a level maths paper 2

Contradiction

A contradiction is a disagreement between two statements, which means that both cannot be true. Proof by contradiction is a powerful technique.

Proof by contradiction

To prove a statement by contradiction you start by assuming it is not true. You then use logical steps to show that this assumption leads to something impossible (either a contradiction of the assumption, or a contradiction of a fact you know to be true). You can conclude that your assumption was incorrect and the original statement was true.

A rational number can be written as

a/b, where a and b are integers.

An irrational number cannot be expressed in

the form a/b, where a and b are integers.

To multiply fractions

cancel any common factors, then multiply the numerators and multiply the denominators.

To divide two fractions

multiply the first fraction by the reciprocal of the second fraction.

To add or subtract two fractions,

find a common denominator.

Splitting a fraction into partial fractions

A single fraction with two distinct linear factors in the denominator can be split into two separate fractions with linear denominators.

Two methods to find the constants of partial fractions

substitution and equating coefficients.

The method of partial fractions ca also be used when

there are more than two distinct linear factors in the denominator.

A single fraction with a repeated linear factor in the denominator can

be split into two or more separate fractions.

Improper algebraic fraction

One whose numerator has a degree equal to or larger than the denominator.

An improper fraction must ... before you can express it in partial fractions

be converted to a mixed fraction.

To connect an improper fraction into a mixed fraction you can either use:

- algebraic division
- or the relationship F(x) = Q(x) x divisor + remainder

Degree of a polynomial

the largest exponent in the expression

You can measure angles in units called

radians

2⫪ radians =

360⁰

⫪ radians =

180⁰

1 radian =

180⁰/⫪

30⁰ =

⫪/6 radians

45⁰ =

⫪/4 radians

60⁰ =

⫪/3 radians

90⁰ =

⫪/2 radians

180⁰ =

⫪ radians

360⁰ =

2⫪ radians

sin(⫪/6) =

1/2

sin(⫪/3) =

√3/2

sin(⫪/4) =

1/√2 = √2/2

cos(⫪/6) =

√3/2

cos(⫪/3) =

1/2

cos(⫪/4) =

1/√2 = √2/2

tan(⫪/6) =

1/√3 = √3/3

tan(⫪/3) =

√3

tan(⫪/4) =

1

sin(⫪ - θ) =

sinθ

sin(⫪ + θ) =

-sinθ

sin(2⫪ - θ) =

-sinθ

cos(⫪ - θ) =

-cosθ

cos(⫪ + θ) =

-cosθ

cos(2⫪ - θ) =

cosθ

tan(⫪ - θ) =

-tanθ

tan(⫪ + θ) =

tanθ

tan(2⫪ - θ) =

-tanθ

Using radians greatly simplifies the formula for

arc length.

To find the arc length l of a sector of circle

use the formula l = rθ, where r is the radius of the circle and θ is the angle, in radians, contained by the sector.

Using radians also greatly simplifies the formula for the area of a

sector

To find the area A of a sector of a circle

use the formula A=(1/2)r²θ, where r is the radius of the circle and θ is the angle, in radians, contained by the sector.

Minor sector

The smaller area enclosed by a radii and an arc

Major sector

The larger area enclosed by a radii and an arc

You can find the area of a segment by

subtracting the area of triangle OPQ from the area of sector OPQ

The area of a sefment in a circle of radius r is

A = (1/2)r²(θ - sinθ)

You can use radians to find ... for the values of sinθ, cosθ and tanθ

approximations

When θ is small and measured in radians: sinθ ≈

θ

When θ is small and measured in radians: tanθ ≈

θ

When θ is small and measured in radians: cosθ ≈

1 - θ²/2

There are n+1 terms, so this formula produces a ... number of terms

finite

If n is a fraction of a negative number you need to use a different version of the

binomial expansion

This form of the binomial expansion can be applied to negative or fraction values of n to obtain an infinite series.

(1 + x)ⁿ = 1 + nx + ((n(n - 1))/2!)x² + ((n(n-1)(n-2))/3!)x³ + ...

The expansion for negative or fractional values of n in a binomial expansion is valid when

IxI < 1, n ∈ R

When n is not a natural number, the 1 over version of the binomial expansion produces an

infinite number of terms.

The binomial expansion is valid for (n)

any real value of n

The binomial expansion is only valid for (x)

values of x that satisfy IxI < 1, or in other words, when -1 < x < 1.

The expansion of (1 + bx)ⁿ, where n is negative or a fraction is valid for

IbxI < 1, or IxI < 1/IbI

The expansion of (a + bx)ⁿ, where n is negative or a fraction, is valid for

I(b/a)xI < or IxI < Ia/bI

When doing binomial expansion on partial fractions

you need to find the range of values of x that satisfy both inequalities.

The distance from the origin to the point (x, y, z) is

√(x² + y² + z²)

The distance between the points (x1, y1, z1) and (x2, y2, z2) is

√((x1 - x2)² + (y1 - y2)² + (z1 - z2)²)

You can represent 3D vectors using the unit vectors

i, j, and k

The unit vectors along the x-, y- and z- axes are denoted by

i, j, and k respectively

i =

(1 0 0)

j =

(0 1 0)

k =

(0 0 1)

For any 3D vector pi + qj + rk =

(p q r)

If the vector a = xi + yj + zk makes an angle θx with the positive x-axis then cosθx =

x/IaI and similarly for the angles θy and θz

AB refers to the ... between A and B

line segment, or its length

AB→ refers to the

vector from A to B

if a and b are two non-parallel vectors and pa + qb = ra + sb then

p = r and q = s

In two dimensions with two vectors you can

compare coefficients

Coplanar vectors

Vectors which are in the same plane

Non-coplanar vectors

vectors which are not in the same plane

If a, b and c are vectors in three dimensions which do not all lie on the same plane then

you can compare their coefficients on both sides of an equation

Since the vectors i, j, and k are non-coplanar, if pi + qj + rk = ui + vi + wk then

p = u, q = v, and r = w

Modulus of a number

it's non-negative numerical value

Modulus function is also known as the

absolute value

A modulus function is, in general, a function of the type

y = |f(x)|

When f(x) >/= 0, |f(x) =

f(x)

When f(x) < 0, |f(x)| =

-f(x)

To sketch the graph of y = |ax + b|

sketch y = ax + b and then reflect the section of the graph below the x-axis and in the x-axis

The function inside the modulus is called the

argument of the modulus

Mapping

A mapping transforms one set of numbers into a different set of numbers.

A mapping is a function if

every input has a distinct output

Functions can either be

one-to-one or many-to-one

Domain

The domain is the set of all possible inputs for a mapping

Range

The range is the set of all possible outputs for the mapping.

Piecewise-defined function

Consists of two parts, one linear and one quadratic

Composite function

Two or more functions can be combined to make a new function. The new function is called a composite function.

fg(x) means

apply g first, then apply f

fg(x) =

f(g(x))

Inverse functions

The inverse of a function performs the opposite operation to the original function.

Functions f(x) and f-¹(x) are

inverses of each other