Trigonometry & Complex Numbers
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.
Combinatorics and Probability
Proof of Cauchy-Schwarz Inequality
Expand out and we get
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].
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
If you are asked to expand a function to x^2, such as
Sequences & Series
Quick and dirty introduction to solving simple recurrence relations
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.
- 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.
- Using abbreviations can save a great deal of writing
- The parts of a question are often linked together, but sometimes with slight modifications.
- To show a statement is true, give a formal proof; to show one is false, give a (if possible, simple) counterexample.
- It doesn't matter if you start from the given answer and work backwards - it is still a mathematical proof and any proof will get the marks.
- A geometric understanding of modulus questions can help when examining the different cases.
- If you are unsure what to do some way into a question, examine what you have already demonstrated. STEP often teaches small tricks in the first part of the question then gets you to use this method by yourself.