Curves g(x)(n-shape) and f(x)
(u-shape).
𝑏
𝑏
−[∫ 𝑓 (𝑥)𝑑𝑥 − ∫ 𝑔(𝑥)𝑑𝑥]
𝑎
𝑎
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A-Math Formula List - Additional Math (4047)
Area between the Curves g(y) (inverted c-shape) and
shape).
𝑏
f(y) (c-
𝑏
∫ 𝑔(𝑦)𝑑𝑦 − ∫ 𝑓 (𝑦)𝑑𝑦
𝑎
𝑎
Area between the Curves f(y) (c-shape) and g(y)
(inverted c-shape).
𝑏
𝑏
−[∫ 𝑓(𝑥 )𝑑𝑥 − ∫ 𝑔(𝑥 )𝑑𝑥]
𝑎
𝑎
Kinematics
Velocity is the RATE of CHANGE of
Acceleration is the RATE of CHANGE
Displacement
of Velocity
v
ds
dt
a
or
v   a dt  
dv
dt
dt
where v:velocity, s:displacement,
t:time
dv
dt
where a:accleration
Some other methods to find
acceleration
d 2s
a 2
dt
a
dv ds

ds dt
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A-Math Formula List - Additional Math (4047)
𝑑𝑠
𝑑𝑠
𝑑𝑡
Displacement (s)
𝑑𝑡
Velocity (v)
∫∫𝑣 𝑣𝑑𝑡𝑑𝑡
𝑑𝑣
𝑑𝑡
Acceleration (a)
∫ 𝑎 𝑑𝑡
The End
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