HSC 2015 MX1 Marathon (archive) (1 Viewer)

rand_althor · Oct 5, 2015

Re: HSC 2015 3U Marathon

InteGrand said:
I think it's a circle of radius 1. (Incidentally, a mathematical "ball" need not be 3-D: https://en.wikipedia.org/wiki/Ball_(mathematics))

Yep that's what I meant.

psyc1011 · Oct 5, 2015

Re: HSC 2015 3U Marathon

photastic said:
Take 2 and 3 back to the 2U marathon

2. Is 3U m8. it requires subsittutions

Paradoxica · Oct 5, 2015

Re: HSC 2015 3U Marathon

rand_althor said:
$$1. Let $f(x)$ be a function such that $f(xy) = f(x)+f(y)$. Prove by induction that $f(a^n) = n \times f(a)$ for all $n\geq1$.$
$$2. (a) Consider the circle $x^2+(y-k)^2=1$. For what value(s) of $k$ is the curve $y=x$ a tangent to the circle? \\ \indent (b) A ball of radius $1$ is dropped inside the curve $y=|x|$. Find the area bounded by the ball and the curve.$

$Second clarification required. What does it mean for the circle to be 'inside' the curve? It's an open curve, there is no 'inside' and 'outside'$

davidgoes4wce · Oct 5, 2015

Re: HSC 2015 3U Marathon

Induction Question from El Hosri Paper 2

Its worth 3 marks, someone critique my working out and let me know what they would give out of 3. I have tried to blend in the explanations from Kinney Lewis' videos and Fitzpatrick textbook explanation when writing my answer.

Drsoccerball · Oct 5, 2015

Re: HSC 2015 3U Marathon

Paradoxica said:
$Second clarification required. What does it mean for the circle to be 'inside' the curve? It's an open curve, there is no 'inside' and 'outside'$

Bruh... how can it be so hard to understand theres the curve y=|x| and theres a circle in squished between the two lines...

rand_althor · Oct 5, 2015

Re: HSC 2015 3U Marathon

Hope this clears it up:

rand_althor · Oct 5, 2015

Re: HSC 2015 3U Marathon

davidgoes4wce said:
Induction Question from El Hosri Paper 2

Its worth 3 marks, someone critique my working out and let me know what they would give out of 3. I have tried to blend in the explanations from Kinney Lewis' videos and Fitzpatrick textbook explanation when writing my answer.

I'm not sure if your proof for n=k+1 is correct. You say you want to prove $4^{k+1}(k+4)>4^{k+2}$ , but you haven't really done so. I think what you should do is say RTP $4^{k+1}(k+4)>4^{k+2}$ , and then say "Consider the difference". Then simplify, show that the expression (you end up with $4k \cdot 4^k$ ) is >0, and then go back to what you started with to show $4^{k+1}(k+4)>4^{k+2}$ and hence $(k+4)!>4^{k+2}$ . If I'm wrong someone please correct me.

InteGrand · Oct 5, 2015

Re: HSC 2015 3U Marathon

What he has done is technically correct because his the statements in his steps are biconditional, which basically mean they go both ways. He should have made this clearer by adding $\Longleftrightarrow$ signs between the steps. Even though this is correct though, I think it is despised or not approved of by markers sometimes...it is safest in the HSC to just reverse his steps lol.

rand_althor · Oct 5, 2015

Re: HSC 2015 3U Marathon

InteGrand said:
What he has done is technically correct because his the statements in his steps are biconditional, which basically mean they go both ways. He should have made this clearer by adding $\Longleftrightarrow$ signs between the steps. Even though this is correct though, I think it is despised or not approved of by markers sometimes...it is safest in the HSC to just reverse his steps lol.

What do you mean 'reverse his steps'? Start from k>0?

InteGrand · Oct 5, 2015

Re: HSC 2015 3U Marathon

rand_althor said:
What do you mean 'reverse his steps'? Start from k>0?

$Yes, he could start at the $k>0$ part and just work his way from the bottom of his current working to the top. He would end up with $4^{k+2}< 4^{k+1} (k+4)$, and he could say that the R.H.S. here is less than $(k+4)!$ because $4^{k+1}< (k+3)!$ by the inductive hypothesis, which means that $4^{k+2}< (k+4)!$ (by transitivity of `less than'). This would complete the induction step.$

davidgoes4wce · Oct 5, 2015

Re: HSC 2015 3U Marathon

Induction is like an essay, I think I have watched and read about 5 different methods when evaluating the 'inequality induction'.

All slight variations but I like Kinney-Lewis way of explanation (you can view it on Youtube) and Fitzpatrick for divisibility and sums/products.

For all you current HSCers, I just did a quick check in previous years, the inequality induction has popped up 3 times in the past 17 years.

davidgoes4wce · Oct 5, 2015

Re: HSC 2015 3U Marathon

InteGrand said:
$Yes, he could start at the $k>0$ part and just work his way from the bottom of his current working to the top. He would end up with $4^{k+2}< 4^{k+1} (k+4)$, and he could say that the R.H.S. here is less than $(k+4)!$ because $4^{k+1}< (k+3)!$ by the inductive hypothesis, which means that $4^{k+2}< (k+4)!$ (by transitivity of `less than'). This would complete the induction step.$

Would you give that full marks Integrand? (out of 3). I think inequality is the trickiest of the main 4 induction types (divisiblity, series, sums/products, inequalities)

InteGrand · Oct 5, 2015

Re: HSC 2015 3U Marathon

davidgoes4wce said:
Would you give that full marks Integrand? (out of 3). I think inequality is the trickiest of the main 4 induction types (divisiblity, series, sums/products, inequalities)

Well I probably would, but I'm fairly sure a lot of markers wouldn't, so it is safest for HSC purposes to write it differently.

Paradoxica · Oct 5, 2015

Re: HSC 2015 3U Marathon

Drsoccerball said:
Bruh... how can it be so hard to understand theres the curve y=|x| and theres a circle in squished between the two lines...

I don't associate gravity with the cartesian plane. So EXCUSE ME if I keep HS physics and HS mathematics separated.

davidgoes4wce · Oct 5, 2015

Re: HSC 2015 3U Marathon

Time for a break, Im heading to Maccas to get some fat food. See you guys soon.

braintic · Oct 5, 2015

Re: HSC 2015 3U Marathon

Paradoxica said:
I don't associate gravity with the cartesian plane.

So you haven't done Ext 1 Projectile Motion yet?

braintic · Oct 5, 2015

Re: HSC 2015 3U Marathon

davidgoes4wce said:
Time for a break, Im heading to Maccas to get some fat food. See you guys soon.

Did you leave the 's' out of 'fat'? (Or perhaps it is 'r' ?)

dan964 · Oct 5, 2015

Re: HSC 2015 3U Marathon

davidgoes4wce said:
Induction is like an essay, I think I have watched and read about 5 different methods when evaluating the 'inequality induction'.

All slight variations but I like Kinney-Lewis way of explanation (you can view it on Youtube) and Fitzpatrick for divisibility and sums/products.

For all you current HSCers, I just did a quick check in previous years, the inequality induction has popped up 3 times in the past 17 years.

Induction: View attachment Mathematical Induction.pdf
Excuse the light writing in sections (I used aqua pen i think)

davidgoes4wce · Oct 5, 2015

Re: HSC 2015 3U Marathon

InteGrand said:
$Yes, he could start at the $k>0$ part and just work his way from the bottom of his current working to the top. He would end up with $4^{k+2}< 4^{k+1} (k+4)$, and he could say that the R.H.S. here is less than $(k+4)!$ because $4^{k+1}< (k+3)!$ by the inductive hypothesis, which means that $4^{k+2}< (k+4)!$ (by transitivity of `less than'). This would complete the induction step.$

It may have been the first time I have heard of the 'inductive hypothesis', that to me sounds like the 'Assumed step' in this process.

InteGrand · Oct 5, 2015

Re: HSC 2015 3U Marathon

davidgoes4wce said:
It may have been the first time I have heard of the 'inductive hypothesis', that to me sounds like the 'Assumed step' in this process.

Yes, the thing you assume is called the inductive hypothesis (or inductive assumption, etc.).

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