Tips and Tricks for STEP Questions
A collection of simple and broad tips and tricks to help you solve STEP questions.

Some parts of STEP questions can be dramatically simplified with the use of a few simple tricks. The tricks below are ones which I came across as I went through my preparation.

But you should form your own list too! As you go through past paper questions, keep a list of methods you used or were led through by the question. These will often come in handy for other questions later on.

Draw it

Drawing the objects in a question, be they functions, numbers, objects, or forces, can be a very effective way to see shortcuts or symmetries in answering a question.

Integration

One instance when drawing graphs can be useful is when finding or approximating integrals.

Sketching the functions can show symmetries, especially for trigonometric functions, which can be exploited with substitutions.

Also, it can be helpful with arbitrary integrals. For instance, I recall a question exploiting the relationship between the integral of function and its inverse, which was much more easily understood with a diagram. 

Complex Numbers

Drawing complex numbers on an Argand diagram can be extremely useful for visualising things like the triangle inequality and roots of unity.

Learn your identities and inequalities and substitutions

Identities

It is vital that you know, or at least can derive, the most common identities in trigonometry, but some of the rarer ones can also be very useful at times. Also, identities for factorisation can also save a lot of time in algebra based questions, such as the difference of two cubes.

Inequalities

Knowing inequalities such as the Arithmetic Mean-Geometric Mean inequality or Cauchy-Schwarz and their proofs can often be used to dramatically simplify proofs of other inequalities.

Substitutions

Many STEP questions end with an integral requiring a substitution. Often these questions can be solved just by trying things until something works, but it can also help if you know some standard substitutions which can easily be adapted. One of the best examples is the t-substitution, which comes up in STEP every now and then. My suggestion would be to write down separately any substitutions you discover (or are led to) when doing past paper questions and looking back over them later, as they might well come up again.

Proof Techniques

Counterexamples

To prove something false all you need is one counter-example. This can save a lot of time in a STEP question. Instead of spending time searching for a proof of something which is false or looking for an explanation why something isn’t true, just try to find a simple counter-example. Looking, and not finding, counter-examples can also help with a proof as you can get a sense of why something is true.

Work from the end

When doing a “Show that” question involving simplifying algebra, it can often be helpful to work from the end backwards as well as from the front forwards. This can sometimes allow you to meet in the middle or recognise a helpful simplification. At the least, it can help to recognise where different sections of the result come from, for example the  term might come from the product of and .

Ifs and only ifs

Typically, to do an ‘if and only if’ proof (iff), you have to provide two proofs, one for the ‘if’, the other for the ‘only if’. However, in an exam, you can often save time and effort by just doing the proof one way and then checking whether the steps are reversible. If only a couple are not reversible, you only have to prove those steps in reverse, and not the whole argument.

Only do what is necessary

It is easy to accidentally end up doing much more work than is necessary, and waste a lot of time, potentially making mistakes in the process. It is important to do only what the question asks as much as possible, taking shortcuts where possible.

For an extreme example, in this question it is tempting to start expanding the whole expression, while some thought at the start can make that unnecessary.

Binomial Coefficients

Questions involving binomial coefficients can be done in many ways, with different interpretations of their meaning. Often, one of these interpretations make the question significantly easier to understand than the others.

For instance, you can do them purely algebraically (using many factorials), combinatorically (considering subsets of sets), or using Pascal’s triangle.

Conclusion

In conclusion, there are many different tips and tricks you can learn on the way to help you solve STEP questions. However, it is important you make up and write down your own as you do past paper questions. So, keep a sheet next to you when doing past papers and note down techniques you use which might come in useful in later questions.