# Step tips and tricks

## Identities

### Algebra

#### Identity 1

Essentially

Special case for squaring:

### Trigonometry & Complex Numbers

#### Identity 1

In particular, note that taking the modulus gives .

Equivalently, , which comes up a lot when calculating the distance between two points on the circumference of a circle

#### Identity 2

This trick is particularly useful in summing trigonometric series or any other situations which involve extracting real or imaginary parts out of a complex expression.

For the case of you can do similiar factorisation, which simplifies the algebra considerably.

= = =

This useful summation isn't difficult to memorise or derive if you want to now.

### Combinatorics and Probability

#### Identity 1

Proof: consider the coefficient of in the identity

The most famous case is when n=m=r:

## Methods

### Algebra

#### Proof of Cauchy-Schwarz Inequality

Prove that .

Consider . Then for all .

Expand out and we get

. But then since for all , we must have , so . Hence result.

For interest, there's equality iff for some value of . In other words, g is just a multiple of f (or f = 0 and g is anything you like).

Exactly the same proof shows etc.

### Calculus

#### Integral Calculus

The fundamental idea here is that an integral measures the area under a curve. Lots of consequences, such as if for all , then .

In general, when they ask you to prove something like this, you just have to draw a vague sketch to justify yourself!

A particular case that comes up often is that if f(x) is a decreasing function, then , as can be seen by sketching the graph of f and drawing "staircase" functions that are constant on each interval [k,k+1].

A particular case of the particular case(!) is to consider (with a little care about what happens at x=0) to deduce results such as .

Note I haven't explained this terribly well, because you really need the diagrams, but this concept of relating a sum to an integral is a pretty important one, and so you should spend a little time going over this.

##### Fundamental theorem of calculus

. This is isn't necessarily true if f isn't continuous at x, but don't worry too much about that. Again, if you need to prove it, you just need to draw a vague sketch justifying that

#### Maclaurin Expansion

If you are asked to expand a function to x^2, such as

Let

Compare coefficients

### Sequences & Series

#### Quick and dirty introduction to solving simple recurrence relations

To solve a recurrence relation of the form , assume a solution of form and deduce

You end up with a general solution where are the roots of the quadratic . (If the roots are repeated, the general solution takes the form ).

To solve , first look for a particular solution, typically of the form , and then make it general by adding solutions to . (All analogous to what you'd do for a linear diff equation)

### Statistics

#### Indicator functions and Expectation

All an indicator function is is a function that is 1 when an event happens and 0 when it doesn't. What's neat about indicator functions is that you can often use them to break a more complicated function down into something almost trivial.

E.g. Suppose and we want to prove the formulas for the mean and variance of X.

Instead of doing calculations based on finding etc, define the indicator function to be 1 if the k'th trial is a success, 0 otherwise.

Then .

But , so . (note that we can get this far without requiring the to be independent, which is often very useful).

Since the are independent, we also have , so .

• The examination requires intensity and persistence.
• Expect to take a long time answering questions at first.
• Answering a question yourself then checking is much more pleasing than giving up and looking at the answer.
• Questions require a systematic approach.
• Checking will improve the work of many candidates.
• The fluent, confident and correct handling of mathematical symbols is necessary and expected.
• Set out a well-structured answer.
• Sometimes a fresh start to a question is needed.
• Sound technique is necessary, and checking required.
• Working to be legible.
• Aim for thoughtful and well set-out work.
• Arithmetic and algebraic accuracy would most improve marks.
• It is not a good idea to plunge into the algebra without thinking about alternative methods.