HSC 2012 MX2 Marathon (archive) (2 Viewers)

IamBread · Feb 24, 2012

Re: 2012 HSC MX2 Marathon

Trebla said:
Find

$\displaystyle\int\dfrac{1-\sin^{12}x}{\cos x}\,dx$

$\int\frac{1-\sin^{12}x}{\cos x}\,dx = \int secx - \frac{\sin^{12}x}{cosx}dx \\\\ = \int secx - \frac{(1-\cos^2x)\sin^{10}x}{cosx} dx \\\\ = \int cosx\sin^{10}x dx = \frac{1}{11}\sin^{11}x + c$

Nooblet94 · Feb 24, 2012

Re: 2012 HSC MX2 Marathon

Carrotsticks said:
Furthermore, it doesn't hurt to look up a few out-of-syllabus things, as they often appear in the HSC.

I occasionally do this just for fun, but obviously it's impossible cover everything that might come up. Is there anything specific you'd recommend having a look at?

seanieg89 · Feb 24, 2012

Re: 2012 HSC MX2 Marathon

A simpler solution to the integration question:

https://docs.google.com/open?id=0B5x34d2OjtrQWlgwblItMGFUc0MzYUw3eG4zLWNodw

someth1ng · Feb 24, 2012

Re: 2012 HSC MX2 Marathon

All of you guys are so..."keen"! I can't do any of these marathon questions...

IamBread · Feb 24, 2012

Re: 2012 HSC MX2 Marathon

someth1ng said:
All of you guys are so..."keen"! I can't do any of these marathon questions...

None of those integration questions I put up? D:

SpiralFlex · Feb 24, 2012

Re: 2012 HSC MX2 Marathon

Remember Spiral's three levels? Easy, Medium, Hard, Elite. That's 4...

IamBread · Feb 24, 2012

Re: 2012 HSC MX2 Marathon

SpiralFlex said:
Remember Spiral's three levels? Easy, Medium, Hard, Elite. That's 4...

Are you saying they were elite?

SpiralFlex · Feb 24, 2012

Re: 2012 HSC MX2 Marathon

IamBread said:
Are you saying they were elite?

Lolcopters. No. I suggested to make 4 levels.

IamBread · Feb 24, 2012

Re: 2012 HSC MX2 Marathon

SpiralFlex said:
Lolcopters. No. I suggested to make 4 levels.

Ok lol. You had me confused there.

SpiralFlex · Feb 24, 2012

Re: 2012 HSC MX2 Marathon

^This is to make other students welcomed.

IamBread · Feb 24, 2012

Re: 2012 HSC MX2 Marathon

Well here's another question
$$Find $ \\\\ \int\frac{x}{\sqrt{1-x^2}}dx$

SpiralFlex · Feb 24, 2012

Re: 2012 HSC MX2 Marathon

Ewwww

$\int -\frac{-2x}{2\sqrt{1-x^2}} \; dx$

$-\sqrt{1-x^2}+C$

IamBread · Feb 24, 2012

Re: 2012 HSC MX2 Marathon

SpiralFlex said:
Ewwww

$\int -\frac{-2x}{2\sqrt{1-x^2}} \; dx$

$-\sqrt{1-x^2}+C$

Haha I didn't think about doing it that way, turns out was easier then I expected. Here's one that I can't do, let's see if one of you can figure it out.

$\int e^x lnx dx$

SpiralFlex · Feb 24, 2012

Re: 2012 HSC MX2 Marathon

Use da powerfwul serreees

SpiralFlex · Feb 24, 2012

Re: 2012 HSC MX2 Marathon

Some Conics...

$\alpha)\; $Let$\; a\;$ and $\;b \;$ be positive constants.$$

$$Consider the curve E:$ \; y=\frac{b}{a} \sqrt{a^2-x^2}. \; $Prove that the straight line$ \; y=mx+c $\; is a tangent to$ \;E \;$if and only if$ \; m<0 \; $and$ \; c=\sqrt{a^2m^2+b^2}\; \; \; \fbox{4}$

$\beta)\; $Consider the curve$ \; E_{1}: y=\sqrt{27-3x^2}, \; $where$ \; 0<x<3,\; $and the curve$\; E_{2}:y=\sqrt{9-\frac{x^2}{3}}\; $where$ \;0<x<3\sqrt{3}$

$$a) Find the point of intersection of \; $ \;E_{1} \; $and$ \; E_{2}\; \; \; \fbox{2}$

$$b) Let$ \; L\; $be the common tangent to $ \;E_{1} \; $and$ \; E_{2}\; \; \;$ .

$$\;i) Find the equation of$\; L.\; \; \;\fbox{2}$

$$\; ii) Find the area of the region bounded by $\;E_{1} \;, \; E_{2} \; $and$ \; L.\; \; \; \fbox{2}$

Nooblet94 · Feb 24, 2012

Re: 2012 HSC MX2 Marathon

SpiralFlex said:
Some Conics...

$\alpha)\; $Let$\; a\;$ and $\;b \;$ be positive constants.$$

$$Consider the curve E:$ \; y=\frac{b}{a} \sqrt{a^2-x^2}. \; $Prove that the straight line$ \; y=mx+c $\; is a tangent to$ \;E \;$if and only if$ \; m<0 \; $and$ \; c=\sqrt{a^2m^2+b^2}\; \; \; \fbox{4}$

$\beta)\; $Consider the curve$ \; E_{1}: y=\sqrt{27-3x^2}, \; $where$ \; 0<x<3,\; $and the curve$\; E_{2}:y=\sqrt{9-\frac{x^2}{3}}\; $where$ \;0<x<3\sqrt{3}$

$$a) Find the point of intersection of \; $ \;E_{1} \; $and$ \; E_{2}\; \; \; \fbox{2}$

$$b) Let$ \; L\; $be the common tangent to $ \;E_{1} \; $and$ \; E_{2}\; \; \;$ .

$$\;i) Find the equation of$\; L.\; \; \;\fbox{2}$

$$\; ii) Find the area of the region bounded by $\;E_{1} \;, \; E_{2} \; $and$ \; L.\; \; \; \fbox{2}$

For part alpha I can't work out why m<0. I can see it must be the case for the rest of the question because of the domain.

IamBread · Feb 24, 2012

Re: 2012 HSC MX2 Marathon

SpiralFlex said:
Use da powerfwul serreees

Hmm I don't think I know enough on power series to get it

Carrotsticks · Feb 24, 2012

Re: 2012 HSC MX2 Marathon

IamBread said:
Haha I didn't think about doing it that way, turns out was easier then I expected. Here's one that I can't do, let's see if one of you can figure it out.

$\int e^x \lnx dx$

The integral of this cannot be done via elementary methods. I was actually playing around with this very same integral when I was doing MATH1903. One night I was thinking "What will happen if I combine the mutually inverse functions e^x and ln(x) ? Will my math book explode?", so I started working on it.

Using Integration by Parts, we will acquire a very interesting integral:

$\int_{-\infty}^{x}\frac{e^{-t}}{t}$

The power series expansion for this is fascinating, as it involves the Euler-Mascheroni Constant (denoted by a Gamma).

This series can be found by acquiring the Taylor series for the function:

$\frac{e^{-x}}{x}$

Then integrating the series term-by-term. iirc there is justification required for the convergence of the power series to the integral itself, but nothing too profound.

tl;dr can't do the question using HSC methods.

math man · Feb 24, 2012

Re: 2012 HSC MX2 Marathon

pretty sure we will encounter this in complex analysis

IamBread · Feb 24, 2012

Re: 2012 HSC MX2 Marathon

Ah okay, interesting... I have been trying to do this one for a while, never been able to get anywhere, now I know why

.

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