A-Math Formula List - Additional Math (4047) Updated 6th Apr 2015

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 Also known as Unequal Roots

b2 - 4ac = 0 Real and Equal Roots Also known as Repeat Roots or Coincident Roots.

b2 - 4ac < 0 Imaginary roots Also known as Complex Roots.

b2 - 4ac > 0 Real and Distinct Roots Also known as Unequal Roots

b2-4ac = 0 Real and Equal Roots Also known as Repeat Roots or 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 𝛼 + 𝛽 = −

𝑏

Product of Roots is 𝛼𝛽 =

𝑎

The Equation is x2 – (Sum of Roots) x + (Product of Roots) = 0

𝛼 2 + 𝛽 2 = (𝛼 + 𝛽)2 − 2𝛼𝛽

𝑐

𝑎

𝛼 − 𝛽 = ±�(𝛼 − 𝛽)2

𝛼 4 + 𝛽 4 = (𝛼 2 + 𝛽 2 )2 − 2(𝛼𝛽)2

Indices Same Base Number

Same Power

x a × xb = x a +b

a × b =( a × b )

xa = x a −b xb

a a =  m b b

(x )

a b

m

m

m

Same Base Number→ Power Add or Subtract

m

Same Power same→ Base Number Multiply or Divide.

m

= x a×b ≠ x a × xb = x a +b

Other Laws of Indices 1 b

1 xa

x−a =

x a y −b =

x = b x1

xa yb

1 = xa x−a a b

x = x b

a

−

1

b x=

x    y −

a b

x=

1 = 1 b x −a

x0 = 1

1 b

x1

 y =  x

1 = a xb

a

1 b

xa

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A-Math Formula List - Additional Math (4047) Surds a a÷ b= b

a × b = a×b

( a )=

m a ×n b = m× n a×b

2

a × a=

a

m a +n a = ( m + n) a

m a −n a = ( m − n) a

Rationalizing Denominator 1 n− a × = n+ a n− a

(n − n −(

1 n+ a × = n− a n+ a

(n + n −(

1 n+

1 n−

a

a

2

2

) a)

(n −

) a)

(n −

a

a

= 2

= 2

a

)

n −a 2

a

)

n −a 2

×

n− a = n− a

( n − a= ) (n − a ) n−a ( n) −( a)

×

n+ a = n+ a

( n + a= ) (n + a ) n−a ( n) −( a)

2

2

2

2

Partial Factions Linear Factor Check if the highest coefficient of the

mx + n A B = + (ax + b)(cx − d ) (ax + b) (cx + d )

NUMERATOR is the SAME or LARGER that the DENOMINATOR. If it is, do LONG DIVISION first before partial

Repeat Factor

fractions.

mx + n A B C = + + 2 (ax + b)(cx − d ) (ax + b) (cx + d ) (cx + d ) 2

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A-Math Formula List - Additional Math (4047) Quadratic Factor mx + n A Bx + C = + 2 (ax + b)(cx − d ) (ax + b) (cx 2 + d )

Logarithm log b 1 = 0

ln1 = 0

log b b = 1

ln e x = x

by = x can be covert to y = log b x

log b bx = x log b m + log b n = log b (m × n) log b m – log b n = log b (m ÷ n)

ln e xa = xa * log b (m × n) ≠ log b m × log b n

ln x = log e x ln e =1 as log e e=1

log b m a = a × log b m log v u =

log a u log a v

log = vu

log u u 1 = log u v log u v

Binomial Expansion n n!  =  r  r !(n − r )!

n n   = Cr r

n n n n n n bx)n   a n (bx)0 +   a n −1 (bx)1 +   a n −2 (bx)2 +   a n −3 (bx)3 ....   a n −r (bx)r +   a n −n (bx)n (a += 0 1  2  3 r n OR

n n n 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 (a + bx) n = 1  2  3 r

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A-Math Formula List - Additional Math (4047) n (n)(n − 1) n − 2 (n)(n − 1)(n − 2) n −3 a n + a n −1 (bx)1 + a (bx) 2 + a (bx)3 .... + (bx) n (a + bx) n = 1 1× 2 1× 2 × 3

n n  n  n −1  n  n−2  n  n −( n −1) n 0 1 2 n −1 n ( kx ) +  1n−n ( kx )  1 ( kx ) +  1 ( kx ) +  1 ( kx ) + ....  1 0 1  2  n − 1 n

kx ) (1 += n

OR

(1 + kx )

n

n n  n  n −1 n 1 2 = 1 +   ( kx ) +   ( kx ) + ....   ( kx ) + ( kx ) 1  2  n − 1

Trigonometry

cos ecθ =

1 sin θ

sin 2 θ + cos 2 θ = 1

sec θ =

1 cos θ

1 + cot 2 θ = cos ec 2θ

cot θ =

1 cos θ = tan θ sin θ

1 + tan 2 θ = sec 2 θ

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

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A-Math Formula List - Additional Math (4047) tan A ± tan B tan ( A ± B ) = 1  tan A tan B

Double Angle Formula

sin 2 A = 2sin A cos A

cos = 2 A cos 2 A − sin 2 A

tan 2 A =

= cos 2 A 2 cos 2 A − 1

2 tan A 1 − tan 2 A

cos 2 A = 1 − 2sin 2 A

Half Angle Formula

sin

A 1 − cos A = ± 2 2

cos

A 1 + cos A = ± 2 2

A = tan 2

1 − cos A 1 − cos A = 1 + cos A sin A

R-Formula aCosθ ± bSinθ = RCos (θ  α )

R Where=

aSinθ ± bCosθ = RSin(θ ± α )

tan α =

a 2 + b2

b a

Co-ordinate Geometry Gradient(m) =

y2 − y1 x2 − x1

Linear Graph Y= m x + c

y2 − y= m( x2 − x1 ) 1 m= gradient

General Equation

c= y-intercept (point on the graph that intersects the y-axis)

Y − y1= m( X − x1 ) where

( x1 , y1 ) is a point on the

graph.

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A-Math Formula List - Additional Math (4047) 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

m1 = m2

gradient i.e. 90° to each other.

m1 = −

1 or m1 × m2 = −1 m2

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

*It does NOT matter if you calculate in a

𝑥2 𝑥3 𝑦2 𝑦3

𝑥4 𝑦4

𝑥1 𝑦1 �

clockwise or anti-clockwise direction. The is a |modulus| in the formula

1 |(𝑥 𝑦 + 𝑥2 𝑦3 + 𝑥3 𝑦4 + 𝑥4 𝑦1 ) − (𝑥2 𝑦1 + 𝑥3 𝑦2 + 𝑥4 𝑦3 + 𝑥1 𝑦4 )| 2 1 2

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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 or

𝑥 2 + 𝑦 2 + 2𝑔𝑥 + 2𝑓𝑦 + 𝑐 = 0

where 𝑎 = −𝑔, 𝑏 = −𝑓 and 𝑟 = �𝑔2 + 𝑓 2 − 𝑐

Proofs in Plane Geometry Midpoint Theorem

Tangent –Chord Theorem (Alternate Segment Theorem)

sin If X and Y are midpoints, then

Angle W = Angle X

YZ // WX

Angle Y = Angle Z

YZ = ½ WX

Differentiation dy (ax n ) = anx n −1 dx

dy (ax) = a dx

dy (a) = 0 dx

Where’ a’ is a constant and ‘n’ is an integer.

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A-Math Formula List - Additional Math (4047) Chain Rule

Sum/Difference of Function

dy dy du = × dx du dx

d du dv (u ± v) = ± dx dx dx

Product Rule

Quotient Rule

d dv du (= uv) u + v dx dx dx

du dv v −u d u dx dx  = dx  v  v2

d (sin x) = cos x dx

d (cos x) = − sin x dx

d (tan x) = sec 2 x dx

The functions below are obtained using chain rule.

d c )  ab cos(bx + c)  a sin ( bx += dx 

d −ab sin(bx + c)  a cos ( bx + c )  = dx 

d + c )  ab sec 2 (bx + c)  a tan ( bx = dx 

d  a sin n ( bx = + c )  anb sin n −1 (bx + c) cos(bx + c) dx

d  a cos n ( bx + c )  = − anb cos n −1 (bx + c) sin(bx + c) dx  d  ata n n ( bx = + c )  anb tan n −1 (bx + c) sec 2 (bx + c) dx 

Exponential/Natural Logarithm Function d x (e ) = e x dx

d ax +b (e ) = ae ax +b (where a and b are constants) dx

d 1 (ln x) = (where x>0) dx x

d a [ln(ax + b)] = (where ax +b>0) dx ax + b

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A-Math Formula List - Additional Math (4047) Integration n dx ∫ ax=

ax n +1 + c where n≠1 n +1

∫ a dx= ax + c

∫ ( ax + b )

n

( 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

∫ [α f ( x) ± β g ( x)]dx = α ∫ f ( x)dx ± β ∫ g ( x)dx

bx dx ∫ a cos=

where f(x) is a function

a sin bx + c b

a

− cos bx + c ∫ a sin bx dx = b

bx dx ∫ a sec= 2

a tan bx + c b

where x is in radian.

1 dx ln x + c where x>0 ∫x=

∫e

x

dx= e x + c

�

1 ln(𝑎𝑥 + 𝑏) 𝑑𝑥 = +𝐶 𝑎𝑥 + 𝑏 𝑎

�𝑒

𝑎𝑥+𝑏

𝑒 𝑎𝑥+𝑏 𝑑𝑥 = +𝐶 𝑎

�

1 ln(𝑎𝑥 2 + 𝑏𝑥) 𝑑𝑥 = +𝐶 𝑎𝑥 2 + 𝑏𝑥 2𝑎𝑥 + 𝑏

�𝑒

𝑎𝑥 2 +𝑏𝑥

2

𝑒 𝑎𝑥 +𝑏𝑥 𝑑𝑥 = +𝐶 2𝑎𝑥 + 𝑏

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A-Math Formula List - Additional Math (4047)

Integration of Area Area between Curve and X-axis 𝑏

∫𝑎 𝑓(𝑥)𝑑𝑥 Area between Curve and X-axis 𝑏

− ∫𝑎 𝑓(𝑥)𝑑𝑥 Area between Curve and Y-axis 𝑏

∫𝑎 𝑓(𝑦)𝑑𝑦 Area between Curve and Y-axis 𝑏

− ∫𝑎 𝑓(𝑦)𝑑𝑦

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 f(y) (c-shape). 𝑏

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

−[∫𝑎 𝑓(𝑥)𝑑𝑥 − ∫𝑎 𝑔(𝑥)𝑑𝑥]

Kinematics Velocity is the RATE of CHANGE of

Acceleration is the RATE of CHANGE of

Displacement

Velocity

v=

ds dt

a=

where v: velocity,s:displacement, t:time = v

where a:accleration Some other methods to find acceleration

dv

a dt ∫ dt ∫= dt

= a

Displacement (s)

dv dt

𝑑𝑠 𝑑𝑡

Velocity (v)

� 𝑣 𝑑𝑡

d 2s dt 2

= a

𝑑𝑣 𝑑𝑡

dv ds × ds dt

Acceleration (a)

� 𝑎 𝑑𝑡 The End

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