How to lay out a question properly (STEP)

The big one I see is that people fail to connect their lines together. So they'd write something like:

$x^2 +2bx + c = 0$

$(x-b)^2+c-b^2 = 0$

$x-b = \pm \sqrt{c-b^2}$

$x = b \pm \sqrt{c-b^2}$

Now in this case it's not that likely to cause confusion, but in principle an examiner could be looking at this and thinking "Are these 4 separate equations, or are they supposed to be linked?".

You don't need to write much, but something that indicates how the lines are joined together makes it a lot clearer. My personal inclination is to link with words, and lay out horizontally, so you actually need less vertical space overall:

Suppose $x^2 +2bx + c = 0$. Then $(x-b)^2+c-b^2 = 0$, so $x-b = \pm \sqrt{c-b^2}$ and finally $x = b \pm \sqrt{c-b^2}$

Diagrams are also helpful (if difficult to impossible on TSR).

Not really to do with "layout", but make sure you're clear about the difference between "A implies B" and "B implies A". I'm amazed how many people doing STEP drop easy marks by getting confused about this.

Other than that, yeah, if you post a solution I'll give feedback.

I think using arrows => is more convenient

The biggest problem with using $\implies$ is when your are already manipulating statements with implication signs.

So, if you're doing something with limits, you might want to start from the definition of pointwise convergence:

$\forall \epsilon > 0, \exists N {\text s.t. } n > N \implies |a_n - a| < \epsilon$.

If I were then to write:

$\implies \forall \epsilon > 0, \exists N {\text s.t. } n > N \implies |a_n^2 - a^2| < \epsilon |a_n+a|$.

Then it quickly gets confusing about what is implying what. (Particularly if you don't use a new line).

But of course, if you want to use $\implies$ that's fine. Do be aware, however, that there are no prizes for the person with the highest "mathematical symbol to English words" ratio. (Overuse of symbols and underuse of English is, IMHO, another thing people tend to do badly when laying out mathematics).

Also, while writing down my thoughts, such as A => B, then later realise than B => A is also true, and it's needed for my next step, then I would just need to add < to make it <=>. It saved valuable time for me many times

Hence there are no solutions.

x

Please point out anything I could do better.


Step iii 1988 question 1.

sketch the function h, where $h(x) =\frac{ln(x)}{x}$ for x>0
Hence or otherwise find all pairs of distinct integers $n^m=m^n$




Hence there are no solutions.

I remember answering a similar question for my entrance exam - looked it up as Maths II 23rd Nov 1973 Q1. On that occasion it asked "how many positive values of x correspond to a given value of h?" And it asked for a sketch of x^y = y^x.

I recall 2^4 = 4^2.

$x\rightarrow 0$ Therefore $\frac{ln(x)}{x}\rightarrow -\infty$ So h approaches -infinity when x approaches 0.

I think we should justify why h approaches - infinity instead of + infinity

maybe something like this:
when 0<x<1, lnx <0 => lnx/x <0
=> it approaches - infinity when x approaches 0?

Thankyou, yes I agree. I was also worried about justifying ln(x)/x approaches 0 when x=> infinity. Do you know of any fancy formula I can quote.
How do I justify ln(x)/x is => 0 as x=>infinity?

It saves time, but it's worth being careful. There are occasions, usually when the question consists of mere algebraic manipulation, that our arguments can be reversed in order to obtain and 'if and only if' relationship. However, there are many more situations where one cannot, and we require separate consideration of our hypotheses and our desired conclusion. I say this because it's something that becomes very important at university level, whereby one has many theorems floating around in ether, requiring various hypotheses and giving several results that may be stronger, weaker, equivalent, etc., yet such consideration is almost never required at A level. And naturally the examiner is keen to test awareness of this. In my opinion, if you want to maximize your marks, unless it is obvious your argument can be reversed, I would break A iff B down into A -> B and B -> A and give equal care to both. I'm not saying you're guilty of this, your justification being notational, but it's salient to STEP exams.

Thankyou, yes I agree. I was also worried about justifying ln(x)/x approaches 0 when x=> infinity. Do you know of any fancy formula I can quote.
How do I justify ln(x)/x is => 0 as x=>infinity?

Thankyou, yes I agree. I was also worried about justifying ln(x)/x approaches 0 when x=> infinity. Do you know of any fancy formula I can quote.
How do I justify ln(x)/x is => 0 as x=>infinity?

Tbh, unless it's explicitly in the question, I'd say it's fairly obvious... You could always do something along the lines of l'hopital's rule, and differentiate top and bottom, and consider what will be happening to each, but I wouldn't bother justifying it, unless they ask, when you can generally fudge something, or it's genuinely difficult and it took you doing some algebra/calculus to work out the limit. I'm not really a fan of STEP limits questions because of this.. I'm never really sure how thorough they expect you to be, and so I generally try to justify, but be fairly hand-wavey!