# 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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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 ) = 0; perpendicular
Complex Numbers = || = 2 + 2 , , 0
= arg = arctan ( ) tan = , - <
= + = - , ,
= (cos() + sin()), - <
= , - <
1 2 = 1 2 (cos(1 + 2 ) + sin(1 + 2 ))
1
2
= 1 (cos(1 - 2 ) + sin(1 - 2 ))
2
|1 2 | = |1 ||2 |
| |
|1 | = |1 |
2 2
arg(1 2 ) = arg(1 ) + arg(2 )
arg (1 ) = arg(1 ) - arg(2 )
2
= = cos + sin ; Euler's relation

## Other pages in this set

### Page 2

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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
= ()() () = () +
() + () = [()]
+ = = +
2
2 + + = () = . . +. .…read more

### Page 3

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Taylor series
3 5 2+1
sin = - 3!
(-1) (2+1)!,
+ 5! + +
2 4 2
cos = 1 - 2! + 4! + + (-1) (2)! + ,
2
= 1 + + + + + ,
2! !
2 3
ln(1 + ) = - 2 + 3 + + (-1)-1 + , -1 < 1

### Page 4

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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 + 2
= arctan tan =
Find by substituting = arctan
(, ) ( 2 + 2 , arctan )
To convert a polar coordinate to a Cartesian
coordinate or a Cartesian equation to a…read more

### Page 5

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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 inequality
+
< + , , Identify the solutions by drawing a sketch
( + )( + )2 < ( + )2 ( + ) < 0 Find the critical values by using factorisation

### Page 6

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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 = (cos +
+ = 2 cos
sin )
1
- = 2 sin = 1 (cos 1 + sin 1) =
= ( ) = = [(cos + sin )] = (cos + sin )
(cos + sin ); de Moivre's theorem As = , de Moivre's theorem is true

### Page 7

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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 + sin )
1 cos - sin
= (cos + sin ) × cos - sin
cos - sin
=
(cos2 - 2 sin2 )
= - (cos
- sin )
= - [cos(-)

### Page 8

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Find by differentiating the substitution
Find the new equation by substituting and
into the first order differential equation
2
2 + + = () = . . +. .…read more

### Page 9

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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 that if the particular integral appears in the
= + 2 + 3 ; particular integral 1b complementary function, multiply by
= ; particular integral 2a Know that for a complementary function:
= ; particular integral 2b If…read more

### Page 10

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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!
cos , , ln(1 + ) and other simple functions in
2 4 2
cos = 1 - 2! + 4! + + (-1) (2)! + , ascending powers of
2
= 1 + + 2! +…read more