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.
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 in the direction of is ,
which can be pictured as follows: For the exam question, we have , and .
Of course, given and it is standard to find . After a bit of trig and unit vectors, we have (in most useful form)
The exam question, however, is different: the question is, given and , how to find .
The problem with that is, unless the vectors and are appropriately related, the scenario simply cannot occur, meaning cannot exist. Most obviously, the length of must be no greater than the length of . 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 , 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 be perpendicular to . Which implies, once again, that the exam question is ridiculous.
Next, suppose we lucked out and began with perpendicular to . (Of course it is very easy to check whether we’ve lucked out.) How, then, do we find ? The answer is, as is made clear by the picture, “Well, duh”. The possible vectors are simply the (non-zero) scalar multiples of , 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 above and working backwards. And, these equations correctly return no solutions. Moreover, if the relationship between and had been such that there were solutions, then the A equations would have found them. So, completely ridiculous but still ok?
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.
Further Update (26/06/20)
As commenters have noted, the Examination Report has finally appeared. And, as predicted, answer A was deemed correct, with the Report noting
Option A gives the set of equations that can be used to obtain the values of m, n and p. Explicit solution would result in a null set as it is not possible for a result of a vector to be of greater magnitude than the vector itself.
Well, it’s something. Presumably “result of a vector” was intended to be “resolute of a vector”, and the set framing is weirdly New Mathy. But, it’s something. Seriously. As John Friend notes, it is at least a small step along the way to indicating the question is not all hunky-dory.
That step, however, is way too small. We’ll close with two comments, reiterating the points made above.
1. The question is wrong
Read the question again, and read the first sentence of the Report’s comment. The question and report justification are fundamentally stuffed by the definite articles, by the language of existence. All answers should have been marked correct.
2. The question is worse than wrong
Even if the vectors and had been chosen appropriately, the question is utterly devoid of mathematical sense. It suggests a long and difficult method to solve a problem that, if indeed is solvable, is trivial.