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# STEP maths I, II, III 1990 solutions Tweet

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1. Re: STEP maths I, II, III 1990 solutions
It's bizarre how after four days of no posting there are two independent posts within the space of two minutes.
2. Re: STEP maths I, II, III 1990 solutions
Haha, yeah nice timing!

I'm thinking of having a look at a few more questions later, seems to be a few that are doable.
3. Re: STEP maths I, II, III 1990 solutions
Yup.

I think I've managed to do STEP II, Q16 (the amusing probability one), but since I haven't done any statistics whatsoever since my S1 exam in January, I won't risk writing up my solution unless someone else has got the same answer.

I got what the question asked me to show for the first part, and 11/63 for the second part.
4. Re: STEP maths I, II, III 1990 solutions
(Original post by Dystopia)
Yup.

I think I've managed to do STEP II, Q16 (the amusing probability one), but since I haven't done any statistics whatsoever since my S1 exam in January, I won't risk writing up my solution unless someone else has got the same answer.

I got what the question asked me to show for the first part, and 11/63 for the second part.
Same here. This looks tedious to write out; it would at least have been courteous to make the colours start with different letters!

Edit: Q15 is a much nicer question (also rather easy!).
Last edited by DFranklin; 16-06-2007 at 13:45.
5. Re: STEP maths I, II, III 1990 solutions
Bump. Thought people might want to try get a few more of these out .
6. Re: STEP maths I, II, III 1990 solutions
Bump again.

2007 thread's had a bit of work done to it, but this one's had practically nothing; fresh meat! Rabite, you still up for updating the first post?
7. Re: STEP maths I, II, III 1990 solutions
(Original post by generalebriety)
Bump again.

2007 thread's had a bit of work done to it, but this one's had practically nothing; fresh meat! Rabite, you still up for updating the first post?
I still can't be bothered to write up that stats question, but I was going through a 1991 paper the other day and realised that there was a rather nice alternate solution, so I think I'll go and write it up.
8. Re: STEP maths I, II, III 1990 solutions
Well, I've done I/7. Might all be absolute rubbish, I'm a bit tired, and I can't be bothered latexing, but I shall scan.

Edit: oh, forgot the initial conditions. Sigh, tomorrowwwww.
Attached Thumbnails

Last edited by generalebriety; 26-08-2007 at 03:44.
9. Re: STEP maths I, II, III 1990 solutions
Ooh I liked that question. I came out with something monstrous the first time though.

(Original post by The Billy)
Rabite, you still up for updating the first post?
Sure ~
I'll try to clean stuff up later today.
Oh, by the way...was it this thread that was littered with 1987 and 1989 discussion too, though? What do we do with those?
10. Re: STEP maths I, II, III 1990 solutions
EDIT: DAMMIT, just deleted my entire latex posting by clicking back by accident.

Sigh.
Last edited by Square; 26-08-2007 at 11:46.
11. Re: STEP maths I, II, III 1990 solutions
I/5
i)

Consider the quadratic on the denominator

Solve using the quadratic Formula:

Factorise using solutions from Quadratic Formula

Partial Fractions:

Unparseable or potentially dangerous latex formula. Error 4: no dvi output from LaTeX. It is likely that your formula contains syntax errors or worse.
\displaystyle=\text{ln}|\frac{9+ \frac{9}{2}}{6+ \frac{10}{3}}-\text{ln}|{\frac{3+\frac{9}{2}}{ 2+\frac{10}{3}}|

ii) I'm writing up now, not so sure about this bit though.
Last edited by Square; 26-08-2007 at 11:47.
12. Re: STEP maths I, II, III 1990 solutions
*pat pat*
That's why I always hit ctrl+A and ctrl+C every ten seconds.
Every few minutes would suffice I suppose, but I'm neurotic.
13. Re: STEP maths I, II, III 1990 solutions
Hey! II/5 is exactly the same as 2007 III/6, except the 2007 one gives one extra middle step as a hint.

*Insert rant about STEP getting easier*

So er, as there's a solution in

http://www.thestudentroom.co.uk/show...67&postcount=6

maybe you could just link that, Rabite?

Though there's a slight difference in notation, (Seventeen years down the road, has become and has become . Isn't it fascinating how fast maths is progressing?) so perhaps a link and an explanation is in order?
Last edited by ukgea; 26-08-2007 at 13:52.
14. Re: STEP maths I, II, III 1990 solutions
(Original post by Rabite)
Ooh I liked that question. I came out with something monstrous the first time though.

Sure ~
I'll try to clean stuff up later today.
Oh, by the way...was it this thread that was littered with 1987 and 1989 discussion too, though? What do we do with those?
No, I think that was my 1991 thread. Um, I'm ignoring it all to be honest. If anyone wants to start 1987 and 1989 threads (but please god not yet!) and nick stuff from that thread, then meh, why not.
15. Re: STEP maths I, II, III 1990 solutions
I/1

Let . Then

(1)

Consider the lengths of the altitudes dropped

from Y to CD,
from Y to AB and
from X to AD.

They are

,
and
respectively.

Thus the areas of the triangles DCY, DYX, DXA are , and respectively, and the area of the quadrilateral is

(2)

When is fixed, and we wish to maximise this area:

Setting the derivative to zero, we get

Now, both and clearly belong in the interval (the latter because it is the angle marked ), and thus

which is the maximum. This can be seen by working out the second derivative,

which is clearly negative at .

Note that in this case, the maximum area is given by

(3)

If we wish to find the maximum when both and vary, we can proceed by finding the value of for which the is at a maximum. Thus,

Setting the derivative to zero:

Now, both and are in , and thus we can write

from which follows

(4)

Again, to make that that this is a maximum, let us take the second derivative:

which again can be seen to be negative, thus ensuring that (4) indeed gives us the maximum .

Now, insertion into (3) gives us the maximum area as

, i.e.

Just for clarification, note that this happens when

.

Now you could stop reading here, but for the extra educational benefit or something, I couldn't resist pointing out that this question follows very easily from Jensen's inequality, introduced to us avid STEP-solvers in 2007 II/7:

Let f(x) be a concave function on some interval and let lie in that interval. Jensen's inequality states that

and that equality holds if and only if
Now, in 2007 II/7, it was proven that was a concave function on .

Setting, in Jensen's inequality, , , , , and using (2) and (1) we get

with equality iff .

This immediately gives us the maximum.

(Note that what we done here was really use the result in (i) of 2007 II/7, give or take a factor ).
Last edited by ukgea; 26-08-2007 at 15:10.
16. Re: STEP maths I, II, III 1990 solutions
(Original post by Speleo)
1987 III/6

at

Can anyone get me started?
Laplace Transforms should sort this out, however i don't know whether in 1987 it was on the A level course.
Last edited by insparato; 27-08-2007 at 20:31.
17. Re: STEP maths I, II, III 1990 solutions
(Original post by Speleo)
1987 III/6

at

Can anyone get me started?
I might just be incredibly stupid, but can't you just eliminate a variable (thus turning it into a second-order ODE) more or less right away (it's not might not be that right away, but easy to find once you have the idea)? I get

which seems very nice indeed.

I think the method by which you eliminate a variable is the exact same as the method you use to solve the linear equation system after you have Laplace-transformed (in the sense that a Laplace transform takes every step of my solution into a corresponding step in solving a linear equation system).
18. Re: STEP maths I, II, III 1990 solutions
STEP III, Q6.

Spoiler:
Show

So applying the transformations to the vectors (1, 2) and (2, -1), we get (2, 4) and (8, -4), both of which are scalar multiples of the original vectors.

The matrix transforms the general point (x, y) to (X, Y) = (2/5)(9x-2y, 6y-2x).

5X = 18x - 4y
5Y = 12y - 4x

25X2 = 324x2 - 144xy + 16y2
25XY = 232xy - 72x2 - 48y2
25Y2 = 144y2 - 96xy + 16x2

Therefore the function x2 + y2 = 1 is transformed to:

As required.

The area is equal to the original area multiplied by the determinant of the matrix transformation. The original area is ; the determinant is 8. Therefore the area is .

To find the maximum value of X we find

Differentiating implicitly, we get

There is a stationary point when

Treating the curves equation as a quadratic in Y, we get

At the stationary point,

The point (0.6, 0.8) is transformed to the point .

At this point,

The equation of the tangent at this point is therefore .
19. Re: STEP maths I, II, III 1990 solutions
Attempt at STEP III, Q8.

Spoiler:
Show

The integrating factor is .

Unparseable or potentially dangerous latex formula. Error 4: no dvi output from LaTeX. It is likely that your formula contains syntax errors or worse.
\Rightarrow (x^{3}-1)y = -\int{x^{2}e^{x} + \frac{c}{x} \; \mathrm{d}x = -x^{2}e^{x} + 2xe^{x} - 2e^{x} - c\ln{x} + k

Initial conditions: x=2, y=2e2, y'=0

From this I get

Which is utterly horrific, so presumably I've made an error.
Last edited by Dystopia; 27-05-2008 at 16:29.
20. Re: STEP maths I, II, III 1990 solutions
12 + 2 + 2/3 = 44/3
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