A-Level Mathematics (Edexcel) Summary

Proof
Proof By Contradiction Proving a statement by showing the opposite can't be true
:
.




Proof By Deduction Logical argument through logical statements and deductions.
:




Proof By Counterexample :
Use an example to show a statement isn't always truee.

Proof By Exhaustion :
Proving a statement
using smaller , simpler scenarios & proving .
each




Algebra and functions

Discriminants Quadratics
:

Solving :
:

a) + 612 + C
62-49) 0 -

No real Roots CTS : ②
Quadratic formula :
OR

62 49) 0 I repeated root
a(x (a) 162- nac
=
6 +
- -




+ + c 0 ) = m(x x ,)
-




(y y
= -
-


,
x =

2a
62 -uac > O - 2 Distinct roots
↑
used to find the
turning point

* substitution * Elimination

To solve Simultaneous Equations , make lory) the subject of an equation and substitute into another a make the <



Cory) coefficient have the same coefficient from the other , and subtract(only for linear) .




↳Solutions show where the INTERSECT
graphs .




factor Theorem f(x)) is f(p) D2-p) is factor off
It polynomial then 0 &
: =
a ,
a




Sketching Polynomials


&
C intercepts by 0
"I
- =


in even , and o , anco
M
a + a x a ,x + x y intercept y do/constant term
↓ ↓
+
y
y
-

= a anx
:
+... +
, ,


Turning Point to Differentiation
If n= odd , and ot" , anso
h
= (x -

P,)(x -

Pc) ... (x -

Pr)




Reciprocal functions proportionality ~ /INVERSEL How to solve modulus

↓ Square both sides then solve
·




↓ neveodd
.




yxx
,

yax
f(x) =

a ·


Graph & find intersecting points &
y = kx
y
=
x identify a values corresponding to

7gative
(a , n - 0)

= y,x
=

y232
+ ve values of the functions .




Modulus Graphs Defining functions Inverse functions
Domain for "DC) their fix)'s
w
:
Inputs (x) values values
y
f(x)s
=
,



Range :
outputs (y) forf"(2) ,
their values =
f(x)'s y values
·


nu reflected
One-to-one ↳ Reflected IN y =
C

If()l < in the
: each
axis

xdy is unique
Partial fractions
f((x1) reflected Many-to-one : In a

- in they domain () of f , there's f(xC) A B C
axis = + +
also another possible domain (ax + b) ((x + d)2 ax + b a +d (x + d)2
(x2) f(x) =
f(x2) * When should
Improper algebraic division
·


,


be used
First .




Graph Transformations

y
=
f(x) + 6 -
up
by b //
y
=
f(x) -> scale
vertically by 6 //y =
- fD) >
-
reflect in axis


y f
(s) + 6)
= -

LEfTbya//y =
flas) - Scale
horizontally by "a /y =
f(x) >
- reflect in yaxis

↳ order Horizontal
:<
scaling & reflections 3 vertical scaling & reflections
2 Horizontal translations 4 vertical translations

,Coordinate Geometry
Equation ofA Circle Parametric Equations

p(t)
Y single
x describes xay in terms of t.
(y q)
=


(x b)
-
+ -
= r2
& convert to this
form via CTS .
y
=
9 (t) equation is called a cartesion equation
.




Sequences and Series


(b) -
The number of ways Binomial Expansion
of choosing items
(a + bx)" =
an + (2)a" bx + (2)an - b-x +... + (n)an rbx
-




+... + baxh
from a set of .
n

↳ can be used to answer questions such as -

Expand 12-34
n !
-
r ! (n-r) !




General Binomial Expansion

ONLY VALID WHEN KCKI -(EG : 12 3)
-
""

e 2(1 +
titl +
( yaxs -y)) z)
-

+ ...

) converging sequence :




-
-


him Un for
a ll



n(a)x (g)
44 - )
(a + bx)" a1 r + 1)
. (n -




x
ji
= + +... +

-
+...


Arithmetic Sequence Arithmetic Series

Un =
Un-i + d Sn =

=(2a + (n -
1)d]
=
a + (n -
1)d



Geometric
Sequence Geometric Series
common Ratio
Convergent Geometric Series
Un &
Un r 1)
marn If Irk1
arn-
=

run-
-




Sa
,
= =
Un-1 Sn = =
it's
1
r -


1 -
r
convergent




Trigonometry 90r
N




sindo
Converting To Radians :
Converting DegreesTo : Arc Length Sector Area
S
A
1x, y


00
Y


188
= r -
E




T
Orad =

Odegx , 80 Odeg =
Orad x 180 S= - A =
x




-tands o


Area of A
Triangle Sine Rule Cosine Rule
A ↳ C

A = absinc I

sin B
=
sinc
a2 = b2 + c -
26c osA
sin A




Small Angle Approximation irig symmetry periodicity Relating cos & sin
Asymptotes
Singe Sinx Sin(x = 2 x) Sinc cos( *
(C) tanc

nen
Sin (x-x) Sinc
-

= = = 2



cost = 1 -


[02 COS 122 -
x) =
COSIC (os(x2 = 2a) = COSI (osx = Sin (
*
2 -x)
(2k + 1)π
=
x
tano Sin(-x) = -

sinc tan(x = x) =
tanc 2



cos( >() COSSC


-
- =




tan 1-x) = -

tank
same for sec , coses

and eat

, ·
Reciprocals

cose((2
I 1 v
m
=
Secs =

Sins COSx
·



undefined at x = kx
undefined at
12k + 11


-M
x=
2
&

I
(Otx




·
=


tanx


undefined at x = k2




tanx = cot l -

<



Inverse functions

trig Identities

arcsin sin"x & rcCOSC Cos " x arctanx tan"x
in
= = =


tanf =

langle - langle 0= 0= a) langle -0

Sino COS G = 1




m
+




~ ↳
-




lotf =
10sO
SinG
=
coseCeT
reciprocal
Seco
Trig
Identities
V coseco = 1 + cot


seco = 1 + tan@
-
-




I
Simplifying Trig Expressions : Example :



Addition Angle formula
simplify to Rsin(OIC) simplify 3sinO + 4 cost to the form

Sin(A = B) = sinAcosB = COSAsinB ↳
R =
+ b2
Rsin(0 + (C)

Rsin(0 + 1) RsinOcos
Ros
= +
COS/A = B) = COSACOSB SinAsinB ↳ tan" (i) -
c =
↳ 34 =
" 4

to make 3 sing to make 4 cos8
anAItan B
tan(AIB) =
1 tan Atan B simplify to RCOS(OIC
-
>
R =
b2 )
.
3 + 4 = RCOSC) + Rsince =
R(cos + since)


3 + 42 = R2(cos" < + Sin = c) =
R (1)
=


↳ c =
tan" (f)
~
R2 =
32 + 42 = 25


R =
c =
5---- -
tanceSin
a = tan" (b) =
53 1
.




3 sinO 4 cos E
formula
Angle
+
Double Trigequations 1)
=
Ssin 10 53
solve + .




-




Sin (2A) = 2SinACOSA EG : find solutions to sin (2x +
*
2) =
2 1 -
>)
2 = x -




cos(2A) = cos A-sin
? ?
A .
1 Adjust the domain - 2x
+
2a + 7/2
2COSPA-1
=

?
- +
= 1 -
2 sin A


. +an A
2
solvable state sin" (2)
tan (2A) =
.
2
Rearrange to a 2x + = =
=
1 - tan ? A




I
3 and
.
Use
symmetry (x + =

-
periodicity to find
solutions.
x=
52
-