2013 STEP thread mark II

Hmmm ok I think i get it, I see what you mean about if and only if. But surely is a condition is necessary it is in itself sufficient? Because if you need it then of course it will work with it :s

In Q2, am I allowed to use ? (This is what I did :/)

It says "hence", so you must use the previous part. From the exam report:

"Some candidates resorted to the formula book and quoted the standard integral $\displaystyle\int \frac{dx}{\sqrt{a^2+x^2}}$ however, this gained no credit as the question explicitly said hence."

Ugh. Perhaps it didn't go as well as I had hoped, I thought I had 2 almost complete questions, looks like I only have the 1.
I had tried to get it using the previous stuff, with no success, so I was all "Yolo" and whacked out the FP2.

You know when you've posted too much when you need an extra thread because the first became too big

It's probably quicker to use the sub suggested anyhow...

$\int\frac{1}{\sqrt{x^2+2bx+c}}dx$

$\int\frac{1}{\sqrt{(x+b)^2+(c-b^2)}}dx$

$ln (x+b +\sqrt{(x+b)^2+(c-b)^2})$

$ln (x+b+\sqrt{x^2+2bx+c})$

Hardly a slow method.
I couldn't even work out how to use the suggested sub.
Why have I even applied for maths...

I have an odd problem with STEP I '88 Q5...

Not too sure why it's an approx either when I look at it.. I've edited it

Q5 of STEP II, 1991 is the stuff of dreams.

(That said, the final bit is somewhat irritating if you know the easier and faster way of doing it but have to do it similarly anyway. )

Q5 of STEP II, 1991 is the stuff of dreams.

(That said, the final bit is somewhat irritating if you know the easier and faster way of doing it but have to do it similarly anyway. )

My attempt on I 2008 goes as follows:
1: Did the first half, (A, B and C) got a couple of the pairs for the last part (this was the last question I attempted, so I didn't have time to finish it)
2: A complete solution aside from me not using the hence, so I lose all the marks for that bit (which I imagine was the bulk of the marks, as the other bits were straight forward).
3: most of i) done, didn't actually do the introduction bit correctly, couldn't see how to get at ii), so moved on.
4: saw sketching, and was put off straight away, I drew the first diagram as I wasn't quite sure if I could find any more appealing questions.
6: Did it all except the explanation at the end, calculated f(1/2) and g(k), drew a diagram, but didn't make the step to "These areas can add together to form a k by 1/2 rectangle."
10: Did the same as 4 - drew the diagram then decided against it as it looked like an algebraic battle.
13: Had a look at this but didn't get anything meaningful put down, so moved on after 15 mins.

What do you think I would have got?
Had it been the actual exam, and I wasn't restricted to 6 pure questions, I probably would have done better - wouldn't have wasted my time with 4/10/13 and would have just done 5 and 8 instead.

If we have to integrate something like 1/(4+x^2), can we use the formula booklet or do we have to do it from scratch (i.e. substitute tan)? Would the requirement differ depending on whether we're doing I, II or III?

Q6 is to do with special relativity XD

Edit: My papers are labelled wrongly.