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Why does it matter that God could create a universe not finely tuned, it completely side-steps the fact that design implies a designer. Anthropic Principle has been completely discredited buddy Just because only a universe that is fine tuned for observers can have observers in it, then it...

I'm just changing the question, seeing what I can get with the initial conditions (of v's being linear independent)

So what if I altered my statement: \\ $Using the above conditions, the statement$ \\ \\ x \vec{u_1} + y\vec{u_2} + z \vec{u_3} = 0 \\ $implies \\ \alpha \vec{v_1} + \beta \vec{v_2} + \gamma \vec{v}_3 = 0 \\ \text{Which implies} \ \ \alpha = \beta = \gamma = 0

Yep just fixed constants, thank you

\\ $If$ \ \vec{v}_1 , \vec{v}_2 , \vec{v}_3 \ \text{are linear independent} \\ \\ $and$ \ \ a\vec{v}_1 + b\vec{v}_2 + c\vec{v}_3 = x\vec{u}_1 + y\vec{u}_2 + z\vec{v}_3 \\ \\ $then this implies$ \ \vec{u}_1 , \vec{u}_2 , \vec{u}_3 \ \text{is linear indepdendent} Is this true? I am very strongly...

Re: MX2 Integration Marathon \int_0^{\pi/2} \frac{\sin^2(n\theta)}{\sin^2 \theta} \ d\theta

Re: MX2 Integration Marathon \\ x = \sin \theta \\ \Rightarrow \ I = \int \ln(1 + \cos \theta) \cos \theta \ d\theta \\ \\ I = \sin \theta \ln(1+\cos \theta) + \underbrace{\int \frac{\sin^2 \theta}{1+\cos \theta} d\theta}_{J} \\ \\ J = \int \frac{1-\cos^2 \theta}{1+ \cos \theta} d\theta =...

Re: HSC 2014 4U Marathon - Advanced Level Yea I quickly wrote the question up, I thought with part(ii) it would be straightforward induction, but I don't think it is. Here is one (I did this one) \\ $Let$ \ a,b,c \ $be non-negative reals such that$ \ a^2+b^2+c^2 = 8 \\ \\ $Prove that$ \...

Re: HSC 2014 4U Marathon Yea that's pretty much it, not too hard

Re: HSC 2014 4U Marathon Ah yes sorry

Re: HSC 2014 4U Marathon \\ a < b < c \\ \\ $Find$ \ \ \max \left \{a^2(b^2+c^2), \ b^2(a^2+c^2), \ c^2(a^2+b^2) \right \} $Note: $ \ \max \{\dots \} \ \ $refers to the maximum value in the set inside the$ \ \max \ $function$

Re: HSC 2014 4U Marathon - Advanced Level Yea it was supposed to go in the normal level haha

Re: HSC 2014 4U Marathon - Advanced Level \\ $i) Show that for a convex function$ \ f \ $then$ \\ \\ w_1 f(x_1) + w_2f(x_2) \geq f(w_1x_1 + w_2x_2) \\ \\ $Where$ \ \ w_1 + w_2 = 1 \ \ $for non-negative$ \ w_1, w_2 \\ $ii) Hence prove that for some positive integer$ \ n \\ \\ \sum_{j=1}^n w_j...

(1): \ $Let$ \ f(x) = x - \frac{3\sin x}{2 + \cos x} \\ f'(x) = 1 - \frac{3\cos x (2+ \cos x) + \sin x 3\sin x}{(2+\cos x)^2} \\ f'(x) = \frac{(2+\cos x)^2 + 6\cos x + 3}{(2+ \cos x)^2} = \frac{\cos^2 x + 10\cos x + 7}{(2+ \cos x)^2} \\ \\ g(x) = \cos^2 x + 10\cos x + 7 \ \ $prove that$ \ g(x) >...

Re: HSC 2014 4U Marathon \\ a_1,a_2, \dots, a_{n-1}, a_n \ \ $are positive numbers, find the maximum value of$ \\ \\ \frac{a_1 a_2 \dots a_{n-1} a_n}{(1+a_1)(a_1 + a_2) \dots (a_{n-1} + a_n) (a_n + 2^{n+1})}

Re: HSC 2014 4U Marathon - Advanced Level Yes well done :)

Re: HSC 2014 4U Marathon - Advanced Level Bump.

Re: HSC 2014 4U Marathon - Advanced Level \sum_{n=1}^{\infty} (-1)^{\frac{n}{2}(n-1)}\frac{1}{n}

Re: HSC 2014 4U Marathon