Alevel pure mathematics summary sheet

functions The discriminant Circle Theorems
⑮
Angles Angle
o

One to 2 real Segment Centre
b2 (a) > 0 roots
in
at twice
one function
-> o ,
is



Inverse functions are
equal . ⑫ the
ongle
at Circumference
One to to
Switch many not a function root
domain and
doman
Range r
Finding Range
b2 -

49c =
0 repeated
finding
domain
R




Range
domain of (linear cubic) IR Range of near function is




output
quadratic =




real
is




Input
·
, ,




b24ac 40 no
<(x- 2)21
Quadratic

JAK a
= = roots
e




we
y =




X
The
The perpendicular opposite angles
3
rational function set to in
O
to find doman
bisector
·
a
of


reflect 3
=
in
y chord Cyclic quadrilatual total
of a
chord 1800
, splits the
·




composite function
Into

S
two
equal pieces .




100
Set as
-y ,
then make a replace
fg(x) -
>
do g in terms of
Partial Fractions
&(2
Subject
as
y that terms B
· in of
g O




=+
o
Angle between
targent and chord

+
~
(x a) (x + R) Angle in Semicircle [
is equal to
angle subtended by




Graphs
S is 900
the same Chord in alternate
segment.

n 9

Circle equation
Transformations
Vector
(x a) -

+
(y b) - = r Reciprocal Graph
f(x) +
a(a) f(x
a))-) Area of Segment
Geometric

!
+
centre
(a b) radius ~
Formulae (Radians)
= =
,




EQUATIONS
Stretch [ra(0 -S in0)
f(x) Stretchaxis
·

9
midpoint :
( f(ax)Stretcha , Area of
Sector
Arc
length
O
factor a
by




·
t
Perpendiar 0
·


- Reflection factor
o M
,
by O
y= f) f(x) elei + ( x)
:reflectionis
-

Cone Volume
is


equation
m(x x) Sush
of y -y LOG LAWS
=
,
-

modulus
line

( 3)
y =) If(x))belowsup f(x) ·
everything easis In
109e
=
= =
ex +
o
o
Gradient ice
goes lett
loga1 = x va" =

Seg vences
AB
Vectors Factor Theorem
Insy
= (nx + my Binomial Expansion
b a If f(a) then (x-a)
= -




In-ly
In
= 0

arithmetic
=


157 ==
Fit !
,




Toeof Sequence
1d for f(x) and
Pale conditional
-




vica versa
arithmetic Sum Increasing seq
Inx" = Klux
(d)
+
S =


z(2a (n a =
1) b where I constant

( c)
is




·
- + -




inequalities
!
+ = 1 + nx


decreasing Seg In =
-
In
S= =
z(a 1) ,
+
It Unt & Un,
torade
magnitude
-
A
by negative
or X
=
xi +
y1 + zk



any - for(
Geometric Periodic Sequence the
= ↓) to <
Iv - =
(a bx))
181 m ubr Sukches
22 -
y2 Love
tum
+
+ =

Un = ark
-

1) If
Unti =
Un for all - (x a)+




Trigonometry
and its
period/order IS K

Gordi
um
A level Pure Maths - Kyla Robinson
s
Trigonometric Graphs
Radians Exact
Trig Values
Sin 25 3600
1800 Tangent definition



to
1 900 Double formulae
Angle
= = =




Numerical METHODS
~
Sin(0) =
Isinocoso
E 60 =
=
450 I = 300
Los(20) = 1050-SinO

Root Small (20020 I
angle approximations cos(20)
= -




↳achange and function I s a s a Solving equations Reciprocal Trigfunctions 1-2510
COS SinGO cos(20) =




a
12 3



cast
Sin0 ton (201
Cobwebs it
tan
&180 1360
Staircase ta0
=
& x O
-

0




i
y
=




o
LOSO
1- @ 0360-0 1360


"F
1 200 =

-- ! tano o 1 180
Rearranged Double
angle formular

convergeor
aa

addition formulae Integration




I
cofunction
converges
tovre
diesi s
Pythagorean Identities Sio =
-Los2o Coso =

1 + 10520
Starcase
Sin(IB) S
Sin(90 -@)
·

=
SinALOSSISinBlosA . O =
Cos (90-0) COSO =



Sino 1-cos'O
.


Harmonic
Identity
Newton Raphson Method I fails
=



if
Cos (n = 3) =
CusACosS I SinASinB
If Rcosd
Cos = /-Sin "O = a Rsind = b
denominator
tan(A = 3)
-
O

=
Fas
I
=

=
eg + () = 0
Secho = I + tanio
R =b2 + tand =
-a
Coseco = 1 + cot "O




First Principles
DIFFERENTIATION
INTEGRATION
x
f()
f() Rules
-




Trapezium

parameta
Rule Substitution

Jyde "Sirlovessubstantrevaine
Sun rule



th (first last frick)
Quotient

first derivatives + (g() (x) g() h- (x) = + 2 , t

v
+
+ =




Decreasing function f'pc) < 0 product rule
Implicit differentiation Integration by Parts
Potas
tel Extra tips a make things
dy
easier



Stationary Point f (x) = 0 up + ur
f(y)
f(y)x Sov
=
+

&J2 left d /right +
right d(++)
o =

ur-Srir do

Jes put unmat-
Increasing function f
(2)) O function
Integrals Pick
-
Rule


Rio
Chain

Reverse Chain rule
Second Derrature I = Jaxur +
+ C Integrate



f"(x) < 0 :Max Se
do
*
val a Form 2
vex + C
Parametric connected rates of
Change
-"(x) Point inflection
Sk Sk + :(c) [f(x)] dx
=
= 0 of



a= = ( + a




f"(x) 1

Wine [f(x)]v
· +
>
0 be a Rate
Set
y
= In / +(2) Set
y
=




means S coss o Sinx + c
↓

find
by
dos find do
Function >
- derivative ↓ Since 8-cosx + C the
make
A /K =

adjust constant


* dou >
- and d
In Cos -Sinc Ssecis >
- Kanea + C example
example




(G(2x
J 3)
+
3)(x2 30) dx +
=
2(x2 +
+ c




↑ tanx o See's 2Sil
J
=




a ne See talea
Seaton & Sec + <
Si + G) [f(2] des




JLoseciv
↳
(p*]
+ 1



y A
-
=


-
co+ x + c



y (x zx)3
=




InSw
# Cose223
-
Lose Lots & cosecot f -
cosex + C m =
3(x 3) ( =
+ G(2x + 3) ( 3) da+




*
Since - Cosse
Sf(ax b) +
- af(ax +
b) + c
: A = 2