**Further Pure Mathematics 1**

**Luke Storry**

**Proof**

**Algebra**

**Matrices**

Equality

two complex numbers are equal if and only if both the real and imaginary parts are equal.

Conjugate Pairs

The curve meets the X axis twice. The +/- in this formula leads to two CONJUGATE complex numbers, one with a plus and one with a minus. the second has a star superscript:

**Complex Numbers**

Addition, Subtraction, Division & Multiplication

Like Algebra:

Argand Diagrams

Quadratic Equations

Be able to solve any quadratic equation with real coefficients.

For polynomial equations with real coefficients, the complex roots come in pairs, with the others being real roots.

Modulus-Argument form

To simplify division, use complex conjugates:

addition works like vectors:

USING THE SPEC:

http://www.ocr.org.uk/Images/75811-specification.pdf

**Curve Sketching**

Be able to sketch the graph of y=f(x), obtaining information about:

symmetry,

asymptotes parallel to the axes,

intercepts with the coordinate axes,

behaviour near x=0,

behavior for numerically large x.

need to label all of these!

Direction from which a Curve approaches an Asymptote

plug in values either side of the asymptote.

eg, if alpha was the asymptote:

(for rational functions)

Loci

(On the Argand Diagram)

Circles:

|z-a|=r

Half lines:

arg(z-a)=theta

Graphical calculators are awesome, but working still needs to be shown!

**Definitions**

only need to understand the starred ones:

Identity/Equation

An equation is only true for only certain values.

An identity is true for all values of all the variables.

Proof by Induction

VERY USEFUL VIDEO

This question is explicitly stated in the specification!

Difference between a sequence and a series

Sums

Convergence

Be able to manipulate simple algebraic inequalities, to deduce the solution of such an inequality.

Series

**Inequalities**

**Roots**

Appreciate the relationship between

the roots and coefficients of

quadratic, cubic and quartic equations.

Form a new equation whose roots are related to the roots of a given equation

1) Find sum and products of roots of original equation

2) Multiply and sum new roots

3) Substitute numbers from (1) into equations from (2)

4) Simplify to give new equation

by using sum= -b/a, etc

A sequence is an ordered set of "things".

A series is a summation over a sequence

Summing arithmetic progressions over large ranges can be very time consuming.

these formulae only work if we start from 1

eg:

There are formulae that can allow us to calculate the sum more easily:

Know the meaning of the word converge when applied to either a sequence or a series.

For more detail:

http://en.wikibooks.org/wiki/A-level_Mathematics/OCR/FP1/Complex_Numbers

f(x) =(x-a)(x-b)(x-c)

= (x^2-(a+b)x+ab)(x-c)

=(x^3 - (a+b+c)x^2 + (ab+bc+ac)x - abc

Solving inequalities works very similarly to solving equations, but the sign swaps when multiplying or dividing by a negative number

Some Rules

Linear Example:

Quadratic Example:

Draw it!!

solve like an equality,

then sketch it.

When the differences between terms decreases, so the values tend towards a constant.

r < 1

Addition and Multiplication

Zero Matrix

Identity Matrix

Equality

Matrix multiplication is associative but not commutative.

Multiplication Rules

Add

Subtract

Multiply

Matrix * Scalar

Add each cell separately.

Both matrices must have the same order

Same as addition: do each value separately

Again, they must havethe same order

Multiply each individual value of the matrix by the scalar

Orders

It does what 0 does in normal multiplications, but for matrix multiplication

It does what 1 does in normal multiplications, but for matrix multiplication

For A and B to be equal, they must have the same size and shape and they must have the same values in the same spots.

eg:

3y = 33

y = 11

Linear Transformations

Be able to find the matrix associated with a linear transformation and vice-versa

How to find the matrix for a transformation

Combinations of Transformations

(related to matrix multiplication)

Useful ones to know

Don't need to memorize any, just need to be able to calculate them

IN DATA BOOK

Invariance

Invariant Points

Lines of Invariant Points

How to find them

Inverse

Determinant

Equals the area scale factor of the transformation represented by the matrix

Significance of detM=0

This means that the matrix doesn't have an inverse, so is "singular".

An invariant point is one which is left unchanged by a transformation.

Under the identity matrix all points are invariant, and that the point (0,0) is invariant under all transformation matrices.

A line of invariant points is where all points on a certain straight line are invariant for a specific matrix.

Can't divide a matrix, but inverses give a way of doing a similar operation:

What?

How to find?

Product rule:

Then, to solve an equation:

but use the notation:

Since these are the same equation, there must be a line of invariant points:

This shows that if y is equal to -x, then the point (x,y) will be invariant for this matrix, so any point on that line is an invariant point for this matrix.

Solution of Equations

Know how to use matrices to solve linear equations.

Geometrical Interpretation when a 2x2 matrix is singular.

Sense

The sense of the shape changes with the transformation if the determinant is negative.

('sense' = order of labelling)

If no real solutions:

Lines are parallel

If infinitely many solutions:

The two lines are the same line

Every point on each line is shared with the other line.

There are no points shared with both lines.

Do we need to know more?

Not according to specification, but the textbook goes further??