HSC 2016 MX1 Marathon (archive) (1 Viewer)

leehuan · Jan 15, 2016

Re: HSC 2016 3U Marathon

Few suggestions:

1. Use the fact that it's even to make (-1)^n = 1 as early as right after writing the expansion of (1-x)^n instead of at the end, because the fact this question wasn't split into parts meant that the result can be used essentially anytime.
2. Simplify the expansion of (1+x)^n - (1-x)^n earlier
3. Combined with 2 - I would've written nC0 -> nC6 to establish the pattern more clearly.
4. (Most unnecessary suggestion) Just write d^2/dx^2 in front of the LHS and the RHS in your second equation

Correction: Lol, 2x30 does not equal 15x5 (you forgot to rewrite as 12x5)

Though I didn't think too deeply myself to create a solution, so good job!

DatAtarLyfe · Jan 15, 2016

Re: HSC 2016 3U Marathon

Feck me, i corrected the question but still wrote the wrong answer XD

Drsoccerball · Jan 19, 2016

Re: HSC 2016 3U Marathon

$New Question: Paradoxica, Integrand, Ekman and Drsoccerball decide to play a game of 'Bean-boozled.' Taking turns respectively.$

$Assuming there are 4n beans in total and half are barf flavoured, sneeze flavoured, dog food and grass clipping while the other half [are nicer beans(all different).$

$Find the probability that Drsoccerball eats a tainted jelly-bean if when eaten the game ends.$

leehuan · Jan 19, 2016

Re: HSC 2016 3U Marathon

Drsoccerball said:
$New Question: Paradoxica, Integrand, Ekman and Drsoccerball decide to play a game of 'Bean-boozled.' Taking turns respectively.$

$Assuming there are 4n beans in total and half are barf flavoured, sneeze flavoured, dog food and grass clipping while the other half [are nicer beans(all different).$

$Find the probability that Drsoccerball eats a tainted jelly-bean if when eaten the game ends.$

0. Because leehuan stole all the jelly beans before the game started.

Paradoxica · Jan 19, 2016

Re: HSC 2016 3U Marathon

Drsoccerball said:
$New Question: Paradoxica, Integrand, Ekman and Drsoccerball decide to play a game of 'Bean-boozled.' Taking turns respectively.$

$Assuming there are 4n beans in total and half are barf flavoured, sneeze flavoured, dog food and grass clipping while the other half [are nicer beans(all different).$

$Find the probability that Drsoccerball eats a tainted jelly-bean if when eaten the game ends.$

0, because I intentionally ate a tainted jelly bean.

DatAtarLyfe · Jan 22, 2016

Re: HSC 2016 3U Marathon

Can someone post another question coz cbs doing probability

Paradoxica · Jan 23, 2016

Re: HSC 2016 3U Marathon

$$Evaluate $ \dfrac{\displaystyle \sum_{n=1}^{44}\cos n^{\circ}}{\displaystyle \sum_{n=1}^{44}\sin n^{\circ} }$

wu345 · Jan 23, 2016

Re: HSC 2016 3U Marathon

Nice Question! I have a feeling this isn't the quickest way of doing it, and it uses identities not in the 3u course but anyways ( I don't know how to use Latex so I took a picture instead)
http://i.imgur.com/X0pCTvs.jpg?1

wu345 · Jan 23, 2016

Re: HSC 2016 3U Marathon

Can someone please do http://i.imgur.com/kg2tUJs.png ?

Paradoxica · Jan 23, 2016

Re: HSC 2016 3U Marathon

$$\noindent The $x$-value of the midpoint of the two points of intersection is exactly the midpoint of the resultant quadratics in $x$ and $y$ when the two equations are solved simultaneously. Let the line be $y = x\tan{\theta}+2$ and substituting $y$, we have $\sec^2{\theta}x^2 + 4\tan{\theta}x-12 =0 \Leftrightarrow x^2 + 4\sin{\theta}\cos{\theta} -12=0$, and hence, the $x$-value is $-2\sin{\theta}\cos{\theta}=-\sin{2\theta}$. Using the line, the value of $y$ is $2\cos^2{\theta}=1+\cos{2\theta}$. From here, it is clear that $x$ and $y$ are the parametrisations of a circle, which can be solved explicitly for the equation of the circle. This turns out to be $x^2+(y-1)^2=1$. Also, there are no restrictions on the locus, since the chord exists for all values of $\tan{\theta}$, even if the slope is undefined.$

DatAtarLyfe · Jan 23, 2016

HSC 2016 3U Marathon

Another integration question plz
I should just join the 4u integration, wtf am i doing...

Paradoxica · Jan 23, 2016

Re: HSC 2016 3U Marathon

DatAtarLyfe said:
So umm how bout that new question

You can still do it, I'm waiting for an answer that doesn't use Sums to Products.

DatAtarLyfe · Jan 23, 2016

Re: HSC 2016 3U Marathon

Are you talking about the probability question

Paradoxica · Jan 23, 2016

Re: HSC 2016 3U Marathon

DatAtarLyfe said:
Are you talking about the probability question

$$Evaluate $ \dfrac{\displaystyle \sum_{n=1}^{44}\cos n^{\circ}}{\displaystyle \sum_{n=1}^{44}\sin n^{\circ} }$

wu345 · Jan 23, 2016

Re: HSC 2016 3U Marathon

Wow! I didn't realise there was so much to this question... Thanks so much!
(for solving the polynomial q)

wu345 · Jan 23, 2016

Re: HSC 2016 3U Marathon

What about this one?

Also, if anyone is looking for a challenge (use products to sums)
Prove 64sin10sin20sin30sin40sin50sin60sin70sin80sin90=3/4 (angles in degrees)

DatAtarLyfe · Jan 23, 2016

Re: HSC 2016 3U Marathon

Paradoxica said:
$$Evaluate $ \dfrac{\displaystyle \sum_{n=1}^{44}\cos n^{\circ}}{\displaystyle \sum_{n=1}^{44}\sin n^{\circ} }$

i got an answer....it's a fraction with an irrational numerator...

leehuan · Jan 23, 2016

Re: HSC 2016 3U Marathon

DatAtarLyfe said:
i got an answer....it's a fraction with an irrational numerator...

You should at the very least show us your answer lol

DatAtarLyfe · Jan 23, 2016

Re: HSC 2016 3U Marathon

leehuan said:
You should at the very least show us your answer lol

im pree sure i did it wrong, coz he posted the same question in the 4u marathon but made us consider e^itheta, so am not sure how you can use 3u to solve this.
My ways soo abstract its gonna baffle even him XD

InteGrand · Jan 23, 2016

Re: HSC 2016 3U Marathon

DatAtarLyfe said:
im pree sure i did it wrong, coz he posted the same question in the 4u marathon but made us consider e^itheta, so am not sure how you can use 3u to solve this.
My ways soo abstract its gonna baffle even him XD

There's a way to do it via trig. identities, wu345 posted a picture of his solution in a previous post.

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