How much detail is required for if and only if proofs in STEP?

Is it sufficient to put $ \Leftrightarrow $ symbols throughout your working without justification?

In addition when is the above symbol actually valid - for example if $ X=Y \rightarrow X^2 = Y^2 $ however I think the reverse implication is not valid since if
$ X^2 = Y^2 \rightarrow X=Y or X=-Y $

On the other hand is this valid
$ X=-Y or X=Y \Leftrightarrow X^2 = Y^2 $

I assumed there exists positive integers such that

$ C_q (x) \equiv C_r (x) C_s(x) $

However the roots are primitive hence roots can only appear on both sides of equation if q=r or q=s

if q=r it means

$ C_s (x) \equiv 1 $

Since the above statement implies equivalent to 1 for all x, can't I just find say that a primitive root of the LHS will cause the LHS to equal 0 hence the equivalent to statement does not hold - contradiction

In the mark scheme, it just says 'it is not possible for positive s' - what does this mean?

After differentiating * in (3ii),

I obtain y''=0 and

$\ a^2 y' + x(y-xy')\ $ = 0

the above is a differential equation

$\ y' = \frac {xy}{x^2 - a^2}\ $

I get the solution as

$\ y^2 = A (x^2 - a^2) \ $

I have tried substituting the above into * given in question but this has not helped to figure out the constant

How do I show my solution for the differential equation is x^2 + y^2 = a^2 - as given in question ?

Please can someone check if this is the correct method:

I square rooted the expression twice - getting four equations in all. Only one of the initial conditions satisfies each one to ensure the expression under square root is real hence the eqns I obtained are:

$ \sqrt(y^2 -1) $ = $ \pm x+1/2 $

$ \sqrt(1-y^2) $ = $ \pm x+1/2 $

for the first 2 parts 'find y in terms of x...'

How do you remove the modulus sign on the y after integration e.g. for the first part

1/y dy = -1 dx

ln mod(y) = -x + c

mod(y) = Ae^-x

even after substituting the initial conditions, how can I justify in removing the modulus sign on the y?

It's natural to find STEP or other advanced mathematical problem solving hard, but worrying about it doesn't help; the best way to improve is to immerse yourself in the mathematics and keep at it. Share your moments of inspiration, techniques for keeping calm and success stories here!

When I get stuck on a problem I always ask myself a series of questions to try to get myself unstuck:
What do I know? (and write it down)
Can I draw a diagram?
Is this like a problem I've already solved?
Is there any information in the problem that I haven't used yet?

Does anyone else have any good problem solving tips?

I've been doing quite a few STEP questions lately and thought it would be nice to have a thread where people can share their favourites. I enjoyed STEP I 2006 question 8 because although I'm not usually fussed about geometry questions, there were some nice ways of thinking about it and as it was a "show that" question, I knew when I'd got the right answer! Anyone else got a favourite to recommend?

If the MathJax library is installed properly, you should see the square root of x here: $ \sqrt{x} $ and the square root of y here: \(\sqrt{y}\)$$\text{The quadratic formula should appear here: } x = \frac {-b \pm \sqrt {b^2 - 4ac}}{2a}$$\[\text{The cubic equation should appear here: } a x^3\; +\; b x^2\; +\; c x\; +\; d\; =\; 0\]