MitPY 10: Square Roots

This MitPY comes from a student, Jay:

I have a question relating to polynomial equations. For context I have just finished Y11 during which I completed Further 3&4, Methods 1&2 and Specialist 1&2.

This year during my maths methods class we covered the square root graph, however I was confused as to why it only showed the positive solutions. When I asked about it I was told it was because the radical symbol meant only the positive solution.

However since then I have learnt that the graph of \boldsymbol{y=x^{0.5}} also only shows the positive solution of the square root, while \boldsymbol{y^2=x} shows both. I am quite confused by why they aren’t the same. The only reason that I could think of is that it would mean \boldsymbol{y=x^2} would be the same as \boldsymbol{y^2=x^4}, and while the points (-2,-4) and (2,-4) fit the latter they clearly don’t fit former.

Could you please explain why these aren’t the same?

Feeling VCAA’s Draft: Discussion

It seems the VCAA has just released their draft of the new study design for Mathematics:

  • The current (pre-COVID) study design (pdf) is here.
  • The draft for the new study design (word) is here.
  • The key changes overview (work) is here.
  • The link for feedback (until March 9, 2021) is here.

We haven’t yet looked at the draft, because we’re scared. But, don’t let that stop others. May the discussion and the throwing of brickbats begin.

Secret Specialist Business: Exam 2 Discussion

This is our post for teachers and students to discuss Specialist Exam 2 (not online). There are also posts for Methods Exam 1, Methods Exam 2 and Specialist Exam 1.

UPDATE (03/12/2020)

We’ve now gone through the multiple choice component of the exam, and we’ve read the comments below. In brief, and ignoring the screw-ups, most of the questions seemed good, and a number of questions were hard (which is good). We haven’t thought much about the extent to which the questions are trivialised by CAS/Mathematica, although this is of course extremely important; the comments below on this aspect are well worth a careful read.

Here are our question-by-question thoughts:

MCQ1. A decent and non-trivial stationary point question. A pretty mean way to begin.

MCQ2. A contrived and tricky range of function question. A very mean way to continue.

MCQ3. A rather weird piecewise constant acceleration question.

MCQ4. A good and not so easy composition of functions question.

MCQ5. Intrinsically a routine and good complex algebra question, but the presentation is a mess. The notation \boldsymbol{z = a + bi} is introduced, but then plays no role; indeed, the question would have been vastly improved by having the offered answers expressed in terms of \boldsymbol{a} and \boldsymbol{b}. Requiring some extra algebraic manipulation to obtain the correct answer is needless, and a little contrived.

MCQ6. A very easy complex factorisation question.

MCQ7. Ugh! See here.

MCQ8. A nice complex algebra question.

MCQ9. Complete nonsense, as flagged by commenter Red Five, below. See here.

MCQ10. A routine tank mixture problem.

MCQ11. A screw-up, and perhaps a semi-deliberate one, as flagged by commenter John Friend, below. See here.

MCQ12. A straight-forward but nice Euler’s method problem.

MCQ13. A standard linear dependence problem. As noted by commenter John Friend, the problem is trivial with 3 x 3 determinants, which is not on the syllabus but which is commonly taught for this very purpose.

MCQ14. A straight-forward force component question.

MCQ15. A nice parametrised curve question.

MCQ16. A nice dot product and double angle formula question.

MCQ17. A straight-forward acceleration as a function of distance question.

MCQ18. A straight-forward but nice string tension question.

MCQ19. A cricket ball with a mass of 0.02 kg? Otherwise, a nice change of momentum question.

MCQ20. A straight-forward but nice force and acceleration question.

Secret Specialist Business: Exam 1 Discussion

This is our post for teachers and students to discuss Specialist Exam 1 (not online). There are also posts for Methods Exam 1, Methods Exam 2 and Specialist Exam 2.

UPDATE (29/11/2020)

We’ve finally gone through the exam, we’ve read the discussion below, and here are our thoughts.

In brief, the exam is OK but no better, and there are issues. There is some decent testing of skills, but the emphasis (as in the Methods 1 exam) appears to be on fiddly computation rather than deeper concepts. That isn’t great for a 1-hour sprint exam, and commenters have suggested the exam was overly long, but of course a 1-hour sprint exam is intrinsically insane. At a deeper level, some of the questions are contrived and aimless, which is standard, but it feels a little worse this year. And, there are screw-ups.

Here are our question-by-question thoughts:

Q1. The kind of pointless and boring mechanics question whose sole purpose is to make mechanics look bad. Part (a) asks students to compute the normal force, but to no end; the normal force is not required for the rest of the question.

Q2. An intrinsically nice question on integration by substitution, which shoots itself in the foot.

Q3. A routine and nice complex roots question.

Q4. A good inequality inequality question involving absolute values. The question is not difficult but, as commenters have suggested, it seems likely that students will do the question poorly.

Q5. A pretty nice vector resolute (projection) question, sort of a coherent version of last year’s debacle. Part (a) is contrived and flawed by having to choose the integer solution from the two roots of the quadratic; it’s not a hanging offence, but it’s the kind of oddity that would make a thoughtful writer think again.

Q6. A mess. See the comments below, and here.

Q7. An OK if (for a Specialist exam) unusual integration question involving continuity and differentiability of a “hybrid function”. The wording is clumsy, since all that is required is to demand that the function be differentiable; continuity of the function is then automatic, and the demanded continuity of the derivative is irrelevant. Sure, spelling out the continuity may simply be being nice, but including the continuity of the derivative suggests the examiners don’t really get it, or are planning a sleight of hand. We’ll see. Given the most authoritative (Methods) textbook makes a complete hash of this topic, it will be interesting to see if the examination report can get it right. We wouldn’t be betting the house on it.

Q8. An ok but ridiculously contrived volume of revolution question. Asking for the volume to be given in the form \boldsymbol{2\pi(\log_e(a) + b)} where \boldsymbol{a, b \in \mathbb R}  is needless, ill-defined and dumb.

Q9. An OK but ridiculously contrived arclength question. The introduction of the symbol \boldsymbol{s} for the arclength is gratuitous and confusing. And (reviews notes), asking for the arclength to be given in the form \boldsymbol{\log_e(m) + n\log_e(p)} where \boldsymbol{m,n, p \in \mathbb Q}  is needless, ill-defined and dumb.

Hard SEL: The Specialist Error List

This is the home for Specialist Mathematics exam errors. The guidelines are given on the Methods error post, and there is also a Further error post.

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2017 EXAM 1 (Here, and report here)

Q10(c) (added 13/11/20) – discussed here. The intended solution requires computing a doubly improper integral, which is beyond the scope of the subject. The examination report ducks the issue, by providing only an answer, with no accompanying solution.

2017 NHT EXAM 2 (Here, and report here)

Q3(b) (added 13/11/20) – discussed here. The wording of the question is fundamentally flawed, since the “maximum possible proportion” of the function does not exist here, and in any case need not be equal to the “limiting value” of the function. The examination “report” contains nothing but the intended answer.

2006 EXAM 2 (Here, and report here)

MCQ20 (added 24/09/20) The notation F_1, F_2, F_3 refers to the forces in the question being asked, and seemingly also in the diagram for the question, but to the magnitudes of these forces in the suggested answers. The examination report doesn’t acknowledge the error.