Our second WitCH of the day also comes from the 2017 VCE Specialist Mathematics Exam 2. (Clearly an impressive exam, and we haven’t even gotten to the bit about using inverse trig functions to design a brooch.) It is courtesy of the mysterious SRK, who raised it in the discussion of an earlier WitCH.

Question 5 of Section B of the (CAS) exam concerns a boat and a jet ski. Though SRK was concerned with one particular aspect, the entire question is worth pondering:

The Examiner’s Report indicates an average student score of 1.4 on part a, and comments,

**Students plotted the initial positions correctly but significant numbers of students did not label the direction of motion or clearly identify the jet ski and the boat. Both requirements were explicitly stated in the question.**

For part i, the Report indicates an average score of 1.3, and comments,

**Most students found correct expressions for velocity vectors. The most common error was to equate these velocity vectors rather than equating speeds. **

For part ii, the Report gives the intended answer as (3,3). The Report indicates that slightly under half of students were awarded the mark, and comments,

*Some answers were not given in coordinate form.*

For part i, the Report suggests the answer (with the displayed answer adorned by a weird, extra root sign). The report indicates that a little over half of the students were awarded the mark, and comments,

*A variety of correct forms was given by students; many of these were likely produced by CAS technology, including expressions involving double angles. Students should take care when transcribing expressions from technology output as errors frequently occur, particularly regarding the number and placement of brackets. Some incorrect answers retained vectors in the expression.*

For Part ii, the Report indicates the intended answer of 0.33, and that 15% of students were awarded the mark for this question. The Report comments,

**Many students found this question difficult. Incorrect answers involving other locally minimum values were frequent.**

The Report indicates an average score of 1.3 on part d, and comments;

**Most students correctly equated the vector components and solved for t . Many went on to give decimal approximations rather than supplying the exact forms. Students are reminded of the instruction saying that an exact answer is required unless otherwise specified.**

Lots there. Get hunting.

I wonder how many students gave t=0 for part b, since the question omits the “or equal to” for the first time here. As for part c ii, I have long told students that if exact values are not required, to play the VCAA game often means to draw a graph on the CAS and look for key features, in this case a minimum.

Thanks, RF. I’m not sure what you mean about t = 0. As for (c)(ii), I’m not up on what CAS nonsense is intended or advisable in such situations, but this was the part of the question that SRK flagged. My understanding from SRK is that, perhaps depending upon the machine, the approach you suggest can fail.

Depending on the approach, perhaps, when you equate speeds, one solution is sin(t)=0 for t>0. Earlier it was >= 0.

I just wonder how many students just wrote sin(t)=0 so t=0 rather than t=Pi because they missed the subtle change in symbol from earlier.

Part b.ii. I suppose that in some contexts it may be appropriate to distinguish between a linear combination of the standard basis vectors in R^2 and a coordinate pair; this isn’t one of them.

Part c.i. The purpose of this question escapes me, since VCAA did not require any algebraic manipulation of the expression. Complaining about poor transcription from CAS further makes a mockery of the question. Also, it’s not clear why an expression like |rj(t) – rb(t)| should not be acceptable.

Part c.ii. I have nothing to add to my complaints about this question, written as a comment to a different post. I don’t expect people to have read that however, so if you’d like to know what the problem is, then I encourage you to try a few different methods with the TI-nSpire to answer this question.

Part d. Solving rj(t) = rb(t) for t and a, and using these values rounded to four decimal places gives the position of the jet ski at 0.1056i + 3.8943j and the boat at 0.1055i + 3.8944j. So I guess we should use the exact values because the boat and jet ski are unusually small.

Thanks, all. There’s a couple more “Ugh!” moments on the question, but that’s pretty much caught everything. I’ll update the WitCh, and all the WitCHes, when I get a chance to breathe.

John Friend has noted an important error in Part (d). (I honestly can’t remember whether this occurred to me when originally posting the WitCH.) See John’s comment and the update to the exam entry in the Specialist Error list.