Question 5cii on the 2017 exam

[Massimooo123]

I don't know if this was a widespread problem during the exam, but I think it might've been based on the question's average score. Basically, you had to find the minimum to two decimal places of the function sqrt( (sin(t)-2cos(t))^2 + (cos(t)+sin(t)+1)^2 ), where 0<t<2pi. Technically it was just t>0, but it loops after 2pi and I didn't want a general solution (turns out you don't get a general solution when you solve as the CAS has to compute it numerically, but that was my original thought process). Around 55% of students had found this expression or something equivalent, yet only 15% were able to find the minimum, which I thought was weird. On trying the question myself, I took the derivative and set it to 0, getting a few values of t, subbed them back into the function and chose the minimum one. But for some reason, this is wrong on the examiner's report, and graphing the function verifies that for some reason the CAS doesn't give you one of the turning points when you restrict the domain of t (even though the turning point is within that domain). If you DON'T restrict the domain when you set the derivative to 0, it gives you more values of t that are WITHIN the original restrictions on the domain. Does anyone know why this would happen? You can just solve the problem using fMin, but I'd be pretty annoyed if I missed a mark over something like that on the exam, and it seems a lot of other people encountered this problem given there were way harder questions than that on the exam in theory.

[Jackson.Sprigg]

I went back and had a crack at it on my CAS to check and it worked fine for me.. I would just define the function and always do a quick sketch, it takes like half a second and always allows you to see roughly where the turning point should be, especially if it's a funky function with a butt load of turning points. I could see the min was roughly around 4 so I restricted the domain from 2 to 6 and solved for that getting the solution. You've most likely just made a lil slip in your expression somewhere, a lot of people don't graph their functions if they aren't required to sketch it and so may have just subbed in values (as you did) looking for the smallest one with no guarantee it would be correct. Same as how a good proportion of people don't bother to check if a point of inflection changes sign. Quickly defining and then sketching the graph is the safest option imo 

[undefined]

Yeah I tried the question on my calculator and initially had a similar problem. If you plot it then you can see the general t value of the minimum so you can restrict the domain.

Once you restrict the domain realistically you should only get the right t value without having to decide which one is which. Attached is my plot of the |rb(t)-rj(t)|

[Massimooo123]

You guys are right, giving a more realistic restriction on the domain gets you the turning point. Still, pretty weird that it didn't do it correctly from 0 to 2pi. I've attached what I'm talking about. Because it's solving numerically, I guess it just searches for a while and then spits out a few roughly accurate answers that happen to satisfy the equation, not necessarily checking everywhere? I don't really know how computers work.

I'll make sure to graph my functions first in the exam tomorrow. Thanks.

[S_R_K]

This was just a badly vetted question; VCAA really shouldn't be putting questions on exams without checking that all reasonable approaches using the available technologies lead to a correct answer. There have been other instances of this in other exams (see, eg. 2018 NHT Exam 2, Q4; hopefully they do not occur tomorrow.

The shit will really hit the fan if a similar issue arises in a future exam; students input the equation into a graph page to check that their stationary points (or whatever) are correct; but the CAS freezes due to the complexity of the graph. Don't expect any apology or admission of guilt from VCAA either.

[Jackson.Sprigg]

There is a weird domain between 0 and 2pi to ~7 where it won't give the minimum but will give all the other turning points? I'm genuinely interested now as to why that would be the case.

This was just a badly vetted question; VCAA really shouldn't be putting questions on exams without checking that all reasonable approaches using the available technologies lead to a correct answer.

Ye, I've become really pedantic about this because it seems to happen fairly often on company exams especially. My CAS will freeze anywhere from 30 seconds to 10 minutes. If it ever seems like there will be a lot of numerical solutions that may occur I always put some restrictions on so that if it does get stuck due to the complexity, it doesn't have to solve too much before I can try another method.