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# Edexcel A-level Further Mathematics 'Further Pure Mathematics 2 (FP2)' Revision

Contains a list of all the formulae you need for the exam, as well as further notes and tips on the Edexcel specification of Further Pure Mathematics 2 (FP2).

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## Pages in this set

### Page 1

Further Pure Mathematics 2

Formulae

Coordinate Systems = cos cos =

= sin sin =

= 2 + 2 2 = 2 + 2

= arctan tan =

1

= 2

2

1

= 2 2

1

=

2

=0 ( sin ) = 0; parallel

=0

( cos )…

Formulae

Coordinate Systems = cos cos =

= sin sin =

= 2 + 2 2 = 2 + 2

= arctan tan =

1

= 2

2

1

= 2 2

1

=

2

=0 ( sin ) = 0; parallel

=0

( cos )…

### Page 2

1

= - = cos - sin ; Euler's relation

1

+ = 2 cos

1

- = 2 sin

= ( ) = = [(cos + sin )] = (cos + sin ); de Moivre's theorem

1

+ = 2 cos

1

- = 2 sin

Differentiation 1

=…

= - = cos - sin ; Euler's relation

1

+ = 2 cos

1

- = 2 sin

= ( ) = = [(cos + sin )] = (cos + sin ); de Moivre's theorem

1

+ = 2 cos

1

- = 2 sin

Differentiation 1

=…

### Page 3

() ()

() = () + ()( - ) + 2!

( - )2 + + !

( - ) + ; Taylor series

3 5 2+1

sin = - 3!

(-1) (2+1)!,

+ 5! + +

2 4 2

cos = 1 - 2! + 4! + + (-1)…

() = () + ()( - ) + 2!

( - )2 + + !

( - ) + ; Taylor series

3 5 2+1

sin = - 3!

(-1) (2+1)!,

+ 5! + +

2 4 2

cos = 1 - 2! + 4! + + (-1)…

### Page 4

Notes Tips

Coordinate Systems = cos cos = To convert a Cartesian coordinate to a polar

= sin sin = coordinate or a polar equation to a Cartesian

(, ) ( cos , sin ) equation:

= 2 + 2 2 = 2 + 2 Find by substituting = 2…

Coordinate Systems = cos cos = To convert a Cartesian coordinate to a polar

= sin sin = coordinate or a polar equation to a Cartesian

(, ) ( cos , sin ) equation:

= 2 + 2 2 = 2 + 2 Find by substituting = 2…

### Page 5

Find or by substituting the equation into

( cos ) =0

Algebra and Functions < , Remember that , or

- -

( - ) < ( - )2 < 0 To solve an inequality:

( - )[ - ( - )] < 0 Draw a sketch by rearranging the…

( cos ) =0

Algebra and Functions < , Remember that , or

- -

( - ) < ( - )2 < 0 To solve an inequality:

( - )[ - ( - )] < 0 Draw a sketch by rearranging the…

### Page 6

arg (1 ) = arg(1 ) - arg(2 )

2

= = cos + sin ; Euler's relation Know the proof of de Moivre's theorem for +

1

= - = cos - sin ; Euler's relation Basis step by letting = 1:

1 = [(cos + sin )]1 =…

### Page 7

As de Moivre's theorem is true for = 1, then

it has been shown true for all 1 and

+ by mathematical induction

Know the proof of de Moivre's theorem for -

[(cos + sin )] = [(cos + sin )]-

1

= [(cos + sin )]

1

= (cos…

it has been shown true for all 1 and

+ by mathematical induction

Know the proof of de Moivre's theorem for -

[(cos + sin )] = [(cos + sin )]-

1

= [(cos + sin )]

1

= (cos…

### Page 8

Find by differentiating the substitution

Find the new equation by substituting and

into the first order differential equation

2

2 + + = () = . . +. . Remember to re-substitute at the end

To use substitution:

2 + + = 0; auxiliary equation

Find , and by rearranging…

### Page 9

complementary function 3 If two equal real roots, use complementary

= + + - = ( cos + function 2

sin ); complementary function 4 If two imaginary roots, use complementary

function 3

If two complex roots, use complementary

function 4

= + + 2 ; particular integral 1a Know…

= + + - = ( cos + function 2

sin ); complementary function 4 If two imaginary roots, use complementary

function 3

If two complex roots, use complementary

function 4

= + + 2 ; particular integral 1a Know…

### Page 10

() or Taylor series

() = () + ()( - ) + 2!

( - )2 +

()

+ !

( - ) + ; Taylor series

sin = -

3

+

5

+ +

2+1

(-1) (2+1)!, Know how to derive the series expansion of sin ,

3! 5!…

() = () + ()( - ) + 2!

( - )2 +

()

+ !

( - ) + ; Taylor series

sin = -

3

+

5

+ +

2+1

(-1) (2+1)!, Know how to derive the series expansion of sin ,

3! 5!…

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