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Yep there are more... in fact it's an inequality.

Secretz :) If you want one, try the one I gave above!

1. \frac{1 - \frac{1}{\sqrt 2}}{1 + \frac{1}{\sqrt 2}} = \frac{\sqrt 2 - 1}{\sqrt 2 + 1} = \frac{(\sqrt 2 - 1)^2}{1} = 3 - 2 \sqrt 2 2. \frac{5}{\sin ^2 60} = \frac{5}{\frac{3}{4}} = \frac{20}{3} 3. \frac{2 - \sqrt 3}{\frac{1}{\cos ^2 45}} = \frac{2 - \sqrt 3}{2}}

\\ $Consider the function $ f(x) = \sqrt{x+a \sqrt x + b} + \sqrt x -c $ for some real constants $ a,b,c. $ For what values of $ a,b,c $ does the function have infinitely many roots?$

Oh sorry everyone! I made a typo, there should be a 4 in front.

I feel that way sometimes too. I just take 1 day off doing nothing but pure schlomping and the next day I feel ready to do work again.

Yeah

Not really. Most Extension 2 inequality questions are of 3 variables and can actually be defined to be P(a,b,c) or P(x,y,z). It doesn't affect the question at all, it's just a much shorter and easy way to refer to the expression instead of typing it all over again.

Yep, that would be right, but of course we would need to show that with working out and a bit of justification.

Talk to Head Teacher and then Deputy if you can. If you have already demonstrated ability, then it is evident that you are not trying to 'scab marks'.

http://en.wikipedia.org/wiki/Degeneracy_(mathematics)

I want to finish Prelim Chem and hopefully get an essay done for English by the end. And for Maths I'm hoping to be able to finish Fourier Series by then as well.

Yes, it is HSC material. The methodology is purely elementary. Suppose it's not HSC material, does that mean you simply ignore it? I would still do it to try to better myself overall.

1. Acquire some sort of ramp, like a chopping board. 2. Have it slanted at a fixed angle. 3. Pour equal amounts of several fluids from the top at an equal height and measure the time it takes for the fluid to reach the bottom. Use water, honey, oil and a form of alcohol if you can, but then...

Re: HSC 2012 Marathon :) We can break the above series into two sub-series and use the limiting sum for each one.

1. We can use basic Calculus. \\ \frac{df}{dy} = -2y + 4 $ and we let this be equal to 0 to acquire any stationary points. Solving this gives us $ y=2 $ and therefore the maximum value of the function $ f $ is $ f(2) = 17. $ We don't need to verify the nature of the stationary point because it...

The Gradient formula is \frac{y-y_1}{x-x_1} for some constants x_1 and y_1, which we will let be the coordinate given. The case when it passes through the origin contradicts what I said before about m<0.

\\ $Consider the polynomial $ P(x) = x^3 - 7x^2 + 14x -7 $, which has roots $ x_i $ for $ i=1,2,3. $ Show that $ \max (x_i) = 4 \cos ^2 \left ( \frac{\pi}{14} \right )

Umm... are you sure you typed it out correctly? If the following is a function of y, then I presume -x^2 is constant. If so, then the greatest value of the function f(y) is infinity because we have a linear function, which is unbounded from above and below.

Okay, tell me how to prove that tan(105) is -2-root3 as you said? Even OP's expression there is exact by your definition.