WitCH 35: Overly Resolute

This WitCH (arguably a PoSWW) comes courtesy of Damien, an occasional commenter and an ex-student of ours from the nineteenth century. It is from the 2019 Specialist Mathematics Exam 2. We’ll confess, we completely overlooked the issue when going through the MAV solutions.

Update (16/02/20)

What a mess. Thanks to Damo for pointing out the problem, and thanks to the commenters for figuring out the nonsense.

In general form, the (intended) scenario of the exam question is

The vector resolute of \boldsymbol{\tilde{a}} in the direction of \boldsymbol{\tilde{b}} is \boldsymbol{\tilde{c}},

which can be pictured as follows: For the exam question, we have \boldsymbol{\tilde{a}} = \boldsymbol{\tilde{i}} + \boldsymbol{\tilde{j}} - \boldsymbol{\tilde{k}}, \boldsymbol{\tilde{b}} = m\boldsymbol{\tilde{i}} + n\boldsymbol{\tilde{j}} + p\boldsymbol{\tilde{k}} and \boldsymbol{\tilde{c}} = 2\boldsymbol{\tilde{i}} - 3\boldsymbol{\tilde{j}} + \boldsymbol{\tilde{k}}.

Of course, given \boldsymbol{\tilde{a}} and \boldsymbol{\tilde{b}} it is standard to find \boldsymbol{\tilde{c}}. After a bit of trig and unit vectors, we have (in must useful form)

\boldsymbol{\tilde{c} = \left(\dfrac{\tilde{a}\cdot \tilde{b}}{\tilde{b}\cdot \tilde{b}}\right)\tilde{b}}

The exam question, however, is different: the question is, given \boldsymbol{\tilde{a}} and \boldsymbol{\tilde{c}}, how to find \boldsymbol{\tilde{b}}.

The problem with that is, unless the vectors \boldsymbol{\tilde{a}} and \boldsymbol{\tilde{c}} are appropriately related, the scenario simply cannot occur, meaning \boldsymbol{\tilde{b}} cannot exist. Most obviously, the length of \boldsymbol{\tilde{c}} must be no greater than the length of \boldsymbol{\tilde{a}}. This requirement is clear from the triangle pictured, and can also be proved algebraically (with the dot product formula or the Cauchy-Schwarz inequality).

This implies, of course, that the exam question is ridiculous: for the vectors in the exam we have |\boldsymbol{\tilde{c}}| > |\boldsymbol{\tilde{a}}|, and that’s the end of that. In fact, the situation is more delicate; given the pictured vectors form a right-angled triangle, we require that \boldsymbol{\tilde{a}} - \boldsymbol{\tilde{c}} be perpendicular to \boldsymbol{\tilde{c}}. Which implies, once again, that the exam question is ridiculous.

Next, suppose we lucked out and began with \boldsymbol{\tilde{a}}- \boldsymbol{\tilde{c}} perpendicular to \boldsymbol{\tilde{c}}. (Of course it is very easy to check whether we’ve lucked out.) How, then, do we find \boldsymbol{\tilde{b}}? The answer is, as is made clear by the picture, “Well, duh”. The possible vectors \boldsymbol{\tilde{b}} are simply the (non-zero) scalar multiples of \boldsymbol{\tilde{c}}, and we’re done. Which shows that the mess in the intended solution, Answer A, is ridiculous.

There is a final question, however: the exam question is clearly ridiculous, but is the question also stuffed? The equations in answer A come from the equation for \boldsymbol{\tilde{c}} above and working backwards. And, these equations correctly return no solutions. Moreover, if the relationship between \boldsymbol{\tilde{a}} and \boldsymbol{\tilde{c}} had been such that there were solutions, then the A equations would have found them. So, completely ridiculous but still ok?

Nope.

The question is framed from start to end around definite, existing objects: we have THE vector resolute, resulting in THE values of m, n and p. If the VCAA had worded the question to find possible values, on the basis of a possible direction for the resolution, then, at least technically, the question would be consistent, with A a valid answer. Still an utterly ridiculous question, but consistent. But the VCAA didn’t do that and so the question isn’t that. The question is stuffed.

WitCHes in Batches

What we like about WitCHes is that they enable us to post quickly on nonsense when it occurs or when it is brought to our attention, without our needing to compose a careful and polished critique: readers can do the work in the comments. What we hate about WitCHes is that they still eventually require rounding off with a proper summation, and that’s work. We hate work.

Currently, we have a big and annoying backlog of unsummed WitCHes. That’s not great, since a timely rounding off of discussion is valuable. Our intention is to begin ticking off the unsummed WitCHes, which are listed below with brief indications of the topics. Most of these WitCHes have been properly hammered by commenters, though of course readers are always welcome to comment, including after summation. We’ll update this post as the WitCHes get ticked off. Thanks very much to all past WitCH-commenters, and we’re sorry for the delay in polishing off. We’ll attempt to keep on top of future WitCHes.

WitCH 8 (oblique asymptotes – UPDATED 05/02/20)

WitCH 10 (distance function)

WitCH 12 (trig integral)

WitCH 18 (Serena Williams)

WitCH 20 (hypothesis testing)

WitCH 21 (order of algorithms)

WitCH 22 (inflection points)

WitCH 23 (speed functions)

WitCH 24 (functional equations)

WitCH 25 (probability distributions)

WitCH 26 (function composition)

WitCH 27 (function composition)

WitCH 28 (trig graphs)

WitCH 29 (inverse derivatives)

WitCH 30 (Eddie Woo)

WitCH 31 (function composition)

WitCH 32 (PISA)

WitCH 33 (probability distributions)

WitCH 34 (numeracy guide – added 05/02/20)

WitCH 35 (vector resolutes – added 14/02/20 – UPDATED 16/02/20)

 

WitCH 33: Below Average

We’re not actively looking for WitCHes right now, since we have a huge backlog to update. This one, however, came up in another context and, after chatting about it with commenter Red Five, there seemed no choice.

The following 1-mark multiple choice question appeared in 2019 Exam 2 (CAS) of VCE’s Mathematical Methods.

The problem was to determine Pr(X > 0), the possible answers being

A. 2/3      B. 3/4      C. 4/5      D. 7/9      E. 5/6

Have fun.

WitCH 29: Bad Roots

This one is double-barrelled. A strange multiple choice question appeared in the 2019 NHT Mathematical Methods Exam 2 (CAS). We had thought to let it pass, but a similar question appeared in last’s weeks Methods exam (no link yet, but the Study Design is here). So, here we go.

First, the NHT question:

The examination report indicates the correct answer, C, and provides a suggested solution:

\Large\color{blue} \boldsymbol{ g(x)=f^{-1}(x)=\frac{x^{\frac15}-b}{a},\ g'(x) = \frac{x^{-\frac45}}{5a},\ g'(1) = \frac1{5a}}

And, here’s last week’s question (with no examination report yet available):

WitCH 27: Uncomposed

Ah, so much crap …

Tons of nonsense to post on, and the Evil Mathologer is breathing down our neck. We’ll have (at least) three posts on last week’s Mathematical Methods exams. This one is by no means the worst to come, but it fits in with our previous WitCH, so let’s quickly get it going. It is from Exam 1. (No link yet, but the Study Design is here.)