Official BOS Trial 2016 Thread (1 Viewer)

[KingOfActing]

Any particular questions you guys found the hardest or most interesting?

I found the final question for 4u to just be like ??????????

There were so many different letters that my brain was ready to start writing an AoS essay

 

[kashkow]

I found the final question for 4u to just be like ??????????

There were so many different letters that my brain was ready to start writing an AoS essay

Nah dude,
that was the whole point of that question. To test your creative writing skills. How come you didn't write a discovery creative?

 

[BTA]

Did anyone actually understand what question 5 was asking for, and can someone provide an explanation for 9? I was sitting there in the exam with like a "wtf" expression cause of these two lol

 

[BTA]

I found the final question for 4u to just be like ??????????

There were so many different letters that my brain was ready to start writing an AoS essay

Luckily it was like pretty much copied from the earlier prove e is irrational HSC questions, so it wasn't too bad

 

[Trebla]

Did anyone actually understand what question 5 was asking for, and can someone provide an explanation for 9? I was sitting there in the exam with like a "wtf" expression cause of these two lol

Three of the integrals in Q5 will give the same fixed answer regardless of what f(x) actually is...as long it satisfies those properties listed. Only one of them will give a different answer depending on what f(x) actually is.

In Q9, the friction force F is basically fixed at F_max at the given speed. For 0 < v < v_F it is going too slow and is slipping. The question is asking what would be the best actions to take to increase v to v_F where we have control of the mass and the radius (gravity and angle of inclination are fixed).

 

[Carrotsticks]

Luckily it was like pretty much copied from the earlier prove e is irrational HSC questions, so it wasn't too bad

Actually, I wrote the question from scratch.

I first had a completely different proof using the sum of n'th derivatives (similar to Niven's proof) but then borrowed the idea from the 2003 HSC (proof of irrationality of pi) to try to construct my own proof for the irrationality of e^r. Picking the functions/values and generalising it to all rational powers took some time to figure out, but I probably wouldn't say that it was copied.

 

[KingOfActing]

Actually, I wrote the question from scratch.

I first had a completely different proof using the sum of n'th derivatives (similar to Niven's proof) but then borrowed the idea from the 2003 HSC (proof of irrationality of pi) to try to construct my own proof for the irrationality of e^r. Picking the functions/values and generalising it to all rational powers took some time to figure out, but I probably wouldn't say that it was copied.

Yeah, I've seen the HSC question for the irrationality of e, it was nothing like it That one pales in comparison to this one

 

[Carrotsticks]

I found the final question for 4u to just be like ??????????

There were so many different letters that my brain was ready to start writing an AoS essay

The probability one, or the integration one? I think the integration one was fairly tame compared to previous years and most certainly in comparison to the probability one where constructing the bijection is relatively difficult to spot. I thought that most of the students would be okay with the integration one given that it's much closer to what they're familiar with.

 

[KingOfActing]

The probability one, or the integration one? I think the integration one was fairly tame compared to previous years and most certainly in comparison to the probability one where constructing the bijection is relatively difficult to spot. I thought that most of the students would be okay with the integration one given that it's much closer to what they're familiar with.

The integral one, I just could not handle all the letters. Brain was like "sorry brain not found", I don't think I did any of part c. ...Nor much of part b.

 

[InteGrand]

Also Question 11(a) (4U paper) is dodgily worded too. Obviously if g(x) can by any function continuous at alpha (which is what the wording technically means and hence what I thought it meant on first reading), then f(x) need not be identical to (x-alpha)^2 * g(x). E.g. take g(x) to be the zero function, or heaps of things (e^x, sin(x), etc.).

I think what was meant was, show that f(x) can be written as (x-alpha)^2 * g(x), for some function g that is continuous at alpha. In other words, show that f(x)/(x-alpha)^2 has a limit as x -> alpha.

 

[porcupinetree]

Also Question 11(a) (4U paper) is dodgily worded too. Obviously if g(x) can by any function continuous at alpha (which is what the wording technically means and hence what I thought it meant on first reading), then f(x) need not be identical to (x-alpha)^2 * g(x). E.g. take g(x) to be the zero function, or heaps of things (e^x, sin(x), etc.).

I think what was meant was, show that f(x) can be written as (x-alpha)^2 * g(x), for some function g that is continuous at alpha. In other words, show that f(x)/(x-alpha)^2 has a limit as x -> alpha.

I was confused by this too when I read through the exam, I was like "wait what??" until I realised what the question was getting at

 

[InteGrand]

For the Ext. 2 paper, it seems like the second last sub-part of the last part of Question 16 (part (c) (iii)) doesn't have its mark allocation written. (Luckily all the other parts of Question 16 have their mark allocations written, and these sum to 12, so part (c) (iii) should be worth 3 marks, since Question 16 should be worth 15 marks.)

Nice to see Napoleon's theorem appear in the exam (and also a branching chain question).

 

[abecina]

big fan of 13d) - that one took some thought.

 

[Carrotsticks]

Also Question 11(a) (4U paper) is dodgily worded too. Obviously if g(x) can by any function continuous at alpha (which is what the wording technically means and hence what I thought it meant on first reading), then f(x) need not be identical to (x-alpha)^2 * g(x). E.g. take g(x) to be the zero function, or heaps of things (e^x, sin(x), etc.).

I think what was meant was, show that f(x) can be written as (x-alpha)^2 * g(x), for some function g that is continuous at alpha. In other words, show that f(x)/(x-alpha)^2 has a limit as x -> alpha.

I agree that it could be worded better. But given that f(x) is a polynomial, then how can g(x) be e^x etc as you mention?

I will make some changes to that and fix the mark allocation in Question 16. Thanks for pointing that out.

 

[Carrotsticks]

big fan of 13d) - that one took some thought.

Thanks. It is a relatively well-known property of the ellipse, but rephrasing it to make it HSC accessible took some tinkering around with the phrasing and constructions. It's simply proving the fact that T is the center of the excircle of triangle SPQ.

It turned out to be a huge coincidence that an almost identical question was in the IMO entrance exam, and a similar question in the 2016 Sydney Grammar Trial.

 

[InteGrand]

I agree that it could be worded better. But given that f(x) is a polynomial, then how can g(x) be e^x etc as you mention?

I will make some changes to that and fix the mark allocation in Question 16. Thanks for pointing that out.

Yeah I figured out the intended meaning of the Q. pretty quickly because it wouldn't make sense otherwise. (Iirc as it was written, it syntactically meant a "for all" rather than "there exists". Like if we say "Let n be a positive integer. Show n has a prime factorisation.", it means show this for all positive integers n, not simply show there exists a positive integer n with this property. My example with the exponential stuff was to show that the question thus couldn't mean a "for all", despite being phrased like that, in case any students reading it were confused too. So it was more about wording rather than mathematical intent.)

 

[Carrotsticks]

I've adjusted it accordingly. Again cheers for the feedback.

https://www.dropbox.com/s/v8pdypgla0i7jm3/BOS Trials 2016.zip?dl=0

 

[Drsoccerball]

Actually, I wrote the question from scratch.

I first had a completely different proof using the sum of n'th derivatives (similar to Niven's proof) but then borrowed the idea from the 2003 HSC (proof of irrationality of pi) to try to construct my own proof for the irrationality of e^r. Picking the functions/values and generalising it to all rational powers took some time to figure out, but I probably wouldn't say that it was copied.

My discrete maths teacher wrote the hsc question lel

 

[InteGrand]

https://en.wikipedia.org/wiki/Bernoulli's_inequality .

 

[InteGrand]

https://en.wikipedia.org/wiki/Bernoulli's_inequality .

(Of course, proving it for positive numbers is easy without induction – follows straight from the binomial theorem.)