Report on the 2016 Higher Level Leaving Certificate Applied Mathematics paper

by Dominick Donnelly, Bruce College, Cork
http://www.appliedmathematics.ie

Question 1(a)
Straightforward arithmetic velocity/time graph question, very similar to a previously asked question.

Question 1(b)
Standard enough algebraic 2 particle problem. Tricky enough algebra, but standard enough.

Overall on question 1: A good question, pitched at about the right level of difficulty overall. More straightforward than many Q1s have been, which is good.

Question 2(a)
Standard enough interception question. Some candidates may have difficulty rotating the frame of reference. Otherwise fine. Also getting the second solution in part (ii) slightly novel, but not difficult.

Question 2(b)
2(b) (i) Very standard. 2(b) (ii) Absolutely diabolical question. Very difficult. Maximisation / minimisation has only been brought into Q2 once before, and it too was a diabolical question. Many students already struggle with question 2 as it is the most vector based question, there is no need to make it this difficult. I really hope the marks get loaded into 2(a) and 2(b)(i), as I doubt there will be many succeeding with 2(b)(ii). While there is a very elegant solution not involving calculus, I would doubt many if any of the candidates would be aware of it.

Overall on question 2: Part (a) fine, part (b) awful. The hardest question on the paper. Many students were shattered by this question. Hopefully most of them went elsewhere and did a different question. There is no need to make relative velocity this hard, it is hard enough already.

Question 3(a)
A nice question about hitting (or missing) a target. Standard enough, with a slight level of novelty to make it interesting.

Question 3(b)
A tough question involving maximisation of range on the inclined plane. A second question with maximisation / minimisation, albeit in this case in a more familiar context. The trigonometry involved will have caused a serious obstacle to many candidates, particularly when coupled with the calculus. Its saving grace is that it is reasonably similar to a recent question, so if they had practised the trig. there, it is the same here.

Overall on question 3: This question is usually a banker for the vast majority of students, particularly the more moderate students. This year it was at the top end of the range of difficulty for Q3 in general, and may have caused some students to reach out for a different question. They could have found both parts off-putting, part (a) for the novelty of its wording, and part (b) if they weren’t well practised on maximisation, which many won’t have been.

Question 4(a)
A standard enough question involving a pulley system on an inclined plane. There was a similar pulley system in a recent enough question, so the candidates should have been able to get the accelerations right. Of course this will be the main source of errors. Overall a nice question.

Question 4(b)
A very standard moving pulley in the middle question. The main source of error in these is usually mistakes in the accelerations, and I presume it will be the same again here. Part (ii) is an interesting variation for this type of question.

Overall on question 4: A good question, pitched at about the right level of difficulty overall. Again it is a banker for most candidates, particularly the more moderate ones, and they hopefully should have been ok here.

Question 5(a)
It is a long time since there was a pendulum involved in a Q5 collision question, and since many candidates will not have studied question 6, they may not have been aware that they need to use conservation of energy here, twice. Otherwise it is a straightforward enough question, albeit quite long. The given diagram should have included the 60° angle.

Question 5(b)
Quite a straightforward oblique collision question, though with two distinct novelties. The first novelty was that the collision was in the j direction rather than the i. The second is that the question gives v for each particle instead of u. Candidates should have been able to work around these two ok. Once the novelties are dealt with a very straightforward question, and quite short.

Overall on question 5: A good question, pitched at about the right level of difficulty overall. Slight novelties in both parts may have hindered some students, though hopefully not too many, as it is another banker for most students.

Question 6(a)
A very standard motion in a vertical circle question. I imagine the main source of errors would be candidates taking v = 0 as the condition at D, rather than T = 0. Quite long though. Also too similar to question 5(a), with more conservation of energy.

Question 6(b)
A standard vertical Hooke’s Law problem. Again quite long, particularly part (iii).

Overall on question 6: A good question, pitched at about the right level of difficulty overall, though both parts are quite long.

Question 7(a)
This question is, quite literally, impossible. It is utterly shocking that this question made it through the checks and balances that surely exist, and actually made it onto the paper. There must be an internal inquiry within the SEC as to how this happened, and the internal procedures must be strengthened to avoid this happening again. The question is impossible because as the problem is written it cannot be in equilibrium. There are only three forces on the rod, and two are definitely vertical (the reaction at A and the weight), and since it is stated that the third (the tension at B) is not vertical therefore there is a resultant horizontal force and the rod is accelerating to the right. To rectify this question, either the horizontal surface should have been rough instead of smooth, or there should have been a hinge at A, or the string at B should have been vertical. It could, and should, have been a standard enough question. It is possible to get answers to this question, but they are in reality nonsense. How this is going to be marked must be worked out very carefully so as not to disadvantage those who attempted it. The examiners made the mistake here, not the candidates, and the candidates must be treated fairly.

Question 7(b)
A standard enough double ladder question. Part (i) very straightforward, part (ii) a bit trickier. Overall quite long.

Overall on question 7: The most important thing for now is that the candidates who tried this question are treated fairly and not disadvantaged. Hopefully there aren’t too many of them. Then the SEC must act to ensure that this does not happen again, and they must be open enough in their processes to reassure us all that they have taken the necessary measures, which unfortunately is not usually the way they operate. We all make mistakes, to err is human. However in a process as vital as writing a national examination paper, there should be sufficient checks and balances in place to prevent something like this happening. Obviously there are not.

Question 8(a)
Standard proof of moment of inertia of a rod about its midpoint.

Question 8(b)
Nice question about getting the moment of inertia of a wheel comprised of various components. In part (i) the likely errors will occur in not using the Parallel Axis Theorem correctly, or not using it at all, to find I for the spokes. Part (ii) should be straightforward, though they may forget to use both forms of kinetic energy. Part (iii) is conservation of energy again, of which there is probably too much in one paper, Quite a long question all told.

Overall on question 8: A good question, pitched at about the right level of difficulty overall. Part (b) quite long, involving more work than usual.

Question 9(a)
An interesting and novel twist on a typical U-tube question. Once you get past the novelty a very straightforward and quick question. My only concern is that this question sets a precedent for future papers to go further into the area of hydraulics. The course is long enough!

Question 9(b)
A standard inclined rod question.

Overall on question 9: A good question, pitched at about the right level of difficulty overall. One of the shorter questions, which is good, as this paper was overall on the long side.

Question 10(a)
A standard question involving variable acceleration. I imagine a large proportion of candidates will give displacement instead of distance for part (ii).

Question 10(b)
A standard enough question, also involving variable acceleration. Basically this is the proof of a couple of SHM equations by integration.

Overall on question 10: A good question, pitched at about the right level of difficulty overall.

Overall impressions of whole paper
Eight of the ten questions (Qs. 1, 3, 4, 5, 6, 8, 9 &10) were pitched within the normal range of questions, and each one on its own was fine. However collectively I feel not enough of them were straightforward enough, with a large range of either novelty or complexity involved. This seems to have demoralised a number of candidates, particularly the weaker ones. Also many questions were longer than usual, and I am sure many candidates were under severe time pressure. However if students chose their 6 questions from within this 8, then there should have been sufficient scope for them to show off their problem solving skills successfully.

While question 2(a) was fine, question 2 (b) is what I describe to my students as an “avoid” question. Applying a max / min problem to a new, and difficult, area, that probably no candidate would have ever seen before is stretching the syllabus too far, in my opinion. We have had a number of these “avoid” questions over the years, and I really do think that they do our fabulous subject no favours.

As it is we have a very poorly defined syllabus, and all these novel and “avoid” questions stretch the edges of the syllabus, and push its boundaries wider. Any teacher or student wishing to prepare comprehensively and properly for the exam actually faces an impossible task, as all they can do is speculate where it might go next. We really need tightly defined limits within which the syllabus operates. These limits do not need to be narrow and simplistic, just defined and clear. In this paper alone we have varying degrees of novelty in the following areas: (i) in Q2(b) applying a max / min problem to a new area, (ii) in Q3(a) bringing in the idea of missing a target (while this question itself was fine, it is the precedent I am worried about), (iii) in Q5(b) there were two novelties in a fairly straightforward question, namely the collision being in the j direction, and giving v instead of u, (iv) in Q8(b) applying the Parallel Axis Theorem to the rods in the way required is novel, (v) in Q9(a) while it was in reality a fairly straightforward question, the novelty of this variation on U-tubes is a possible precedent to a whole new area of hydraulics. While some novelty in any paper is welcome and interesting, I feel there was too much here in one paper, and also there is the fear that each novelty is later used to continually broaden the scope of the syllabus: “we’ve done it before so we can do it again.”

The startling departure in this paper was the glaring error in Q7(a), which is unforgivable. We really must get meaningful reassurance from the SEC on this, that they have tightened their procedures to ensure this cannot happen again ever. Something like this can be extremely upsetting for a candidate, as in the middle of an exam they can have their confidence really shaken. The candidates will always doubt themselves, not the examiner. When I first read this question I had to check with colleagues online to see if I was missing something, the mistake seemed that glaring to me. I presumed I had to be wrong, and that I was missing something, as it seems unconscionable to me that there could be such a fundamental error on a paper. I have been teaching this subject for 26 years now, having studied it at school and college for another 6 or 7 before that, and I was in doubt with this question on first reading it. What it might have done to some candidates’ confidence during the exam is a very serious error indeed. Never again please.

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