This one comes from the 2021 Specialist Mathematics Exam 2 (not yet online). The general discussion on the exam is here, where some concerns on this question have been raised. It’s bad enough to be a WitCH, however, and so here we are.
The main issue, of course, is that Part (a) is completely stuffed. Telling us that one root of a cubic is real and that the other two are complex tells us absolutely nothing about the complex roots. In particular, the three roots may still all be real, and anything, meaning there is simply no “relationship” to determine between any two roots. Specifically, Part (a)(ii) has two solutions, one of the form and one of the form .
There is plenty more in the question to criticise, but let’s stop to ask: how can such a screw-up occur? How can these screw-ups continually, predictably occur? There is simply no way that a competent mathematician would read Part (a) and not immediately see that the question was stuffed. The logical conclusion is that VCAA does not have its exams vetted by competent mathematicians. Which is insane. The exams should be written by competent mathematicians. VCAA’s monumental indifference is inexcusable.
Now, the other criticisms, which are not minor, but only appear minor in comparison to the Everest of idiocy we’ve just described.
- Let’s begin at the beginning, with the godawful subscripts. One sentence in and already the question is a muddy mess. Why the hell not call the roots of the polynomial α and β and γ, like normal human beings? Oh, yeah, because you clowns decided to use α and β and γ as the polynomial coefficients. Why? Notation matters and you should always have a reason for using one form or the other. But, anyway, call the roots a and b and c, or u and v and w. Whatever. But give the damn things some proper damn names. Why is this so hard?
- Don’t write . It is pompous, it is unnecessary and it is confusing. Just state that the polynomial has real coefficients or something.
- If you are asking for the “relationship” between two numbers then you are almost certainly asking the wrong question. In this instance you have forced yourself to use cutely meaningless language because you are testing – falsely, as it happens – a triviality.
- Writing is pointless and way, way too cute. Test proper mathematical thought, not the detection of cheap tricks.
- Seriously, you can’t think of a single coherent complex numbers question worth 9 marks?
- OK, so you have two, boring, questions in one. Why on earth refer to the complex number in the second question as ? Remember that thing about notation mattering?
- The second question is a mess. There is no particular reason for students to hunt for an exact imaginary intercept in Part (b), nor to assume that the intercept is anything nice. If you want it, ask for it.
- Part (c) then requires the imaginary intercept you didn’t ask for, and the absence of which you’ll probably blame on the students in your typical smugly incompetent manner. The imaginary intercept is apparent, but it is unclear whether this was required (even though it was required in 2(c)).
What a mess.
The exam is here and, finally, the exam report is here. Of course the report simply lies about the answer to 2(a). It eventually notes there is “an alternative solution” , but this is way too little, way too late and way too dishonest. It is still unclear what detail was expected in the answer to 2(b).