4 Unit Revising Marathon HSC '10 (1 Viewer)

NewiJapper · Jan 16, 2010

This maybe slightly off topic, but I admire you Ext 2 guys. You make it look so easy

and here I am working my way through 2 Unit stuff... lol

gurmies · Jan 16, 2010

NewiJapper said:
This maybe slightly off topic, but I admire you Ext 2 guys. You make it look so easy and here I am working my way through 2 Unit stuff... lol

To each his own buddy =]

untouchablecuz · Jan 16, 2010

Trebla said:
Don't think I can match the interesting results of my previous questions, yet. So I'll just settle with this...enjoy!

$$Suppose that $T_n(x)= \cos (n\cos^{-1}x)\\\\$Show that $\\\int_{-1}^{1}\dfrac{T_m(x)T_n(x)}{\sqrt{1-x^2}}\,dx = \begin{cases} 0& $ if $ m \neq n \\\\\dfrac{\pi}{2} &$ if $ m = n \neq 0\\\\\pi & $ if $ m = n = 0 \end{cases}$

$\\\int_{-1}^{1}\dfrac{T_m(x)T_n(x)}{\sqrt{1-x^2}}\,dx=\int_{0}^{\pi}\cos(mx)\cos(nx)\, dx\\$If $m\neq n,\\=\int_{0}^{\pi}\cos(mx)\cos(nx)\, dx\\=\frac{1}{2}\int_{0}^{\pi}(\cos(m+n)x+\cos(m-n)x)\, dx\\=0\\$If $ m=n\neq 0,\\\int_{0}^{\pi}\cos(mx)\cos(nx)\, dx\\=\int_{0}^{\pi}\cos^2(mx)\, dx\\=\frac{1}{2}\int_{0}^{\pi}(\cos(2mx)+1)\, dx\\=\frac{\pi}{2}\\$If $m=n=0,\\\int_{0}^{\pi}\cos(mx)\cos(nx)\, dx\\=\int_{0}^{\pi}dx\\=\pi$

yes im not sleepy

Shadowdude · Jan 16, 2010

I may look at this and go "WTF I DON'T GET IT", but I'm sure after the proper methods have been taught in class - I too can peer upon this and laugh.

/me gets back to his 4u circle geometry in an Argand diagram/complex numbers question he's been working on for two hours.... <_<

gurmies · Jan 16, 2010

Shadowdude said:
I may look at this and go "WTF I DON'T GET IT", but I'm sure after the proper methods have been taught in class - I too can peer upon this and laugh.

/me gets back to his 4u circle geometry in an Argand diagram/complex numbers question he's been working on for two hours.... <_<

Post here xD

cutemouse · Jan 16, 2010

untouchablecuz · Jan 16, 2010

gurmies said:
Post here xD

boredofgurmies

buchanan · Jan 16, 2010

Solution:

http://users.tpg.com.au/nanahcub/1989.pdf

Next Question:

Do you know what Coroneos said about it in 1989?

cutemouse · Jan 16, 2010

buchanan said:
Next Question:

Do you know what Coroneos said about it in 1989?

That it was too long and hard to be included as a fair test of the 4U course.

buchanan · Jan 16, 2010

Correct!

This is in my longer pdf file:

http://users.tpg.com.au/nanahcub/1916-1989-2001.pdf

addikaye03 · Jan 16, 2010

buchanan said:
Correct!

This is in my longer pdf file:

http://users.tpg.com.au/nanahcub/1916-1989-2001.pdf

Got a question buchanan?

Trebla · Jan 16, 2010

untouchablecuz said:
$\\\int_{-1}^{1}\dfrac{T_m(x)T_n(x)}{\sqrt{1-x^2}}\,dx=\int_{0}^{\pi}\cos(mx)\cos(nx)\, dx$

Please elaborate on how you got to that line. After all...it is a "show" question lol

untouchablecuz · Jan 16, 2010

Trebla said:
Please elaborate on how you got to that line. After all...it is a "show" question lol

oh, i thought it was quite obvious

reverse chain rule

OR

let u=cos^-1x and since u is a dummy variable, you can change the variable back to x

shaon0 · Jan 16, 2010

jetblack2007 said:
But see, you're approaching it after having covered the complete 4 unit course with hours of practice under your belt. The 10ers don't have this amount of experience

Last time i checked, there were very few current MX2 students actually posting on this thread.

Anymore questions?

vafa · Jan 16, 2010

a) Factorise a^3+b^3+c^3-3abc

b) Using a) or otherwise, show that (x+y+z)/3>=(xyz)^(1/3)

Koroneos · Jan 17, 2010

The Nomad · Jan 17, 2010

$\\ a^2 + b^2 \geq 2ab\\ c^2 + d^2 \geq 2cd\\ a^2 + b^2 + c^2 + d^2 \geq 2(ab + cd)\\ $From a^2 + b^2 \geq 2ab$, $ it follows that $ab + cd \geq 2\sqrt{abcd}\\ a^2 + b^2 + c^2 + d^2 \geq 4\sqrt{abcd}\\ $Let $abc = d^3,$ so $ a^2 + b^2 + c^2 + d^2 \geq 4\sqrt{d^4}\\ a^2 +b^2 + c^2+d^2\geq 4d^2\\ a^2 + b^2 + c^2 \geq 3d^2 = 3(abc)^{\frac{2}{3}}\\ $Let $ a = \sqrt{x}, b = \sqrt{y}, c = \sqrt{z}\\ \frac{x+y+z}{3}\geq \sqrt[3]{xyz}$

Alternative method.

cutemouse · Jan 17, 2010

Assess the impacts of Leibniz' and Newton's contributions to calculus on society....

OR

Consider the word SOCCER.

How many:

(i) six-letter different arrangements [1]

(ii) selections of 4 letters [2]

can be made from the word SOCCER?

Trebla · Jan 17, 2010

Cazic · Jan 17, 2010

Damn, I think that question has been made this forums bitch

4 Unit Revising Marathon HSC '10 (1 Viewer)

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