STEP 2018 Solutions

If you get the right answer but the wrong method in a STEP question but the wrong method, and it isn't a 'show that' question, do you get all the marks?
For the last part of q1 I thought that the range of Beta values was unchanged, and so Beta is less than or equal to one, so g(Beta) max is 3, which is the right result but wrong method.

Reply 221

123Master321

14

Original post by mas252

If you get the right answer but the wrong method in a STEP question but the wrong method, and it isn't a 'show that' question, do you get all the marks?
For the last part of q1 I thought that the range of Beta values was unchanged, and so Beta is less than or equal to one, so g(Beta) max is 3, which is the right result but wrong method.

Unfortunately you wont really be getting any of the marks, if you look at markschemes they do have M marks and A marks, and you cant get A marks unless the method is there

Reply 222

DFranklin

18

Original post by Integer123

Paper 2, Question 12

Spoiler

It seems to me you made somewhat heavy weather over the maximization here (by not using the fact that h is an integer):

Spoiler

Reply 223

123Master321

14

Original post by DFranklin

It seems to me you made somewhat heavy weather over the maximization here (by not using the fact that h is an integer):

Spoiler

Would solving the inequality

(h+1)p^{h+1} \geq hp^h

giving

h \leq N

and then stating the maximum must occur at h=N be fine?

Reply 224

Sequin Rugby

7

If anyone wants a tenner just scan the STEP III paper and PM me it to me. I really wanna have a go '-'

Reply 225

Zacken

OP

22

Original post by A02

Does this mean that those who didn't sit the paper won't have access to it for several months?

Yes.

Original post by JudeH2001

How bad is it that I think STEP I was bad if I am year 12 (I'm actually year 13 in northern Ireland but our year 13 is equivalent to the English year 12). I was scoring 1 or S in most of my past papers but I doubt I got higher than a 2 in this paper. Will this reflect badly on an application to Oxford considering I had not learned the entire maths A level course so parts were self taught?

The fact that you sat it early and didn't learnt all the relevant material is squarely on you, universities won't care - if you sat the exam, they (or at least Cambridge) presume that you were ready for it and treat you as such. I don't know how Oxford works, but it would certainly detract from a Cambridge application.

Reply 226

username1162972

18

Original post by Sequin Rugby

If anyone wants a tenner just scan the STEP III paper and PM me it to me. I really wanna have a go '-'

Better off paying to go lose ur virginity mate

Reply 227

Zacken

OP

22

Original post by I hate maths

Yeah I'm in year 13 with an offer. I did all right in step 2, I'll just mention the bits of the questions I didn't do. Of the six questions I did, I didn't do the last part of q4 (think I might have got a quadratic which I didn't solve), didn't explain my sketch in q2 stem (did have a sketch though, rest of q was ok), didn't do n=6 for q6, didn't find sufficient conditions in q1 (though I had working towards it I think?). Q5 and q8 should hopefully be around 20.

Step 3... I don't think I had a single full to be honest. Had decent jabs at q1, q4, q5, did some stem stuff and part i integral for q8 and had a little bit of working for the latter "deduce" bit of stem, awful partials on q2 and q3 even though I thought these questions would be all right. Wish I did q7 instead of one of these, I felt lured lol.

Basically I don't think I'm getting in, but hey nought I can do about that now.

Original post by I hate maths

Post 184 by Integer has a solution to q8.

I also have a very small hope that I got an S in step 2, if the holes I had are small enough and people found it hard enough so hope and pray for me as well! But at this point, I've resigned myself to my impending doom on results day.

I was identical to this two years ago (although worried about STEP II and hopeful for an S in III), spent ages looking at stats and hoping for a 2,S and hoping I'd get taken in anyway. It all worked out fine and I got a 1,1. My STEP II grade was about 20 marks higher than I expected; I don't think I had a proper full solution in that as well. Basically, I think you're in a weakly isomorphic position to me, would be very surprised to hear you didn't get in.

Reply 228

username3555092

14

Original post by Zacken

I was identical to this two years ago (although worried about STEP II and hopeful for an S in III), spent ages looking at stats and hoping for a 2,S and hoping I'd get taken in anyway. It all worked out fine and I got a 1,1. My STEP II grade was about 20 marks higher than I expected; I don't think I had a proper full solution in that as well. Basically, I think you're in a weakly isomorphic position to me, would be very surprised to hear you didn't get in.

You're far too kind - but I very much appreciate your thoughtful post.

Reply 229

Zacken

OP

22

Original post by I hate maths

You're far too kind - but I very much appreciate your thoughtful post.

I had plenty of people telling me, back then, that I'd underestimated myself and it'd all work out. Was kind of annoying at the time because I thought I knew how I did, I'm sure you're feeling something similar. Just try to enjoy your summer and you can buy me a drink when you come to Cambridge in October. :tongue:

Reply 230

username1357565

15

Original post by carpetguy

STEP III 2018 Q4

The question went something like:

P(a\sec\theta,b\tan\theta)

is a general point on the hyperbola

\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1

.

Firstly, show that the equation of the tangent to the hyperbola at P is

bx - ay\sin\theta = ab\cos\theta

.

Next, find the two coordinates (and/or their midpoint) where this tangent meets the lines

\frac{x}{a} = \pm\frac{y}{b}

.

Then, the tangents to the hyperbola at

P

and

Q(a\sec\phi,b\tan\phi)

are perpendicular. Find

x

and

y

as the coordinates of the intersection of these tangents in terms of

a,\,\theta,\,\phi

only.
Lastly, show that

x^2 + y^2 = a^2 - b^2

.

I'll start assuming the first result was reached and the midpoint of the two meeting points was shown to be at

P

.
So we have the equation of the tangent to the hyperbola at

P

is

bx - ay\sin\theta = ab\cos\theta

and by similarities we reach the tangent at

Q

being

bx - ay\sin\phi = ab\cos\phi

.

From the perpendicular requirement,

\frac{b}{a\sin\theta} \times \frac{b}{a\sin\phi} = -1

\therefore b^2 = -a^2\sin\theta\sin\phi

Subtracting the two tangent equations from each other and rearranging:

y = \frac{b\,(\cos\phi \,-\, \cos\theta)}{\sin\theta \,-\, \sin\phi} = \frac{\sqrt{-a^2\sin\theta\sin\phi}\,(\cos \phi \,-\, \cos\theta)}{\sin\theta \,-\, \sin\phi}[br]

Multiplying the tangent equation for

P

by

\sin\phi

and the tangent equation for

Q

by

\sin\theta

, subtracting the two new equations and rearranging:

x = \frac{a\,(\cos\phi\sin\theta \,-\, \cos\theta\sin\phi)}{\sin\theta \,-\, \sin\phi}

\therefore x^2 + y^2 = \frac{a^2}{(\sin\theta \,-\, \sin\phi)^2}\,\{(\cos\phi \sin \theta \,-\, \cos\theta\sin\phi)^2 \,- \, \sin\theta\sin\phi\,(\cos\phi \,-\, \cos\theta)^2 \}

Now yeah that does look ugly, but you can see that the

-2\cos\phi\cos\theta\sin\phi\sin \theta

term is generated from both squares and is cancelled out - so inside the big bracket you're left with:

\cos^2\phi\sin^2\theta + \cos^2\theta\sin^2\phi - \sin\theta\sin\phi\cos^2\phi - \sin\theta\sin\phi\cos^2\theta

Note that this can be factorised to:

(\sin\theta \,-\, \sin\phi)(\cos^2\phi\sin\theta \,-\, \cos^2\theta\sin\phi)

and so

x^2 + y^2 = \frac{a^2}{(\sin\theta \,-\, \sin\phi)}\,(\cos^2\phi\sin \theta \,-\, \cos^2\theta\sin\phi)

Apply trig Pythagoras' and this becomes

\frac{a^2}{(\sin\theta \,-\, \sin\phi)}\,(\sin\theta \,-\, \sin\phi \,+\, \sin^2\theta\sin \phi \,-\, \sin^2\phi\sin \theta)

\therefore x^2 + y^2 = \frac{a^2}{(\sin\theta \,-\, \sin\phi)}\,(\sin\theta \,-\, \sin\phi)(1 + \sin\theta\sin\phi)

= a^2 + a^2\sin\theta\sin\phi = a^2 - b^2

Phew! You could probably introduce some shorthand or something if you wanted to save on the writing but that's pretty much a slog.

(edited 5 years ago)

Reply 231

memequeen69

6

how many marks would i get for Q8 STEP III for doing everything apart from the last integral (integral from 0 to 1 of fractional part of 2x^-1, all over x^2+1)?

Reply 232

Electric-man7

8

Original post by mas252

If you get the right answer but the wrong method in a STEP question but the wrong method, and it isn't a 'show that' question, do you get all the marks?
For the last part of q1 I thought that the range of Beta values was unchanged, and so Beta is less than or equal to one, so g(Beta) max is 3, which is the right result but wrong method.

I don't think you'll get marks for the last part, because it is based on incorrect maths.

Reply 233

Electric-man7

8

Original post by memequeen69

how many marks would i get for Q8 STEP III for doing everything apart from the last integral (integral from 0 to 1 of fractional part of 2x^-1, all over x^2+1)?