MitPY 11: Asymptotes and Wolfram Alpha

This MitPY comes from frequent commenter, John Friend:

Dear colleagues,

I figured this was as good place as any to ask for help. I’m writing a small test on rational functions. One of my questions asks students to consider the function \displaystyle f(x) = \frac{x^3 + x}{x^2 + ax - 2a} where a \in R and to find the values of a for which the function intersects its oblique asymptote.

The oblique asymptote is y = x - a so they must first solve

\displaystyle \frac{x^3 + x}{x^2 + ax - 2a} = x - a … (1)

for x. The solution is \displaystyle x = \frac{2a^2}{(a+1)^2} and there are no restrictions along the way to getting this solution that I can see. So obviously a \neq -1.

It can also be seen that if a = 0 then equation (1) becomes \displaystyle \frac{x^3 + x}{x^2} = x which has no solution. So obviously a \neq 0.

When I solve equation (1) using Wolfram Alpha the result is also \displaystyle x = \frac{2a^2}{(a+1)^2}. But here’s where I’m puzzled:

Wolfram Alpha gives the obvious restriction a + 1 \neq 0 but also the restriction 5a^3 + 4a^2 + a \neq 0.

a \neq 0 emerges naturally (and uniquely) from this second restriction and I really like that this happens as a natural part of the solution process. BUT ….

I cannot see where this second restriction comes from in the process of solving equation (1)! Can anyone see what I cannot?

Thanks.

WitCH 49: Trigged Again

The question below is from the second 2020 Specialist exam (not online), and was flagged by commenter John Friend in the discussion here. John has spelled out the problems, but the question is bad enough to warrant its own post, and there’s arguably a little more to be said.

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

UPDATE (31/12/20) The exam is now online.

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.

 

UPDATE (04/12/2020)

We’ve now gone through Section B (extended question) of the exam, and we’ve read the comments below. There do not appear to be any significant screw-ups, but most of it is pretty poor. In the main, the questions are aimless and badly written, with CAS washing away the potentially good effect of any decent content. Nothing is quite a WitCH or PoSWW, but almost everything is close.

Here are our question-by-question thoughts:

Q1. A strikingly aimless parametrised motion question. Seriously, who gives a shit about any of it? Part (b)(i) asks for dy/dx as a function of t, to “hence” obtain the equation of the tangent at t = π, when it is more natural and simpler to first evaluate dy/dt and dx/dt at π. Then, (b)(ii) asks for the velocity at π, for which you need … This is stupid with a capital stupid.

Q2. An OK complex geometry question, which begins thusly:

Two complex numbers, u and v are defined as \boldsymbol{u = -2 -i} and \boldsymbol{v = -4 -3i}.

Jesus. What’s wrong with “Let \boldsymbol{u = -2 -i} and \boldsymbol{v = -4 -3i}“? The symbols \boldsymbol{u} and \boldsymbol{v} are pretty crappy choices for fixed complex numbers, and the later choice of \boldsymbol{z_c} for the centre of a circle is really crappy. Part (d), finding the centre and radius of this circle, would be a nice question in a CAS-free world.

Q3. The best question, graphing \boldsymbol{f(x) = x^2e^{-x}} and then finding the number of inflection points of \boldsymbol{g(x) = x^ne^{-x}} for \boldsymbol{n\in\mathbb Z}. Much of the goodness is killed by CAS. It is not entirely clear what is meant by “asymptotes” in part (b). (See the discussion here.)

Q4. Another parametrised motion question, this one involving a pilot seemingly unaware of the third dimension. Pointless and boring CAS nonsense.

Q5. An absolute mess of a dynamics question. The diagram is shoddy. The appropriate range of the frictional parameter \boldsymbol{k} should be given or determined before asking students to compute a Fantasyland acceleration. Part (e), which feels like an afterthought, involves a jarring and needless switch from the algebraic to numeric, with a specific velocity and implausible force plucked from thin air.