PURE OCR A LEVEL MATHS

Proof:

this means

a implies b

Proof:

This means

a is true if and only if b is true

Proof by exhaustion:

use when

There is a set number of scenarios small enough to consider

Proof by exhaustion:

Prove n is a prime number

n/2= a decimal so 2 is not a factor

3,5,7,11,… check the prime numbers until you reach a value close to the sqrt of n

I.e n=97 sqrt97 is roughly 10 so dont check a factor above 10

Conclude: as there are no factors <sqrtn, there are no factors > sqrtn

Proof by exhaustion:

No square number ends in ‘n’

If a number ends in a 1, number squares will end in a 1 as …1²=1

Check all single digits

2²=4 3²= 9 4²=6 5²=5 6²=6 7²=9 8²=4 9²= 1 0²=0

So none of these values end in n

Proof by exhaustion:

Every integer can be written as these general forms

n=3k

n=3k-1

n=3k-2

Proof be deduction:

Use when

Showing a conjecture must be true

Mostly using algebra

Proof by deduction:

Show k³-k is divisible by (for example) 6 for all integers n>1

k(k-1)(k+1)

Three consecutive integers will either be

Odd, even, odd

Even, odd, even

Therefore as it will always include an even, its must be divisible by 2

As it will always include a multiple of three, it must be divisible by 3

Therefore it is divisible by 6

Disprove:

Counter example is used when

You can think of a single example that disproves a statement, generally by inspection

Proof by contradiction:

Is used when

You can state the opposite statement

Prove it is false

And therefore show the original statement was true

Proof by contradiction:

Prove sqrt n is irrational

Assume sqrt n is rational so sqrt n= a/b where a,b are whole numbers in their simplest form, therefore a and b cannot be even

Square both sides

Multiply both sides by b²

Therefore a² is a multiple of n, hence a must be a multiple of n where a=nk

Substitute a=nk

Therefore b² is a multiple of n, hence a must be a multiple of n where b=nk

Therefore the assumption has been contradicted as if they are both of the multiple n, a/b is not in its simplest form

Proof by contradiction:

Prove 2^(3/2) is irrational

Assume 2³/2 is rational so 2³/2= a/b where a,b are integers in their simplest form

2=a³/b³ so a³ has a factor of 2 so a=2k

2b³=8k³ so b³ has a factor of 2 so b=2k

If both a and b have a factor of 2, a/b not in simplest form therefore a contradiction

Proof be contradiction:

There are infinitely many primes

Assume there is a finite list of primes where p is the product of all primes up to pn+1

If p is prime, p is not in the list

If p isn’t prime, p has a prime factor which must be a factor of 1

This is impossible as P is a whole number, so this is a contradiction

Proof by contradiction:

Contradictions to look out for

If a and b are integers, a+or-b must be an integer also

One side of the general equation is even, one side is odd, an odd number cant be equal to an even number

Straight lines:

midpoint between two points

(x1+x2/2, y1+y2/2)

Straight lines:

Distance between two points

sqrt ( (x1-x2)²+(y1-y2)² )

Straight lines:

Gradient of the line joining two points

(y1-y2)/(x1-x2)

Straight lines:

The equations of straight lines passing through two points

y=mx+c

y-y1=m(x-x1)

Straight lines:

If they are parallel..

The gradients will be the same

Straight lines:

If they are perpendicular…

m1 x m2= -1

Or the negative reciprocal

Straight lines:

Find the perpendicular bisector

1. Find the midpoint

2. Find the gradient

3. Find the perp gradient

4. Sub into the straight line equation

Straight lines:

The area of a trapezium

½ x h x (a+b)

Quadrilaterals:

Definition of a trapezium

One pair of parallel sides

Quadrilaterals:

The definition of a parallelogram

2 pairs of parallel sides

2 pairs of sides of the same length

Quadrilaterals:

Definition of a rhombus

All sides are the same length

Diagonals meet at 90’

Quadrilaterals:

Definition of a Kite

2 pairs of equal sides

Diagonals meet at 90’

Circles:

The equation of a circle

(x-a)² + (y-b)² = r²

Has centre (a,b) with radius r

Circles:

Determine if a line and a circle intersect

Substitute y or x and determine if the roots given are real, if so then it intersects

Alternatively can be proven with discriminant, if b²-4ac<0 then there are no real roots

Circles:

Find the points of intersection of two circles

find the centre and radius for both circles

Expand both equations and set to 0

Make the equations equal to each other and set equal to y

Substitute this into one of the original circle equations to find either x and y

Sub the value of x or y to find the other

Circle theorems:

The angles in a semi circle are

Right angles

Circle theorems:

The use of the semi circle theorem to prove AC is a diameter with 3 points

Draw a diagram

Find the gradient of AB

Find the gradient of BC

show the gradients are perpendicular through m1 x m2= -1

Hence through the circle theorem AC is a diameter as AB and AC are perpendicular with A,B,C on the circumference

Circle theorems:

The perpendicular bisector of two points on the circumference..

Will always cross the centre

Circle theorems:

Finding the centre and radius of a circle with the perpendicular bisector theorem, given three points on the circumference

Find 2 perpendicular bisectors

Find the intersection of the two perp bisectors (the centre of the circle)

Find the radius by measuring the centre point to one point on the circumference

Circles:

The normal is..

The tangent is…

The line that passes through the circumference and the centre

The line that touches the edge of the circumference once

They are perp to each other

Circles:

Circles not touching..

distance between centres> r1+ r2

Circles:

Circle found floating in a circle..

distance between centres +r1<r2

Circles:

Co-centric circles when

They have a common centre but different radiuses

Circles:

The circle edges are touching

d=r1+r2

Circles:

The circles are overlapping (venn diagram)

d<r1+r2

Parametric equations:

Meaning of parametric equation

Multiple equations describe the curve

Parametric equations:

Define x, y and t

x= f(t)

y=g(t)

t is the parameter

Parametric equations:

Write cartesian as parametric

Let x=t

Substitute into the y=f(x) equation

Parametric equations:

Write parametric as cartesian

Find x+y= f(t) 1

Find x-y= f(t) 2

Perform an operation (add, subtract, multiply, divide) to remove t and gain a cartesian

Or set both equal to t and and set equal to each other

Parametric equations:

Draw the parametric equation given x=f(t) y=g(t)

Create a table and sub in values of t to find (x,y) coordinates

Parametric equations:

Define an circle parametrically

x=r costheta

y=r sintheta

Parametric equations:

Define an ellipse parametrically,

And then write it in cartesian form

x=a costheta

y=b sintheta

x/a=costheta y/b=sintheta

Cos²theta+ Sin²theta= 1 so

(x/a)² + (y/b)² = 1

Parametric equations applied:

When finding a length of time t, for which y or x is equated to something, use these equations

r=(u costheta t)i+ (u sintheta t - ½ gt²)j

where x=u costheta t

y=u sintheta t

Quadratics:

Completing the square

Get in the form a( x²+ b/ax+ c/a )

a( (x+half the x coefficient)² -(half the x coefficient)² - the constant )

The vertex is therefore at ( -(whats in the bracket), outside the bracket)

Quadratics:

Finding the line if symmetry of a quadratic curve

Complete the square of the line

The negative of the constant value inside the bracket is the line of symmetry

i.e (x-2)²+5 → x=2 is the line of symmetry

Quadratics:

What is the discriminant

b²-4ac

Quadratic

If b²-4ac<0

There are no real roots to the quadratic equation since in the quad formula you cant sqrt a negative number

Quadratics:

b²-4ac=0

There is one real repeated root

Quadratics:

b²-4ac>0

There are two distinct real roots

Inequalities:

Set notation

{ xER: inequality}

Inequalities

Interval notation rules

If x<a

xE (-infinity, a)

If a<=x<=b

xE [a,b]

If x<a or a<=x<=b

xE (-infinity, a) U [a,b]

Inequalities:

draw an inequality on a line

Full circle includes

Empty circle does not include

This is useful to see the overlap of inequalities on a number line

Polynomials:

Bus stop division with quadratics

Polynomials:

Factor Theorem

If (x-a) is a factor of f(x) then f(a)=0

Polynomials:

remainder theorem

Function= linear x quotient + remainder

F(x)= (x-a)(q) + R

Graphs:

The modulus function is also known as

The absolute value of x

Graphs:

Sketching y=|x|

Modulus:

Find the vertex of y=a+b|x-c|

Think of completed square form

(The negative of whats in the modulus, whats outside the modulus)

So (c,a)

Modulus:

Find where the modulus equation crosses the axis

When x=0 to find one y value

Use symmetry if it crosses the axis again

Modulus:

Solving modulus equations with no y value

Get into the form a|x-b| to find the vertex

Set both sides to y and draw the lines

Notice the points of intersection

Solve the equation for the positive/negative gradient

Either ax-ab (positive) or -ax+ab (negative)

Modulus:

write a modulus as an inequality without the modulus

|x-a|<b

-b<x-a<b

Modulus:

Write an inequality as a modulus

‘a<x<b’ or ‘x<-a , x>b’

a+b/2

a-(a+b/2)<x-(a+b/2)<b-(a+b/2)

Find an operation that makes

|a-(a+b/2)|=|b-(a+b/2)|

|x-(a+b/2)|<|b-(a+b/2)|

Aka

2<x<10 2+10/2=6 -4<x-6<4 so |x-6|<4

Proportion:

y is directly proportional to x is written as

y=kx

Proportion:

y is inversely proportional to x is written as

y= k/x

Proportion:

Show two measurements are proportional

Use an equation involving the two measurements, i.e mass=density x volume

Sub your two measurements into the equation and get into the form y=kx or y=k/x depending on what you’re proving

Show that this proportional equation is true for all given measurements and conclude

Proportion:

When drawing proportional graphs what should you be wary of

If the units of measurement can be negative, i.e time cant be <0 so dont draw that part of the graph

Functions:

The range is

The scope of the y values

Functions:

The domain is

The scope of the x-axis

Functions:

a one-to-one function

Has one x, for one value of y

Functions:

a many-to-one function

one x, for many y values

Functions:

A One-to-many non-function

Many values of x for multiple y values

Functions:

A Many-to-many non-function

Multiple values of x for multiple values of y

Function:

What is a function

Giving in a value of x gives only one value of y

Only functions can have an inverse, so one-to-many and many-to-many graphs can have no inverse

Functions:

Definition of an even function

F(-x)= F(x)

This is because of a line of symmetry along the y-axis

Functions:

Definition of an odd function

F(-x)= -F(x)

The graph has rotational symmetry about the origin by 180’ (if u rotate part of the graph 180’, it will map onto itself)

Functions

f²(x) means

ff(x)

Functions:

Finding the domain of a composite function

Find the domain for all involved functions in the composite

The domain of the composite will be

-the domain of the composite function created

-and the domain of the first function (i.e for hgf(x), f(x) is the first function)

Functions:

Find and inverse function

Set f(x) to y

Swap x and y

Solve for y

Write y as f^-1 (x)

Functions:

What will the answer always be for

f(f^-1(x))

f^-1(f(x))

x

This shows e and ln are inverse functions

Functions:

Find the range and domain of the inverse function

The range is the domain of the original function

The domain is the range of the original function

Transformations:

f(x)+_a

A translation up or down by a

Transformations:

f(x+_a)

A translation right or left by -(a)

Transformations:

y=k(f(x))

A stretch parallel to the y-axis by s.f k

Transformations:

y=f(kx)

Stretch by s.f 1/k parallel to the x-axis

Transformations:

y=-f(x)

Reflection in the x-axis (y=0)

Transformations:

y=f(-x)

Reflection in the y-axis (x=0)

Transformations:

Find the equation of the graph after the transformation

y=f(x-a)+b

replace x with (x-a)

replace y with (y-b)

Transformations:

Find the equation of the graph after the transformation

y=(1/k x)

Replace x with (1/k x)

Transformations:

Find the equation of the graph after the transformation

y=k(f(x))

Replace y with 1/k y

Transformations:

Find the equation of the graph after the transformation

y=f(-x)

Replace x with -x

EVEN FUNCTION

Transformations:

Find the equation of the graph after the transformation

y=-f(x)

Replace y with -y

ODD FUNCTION

Transformations:

Rules for the order of graph transformations

If you do a transformation effecting y followed by a transformation effecting x the order does not matter

If you do a transformation effecting the same axis twice, order does matter

Use BIDMAS if both translations are outside when deciding the order to do them in

Use SAMDIB if both translations are inside the brackets when deciding the order

Partial Fractions:

Form for x/x²

A/(x..) + B/(x..)

Partial fractions:

Form for x/x(x+b)²

A/(x..) + B/(x+b) + C/(x+B)²

Partial fraction:

Form for x²/x²

A/(x…)+B(x..)+C

Partial fractions:

Form for x³/x² or x^4/x²

A/(x..)+B/(x..)+cx+D

A/(x..)+B/(x..)+cx²+Dx+E

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