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STEP III 2012 Discussion Thread

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    (Original post by DFranklin)

    MP: I've always thought that saying two curves intersect at a point just meant that that point is on both curves - i.e. that it is in the intersection of the two curves, considered as subsets of the plane. Obviously I'm wrong, but before this discussion it had never occurred to me that anyone put any more conditions than that on curves intersecting.

    I tend to agree in the abstract, but in this context it's kind of clear that "intersect but not touch" is what was being asked (otherwise "do these curves intersect or touch" makes no sense).
    I tend to think that definitions shouldn't depend on context. I thought I was agreeing with TheMagicMan that, whatever one said about those two curves touching, they quite definitely do intersect at that point (because that point is in the intersection of the two curves!) It is better to challenge the question "do these curves intersect or touch", rather than accept it as setting the context, because it suggests it is a case of either/or; in fact (at least, according to what I have always understood were the accepted definitions), it is a case of neither, intersect or both.

    (Original post by DFranklin)
    Do the curves y=x^3 and y=x^4 cut or touch at x = 0? (I would say cut, your definition seems to say touch).
    For me, definitely touch (and cross), not cut. I'm getting quite worried now, finding that people I have a great deal of respect for use these sorts of words differently from me. Perhaps we really should be pressing for the exam boards to decide on standard definitions to be included in syllabusses
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    (Original post by MAD Phil)
    whatever one said about those two curves touching, they quite definitely do intersect at that point (because that point is in the intersection of the two curves!)
    The problem is that the word touching is ambiguous - the sentence "these two curves touch", in the English language, could mean "they share a common point", which is not what it is supposed to mean. I thought "touching = common point but no intersection" - but apparently that is also ambiguous... Anyhow, what we do know is that intersecting definitely means "to cut through" so I don't see how the curves could touch and intersect?

    (Original post by MAD Phil)
    For me, definitely touch (and cross), not cut. I'm getting quite worried now, finding that people I have a great deal of respect for use these sorts of words differently from me. Perhaps we really should be pressing for the exam boards to decide on standard definitions to be included in syllabusses
    How do you define crossing and cutting? Are differentiating the cases f'(a)= g'(a) and f'(a)\neq g'(a) respectively? Sort of makes sense!

    Blah, stupid terminology :rolleyes:
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    (Original post by Lord of the Flies)
    The problem is that the word touching is ambiguous - the sentence "these two curves touch", in the English language, could mean "they share a common point", which is not what it is supposed to mean. I thought "touching = common point but no intersection" - but apparently that is also ambiguous... Anyhow, what we do know is that intersecting definitely means "to cut through" so I don't see how the curves could touch and intersect?
    Yes, ordinary English usage certainly doesn't follow what I had thought was standard mathematical usage - e.g. it seems fine to say (as you suggested earlier, I think) that y=x and y= \lfloor x\rfloor touch at (0,0), but I'd say that that's just the ordinary English usage.

    Similarly, I guess "intersect" has an etymology that means "cut through", but the set-theoretic meaning is now so central to maths that that fact is of only historical significance (I thought!), and now any member of two sets is a member of the intersection of the sets, and so, (if it is a point) is a point of intersection.


    (Original post by Lord of the Flies)
    How do you define crossing and cutting? Are differentiating the cases f'(a)= g'(a) and f'(a)\neq g'(a) respectively? Sort of makes sense!:
    That's what I'd say.

    EDIT: No it isn't - given that it's a point of intersection, I'd say that that was the distinction between touching and cutting. Crossing is a property that is fairly easy to recognise, but harder to define. Maybe something like this: if you can find arbitrarily small neighbourhoods of the point of intersection such that the neighbourhood minus one of the curves falls into two unnconnected regions, both of which intersect with the other curve, then they cross. (I haven't worked out the details!) That way, curves that cut (in my understanding of the term) also cross, unless one of them ends at the point of intersection of the curves. That feels right to me.

    (Original post by Lord of the Flies)
    Blah, stupid terminology :rolleyes:
    Yep
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    (Original post by MAD Phil)
    Similarly, I guess "intersect" has an etymology that means "cut through", but the set-theoretic meaning is now so central to maths that that fact is of only historical significance (I thought!), and now any member of two sets is a member of the intersection of the sets, and so, (if it is a point) is a point of intersection.
    Aaah I now see where you were coming from! I guess we'll have to live with inadequacy/ambiguity for now.
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    (Original post by MAD Phil)
    EDIT: No it isn't - given that it's a point of intersection, I'd say that that was the distinction between touching and cutting. Crossing is a property that is fairly easy to recognise, but harder to define. Maybe something like this: if you can find arbitrarily small neighbourhoods of the point of intersection such that the neighbourhood minus one of the curves falls into two unnconnected regions, both of which intersect with the other curve, then they cross. (I haven't worked out the details!) That way, curves that cut (in my understanding of the term) also cross, unless one of them ends at the point of intersection of the curves. That feels right to me.
    Spoiler:
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    Wait, can we use "common point" instead of point of intersection? Because the latter in my head is "to cut through".

    This is my view on things (for continuous functions):

    Touch (f'(0)=g'(0),\;f''(0)\neq g''(0)):

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    Intersect, cross (f'(0)=g'(0),\;f''(0)=g''(0)):

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    Intersect, cut (f'(0)\neq g'(0)):

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    Anything you disagree with?
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    (Original post by Lord of the Flies)
    Spoiler:
    Show

    Wait, can we use "common point" instead of point of intersection? Because the latter in my head is "to cut through".

    This is my view on things (for continuous functions):

    Touch (f'(0)=g'(0),\;f''(0)\neq g''(0)):

    Click image for larger version. 

Name:	Screen shot 2012-07-05 at 2.30.29 PM.png 
Views:	39 
Size:	9.2 KB 
ID:	161536

    Intersect, cross (f'(0)=g'(0),\;f''(0)=g''(0)):

    Click image for larger version. 

Name:	Screen shot 2012-07-05 at 2.31.51 PM.png 
Views:	44 
Size:	9.8 KB 
ID:	161537

    Intersect, cut (f'(0)\neq g'(0)):
    Click image for larger version. 

Name:	Screen shot 2012-07-05 at 2.32.17 PM.png 
Views:	40 
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ID:	161538

    Anything you disagree with?

    Yes, I'd agree with both versions of your labelling of the diagrams. (Though of course I'd still say that the first diagram shows an intersection as well as a touching, and the curves touch in the second one, as well as crossing.)

    But I wouldn't say that the two ways you have labelled the diagrams were equivalent to each other. In particular, (f'(0)=g'(0),\;f''(0)=g''(0)) could also be touching without crossing, as with y = 0 and y = x^4. As DFranklin pointed out earlier (though using different terminology from mine), to know whether they cross or not you may have to differentiate indefinitely often. What matters is whether the first nonequal derivatives are found by an odd or even number of differentiations.

    That would be the sensible test for crossing or not, but I'd still go for something like the topological property I roughed out earlier as a definition.
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    (Original post by MAD Phil)
    As DFranklin pointed out earlier (though using different terminology from mine), to know whether they cross or not you may have to differentiate indefinitely often. What matters is whether the first nonequal derivatives are found by an odd or even number of differentiations.

    That would be the sensible test for crossing or not, but I'd still go for something like the topological property I roughed out earlier as a definition.
    Yes I see my mistake now. Thanks for pointing that out!
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    (Original post by MrDD)
    Solution to assumed version of Q1
    heyy i did exactly this (the ans on here for Q.1) !! Would this be counted as a full solution/how many marks do you think this would get given the misprint and stuff?
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    (Original post by joievee)
    heyy i did exactly this (the ans on here for Q.1) !! Would this be counted as a full solution/how many marks do you think this would get given the misprint and stuff?
    If that's what you got then close to 20 marks. Even with the misprint you can't get more than 20!
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    Does anyone know if there have been any further updates on the q1 thing?


    This was posted from The Student Room's iPhone/iPad App
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    (Original post by TheMagicMan)
    Does anyone know if there have been any further updates on the q1 thing?


    This was posted from The Student Room's iPhone/iPad App
    There probably won't be until all scripts have been marked


    This was posted from The Student Room's iPhone App
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    (Original post by msmith2512)
    If that's what you got then close to 20 marks. Even with the misprint you can't get more than 20!
    But you should...

    If Q1 was carefully designed so that completing it represented 20 marks worth of 'work' in the exam, then completing it when it had been made harder by the omission of key information should represent more than 20 marks.

    I'm not saying they *will* treat it that way, but it seems a logical argument to me. If you'd been given the first part (as intended) then you would have potentially had time to pick up more marks on a different question. You could argue that you did the actual question, PLUS a small part of another question (which happened to contribute to Q1).

    It's a really, really interesting problem they are left with. Based on previous stats, roughly 700 people will have sat STEP III, around 450 of them hold offers for Cambridge, and around 250 of them will get in.

    Presumably the 200 who don't hold Cam offers are either intending to apply to Cam retrospectively, or have an offer from Warwick that includes STEP III, or are hedging their bets on the 'any STEP paper' offers.

    Given that - if I remember correctly - STEP II was an unusually narrow paper (heavy on function analysis, low on interesting pure questions), STEP II on its own is a fairly blunt instrument for dividing those 450 offer-holders. They need STEP III for that, and more so than in previous years I'd say: it seems like STEP I and II were slightly reshaped for the benefit of the non-Cam courses using it for offers this year. Unfortunately the misprint has contaminated the STEP III data.

    Now, if this was the department of psychology, english lit, classics - in fact, pretty much any other department - they'd all sit around and discuss it and eventually generate enough group-think to collectively post-rationalise a strategy as being 'fair'. But this is the *Maths* department. These people are some of the top logicians around. They won't be able to just accept that A > B where neither A nor B is measurable or well defined.

    So - the only consolation I can offer those of you who are about to have your futures decided is that, just as you lost sleep over STEP, the folk who have to decide how they handle this are now probably losing sleep over it too!
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    Personally, I think that nothing needs to be modified in the marking because of Q1. I know that candidates will have been affected by different amounts, but maybe that is best sorted out by 'flagging up' the people who seemed to spend a substantial amount of time getting nowhere with Q1. The fact that some people were able to get past the omission and complete the question means that, in my opinion, there should be no change in the marking for this question. And this comes from someone who did spend 15 minutes and 1 page of writing getting nowhere with it. It would seem more appropriate to deal with this problem qualitatively (i.e. flag scripts up) rather than quantitatively (i.e. giving compensatory marks).

    I do, however, fully respect what others are saying. I just hope, as a person who believes that they did really badly in II but reasonably well in III, that this does not put extra pressure on the grade achieved in II. If anything, getting a score of around 70 on III should be seen as an even greater achievement than normal, considering the difficulties with Q1.
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    (Original post by RacingPro97)
    the people who seemed to spend a substantial amount of time getting nowhere with Q1.
    ... the *seemed* is important. If someone wrote 1)... and little else (thinking before jumping in, as Silkos advises), there is no way of knowing whether they spent 2 minutes or 20 before they moved on.

    It's not unreasonable to think that some people were probably affected by 10 marks - which in 2010 was the difference between the 2/3 boundary and the 1/2 boundary. :/

    I think you're right that they'll deal with it qualitatively - being able to see the scripts is one reason they cite for liking STEP.

    If they do take that approach I think they will have to set the grade boundary higher than normal, so that more people miss their offers and they have greater flexibility in hand picking candidates. Which is a shame given that, no matter what the reason, lots of people who've worked really hard will be disappointed with a 2 even if they subsequently get accepted. I suppose the grade curve and boundaries will give a good hint as to the approach they've taken, though my guess is that they'll try to be transparent about it.
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    (Original post by Stray)
    ... the *seemed* is important. If someone wrote 1)... and little else (thinking before jumping in, as Silkos advises), there is no way of knowing whether they spent 2 minutes or 20 before they moved on.

    It's not unreasonable to think that some people were probably affected by 10 marks - which in 2010 was the difference between the 2/3 boundary and the 1/2 boundary. :/

    I think you're right that they'll deal with it qualitatively - being able to see the scripts is one reason they cite for liking STEP.

    If they do take that approach I think they will have to set the grade boundary higher than normal, so that more people miss their offers and they have greater flexibility in hand picking candidates. Which is a shame given that, no matter what the reason, lots of people who've worked really hard will be disappointed with a 2 even if they subsequently get accepted. I suppose the grade curve and boundaries will give a good hint as to the approach they've taken, though my guess is that they'll try to be transparent about it.
    Maybe this is over-thinking it. I'm not in any way condoning the error (in fact, its bloody ridiculous that people as intelligent as those involved in the STEP process let something like this slip), but printing errors have been made before (such as the ending of a question where the range of a trigonometric equation).

    Yes its bad that the omission was on Q1. But some people managed to get through the question (goes to show how clever some of you lot are ). And there were other questions on the paper which were accessible. I know Cambridge / Warwick / good STEP grades are important to you all, but let us trust Cambridge Assessment to try to adjust equitably.

    (If people need help appealing after results, I've been told I'm quite good at arguing . But I would be shocked if many here will need that kind of help. Please relax - try Jack's thread if you want to do something productive over the summer. Or maybe this will help you prep for next year.)
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    (Original post by DFranklin)
    Q12. Once you get your head round it, this is a really, really nice question. (Or possibly that it's got a nasty trick in the 2nd part, but I'm assuming not).

    Spoiler:
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    (i) Triangle ABC has height 1 => triangle ABC has area 1 (base x height). Similarly, APB, BPC, CPA have areas x_1, x_2, x_3. But these areas must equal the area of ABC. So x_1 + x_2 + x_3 = 1.

    I won't bother doing the sketch, but you basically shade the triangle ABG, where G is the centroid of the triangle.

    By symmetry (referring to the sketch), P(X < x) = P(X < x | X = X1). Note that we must have x <= 1/3. If we shade the region P(X>=x | X = X1) we get a triangle similar to ABG but scaled down by (1-3x). So P(X>=x) = (1-3x)^2. So P(X<x) = 1-(1-3x)^2, and so the PDF is the derivative of this: 6(1-3x).

    E[X] =  \int_0^{1/3} 6x(1-3x)\,dx = [3x^2 -6x^3]_0^{1/3} = 3/9 - 6/27 = 1/9

    (ii) Now let F1, F2, F3, F4 be the faces of the tetrahedron, and G the centroid. Define X1, X2, X3, X4 to be the distances from the relevant face. Again, by considering total volume we find that X1+X2+X3+X4 = 1. Clearly the minimum of X1, X2, X3, X4 cannot exceed 1/4.

    The region where x1 is the smallest is going to be the tetrahedron T with base F1 and peak G.

    The same symmetry arguments apply, so that P(X >= x) = P(X>= x | X = X1). When X=X1, X>=x gives us a tetrahedron similar to T but scaled down by (1-4x). So P(X>= x) = (1-4x)^3. So P(X < x) = 1 - (1-4x)^3, and the PDF is going to be 12(1-4x)^2.

    So the expected value for the distance is

    12 \int_0^{1/4} x(1-4x)^2\,dx = 12 \int_0^{1/4} x - 8x^2 + 16x^3 \,dx = [6x^2 - 32 x^3 + 48 x^4]_0^{1/4}

     = 6/16 - 32/64 + 48/256 = 6/16 - 8/16 + 3/16 = 1/16.


    FWIW I believe the result for expected value for the distance generalises to arbitrary dimensions, E[x]=(d+1)^-2 .

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