Contents
Formulae in other document but
otherwise should be all covered...
From additional mathematics for OCR book
 Algebra 1
 Linear expressions
 Quadratic
expressions
 Completing the
square
 Simultaneous
equations
Contents
 Algebra 4
 Binomial expansion
 Binomial distribution
 Geometry 1
 Gradient of a line
 Different lines and
 Calculus 1




Introduction
Gradient of a curve
Differentiation
Tangents and
normals
 Stationary points and
Higher Derivatives
calculating distances
 Equation of a straight
 Algebra 2
 Calculus 2
line
 Linear inequalities
 Circle
 Integration
 Quadratic
 Definite integrals
 Geometry 2
inequalities
 Area between curves
 Inequalities
 Algebraic fractions
 Linear programming  Calculus 3
 Expressions
 Motion in a straight
containing a square  Trigonometry 1
line
root
 The Basics
 Constant
 Sine rule
 Algebra 3
acceleration

Cosine
rule
 Polynomials
 Velocity
 Trigonometry 2
 Factor theorem
 Formulae (other
 Remainder theorem
 3d work
document)
Linear expressions
Quadratic expressions
Completing the square
Simultaneous equations
Ex 1A, B & C
Contents
 This is basically GCSE if
not SAT revision
 When simplifying
remember to:
 Collect like terms
 Remove brackets
 Factorise
 Find a common
denominator when
involving fractions
 When solving an
equation remember to:
 Simplify
 Do the same on both
sides so that it remains
the same equation
 They sometimes ask to
rearrange an expression
in which case be careful
to do so correctly
Ex 1D
Contents
 This is when the highest
power is 2
 It often involves
expanding and
factorising
 Before starting to solve
a quadratic equation,
make sure that all
terms of the quadratic
are on the left hand
side of the equation
 There are 3 ways to
solve a quadratic
equation:
Factorise
2. Completing the square
3. Using the Quadratic
Formula
1.
 Remember that the
formula is:


You use this when you
are in a calculator test
and cannot factorise
Ex 1E
Contents
 Method:
 It is used
 Method:
when you
 X²-8x=-3*
 Y = x²-8x +3*
 Take the coefficient
cannot
 This can be
of x-8
written as y=(xfactorise a
 Half it
4)²-13
quadratic
-4
 See

Square
the
answer
 It is also
previous
+16
useful when
example
 Add this to both sides
and its
sketching a
 Factorise where
answer
graph as it
possible
 Therefore the
 Take the square root
identifies the
line of symmetry
of both sides
line of
is x=4 and the
 Add the constant to
symmetry and
vertex is (4,-13)
both sides
the vertex
 Find the answer
Ex 1F
Contents
 This is when there is
more than one variable
 ...by substitution
 Suitable for when y is
the subject
 Take the expression for y
from the equation and
substitute it in the other
equation – then solve as
before
 ...by elimination
 Suitable for when y is
not the subject or either
equation
 Multiply the equations
as so that when they are
subtracted/added from
the other they eliminate
variables
 Substitute this into the
first equation and thus
solve
Linear inequalities
Quadratic in equalities
Algebraic fractions
Expressions containing a square root
Ex 2A
Contents
 Like simplifying linear
expressions, you do the
same to both sides
 However, remember to
have the inequality sign
the right way and
whether it is equal to or
not
 You may also be asked
to show the answer on
a number line
 In this case, remember
that open circles at the
end of the line show
that the number is not
included
 Closed circles mean the
figure is included in
your answer
 Both can be used in a
single answer
Ex 2B
Contents
 There are two methods:
 Sketching a graph to
show the answer
 Or drawing up a table
showing the values of x
 But remember that if it
has terms on both sides
these must be collected
to one side
 These quadratic
inequalities will be able
to be factorised
 Remember to be careful
in reading and working
the question especially
when using a graph
Ex 2C & D
 Algebraic fractions follow
the same rules as the
fractions in arithmetic
 The common
denominator should be
the lowest common
multiple of the original
denominators
Contents
 Other than being asked to
simplify an algebraic
fraction you may be asked
to solve an equation
involving fractions
 This is done in the same
way as before but also
having to simplify fractions
 Remember that when you
multiply a fraction you only
multiply its numerator
Ex 2E
Contents
It is often easier to use surds when working
with square roots to get a more accurate
answer than just working out the numerical
value
You should try to make the number that is
under the square root sign as small as possible
or as easy to work with as possible
Rationalising the denominator is an important
technique to be aware of
Triangles
Sine rule
Cosine rule
Contents
 Just remember the hyp, app and adj. And that θ
is used for the angle
 The Trigonometric ratios are:
 Remember:
 sinθ (etc.) will give you the ratio
-1
 sin known side (etc.)will give you the angle
 side known X sinθ (etc.) will give you the side’s length
Contents
Sin is Opposite divided by Hypotenuse.
Opposite is a helpful way of remembering it.
S
• skiving
O
• off
H
• homework
Contents
Tan is Opposite divided by Adjacent
An easy way to remember is it doesn’t have the hyp and opp is always on top
Opposite is a helpful way of remembering it.
T
O
A
Contents
Cos is Adjacent divided by Hypotenuse
Opposite is a helpful way of remembering it.
C
A
H
Contents
It is an extension of
Pythagoras’ theorem which
allows it to be applied to any
triangle
AB = AC + BC - 2(AC)(BC) cos C
c = b + a - 2ab cos c
a = b + c - 2bc cos a
b = a + c 2ac cos b
OR
cos A = b + c - a / 2bc
Contents
It is based on that fact that in any triangle the length of any edge is proportional to the sine of the angle
opposite to that edge
a / sin A
=
=
b / sin B
c / sin C
Contents
This formula (which is cyclic) is for finding the area of a triangle when the lengths of 2 edges are known and also
the size of the angle between them
Area
½ bc sin A
=
=
½ ab sin C
½ ca sin B
=
3d work
Contents
 Anti-clockwise is always positive
 Clockwise is negative
 Always go from the x axis
 Cosine and Sine are between -1 and 1 whereas Tangent is
over 1
Silver
Sin
Cos
Tan
Sin
Cos
Tan
+
+
Tea
All
Sin
Cos
Tan
+
+
+
Sin Cos +
Tan Cups
 All
 Silver
 Tea
 Cups
It is like have a
circle of one unit
1
1
-1
-1
Contents
 It is really important to
draw good diagrams
 There are two types:
 Representations of 3D
objects
 True shape diagrams of
2D sections in a 3D
object
Contents
The two main identities that need to be
learnt:


Introduction – Curves, Tangents, and Normals
Gradient of a curve
Differentiation
Tangents and normals
Stationary points and Higher Derivatives
Contents
 Cord: joins two points on
the curve
 Tangent: touches the
curve at a point of
contact
 Normal: perpendicular to
the tangent at the point
of contact
 The tangent to a curve
can be considered as the
limit position of a chord
Curved line
Contents
 As B gets closer to A we can say that B tends to A (written as BA)
 The gradient of the cord AB  the gradient of the tangent at A
 E.g. y = x²
A = (xA ,yA)
B = (xB, yB)
Mab =
Yb - Ya
Xb - Xa
Xa
Ya = (Xa)²
Xb
Yb = (Xb)²
2
4
3.5
12.25
5.5
2
4
3
9
5
2
4
2.5
6.25
4.5
2
4
2.25
5.0625
4.25
2
4
2.1
4.41
4.1
2
4
2.05
4.2025
4.05
2
4
2.001
4.004001
4.001
 From the table, we can assume that the gradient of the
tangent to the graph y = x² at A(2,4) is 4
Contents
y = x²
y = x³
X
Gradient
X
Gradient
1
2
1
3
2
4
2
12
3
6
3
27
4
8
4
48
5
10
5
75
x
2x
x
3x²
y = x⁴
X
Gradient
1
4
2
32
3
108
4
256
5
500
x
4x³
Contents
 The value of the gradient
of the chord AB as B tends  Note that ‘d’ has no
independent meaning
to A is called the
and must never be
differential coefficient of
regarded as a factor. The
y with respect to x or the
d
complete symbol dx
derivative of y with
means ‘the derivative
respect to x. The limit is
dy
with respect to x of
denoted by the symbol dx
[previous expression]’
(read as ‘dy by dx’)
dy

We
may
also
write
 The process of obtaining
dx
when y is a function of x
the differential coefficient
as f’(x) or y’
or derivative of a function
is called differentiation.
Contents
Contents
d
e.g. dx
d
dx
(x⁷) =
7x⁶
(xⁿ) = nxⁿ⁻¹
d
dx
d
dx
e.g. √x=
(x^½) = ½x^-½
1
=
2√x
Contents
y
y=c
0
x
 Let y=c
 Graphically this is a
horizontal straight line
and its gradient is zero
 Therefore
differentiating a
constant will give you
zero
 i.e.
d
dx
(c) = 0
Contents

d
dx
(axⁿ) = a
d
dx
(xⁿ) = anxⁿ⁻¹
 Where ‘a’ is a constant
d
 i.e. dx (axⁿ) = anxⁿ⁻¹
For example:

d
dx
(3x⁶) = 18x⁵
Contents
We differentiate each term and
then add or/and subtract the terms
as necessary
For example:

d
dx
(x⁷ + 5x² - 3x + 4)
=
7x⁶ + 10x - 3
Contents
The gradient of the chord AB as it tends to
the point A, is the value of the derivative
at that point A.
We can use this to find the equation of the
tangent and/or of the normal to a curve at
a given point
Contents
Q. Find the equation of the tangent and of the normal to the curve:
y = x² + 3x – 10
at the point (1, -6)
 First differentiate the equation to give:

d = 2x + 3
dx
at x = 1
 Thus:
 m=2X1+3=5
 Using: y – y₁ = m(x – x₁)
 substitute the known values
 y + 6 = 5(x – 1)
 y = 5x – 5 – 6
 y = 5x – 11
 equation of tangent
 Then to find the equation of the normal:
 m = 5 so m¹ = -⅕
 y + 6 = -⅕ (x-1)
 use previous method but using -⅕ instead of 5
 5y + 30 = -x + 1
 x + 5y + 29 = 0
 equation of the normal
Contents
This is basically doing the second derivative
This is just differentiating what you already
have differentiated
It can be used to find stationary points in
increasing and decreasing functions
Contents
 Increasing is from A to B
and from C – represented
with the +
d
y
+
+ 0 -
+
A
0
B
positive
+
-
+
+
 This means that dx is
0
-
-
-
+
+
+
C
+
x
 Decreasing is from B to C
– represented with the  This means that d is
negative
dx
 Stationary point are A, B
and C – represented by
the zero
d
 This means that dx = 0
Contents
This I where the gradient is zero
 They can be maximum points, minimum points, or
points of inflection
To find stationary points:
 Differentiate and find the value(s) of when this = 0
 Substitute these values into the original equation to
find y
 To find the nature of the stationary points work out
the second derivative and then substitute the value(s)
of x found before to decide if they are a min/max
points or points of inflection
Contents
At point P
dy
dx
dy
dx
dy
dx
=0
=0
=0
d²y
dx²
d²y
dx²
Maximum point
<0
Minimum point
>0
d²y
d³y
= 0 and
≠0
dx²
dx³
(doesn’t change sign on
either said of P)
Point of inflection
Remember to physically do and say each step in a question including saying
if a certain point is a max., min. or point of inflection.